diff --git a/books/bookvol7.1.pamphlet b/books/bookvol7.1.pamphlet new file mode 100644 index 0000000..b8e4cf8 --- /dev/null +++ b/books/bookvol7.1.pamphlet @@ -0,0 +1,125229 @@ +\documentclass[dvipdfm]{book} +\usepackage{hyperref} +\usepackage{axiom} +\usepackage{makeidx} +\makeindex +\usepackage{graphicx} +\begin{document} +\begin{titlepage} +\center{\includegraphics{ps/axiomfront.ps}} +\vskip 0.1in +\includegraphics{ps/bluebayou.ps}\\ +\vskip 0.1in +{\Huge{The 30 Year Horizon}} +\vskip 0.1in +$$ +\begin{array}{lll} +Manuel\ Bronstein & William\ Burge & Timothy\ Daly \\ +James\ Davenport & Michael\ Dewar & Martin\ Dunstan \\ +Albrecht\ Fortenbacher & Patrizia\ Gianni & Johannes\ Grabmeier \\ +Jocelyn\ Guidry & Richard\ Jenks & Larry\ Lambe \\ +Michael\ Monagan & Scott\ Morrison & William\ Sit \\ +Jonathan\ Steinbach & Robert\ Sutor & Barry\ Trager \\ +Stephen\ Watt & Jim\ Wen & Clifton\ Williamson +\end{array} +$$ +\center{\large{Volume 7.1: Axiom Hyperdoc Pages}} +\end{titlepage} +\pagenumbering{roman} +\begin{verbatim} +Portions Copyright (c) 2005 Timothy Daly + +The Blue Bayou image Copyright (c) 2004 Jocelyn Guidry + +Portions Copyright (c) 2004 Martin Dunstan + +Portions Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd. +All rights reserved. + +This book and the Axiom software is licensed as follows: + +Redistribution and use in source and binary forms, with or without +modification, are permitted provided that the following conditions are +met: + + - Redistributions of source code must retain the above copyright + notice, this list of conditions and the following disclaimer. + + - Redistributions in binary form must reproduce the above copyright + notice, this list of conditions and the following disclaimer in + the documentation and/or other materials provided with the + distribution. + + - Neither the name of The Numerical ALgorithms Group Ltd. nor the + names of its contributors may be used to endorse or promote products + derived from this software without specific prior written permission. + +THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS +IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED +TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A +PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER +OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, +EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, +PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR +PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF +LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING +NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS +SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. + +\end{verbatim} + +Inclusion of names in the list of credits is based on historical +information and is as accurate as possible. Inclusion of names +does not in any way imply an endorsement but represents historical +influence on Axiom development. +\vfill +\eject +\begin{tabular}{lll} +Cyril Alberga & Roy Adler & Richard Anderson\\ +George Andrews & Henry Baker & Stephen Balzac\\ +Yurij Baransky & David R. Barton & Gerald Baumgartner\\ +Gilbert Baumslag & Fred Blair & Vladimir Bondarenko\\ +Mark Botch & Alexandre Bouyer & Peter A. Broadbery\\ +Martin Brock & Manuel Bronstein & Florian Bundschuh\\ +William Burge & Quentin Carpent & Bob Caviness\\ +Bruce Char & Cheekai Chin & David V. Chudnovsky\\ +Gregory V. Chudnovsky & Josh Cohen & Christophe Conil\\ +Don Coppersmith & George Corliss & Robert Corless\\ +Gary Cornell & Meino Cramer & Claire Di Crescenzo\\ +Timothy Daly Sr. & Timothy Daly Jr. & James H. Davenport\\ +Jean Della Dora & Gabriel Dos Reis & Michael Dewar\\ +Claire DiCrescendo & Sam Dooley & Lionel Ducos\\ +Martin Dunstan & Brian Dupee & Dominique Duval\\ +Robert Edwards & Heow Eide-Goodman & Lars Erickson\\ +Richard Fateman & Bertfried Fauser & Stuart Feldman\\ +Brian Ford & Albrecht Fortenbacher & George Frances\\ +Constantine Frangos & Timothy Freeman & Korrinn Fu\\ +Marc Gaetano & Rudiger Gebauer & Kathy Gerber\\ +Patricia Gianni & Holger Gollan & Teresa Gomez-Diaz\\ +Laureano Gonzalez-Vega& Stephen Gortler & Johannes Grabmeier\\ +Matt Grayson & James Griesmer & Vladimir Grinberg\\ +Oswald Gschnitzer & Jocelyn Guidry & Steve Hague\\ +Vilya Harvey & Satoshi Hamaguchi & Martin Hassner\\ +Ralf Hemmecke & Henderson & Antoine Hersen\\ +Pietro Iglio & Richard Jenks & Kai Kaminski\\ +Grant Keady & Tony Kennedy & Paul Kosinski\\ +Klaus Kusche & Bernhard Kutzler & Larry Lambe\\ +Frederic Lehobey & Michel Levaud & Howard Levy\\ +Rudiger Loos & Michael Lucks & Richard Luczak\\ +Camm Maguire & Bob McElrath & Michael McGettrick\\ +Ian Meikle & David Mentre & Victor S. Miller\\ +Gerard Milmeister & Mohammed Mobarak & H. Michael Moeller\\ +Michael Monagan & Marc Moreno-Maza & Scott Morrison\\ +Mark Murray & William Naylor & C. Andrew Neff\\ +John Nelder & Godfrey Nolan & Arthur Norman\\ +Jinzhong Niu & Michael O'Connor & Kostas Oikonomou\\ +Julian A. Padget & Bill Page & Jaap Weel\\ +Susan Pelzel & Michel Petitot & Didier Pinchon\\ +Claude Quitte & Norman Ramsey & Michael Richardson\\ +Renaud Rioboo & Jean Rivlin & Nicolas Robidoux\\ +Simon Robinson & Michael Rothstein & Martin Rubey\\ +Philip Santas & Alfred Scheerhorn & William Schelter\\ +Gerhard Schneider & Martin Schoenert & Marshall Schor\\ +Fritz Schwarz & Nick Simicich & William Sit\\ +Elena Smirnova & Jonathan Steinbach & Christine Sundaresan\\ +Robert Sutor & Moss E. Sweedler & Eugene Surowitz\\ +James Thatcher & Baldir Thomas & Mike Thomas\\ +Dylan Thurston & Barry Trager & Themos T. Tsikas\\ +Gregory Vanuxem & Bernhard Wall & Stephen Watt\\ +Juergen Weiss & M. Weller & Mark Wegman\\ +James Wen & Thorsten Werther & Michael Wester\\ +John M. Wiley & Berhard Will & Clifton J. Williamson\\ +Stephen Wilson & Shmuel Winograd & Robert Wisbauer\\ +Sandra Wityak & Waldemar Wiwianka & Knut Wolf\\ +Clifford Yapp & David Yun & Richard Zippel\\ +Evelyn Zoernack & Bruno Zuercher & Dan Zwillinger +\end{tabular} +\eject +\tableofcontents +\vfill +\eject +\setlength{\parindent}{0em} +\setlength{\parskip}{1ex} +{\Large{\bf New Foreword}} +\vskip .25in + +On October 1, 2001 Axiom was withdrawn from the market and ended +life as a commercial product. +On September 3, 2002 Axiom was released under the Modified BSD +license, including this document. +On August 27, 2003 Axiom was released as free and open source +software available for download from the Free Software Foundation's +website, Savannah. + +Work on Axiom has had the generous support of the Center for +Algorithms and Interactive Scientific Computation (CAISS) at +City College of New York. Special thanks go to Dr. Gilbert +Baumslag for his support of the long term goal. + +The online version of this documentation is roughly 1000 pages. +In order to make printed versions we've broken it up into three +volumes. The first volume is tutorial in nature. The second volume +is for programmers. The third volume is reference material. We've +also added a fourth volume for developers. All of these changes +represent an experiment in print-on-demand delivery of documentation. +Time will tell whether the experiment succeeded. + +Axiom has been in existence for over thirty years. It is estimated to +contain about three hundred man-years of research and has, as of +September 3, 2003, 143 people listed in the credits. All of these +people have contributed directly or indirectly to making Axiom +available. Axiom is being passed to the next generation. I'm looking +forward to future milestones. + +With that in mind I've introduced the theme of the ``30 year horizon''. +We must invent the tools that support the Computational Mathematician +working 30 years from now. How will research be done when every bit of +mathematical knowledge is online and instantly available? What happens +when we scale Axiom by a factor of 100, giving us 1.1 million domains? +How can we integrate theory with code? How will we integrate theorems +and proofs of the mathematics with space-time complexity proofs and +running code? What visualization tools are needed? How do we support +the conceptual structures and semantics of mathematics in effective +ways? How do we support results from the sciences? How do we teach +the next generation to be effective Computational Mathematicians? + +The ``30 year horizon'' is much nearer than it appears. + +\vskip .25in +%\noindent +Tim Daly\\ +CAISS, City College of New York\\ +November 10, 2003 ((iHy)) +\vfill +\eject +\pagenumbering{arabic} +\setcounter{chapter}{0} % Chapter 1 +\chapter{Release Notes} +\section{releasenotes.ht} +\subsection{What's New in Axiom} +\label{releaseNotes} +\index{pages!releaseNotes!releasenotes.ht} +\index{releasenotes.ht!pages!releaseNotes} +\index{releaseNotes!releasenotes.ht!pages} +<>= +\begin{page}{releaseNotes}{0. What's New in Axiom} +\beginscroll +\beginmenu + \menudownlink{Online information}{onlineInformation} + \menudownlink{February 2005}{feb2005} +\endmenu +\endscroll +\autobuttons +\end{page} + +@ +\subsection{Online Information} +\label{onlineInformation} +\index{pages!onlineInformation!releasenotes.ht} +\index{releasenotes.ht!pages!onlineInformation} +\index{onlineInformation!releasenotes.ht!pages} +<>= +\begin{page}{onlineInformation}{Online information} +\beginscroll +Axiom information can be found online at +{http://axiom.axiom-developer.org} +{http://savannah.nongnu.org/projects/axiom} +{http://sourceforge.net/projects/axiom} +\endscroll +\autobuttons +\end{page} + +@ +\subsection{Feature Complete Release Feb 2005} +\label{feb2005} +\index{pages!feb2005!releasenotes.ht} +\index{releasenotes.ht!pages!feb2005} +\index{feb2005!releasenotes.ht!pages} +<>= +\begin{page}{feb2005}{Feature Complete Release Feb 2005} +\beginscroll +The February 2005 release is the first complete release of the +Axiom system since it was first made available as open source. + +This release includes the full complement of algebra, the graphics +subsystem, and the hyperdoc system. + +This full release runs on Linux and Solaris 9. +The algebra runs on Windows. +\endscroll +\autobuttons +\end{page} + +@ +\chapter{Special hyperdoc pages} +\section{util.ht} +This file contains macros for the Axiom HyperDoc hypertext facility. +Most of the macros for the system are here though there may be some in +individual .ht files that are of a local nature. +\index{files!util.ht} +\index{util.ht files} +\subsection{Names of software and facilities} +<>= +\newcommand{\Browse}{Browse} +\newcommand{\Language}{Axiom} +\newcommand{\SpadName}{\Language} +\newcommand{\LangName}{\Language} +\newcommand{\HyperName}{HyperDoc} +\newcommand{\axiomxl}{Aldor} +\newcommand{\anatural}{Aldor} +\newcommand{\Clef}{Clef} +\newcommand{\Lisp}{Common LISP} +\newcommand{\naglib}{NAG Foundation Library} +\newcommand{\GoBackToWork} +{\vspace{2}\newline +{Click on \ \UpButton{} \ to go back to what you were doing.}} + +@ +\subsection{Special hooks to Unix} +All unix commands should be done as macros defined here so we don't +have to go hunting when moving between Unix versions. +<>= +\newcommand{\newspadclient}[1]{xterm -n "#1" -e \$SPAD/bin/clef \$SPAD/bin/server/spadclient} +\newcommand{\searchwindow}[2]{\unixwindow{#1}{\$SPAD/lib/htsearch "#2"}} +\newcommand{\unixwindow}[2]{\unixlink{#1}{#2}} +\newcommand{\menuunixlink}[2]{\item\unixlink{\menuitemstyle{#1}}{#2}} +\newcommand{\menuunixcommand}[2]{\item\unixcommand{\menuitemstyle{#1}}{#2}} +\newcommand{\menuunixwindow}[2]{\item\unixwindow{\menuitemstyle{#1}}{#2}} + +@ +\subsection{HyperDoc menu macros} +<>= +% Example: +% +% \beginmenu +% \menulink{Thing One}{PageOne} la da di da da ... +% \menulink{Thin Two}{PageTwo} do da day ... +% \item \ACmdMacro{\menuitemstyle{Thing Three}} la di da ... +% \endmenu + +% The menu environment + +\newcommand{\beginmenu} {\beginitems[\MenuDotBitmap]} +\newcommand{\endmenu} {\enditems} + +% This is the usual format for a menu item. + +\newcommand{\menuitemstyle}[1] {{\MenuDotBitmap}#1} + +% Often-used menu item forms + +% These two simply do links +\newcommand{\menudownlink}[2] {\item\downlink{\menuitemstyle{#1}}{#2}} +\newcommand{\menulink}[2] {\menudownlink{#1}{#2}} + +% This will cause lower level links to have a HOME button +\newcommand{\menumemolink}[2] {\item\memolink{\menuitemstyle{#1}}{#2}} + +% This opens a new window for the linked page. +\newcommand{\menuwindowlink}[2] {\item\windowlink{\menuitemstyle{#1}}{#2}} + +% These execute lisp commands in various flavors +\newcommand{\menulispcommand}[2] {\item\lispcommand{\menuitemstyle{#1}}{#2}} +\newcommand{\menulispdownlink}[2]{\item\lispdownlink{\menuitemstyle{#1}}{#2}} +\newcommand{\menulispmemolink}[2]{\item\lispmemolink{\menuitemstyle{#1}}{#2}} +\newcommand{\menulispwindowlink}[2]{\item\lispwindowlink{\menuitemstyle{#1}}{#2}} + +% This executes a unix command +\newcommand{\menuunixcmd}[2] {\item\unixcommand{\menuitemstyle{#1}}{#2}} +\newcommand{\searchresultentry}[3]{\tab{3}\item#3\tab{8}\downlink{\menuitemstyle{#1}}{#2}\newline} +\newcommand{\newsearchresultentry}[3]{\tab{3}\item#1\tab{8}\downlink{\menuitemstyle{#2}}{#3}\newline} + +@ +\subsection{Bitmaps and bitmap manipulation macros} +<>= +\newcommand{\htbmdir}{\env{AXIOM}/doc/hypertex/bitmaps} +\newcommand{\htbmfile}[1]{\htbmdir /#1.bitmap} +\newcommand{\htbitmap}[1]{\inputbitmap{\htbmfile{#1}}} +\newcommand{\ControlBitmap}[1]{\controlbitmap{\htbmfile{#1}}} + +% next group of bitmaps frequently appear in the titlebar +\newcommand{\ContinueBitmap} {\ControlBitmap{continue}} +\newcommand{\DoItBitmap} {\ControlBitmap{doit}} +\newcommand{\ExitBitmap} {\ControlBitmap{exit3d}} +\newcommand{\HelpBitmap} {\ControlBitmap{help3d}} +\newcommand{\ReturnBitmap} {\ControlBitmap{home3d}} +\newcommand{\NoopBitmap} {\ControlBitmap{noop3d}} +\newcommand{\UpBitmap} {\ControlBitmap{up3d}} + +\newcommand{\MenuDotBitmap}{\htbitmap{menudot}} + +% Including control panel pixmaps for help pages: + +\newcommand{\helpbit}[1]{\centerline{\inputpixmap{\env{AXIOM}/doc/hypertex/pixmaps/{#1}}}} + +@ +\subsection{HyperDoc button objects} +<>= +\newcommand{\ContinueButton}[1]{\downlink{Click here}{#1} to continue.} +\newcommand{\ExitButton}[1]{\memolink{\ExitBitmap}{#1}} +\newcommand{\HelpButton}[1]{\memolink{\HelpBitmap}{#1}} +\newcommand{\StdHelpButton}{\HelpButton{ugHyperPage}} +\newcommand{\StdExitButton}{\ExitButton{ProtectedQuitPage}} +\newcommand{\UpButton}{\upbutton{\UpBitmap}{UpPage}} +\newcommand{\ReturnButton}{\returnbutton{\ReturnBitmap}{ReturnPage}} +\newcommand{\on}[1]{{\inputbox[1]{#1}{\htbmfile{pick}} + {\htbmfile{unpick}}}} +\newcommand{\off}[1]{{\inputbox[0]{#1}{\htbmfile{pick}} + {\htbmfile{unpick}}}} + +@ +\subsection{Standard HyperDoc button configurations} +<>= +\newcommand{\autobutt}[1]{\helppage{#1}} +\newcommand{\autobuttons}{} +\newcommand{\exitbuttons}{} + +\newcommand{\autobuttLayout}[1]{\centerline{#1}}} +\newcommand{\autobuttMaker}[1]{\autobuttLayout{\HelpButton{#1}}} +\newcommand{\riddlebuttons}[1]{\autobuttLayout{\link{\HelpBitmap}{#1}}} + +% Macro for downward compatibility (?). + +\newcommand{\simplebox}[2]{\inputbox[#1]{#2}{\htbitmap{xbox}}{\htbitmap{xopenbox}}} + +@ +\subsection{HyperDoc graphics macros} +<>= +% Including viewport bitmaps within \HyperName pages: + +\newcommand{\viewport}[1]{\inputimage{{#1}.view/image}} +\newcommand{\axiomViewport}[1]{\inputimage{\env{AXIOM}/doc/viewports/{#1}.view/image}} +\newcommand{\spadviewport}[1]{\axiomViewport{#1}} + +% Creating a real live viewport: + +\newcommand{\viewportbutton}[2]{\unixcommand{#1}{viewalone #2}} +\newcommand{\axiomViewportbutton}[2]{\unixcommand{#1}{viewalone \$AXIOM/doc/viewports/{#2}}} +\newcommand{\spadviewportbutton}[2]{\axiomViewportbutton{#1}{#2}} + +% Making active viewport buttons: + +\newcommand{\viewportasbutton}[1]{\unixcommand{\inputimage{{#1}.view/image}}{viewalone {#1}}} +\newcommand{\axiomViewportasbutton}[1]{\unixcommand{\inputimage{\env{AXIOM}/doc/viewports/{#1}.view/image}}{viewalone \$AXIOM/doc/viewports/{#1}}} +\newcommand{\spadviewportasbutton}[1]{\axiomViewportasbutton{#1}} + +@ +\subsection{TeX and LaTeX compatibility macros} +<>= +%% Begin macros that are needed because HD uses the wrong names + +\newcommand{\center}[1]{\centerline{#1}} +\newcommand{\box}[1]{\fbox{#1}} + +%% End macros that are needed because HD uses the wrong names + +\newcommand{\LARGE}{} +\newcommand{\LaTeX}{LaTeX} +\newcommand{\Large}{} +\newcommand{\TeX}{TeX} +\newcommand{\allowbreak}{} +\newcommand{\aleph}{\inputbitmap{\htbmdir{}/aleph.bitmap}} +\newcommand{\alpha}{\inputbitmap{\htbmdir{}/alpha.bitmap}} +\newcommand{\angle}{\inputbitmap{\htbmdir{}/angle.bitmap}} +\newcommand{\backslash}{\inputbitmap{\htbmdir{}/backslash.bitmap}} +\newcommand{\beta}{\inputbitmap{\htbmdir{}/beta.bitmap}} +\newcommand{\bigbreak}{\newline\newline} +\newcommand{\bot}{\inputbitmap{\htbmdir{}/bot.bitmap}} +\newcommand{\bullet}{\inputbitmap{\htbmdir{}/bullet.bitmap}} +\newcommand{\caption}[1]{\newline\centerline{#1}\newline} +\newcommand{\chi}{\inputbitmap{\htbmdir{}/chi.bitmap}} +\newcommand{\cite}[1]{bibliography entry for {\it #1}} +\newcommand{\cleardoublepage}{} +\newcommand{\coprod}{\inputbitmap{\htbmdir{}/coprod.bitmap}} +\newcommand{\del}{\inputbitmap{\htbmdir{}/del.bitmap}} +\newcommand{\delta}{\inputbitmap{\htbmdir{}/delta.bitmap}} +\newcommand{\Delta}{\inputbitmap{\htbmdir{}/delta-cap.bitmap}} +\newcommand{\div}{\inputbitmap{\htbmdir{}/div.bitmap}} +\newcommand{\dot}{\inputbitmap{\htbmdir{}/dot.bitmap}} +\newcommand{\ell}{\inputbitmap{\htbmdir{}/ell.bitmap}} +\newcommand{\emptyset}{\inputbitmap{\htbmdir{}/emptyset.bitmap}} +\newcommand{\epsilon}{\inputbitmap{\htbmdir{}/epsilon.bitmap}} +\newcommand{\epsffile}{} +\newcommand{\eta}{\inputbitmap{\htbmdir{}/eta.bitmap}} +\newcommand{\exists}{\inputbitmap{\htbmdir{}/exists.bitmap}} +\newcommand{\forall}{\inputbitmap{\htbmdir{}/forall.bitmap}} +\newcommand{\footnote}[1]{ {(#1)}} +\newcommand{\frenchspacing}{} +\newcommand{\gamma}{\inputbitmap{\htbmdir{}/gamma.bitmap}} +\newcommand{\Gamma}{\inputbitmap{\htbmdir{}/gamma-cap.bitmap}} +\newcommand{\hbar}{\inputbitmap{\htbmdir{}/hbar.bitmap}} +\newcommand{\hbox}[1]{{#1}} +\newcommand{\hfill}{} +\newcommand{\hfil}{} +\newcommand{\huge}{} +\newcommand{\Im}{\inputbitmap{\htbmdir{}/im-cap.bitmap}} +\newcommand{\imath}{\inputbitmap{\htbmdir{}/imath.bitmap}} +\newcommand{\infty}{\inputbitmap{\htbmdir{}/infty.bitmap}} +\newcommand{\int}{\inputbitmap{\htbmdir{}/int.bitmap}} +\newcommand{\iota}{\inputbitmap{\htbmdir{}/iota.bitmap}} +\newcommand{\index}[1]{} +\newcommand{\jmath}{\inputbitmap{\htbmdir{}/jmath.bitmap}} +\newcommand{\kappa}{\inputbitmap{\htbmdir{}/kappa.bitmap}} +\newcommand{\label}[1]{} +\newcommand{\lambda}{\inputbitmap{\htbmdir{}/lambda.bitmap}} +\newcommand{\Lambda}{\inputbitmap{\htbmdir{}/lambda-cap.bitmap}} +\newcommand{\large}{} +\newcommand{\ldots}{...} +\newcommand{\le}{<=} +\newcommand{\marginpar}[1]{} +\newcommand{\mu}{\inputbitmap{\htbmdir{}/mu.bitmap}} +\newcommand{\neg}{\inputbitmap{\htbmdir{}/neg.bitmap}} +\newcommand{\newpage}{} +\newcommand{\noindent}{\indent{0}} +\newcommand{\nonfrenchspacing}{} +\newcommand{\nabla}{\inputbitmap{\htbmdir{}/nabla.bitmap}} +\newcommand{\nu}{\inputbitmap{\htbmdir{}/nu.bitmap}} +\newcommand{\omega}{\inputbitmap{\htbmdir{}/omega.bitmap}} +\newcommand{\Omega}{\inputbitmap{\htbmdir{}/omega-cap.bitmap}} +\newcommand{\pageref}[1]{???} +\newcommand{\parallel}{\inputbitmap{\htbmdir{}/parallel.bitmap}} +\newcommand{\partial}{\inputbitmap{\htbmdir{}/partial.bitmap}} +\newcommand{\phi}{\inputbitmap{\htbmdir{}/phi.bitmap}} +\newcommand{\Phi}{\inputbitmap{\htbmdir{}/phi-cap.bitmap}} +\newcommand{\pi}{\inputbitmap{\htbmdir{}/pi.bitmap}} +\newcommand{\Pi}{\inputbitmap{\htbmdir{}/pi-cap.bitmap}} +\newcommand{\prime}{\inputbitmap{\htbmdir{}/prime.bitmap}} +\newcommand{\prod}{\inputbitmap{\htbmdir{}/prod.bitmap}} +\newcommand{\protect}{} +\newcommand{\psi}{\inputbitmap{\htbmdir{}/psi.bitmap}} +\newcommand{\Psi}{\inputbitmap{\htbmdir{}/psi-cap.bitmap}} +\newcommand{\quad}{\inputbitmap{\htbmdir{}/quad.bitmap}} +\newcommand{\Re}{\inputbitmap{\htbmdir{}/re-cap.bitmap}} +\newcommand{\rho}{\inputbitmap{\htbmdir{}/rho.bitmap}} +\newcommand{\sc}{\rm} +\newcommand{\sf}{\bf} +\newcommand{\sigma}{\inputbitmap{\htbmdir{}/sigma.bitmap}} +\newcommand{\Sigma}{\inputbitmap{\htbmdir{}/sigma-cap.bitmap}} +\newcommand{\small}{} +\newcommand{\sum}{\inputbitmap{\htbmdir{}/sum.bitmap}} +\newcommand{\surd}{\inputbitmap{\htbmdir{}/surd.bitmap}} +\newcommand{\tau}{\inputbitmap{\htbmdir{}/tau.bitmap}} +\newcommand{\theta}{\inputbitmap{\htbmdir{}/theta.bitmap}} +\newcommand{\Theta}{\inputbitmap{\htbmdir{}/theta-cap.bitmap}} +\newcommand{\times}{\inputbitmap{\htbmdir{}/times.bitmap}} +\newcommand{\top}{\inputbitmap{\htbmdir{}/top.bitmap}} +\newcommand{\triangle}{\inputbitmap{\htbmdir{}/triangle.bitmap}} +\newcommand{\upsilon}{\inputbitmap{\htbmdir{}/upsilon.bitmap}} +\newcommand{\Upsilon}{\inputbitmap{\htbmdir{}/upsilon-cap.bitmap}} +\newcommand{\vbox}[1]{{#1}} +\newcommand{\wp}{\inputbitmap{\htbmdir{}/wp.bitmap}} +\newcommand{\xi}{\inputbitmap{\htbmdir{}/xi.bitmap}} +\newcommand{\Xi}{\inputbitmap{\htbmdir{}/xi-cap.bitmap}} +\newcommand{\zeta}{\inputbitmap{\htbmdir{}/zeta.bitmap}} +\newcommand{\bs}{\\} + +@ +\subsection{Book and .ht page macros} +<>= +\newcommand{\beginImportant}{\horizontalline} +\newcommand{\endImportant}{\horizontalline} +% +% following handles things like "i-th" but uses "-th" +\newcommand{\eth}[1]{{#1}-th}} +% +\newcommand{\texnewline}{} +\newcommand{\texbreak}{} +\newcommand{\Gallery}{{AXIOM Images}} +\newcommand{\exptypeindex}[1]{} +\newcommand{\gotoevenpage}{} +\newcommand{\ignore}[1]{} +\newcommand{\ind}{\newline\tab{3}} +\newcommand{\labelSpace}[1]{} +\newcommand{\mathOrSpad}[1]{{\spad{#1}}} +\newcommand{\menuspadref}[2]{\menudownlink{#1}{#2Page}} +\newcommand{\menuxmpref}[1]{\menudownlink{`#1'}{#1XmpPage}} +\newcommand{\noOutputXtc}[2]{\xtc{#1}{#2}} % comment and then \spadcommand or spadsrc +\newcommand{\not=}{\inputbitmap{\htbmdir{}/not=.bitmap}} +\newcommand{\notequal}{\inputbitmap{\htbmdir{}/notequal.bitmap}} +\newcommand{\nullXtc}[2]{\xtc{#1}{#2}} % comment and then \spadcommand or spadsrc +\newcommand{\nullspadcommand}[1]{\spadcommand} +\newcommand{\pp}{\newline} % Use this instead of \par for now. +\newcommand{\psXtc}[3]{\xtc{#1}{#2}} % comment and then \spadcommand or spadsrc +\newcommand{\ref}[1]{(see the graph)} +\newcommand{\showBlurb}[1]{Issue the system command \spadcmd{)show #1} to display the full list of operations defined by \spadtype{#1}.} +\newcommand{\smath}[1]{\mathOrSpad{#1}} +\newcommand{\spadFileExt}{.spad} +\newcommand{\spadkey}[1]{} +\newcommand{\spadref} [1]{{\it #1}} +\newcommand{\spadsig}[2]{\spadtype{#1} {\tt ->} \spadtype{#2}} +\newcommand{\axiomSig}[2]{\axiomType{#1} {\tt ->} \axiomType{#2}} +\newcommand{\subscriptIt}[2]{{\it {#1}\_{#2}}} +\newcommand{\subscriptText}[2]{{\it {#1}\_{\it #2}}} +\newcommand{\subsubsection}[1]{\newline\indent{0}{\bf #1}\newline\newline} +\newcommand{\syscmdindex}[1]{} % system command index macro +\newcommand{\threedim}{three-dimensional} +\newcommand{\twodim}{two-dimensional} +\newcommand{\unind}{\newline} +\newcommand{\void}{the unique value of \spadtype{Void}} +\newcommand{\xdefault}[1]{The default value is {\tt "#1"}.} +\newcommand{\xmpLine}[2]{{\tt #1}\newline} % have to improve someday +\newcommand{\xmpref}[1]{\downlink{`#1'}{#1XmpPage}} +\newcommand{\xtc}[2]{#1 #2} % comment and then \spadcommand or spadsrc + +% glossary terms +\newcommand{\axiomGloss}[1]{\lispdownlink{#1}{(|htGloss| '|#1|)}} +\newcommand{\axiomGlossSee}[2]{\lispdownlink{#1}{(|htGloss| '|#2|)}} +\newcommand{\spadgloss}[1]{\axiomGloss{#1}} +\newcommand{\spadglossSee}[2]{\axiomGlossSee{#1}{#2}} + +% use this for syntax punctuation: \axiomSyntax{::} +\newcommand{\axiomSyntax}[1]{``{\tt #1}''} +\newcommand{\spadSyntax}[1]{\axiomSyntax{#1}} + +% constructors +\newcommand{\axiomType}[1]{\lispdownlink{#1}{(|spadType| '|#1|)}} +\newcommand{\spadtype}[1]{\axiomType{#1}} +\newcommand{\nonLibAxiomType}[1]{{\it #1}} % things that browse can't handle +\newcommand{\pspadtype}[1]{\nonLibAxiomType{#1}} + +\newcommand{\axiom} [1]{{\tt #1}} % note font +\newcommand{\spad} [1]{\axiom{#1}} +\newcommand{\spadvar} [1]{\axiom{#1}} % exists in ++ comments; to be removed +\newcommand{\s} [1]{\axiom{#1}} + +\newcommand{\httex}[2]{#1} +\newcommand{\texht}[2]{#2} + +% Function names: +% +% The X versions below are used because AXIOM function names that end +% in ``!'' cause problems for makeindex for had-copy. +% +% Example: \spadfunFromX{reverse}{List} prints as reverse! +% +% In the "From" versions, the first arg is function name, second is constructor +% where exported. +% +% Use the "op" flavors of "-", "+", "*" etc, otherwise the "fun" flavors + +\newcommand{\userfun} [1]{{\bf #1}} % example, non-library function names + +\newcommand{\fakeAxiomFun}[1]{{\bf #1}} % not really a library function +\newcommand{\pspadfun} [1]{\fakeAxiomFun{#1}} + +\newcommand{\axiomFun} [1]{\lispdownlink{#1}{(|oPage| '|#1|)}} +\newcommand{\spadfun} [1]{\axiomFun{#1}} +\newcommand{\axiomFunX}[1]{\axiomFun{#1!}} +\newcommand{\spadfunX}[1]{\axiomFun{#1!}} + +\newcommand{\axiomFunFrom}[2]{\lispdownlink{#1}{(|oPageFrom| '|#1| '|#2|)}} +\newcommand{\spadfunFrom}[2]{\axiomFunFrom{#1}{#2}} +\newcommand{\axiomFunFromX}[2]{\axiomFunFrom{#1!}{#2}} +\newcommand{\spadfunFromX}[2]{\axiomFunFrom{#1!}{#2}} + +\newcommand{\axiomOp} [1]{\lispdownlink{#1}{(|oPage| '|#1|)}} +\newcommand{\spadop} [1]{\axiomOp{#1}} +\newcommand{\axiomOpX}[1]{\axiomOp{#1!}} + +\newcommand{\axiomOpFrom}[2]{\lispdownlink{#1}{(|oPageFrom| '|#1| '|#2|)}} +\newcommand{\spadopFrom} [2]{\axiomOpFrom{#1}{#2}} +\newcommand{\axiomOpFromX}[2] {\axiomOpFrom{#1!}{#2}} +\newcommand{\spadopFromX}[2] {\axiomOpFrom{#1!}{#2}} + +% redundant defns for system commands +\newcommand{\syscom}[1]{\lispdownlink{)#1}{(|syscomPage| '|#1|)}} +% +\newcommand{\spadsyscom}[1]{{\tt #1}} +\newcommand{\spadcmd}[1]{\spadsyscom{#1}} +\newcommand{\spadsys}[1]{\spadsyscom{#1}} + + +% Following macros should be phased out in favor of ones above: + +\newcommand{\gloss}[1]{\axiomGloss{#1}} +\newcommand{\spadglos}[1]{\axiomGloss{#1}} +\newcommand{\glossSee}[2]{\axiomGlossSee{#1}{#2}} + +@ +\subsection{Browse macros} +<>= +\newcommand{\undocumented}[0]{is not documented yet} +\newcommand{\aliascon}[2]{\lispdownlink{#1}{(|conPage| '|#2|)}} +\newcommand{\aliasdom}[2]{\lispdownlink{#1}{(|conPage| '|#2|)}} +\newcommand{\andexample}[1]{\indent{5}\spadcommand{#1}\indent{0}\newline} +\newcommand{\blankline}{\vspace{1}\newline } +\newcommand{\con}[1]{\lispdownlink{#1}{(|conPage| '|#1|)}} + +\newcommand{\conf}[2]{\lispdownlink{#1}{(|conPage| '{#2})}} +% generalizes "con" to allow arbitrary title and form + +\newcommand{\ops}[3]{\lisplink{#1}{(|conOpPage| #2 '{#3})}} +% does lisplink to operation page of a constructor or form +% #1 is constructor name or form, without fences, e.g. "Matrix(Integer)" +% #2 is page number, extracted from $curPage (see fromHeading/dbOpsForm) +% #3 is constructor name or form, with fences, e.g. "(|Matrix| (|Integer|))" + +\newcommand{\dlink}[2]{\downlink{#2}{#1}} +\newcommand{\dom}[1]{\lispdownlink{#1}{(|conPage| '|#1|)}} +\newcommand{\example}[1]{\newline\indent{5}\spadcommand{#1}\indent{0}\newline} +\newcommand{\lisp}[2]{\lispdownlink{#2}{#1}} +\newcommand{\spadatt} [1]{{\it #1}} +\newcommand{\indented}[2]{\indentrel{#1}\newline #2\indentrel{-#1}\newline} +\newcommand{\keyword}[1]{\lispdownlink{#1}{(|htsn| '|#1|)}} +\newcommand{\op}[1]{\lispdownlink{#1}{(|htsn| '|#1|)}} +\newcommand{\spadignore}[1]{#1} + +@ +\subsection{Support for output and graph paste-ins} +<>= +\newcommand{\axiomcommand}[1]{\spadcommand{#1}} +\newcommand{\axiomgraph}[1]{\spadgraph{#1}} + +\newcommand{\pastecommand}[1]{\spadpaste{#1}} +\newcommand{\pastegraph}[1]{\graphpaste{#1}} + +\newcommand{\showpaste}{\htbitmap{sdown3d}} +\newcommand{\hidepaste}{\htbitmap{sup3d}} +\newcommand{\spadpaste}[1]{ + \newline + \begin{paste}{\pagename Empty \examplenumber}{\pagename Patch \examplenumber} + \pastebutton{\pagename Empty \examplenumber}{\showpaste} + \tab{5}\spadcommand{#1} + \end{paste} +} + +\newcommand{\graphpaste}[1]{ + \newline + \begin{paste}{\pagename Empty \examplenumber}{\pagename Patch \examplenumber} + \pastebutton{\pagename Empty \examplenumber}{\showpaste} + \tab{5}\spadgraph{#1} + \end{paste} +} + +@ +\subsection{Hook for including a local menu item on the rootpage} +<>= +\newcommand{\localinfo}{} + +@ +\section{Special Hyperdoc pages} +\subsection{Not Connected to Axiom} +\index{pages!SpadNotConnectedPage!util.ht} +\index{util.ht!pages!SpadNotConnectedPage} +\index{SpadNotConnectedPage!util.ht!pages} +<>= +\begin{page}{SpadNotConnectedPage}{Not Connected to Axiom} +\beginscroll +\HyperName{} isn't connected to Axiom, therefore cannot execute +the button you pressed. +% +\GoBackToWork{} +\endscroll +\end{page} + +@ +\subsection{Do You Really Want to Exit?} +\index{pages!ProtectedQuitPage!util.ht} +\index{util.ht!pages!ProtectedQuitPage} +\index{ProtectedQuitPage!util.ht!pages} +<>= +\begin{page}{ProtectedQuitPage}{Do You Really Want to Exit?} +\beginscroll +{Click again on \ \ExitButton{QuitPage} \ to terminate \HyperName{}.} +\vspace{1}\newline +\centerline{OR} +\GoBackToWork{} +\endscroll +\autobuttons +\end{page} + +@ +\subsection{Missing Page} +\index{pages!UnknownPage!util.ht} +\index{util.ht!pages!UnknownPage} +\index{UnknownPage!util.ht!pages} +<>= +\begin{page}{UnknownPage}{Missing Page} +\beginscroll +\pp +The page you requested was not found in the \HyperName{} database. +\GoBackToWork{} +\endscroll +\end{page} + +@ +\subsection{Something is Wrong} +\index{pages!ErrorPage!util.ht} +\index{util.ht!pages!ErrorPage} +\index{ErrorPage!util.ht!pages} +<>= +\begin{page}{ErrorPage}{Something is Wrong} +\beginscroll +{For some reason the page you tried to link to cannot be formatted.} +\GoBackToWork{} +\endscroll +\autobuttons +\end{page} + +@ +\subsection{Sorry!} +\index{pages!Unlinked!util.ht} +\index{util.ht!pages!Unlinked} +\index{Unlinked!util.ht!pages} +<>= +\begin{page}{Unlinked}{Sorry!} +\beginscroll +{This link is not implemented yet.} +\GoBackToWork{} +\endscroll +\autobuttons +\end{page} + +@ +\chapter{Hyperdoc pages} +The hyperdoc pages can be viewed as a forest of rooted pages. +The main routine in hypertex will look for a page called ``RootPage''. +See RootPage \ref{RootPage} on page~\pageref{RootPage} +\section{algebra.ht} +@ +\subsection{Abstract Algebra} +\label{AlgebraPage} +\begin{itemize} +\item NumberTheoryPage \ref{NumberTheoryPage} on +page~pageref{NumberTheoryPage} +\item GroupTheoryPage \ref{GroupTheoryPage} on +page~pageref{GroupTheoryPage} +\end{itemize} +\index{pages!AlgebraPage!algebra.ht} +\index{algebra.ht!pages!AlgebraPage} +\index{AlgebraPage!algebra.ht!pages} +<>= +\begin{page}{AlgebraPage}{Abstract Algebra} +\beginscroll +Axiom provides various facilities for treating topics in abstract +algebra. +\beginmenu +\menulink{Number Theory}{NumberTheoryPage} \newline +Topics in algebraic number theory. +%\menulink{Algebraic Geometry}{AlgebraicGeometryPage} \newline +%Computational algebraic geometry: Groebner bases, integral bases, +%divisors on curves. +\menulink{Group Theory}{GroupTheoryPage} \newline +Permutation groups; representation theory. +\endmenu +\endscroll +\autobuttons +\end{page} + +@ +\subsection{Number Theory} +\label{NumberTheoryPage} +\begin{itemize} +\item ugProblemGaloisPage \ref{ugProblemGaloisPage} on +page~pageref{ugProblemGaloisPage} +\item IntegerNumberTheoryFunctionsXmpPage +\ref{IntegerNumberTheoryFunctionsXmpPage} on +page~pageref{IntegerNumberTheoryFunctionsXmpPage} +\end{itemize} +\index{pages!NumberTheoryPage!algebra.ht} +\index{algebra.ht!pages!NumberTheoryPage} +\index{NumberTheoryPage!algebra.ht!pages} +<>= +\begin{page}{NumberTheoryPage}{Number Theory} +\beginscroll +Here are some sample computations using Axiom's algebraic number +facilities. +\beginmenu +\menulink{Galois Groups}{ugProblemGaloisPage} \newline +Computation of Galois groups using factorizations over number fields. +\menulink{Number Theory Functions}{IntegerNumberTheoryFunctionsXmpPage}\newline +Some functions of interest to number theorists. +\endmenu +\endscroll +\autobuttons +\end{page} + +@ +\section{alist.ht} +<>= +\newcommand{\AssociationListXmpTitle}{AssociationList} +\newcommand{\AssociationListXmpNumber}{9.1} + +@ +\subsection{AssociationList} +\label{AssociationListXmpPage} +\begin{itemize} +\item TableXmpPage \ref{TableXmpPage} on page~\pageref{TableXmpPage} +\item ListXmpPage \ref{ListXmpPage} on page~\pageref{ListXmpPage} +\end{itemize} +\index{pages!AssociationListXmpPage!alist.ht} +\index{alist.ht!pages!AssociationListXmpPage} +\index{AssociationListXmpPage!alist.ht!pages} +<>= +\begin{page}{AssociationListXmpPage}{AssociationList} +\beginscroll + +The \spadtype{AssociationList} constructor provides a general structure for +associative storage. +This type provides association lists in which data objects can be saved +according to keys of any type. +For a given association list, specific types must be chosen for the keys and +entries. +You can think of the representation of an association list as a list +of records with key and entry fields. + +Association lists are a form of table and so most of the operations available +for \spadtype{Table} are also available for \spadtype{AssociationList}. +They can also be viewed as lists and can be manipulated accordingly. + +\xtc{ +This is a \pspadtype{Record} type with age and gender fields. +}{ +\spadpaste{Data := Record(monthsOld : Integer, gender : String) \bound{Data}} +} +\xtc{ +In this expression, \spad{al} is declared to be an association +list whose keys are strings and whose entries are the above records. +}{ +\spadpaste{al : AssociationList(String,Data) \free{Data}\bound{al}} +} +\xtc{ +The \spadfunFrom{table}{AssociationList} operation is used to create +an empty association list. +}{ +\spadpaste{al := table() \free{al}\bound{al1}} +} +\xtc{ +You can use assignment syntax to add things to the association list. +}{ +\spadpaste{al."bob" := [407,"male"]\$Data \free{al1}\bound{al2}} +} +\xtc{ +}{ +\spadpaste{al."judith" := [366,"female"]\$Data \free{al2}\bound{al3}} +} +\xtc{ +}{ +\spadpaste{al."katie" := [24,"female"]\$Data \free{al3}\bound{al4}} +} +\xtc{ +Perhaps we should have included a species field. +}{ +\spadpaste{al."smokie" := [200,"female"]\$Data \free{al4}\bound{al5}} +} +\xtc{ +Now look at what is in the association list. +Note that the last-added (key, entry) pair is at the beginning of the list. +}{ +\spadpaste{al \free{al5}} +} +\xtc{ +You can reset the entry for an existing key. +}{ +\spadpaste{al."katie" := [23,"female"]\$Data \free{al5}\bound{al6}} +} +\xtc{ +Use \spadfunFromX{delete}{AssociationList} to destructively remove +an element of the association list. +Use \spadfunFrom{delete}{AssociationList} to return a copy of the +association list with the element deleted. +The second argument is the index of the element to delete. +}{ +\spadpaste{delete!(al,1) \free{al6}\bound{al7}} +} + +For more information about tables, see \downlink{`Table'}{TableXmpPage}\ignore{Table}. +For more information about lists, see \downlink{`List'}{ListXmpPage}\ignore{List}. +\showBlurb{AssociationList} +\endscroll +\autobuttons +\end{page} + +@ +\section{annaex.ht} +\subsection{AXIOM/NAG Expert System} +\label{UXANNA} +\index{pages!UXANNA!annaex.ht} +\index{annaex.ht!pages!UXANNA} +\index{UXANNA!annaex.ht!pages} +<>= +\begin{page}{UXANNA}{AXIOM/NAG Expert System} +\centerline{\tt{\inputbitmap{\htbmdir{}/anna_logo.xbm}}\rm} +\newline +\centerline{This expert system chooses, and uses, NAG numerical routines.} +\begin{scroll} +\blankline +\indent{2} +\beginmenu +\menumemolink{Integration}{UXANNAInt} +\blankline +\menumemolink{Ordinary Differential Equations}{UXANNAOde} +\blankline +\menumemolink{Partial Differential Equations}{UXANNAPde} +\blankline +\menumemolink{Optimization}{UXANNAOpt} +\vspace{10} +\menumemolink{About the AXIOM/NAG Expert System}{UXANNATxt} +%\unixcommand{(Postscript)}{ghostview \ \htbmdir{}/anna.ps} +%\blankline +%\menumemolink{How to use the NAGLINK}{nagLinkIntroPage} +%\blankline +%\menumemolink{Tutorial}{tutorialIntroPage} +%\blankline +%\item \menulispdownlink{Interpolation}{(|interp1|} +\endmenu +\indent{0} +\end{scroll} +%\unixcommand{Netscape}{netscape \ http:\/\/www.bath.ac.uk\/\~masbjd\/anna.html} +\autobutt{HelpContents} +\end{page} + +@ +\subsection{Integration} +\label{UXANNAInt} +\index{pages!UXANNAInt!annaex.ht} +\index{annaex.ht!pages!UXANNAInt} +\index{UXANNAInt!annaex.ht!pages} +<>= +\begin{page}{UXANNAInt}{Integration} +Welcome to the Integration section of {\tt +\inputbitmap{\htbmdir{}/anna.xbm.tiny}}, the {\em AXIOM/NAG Expert +System}. This system chooses, and uses, NAG numerical routines. +\begin{scroll} +\blankline +\indent{2} +\beginmenu +\item \menulispdownlink{Integration}{(|annaInt|)}\space{}\newline +\indent{5} Integrating a function over a finite or infinite range. +\blankline +\item \menulispdownlink{Multiple Integration}{(|annaMInt|)}\space{}\newline +\indent{5} Integrating a multivariate function over a finite space. +The dimensions of the space need to be 2 <= n <= 15. +\blankline +\menudownlink{Examples}{UXANNAIntEx}\space{}\newline +\indent{5} Examples of integration. These examples cover all of the major +methods. Parameters can be changed to investigate the effect +on the choice of method. +\endmenu +\indent{0} +\end{scroll} +\autobutt{MainHelp} +\end{page} + +@ +\subsection{Ordinary Differential Equations} +\label{UXANNAOde} +\index{pages!UXANNAOde!annaex.ht} +\index{annaex.ht!pages!UXANNAOde} +\index{UXANNAOde!annaex.ht!pages} +<>= +\begin{page}{UXANNAOde}{Ordinary Differential Equations} +Welcome to the Ordinary Differential Equations section of {\tt +\inputbitmap{\htbmdir{}/anna.xbm.tiny}}, the +{\em AXIOM/NAG Expert System}. +This system chooses, and uses, NAG numerical routines. +\begin{scroll} +\blankline +\blankline +\indent{2} +\beginmenu +\item \menulispdownlink{Ordinary Differential Equations}{(|annaOde|)}\space{}\newline +\indent{5} Finding a solution to an Initial Value Problem of a set of Ordinary Differential Equations. +\blankline +\menudownlink{Examples}{UXANNAOdeEx}\newline +\indent{5} Examples of ODE problems with various features using both stiff +and non-stiff methods. Parameters can be changed to investigate the effect +on the choice of method. +\autobutt{MainHelp} +\endmenu +\indent{0} +\end{scroll} +\end{page} + +@ +\subsection{Optimization} +\label{UXANNAOpt} +\index{pages!UXANNAOpt!annaex.ht} +\index{annaex.ht!pages!UXANNAOpt} +\index{UXANNAOpt!annaex.ht!pages} +<>= +\begin{page}{UXANNAOpt}{Optimization} +Welcome to the Optimization section of {\tt +\inputbitmap{\htbmdir{}/anna.xbm.tiny}}, the {\em AXIOM/NAG Expert System}. +This system chooses, and uses, NAG numerical routines. +\begin{scroll} +\blankline +\indent{2} +\beginmenu +\item \menulispdownlink{Optimization of a Single Multivariate Function} +{(|annaOpt|)}\space{}\newline +\indent{6} Finding the minimum of a function in n variables. +\newline +\indent{6} Linear Programming and Quadratic Programming problems. +\blankline +\indent{4} +\beginmenu +\menudownlink{Examples}{UXANNAOptEx}\newline +\indent{8} Examples of optimization problems with various constraint features. +\endmenu +\blankline +\item \menulispdownlink{Optimization of a set of observations of a data set} +{(|annaOpt2|)}\space{}\newline +\indent{6} Least-squares problems. +\newline +\indent{6} Checking the goodness of fit of a least-squares model. +\blankline +\indent{4} +\beginmenu +\menudownlink{Examples}{UXANNAOpt2Ex}\newline +\indent{8} Examples of least squares problems. +\endmenu +\endmenu +\indent{0} +\end{scroll} +\autobutt{MainHelp} +\end{page} + +@ +\subsection{Partial Differential Equations} +\label{UXANNAPde} +\index{pages!UXANNAPde!annaex.ht} +\index{annaex.ht!pages!UXANNAPde} +\index{UXANNAPde!annaex.ht!pages} +<>= +\begin{page}{UXANNAPde}{Partial Differential Equations} +Welcome to the Partial Differential Equations section of {\tt +\inputbitmap{\htbmdir{}/anna.xbm.tiny}}, the +{\em AXIOM/NAG Expert System}. +\begin{scroll} +\indent{2} +\beginmenu +\menulispdownlink{Second Order Elliptic Partial Differential Equation}{(|annaPDESolve|)} +\newline +\indent{4} Discretizing the PDE: +\newline +\centerline{\inputbitmap{\htbmdir{}/d03eef.xbm}} +defined on a rectangular region with boundary conditions of the form +\centerline{\inputbitmap{\htbmdir{}/d03eef1.bitmap}} +and solving the resulting +seven-diagonal finite difference equations using a multigrid technique. +\blankline +%\menulispdownlink{Helmholtz Equation in 3-D, Cartesian Coordinates}{(|d03fafVars|)} +%\newline +%\indent{4} Descretizing the PDE: +%\newline +%\centerline{\inputbitmap{\htbmdir{}/d03faf.xbm}} +%and solving the resulting +%seven-diagonal finite difference equations using a method based on the Fast +%Fourier Transform. +\endmenu +\end{scroll} +\autobutt{MainHelp} +\end{page} + +@ +\subsection{Examples Using the AXIOM/NAG Expert System} +\label{UXANNAOptEx} +\index{pages!UXANNAOptEx!annaex.ht} +\index{annaex.ht!pages!UXANNAOptEx} +\index{UXANNAOptEx!annaex.ht!pages} +<>= +\begin{page}{UXANNAOptEx}{Examples Using the AXIOM/NAG Expert System} +\begin{scroll} +Select any of these examples and you will be presented with a page which +contains active areas for the function and its parameters. +\blankline +These parameters can be altered by selecting the area and replacing the +default parameters by the new values. +\blankline +\beginmenu +\item \menulispdownlink{Example 1: \newline +\indent{2} Minimize the function: +\centerline{\inputbitmap{\htbmdir{}/opt1.xbm}}}{(|annaOptDefaultSolve1|)}\space{} +\blankline +\item \menulispdownlink{Example 2: \newline +\indent{2} Minimize the function: +\centerline{\inputbitmap{\htbmdir{}/opt2.xbm}}}{(|annaOptDefaultSolve2|)}\space{} +\newline \indent{3} With conditions: +\centerline{\inputbitmap{\htbmdir{}/opt2c.xbm}} +\blankline +\item \menulispdownlink{Example 3: \newline +\indent{2} Minimize the function: +\centerline{\inputbitmap{\htbmdir{}/opt3.xbm}}}{(|annaOptDefaultSolve3|)}\space{} +\newline \indent{3} With conditions: +\centerline{\inputbitmap{\htbmdir{}/opt3c1.xbm}} +\centerline{\inputbitmap{\htbmdir{}/opt3c2.xbm}} +\blankline +\item \menulispdownlink{Example 4: \newline +\indent{2} Minimize the function: +\centerline{\inputbitmap{\htbmdir{}/opt4.xbm}} +}{(|annaOptDefaultSolve4|)}\space{} +\newline \indent{3} With conditions: +\centerline{\inputbitmap{\htbmdir{}/opt4c1.xbm}} +\centerline{\inputbitmap{\htbmdir{}/opt4c2.xbm}} +\centerline{\inputbitmap{\htbmdir{}/opt4c3.xbm}} +\blankline +\item \menulispdownlink{Example 5: \newline +\indent{2} Minimize the function: +\centerline{\inputbitmap{\htbmdir{}/opt5.xbm}} +}{(|annaOptDefaultSolve5|)}\space{} +\newline \indent{3} With conditions: +\centerline{\inputbitmap{\htbmdir{}/opt4c1.xbm}} +\centerline{\inputbitmap{\htbmdir{}/opt4c2.xbm}} +\centerline{\inputbitmap{\htbmdir{}/opt4c3.xbm}} +\blankline +\endmenu +\end{scroll} +\autobutt{Mainhelp} +\end{page} + +@ +\subsection{Examples Using the AXIOM/NAG Expert System} +\label{UXANNAOpt2Ex} +\index{pages!UXANNAOpt2Ex!annaex.ht} +\index{annaex.ht!pages!UXANNAOpt2Ex} +\index{UXANNAOpt2Ex!annaex.ht!pages} +<>= +\begin{page}{UXANNAOpt2Ex}{Examples Using the AXIOM/NAG Expert System} +\begin{scroll} +Select this example and you will be presented with a page which +contains active areas for the function and its parameters. +\blankline +These parameters can be altered by selecting the area and replacing the +default parameters by the new values. +\blankline +\beginmenu +\blankline +\item \menulispdownlink{Example 1: \newline +\indent{2} Calculate a least-squares minimization of the following functions: +\centerline{\inputbitmap{\htbmdir{}/opt61.xbm}} +\centerline{\inputbitmap{\htbmdir{}/opt62.xbm}} +\centerline{\inputbitmap{\htbmdir{}/opt63.xbm}}} +{(|annaOpt2DefaultSolve|)}\space{} +\endmenu +\end{scroll} +\autobutt{MainHelp} +\end{page} + + +@ +\subsection{Examples Using the AXIOM/NAG Expert System} +\label{UXANNAIntEx} +\index{pages!UXANNAIntEx!annaex.ht} +\index{annaex.ht!pages!UXANNAIntEx} +\index{UXANNAIntEx!annaex.ht!pages} +<>= +\begin{page}{UXANNAIntEx}{Examples Using the AXIOM/NAG Expert System} +\begin{scroll} +Select any of these examples and you will be presented with a page which +contains active areas for the function and its parameters. +\blankline +These parameters can be altered by selecting the area and replacing the +default parameters by the new values. In this way you can investigate the +effect of the new parameters on the choice of method. +\blankline +\beginmenu +\item \menulispdownlink{Example 1: \newline +\centerline{\inputbitmap{\htbmdir{}/int1.xbm}}}{(|annaFoo|)}\space{} +\blankline +\item \menulispdownlink{Example 2: \newline +\centerline{\inputbitmap{\htbmdir{}/int2.xbm}}}{(|annaBar|)}\space{} +\blankline +\item \menulispdownlink{Example 3: \newline +\centerline{\inputbitmap{\htbmdir{}/int3.xbm}}}{(|annaJoe|)}\space{} +\blankline +\item \menulispdownlink{Example 4: \newline +\centerline{\inputbitmap{\htbmdir{}/int4.xbm}}}{(|annaSue|)}\space{} +\blankline +\item \menulispdownlink{Example 5: \newline +\centerline{\inputbitmap{\htbmdir{}/int5.xbm}}}{(|annaAnn|)}\space{} +\blankline +\item \menulispdownlink{Example 6: \newline +\centerline{\inputbitmap{\htbmdir{}/int6.xbm}}}{(|annaBab|)}\space{} +\blankline +\item \menulispdownlink{Example 7: \newline +\centerline{\inputbitmap{\htbmdir{}/int7.xbm}}}{(|annaFnar|)}\space{} +\blankline +\item \menulispdownlink{Example 8: \newline +\centerline{\inputbitmap{\htbmdir{}/int8.xbm}}}{(|annaDan|)}\space{} +\blankline +\item \menulispdownlink{Example 9: \newline +\centerline{\inputbitmap{\htbmdir{}/int9.xbm}}}{(|annaBlah|)}\space{} +\blankline +\item \menulispdownlink{Example 10: \newline +\centerline{\inputbitmap{\htbmdir{}/int10.xbm}}}{(|annaTub|)}\space{} +\blankline +\item \menulispdownlink{Example 11: \newline +\centerline{\inputbitmap{\htbmdir{}/int13.xbm}}}{(|annaRats|)}\space{} +\blankline +\item \menulispdownlink{Example 12: \newline +\centerline{\inputbitmap{\htbmdir{}/int11.xbm}}}{(|annaMInt|)}\space{} +\endmenu +\end{scroll} +\autobutt{MainHelp} +\end{page} + +@ +\subsection{Examples Using the AXIOM/NAG Expert System} +\label{UXANNAOdeEx} +\index{pages!UXANNAOdeEx!annaex.ht} +\index{annaex.ht!pages!UXANNAOdeEx} +\index{UXANNAOdeEx!annaex.ht!pages} +<>= +\begin{page}{UXANNAOdeEx}{Examples Using the AXIOM/NAG Expert System} +\begin{scroll} +Analyses the function for various attributes, chooses and +then uses a suitable ODE solver to provide a +solution to the system of n ODEs \center{\htbitmap{d02gaf},} +for i = 1,2,...,n. +\blankline +Select either of these examples and you will be presented with a page which +contains active areas for the function and its parameters. +\blankline +These parameters can be altered by selecting the area and replacing the +default parameters by the new values. In this way you can investigate the +effect of the new parameters on the choice of method. +\blankline +\beginmenu +\item \menulispdownlink{Example 1: \tab{12} +\inputbitmap{\htbmdir{}/ode1.xbm}}{(|annaOdeDefaultSolve1|)} +\blankline with initial conditions: \newline +\tab{12}\inputbitmap{\htbmdir{}/y1.xbm} \space{1} and \space{1} +\inputbitmap{\htbmdir{}/x1.xbm} +\blankline +\blankline +\blankline +\item \menulispdownlink{Example 2: \tab{12} +\inputbitmap{\htbmdir{}/ode2.xbm}}{(|annaOdeDefaultSolve2|)} +\blankline with initial conditions: \newline +\tab{12}\inputbitmap{\htbmdir{}/y2.xbm} \space{1} and \space{1} +\inputbitmap{\htbmdir{}/x1.xbm} +\blankline +\blankline +\blankline +\endmenu +\end{scroll} +\autobutt{MainHelp} +\end{page} + +@ +\subsection{About the AXIOM/NAG Expert System} +\label{UXANNATxt} +\begin{itemize} +\item UXANNAEx \ref{UXANNAEx} on page~\pageref{UXANNAEx} +\item UXANNAIntro \ref{UXANNAIntro} on page~\pageref{UXANNAIntro} +\item UXANNADec \ref{UXANNADec} on page~\pageref{UXANNADec} +\item UXANNAInfer \ref{UXANNAInfer} on page~\pageref{UXANNAInfer} +\item UXANNAMeas \ref{UXANNAMeas} on page~\pageref{UXANNAMeas} +\end{itemize} +\index{pages!UXANNATxt!annaex.ht} +\index{annaex.ht!pages!UXANNATxt} +\index{UXANNATxt!annaex.ht!pages} +<>= +\begin{page}{UXANNATxt}{About the AXIOM/NAG Expert System} +\begin{scroll} +\centerline{\tt\inputbitmap{\htbmdir{}/anna_logo.xbm}\rm} +\vspace{-30}\horizontalline +In applied mathematics, electronic and chemical engineering, the modelling +process can produce a number of mathematical problems which require numerical +solutions for which symbolic methods are either not possible or not obvious. +With the plethora of numerical library routines for the solution of these +problems often the numerical analyst has to answer the question {\em Which +routine to choose?} +\blankline +Some analysis needs to be carried out before the +appropriate routine can be identified i.e. {\em How stiff is this ODE?} and +{\em Is this function continuous?} It may well be the case that more than +one routine is applicable to the problem. So the question may become {\em +Which is likely to be the best?} Such a choice may be critical for both +accuracy and efficiency. +\blankline +An expert system is thus required to make this choice based on the result of +its own analysis of the problem, call the routine and act on the outcome. +This may be to put the answer in a relevant form or react to an apparent +failure of the chosen routine and thus choose and call an alternative. +It should also have sufficient explanation mechanisms to inform on the choice +of routine and the reasons for that choice. +\blankline +\end{scroll} +\downlink{ Examples }{UXANNAEx} +\downlink{ Introduction }{UXANNAIntro} +\downlink{ Decision Agents }{UXANNADec} +\downlink{ Inference Mechanisms }{UXANNAInfer} +\downlink{ Measure Functions }{UXANNAMeas} +\end{page} + +@ +\subsection{Introduction to the AXIOM/NAG Expert System} +\label{UXANNAIntro} +\begin{itemize} +\item UXANNADec \ref{UXANNADec} on page~\pageref{UXANNADec} +\item UXANNAInfer \ref{UXANNAInfer} on page~\pageref{UXANNAInfer} +\item UXANNAMeth \ref{UXANNAMeth} on page~\pageref{UXANNAMeth} +\item UXANNAMeas \ref{UXANNAMeas} on page~\pageref{UXANNAMeas} +\end{itemize} +\index{pages!UXANNAIntro!annaex.ht} +\index{annaex.ht!pages!UXANNAIntro} +\index{UXANNAIntro!annaex.ht!pages} +<>= +\begin{page}{UXANNAIntro}{Introduction to the AXIOM/NAG Expert System} +\begin{scroll} +\centerline{\tt\inputbitmap{\htbmdir{}/anna_logo.xbm}\rm} +\vspace{-30}\horizontalline +Deciding amongst, and then implementing, several possible approaches to +solving a numerical problem can be daunting for a novice user, or tedious for +an expert. Different attributes of the problem need to be +identified and their possible interactions weighed up before a final decision +about which method to use can be made. +\blankline +The implementation is then largely an +automatic, if laborious, process of writing, compiling and linking usually +Fortran code. The aim is to build an expert system which will use computer +algebra to analyse such features of a problem, inference mechanisms and a +knowledge base to choose a numerical method appropriate to the solution of a +given problem. +\blankline +Any interactive system is constrained by the need to provide a reasonable +response time for the user. Given the complexity of some of the analysis our +system will need to do, it is clear that we should only aim to select a good +method, rather than try to identify the best one available. The overall goal +is to provide a ``black-box'' interface to numerical software which allows +non-experts access to its full potential. It will also provide explanation +mechanisms commensurate with its role as a teaching aid. +\blankline +Given, say, an integration to perform (which may or may not be able to be +handled symbolically), the system should choose and apply an appropriate +method, thus mirroring as closely as possible the way that an experienced +numerical analyst would think so, for example, given an integration to +perform:\newline +{\it \centerline{\inputbitmap{\htbmdir{}/int1.xbm}}} +\newline +the experienced analyst would see that the integral is semi-infinite and +transform it by splitting the range and transforming the integral over {\it +[1,\inputbitmap{\htbmdir{}/infty.xbm}]} into an integral over +{\it [0,1] } using the transformation {\it x -> 1/t}. +A different numerical routine might be used over each +sub-region and the results added to give the final answer. +\blankline +It then requires +the translation of the problem into Fortran code which may be extensive. +Even with this simple example, the process is quite involved. +\blankline +\end{scroll} +\autobuttons +\downlink{ Decision Agents }{UXANNADec} +\downlink{ Inference Mechanisms }{UXANNAInfer} +\downlink{ Method Domains }{UXANNAMeth} +\downlink{ Measure Functions }{UXANNAMeas} +\end{page} + +@ +\subsection{Example using the AXIOM/NAG Expert System} +\label{UXANNAEx} +See UXANNAEx2 \ref{UXANNAEx2} on page~\pageref{UXANNAEx2} +\index{pages!UXANNAEx!annaex.ht} +\index{annaex.ht!pages!UXANNAEx} +\index{UXANNAEx!annaex.ht!pages} +<>= +\begin{page}{UXANNAEx}{Example using the AXIOM/NAG Expert System} +\begin{scroll} +\xtc{ +{\bf Example 1}: The integral +{\centerline{\inputbitmap{\htbmdir{}/int1.xbm}}} +\newline +is performed as follows: +\blankline +}{} +\xtc{ +}{ +\spadpaste{ans := integrate((exp(-X^3)+exp(-3*X^2))/sqrt(X),0.0..\%plusInfinity)\bound{ans} } +} +\blankline +\xtc{ +It creates a composite structure for which the field containing the result can be +expanded as required.\blankline +}{ +\spadpaste{ans . 'result\free{ans}} +} +\blankline +\xtc{ +}{ +\spadpaste{ans . 'abserr\free{ans}} +} +\blankline +This system has performed the analysis described above, done the necessary +problem transformation, written any necessary Fortran, called two different +numerical routines, and amalgamated their +results. This whole process was transparent to the user. +\end{scroll} +\autobuttons +\downlink{Example 2}{UXANNAEx2} +%\downlink{Decision Agents}{UXANNADec} +\end{page} + +@ +\subsection{Example using the AXIOM/NAG Expert System} +\label{UXANNAEx2} +See UXANNAEx3 \ref{UXANNAEx3} on page~\pageref{UXANNAEx3} +\index{pages!UXANNAEx2!annaex.ht} +\index{annaex.ht!pages!UXANNAEx2} +\index{UXANNAEx2!annaex.ht!pages} +<>= +\begin{page}{UXANNAEx2}{Example using the AXIOM/NAG Expert System} +\begin{scroll} +\xtc{ +{\bf Example 2}: The ODE +{\centerline{\inputbitmap{\htbmdir{}/ode3.xbm}\space{1}with +\space{1} +{\inputbitmap{\htbmdir{}/y3.xbm}}}} +\newline +could be solved as follows: +\blankline +}{} +\xtc{ +}{ +\spadpaste{ans2 := solve([Y[2],-1001*Y[2]-1000*Y[1]], 0.0, 10.0, +[1.0,-1.0], [2,4,6,8], 1.0e-4)\bound{ans2} } +} +\blankline +\xtc{ +It creates a composite structure for which the field containing the result can be +expanded as required.\blankline +}{ +\spadpaste{ans2 . 'result\free{ans2}} +} +\blankline +\xtc{ +}{ +\spadpaste{ans2 . 'y\free{ans2}} +} +\blankline +\end{scroll} +\autobuttons +\downlink{Example 3}{UXANNAEx3} +%\downlink{Decision Agents}{UXANNADec} +\end{page} + +@ +\subsection{Example using the AXIOM/NAG Expert System} +\label{UXANNAEx3} +See UXANNADec \ref{UXANNADec} on page~\pageref{UXANNADec} +\index{pages!UXANNAEx3!annaex.ht} +\index{annaex.ht!pages!UXANNAEx3} +\index{UXANNAEx3!annaex.ht!pages} +<>= +\begin{page}{UXANNAEx3}{Example using the AXIOM/NAG Expert System} +\begin{scroll} +\xtc{ +{\bf Example 3}: The function +{\centerline{\inputbitmap{\htbmdir{}/opt2.xbm}}} +with simple bounds +{\centerline{\inputbitmap{\htbmdir{}/opt2c.xbm}}} +\newline +could be minimized as follows: +\blankline +}{} +\xtc{ +}{ +\spadpaste{ans3 := optimize((X[1]+10*X[2])**2 + 5*(X[3]-X[4])**2 + +(X[2]-2*X[3])**4 + 10*(X[1]-X[4])**4, [3,-1,0,1], [1,-2,\%minusInfinity,1], +[3,0,\%plusInfinity,3])\bound{ans3} } +} +\blankline +\xtc{ +It creates a composite structure for which the field containing the minimum can be +expanded as required.\blankline +}{ +\spadpaste{ans3 . objf\free{ans3}} +} +\blankline +\xtc{ +}{ +\spadpaste{ans3 . x\free{ans3}} +} +\blankline +\xtc{ +}{ +\spadpaste{ans3 . attributes\free{ans3}} +} +\blankline +\end{scroll} +\autobuttons +\downlink{Decision Agents}{UXANNADec} +\end{page} + +@ +\subsection{Decision Agents} +\label{UXANNADec} +\begin{itemize} +\item UXANNAInfer \ref{UXANNAInfer} on page~\pageref{UXANNAInfer} +\item UXANNAMeth \ref{UXANNAMeth} on page~\pageref{UXANNAMeth} +\item UXANNAMeas \ref{UXANNAMeas} on page~\pageref{UXANNAMeas} +\item UXANNAAgent \ref{UXANNAAgent} on page~\pageref{UXANNAAgent} +\end{itemize} +\index{pages!UXANNADec!annaex.ht} +\index{annaex.ht!pages!UXANNADec} +\index{UXANNADec!annaex.ht!pages} +<>= +\begin{page}{UXANNADec}{Decision Agents} +\begin{scroll} +\blankline +Some features are either present or absent in a problem. Examples of such +binary decisions include {\em is a matrix symmetric?} and {\em is a +function continuous?} However in practice many questions are about the {\em +degree} to which a problem exhibits a property: {\em how much does a +function oscillate?}, or {\em how stiff are these differential equations?} +\blankline +We have therefore created decision agents of two types, reflecting their +property --- {\em Binary Agents} are Boolean functions returning either true +or false and {\em Intensity Functions} are quantitative and return a range of +different values, either numerical or structured types. The framework we are +developing is able to deal with both these forms of information. +\blankline + +In any given problem area (for example solving ordinary differential +equations, optimization etc.) we have a selection of {\em methods}. These +might be to use a particular NAG routine, or they might involve employing a +higher-level strategy such as transforming the problem into an equivalent, +but easier to solve, form. +\blankline +Associated with every method we define a {\em +measure function} which assesses the suitability of that method to a +particular problem. Each measure function has access to a range of symbolic +{\em agents} which can answer questions about the various properties of the +problem in hand. +\blankline +\end{scroll} +\downlink{ Inference Mechanisms }{UXANNAInfer} +\downlink{ Method Domains }{UXANNAMeth} +\downlink{ Measure Functions }{UXANNAMeas} +\downlink{ Computational Agents }{UXANNAAgent} + +\end{page} + +@ +\subsection{Inference Mechanisms} +\label{UXANNAInfer} +\begin{itemize} +\item UXANNAMeth \ref{UXANNAMeth} on page~\pageref{UXANNAMeth} +\item UXANNAMeas \ref{UXANNAMeas} on page~\pageref{UXANNAMeas} +\item UXANNAAgent \ref{UXANNAAgent} on page~\pageref{UXANNAAgent} +\item UXANNAEx \ref{UXANNAEx} on page~\pageref{UXANNAEx} +\end{itemize} +\index{pages!UXANNAInfer!annaex.ht} +\index{annaex.ht!pages!UXANNAInfer} +\index{UXANNAInfer!annaex.ht!pages} +<>= +\begin{page}{UXANNAInfer}{Inference Mechanisms} +\begin{scroll} +\blankline +The inference machine will take the problem description as provided by the +user and perform an initial analysis to verify its validity. It will +consider, in turn, all of the available methods within its knowledge base +which might solve that problem. In doing so it analyses the input problem to +find out about any attributes that could affect the ability of the methods +under consideration to perform effectively. +\blankline +Some of these +measures may use lazy evaluation in the sense that, if a method already +assessed is believed to be a good candidate, and if evaluating the current +measure will be relatively expensive, then that measure will not be evaluated +unless later evidence shows that the selected method is not, in fact, a +successful strategy, for example if it has failed. +\end{scroll} +\downlink{ Method Domains }{UXANNAMeth} +\downlink{ Measure Functions }{UXANNAMeas} +\downlink{ Computational Agents }{UXANNAAgent} +\downlink{ Examples }{UXANNAEx} +\end{page} + +@ +\subsection{Method Domains} +\label{UXANNAMeth} +\begin{itemize} +\item UXANNAMeas \ref{UXANNAMeas} on page~\pageref{UXANNAMeas} +\item UXANNAAgent \ref{UXANNAAgent} on page~\pageref{UXANNAAgent} +\item UXANNAEx \ref{UXANNAEx} on page~\pageref{UXANNAEx} +\end{itemize} +\index{pages!UXANNAMeth!annaex.ht} +\index{annaex.ht!pages!UXANNAMeth} +\index{UXANNAMeth!annaex.ht!pages} +<>= +\begin{page}{UXANNAMeth}{Method Domains} +\begin{scroll} +\blankline +An AXIOM {\em domain} has been created for each method or strategy for +solving the problem. These method domains each implement two functions with +a uniform (method independant) interface. +\blankline {\bf measure:} A function which calculates an estimate of suitability of +this particular method to the problem if there is a possibility that the +method under consideration is more appropriate than one already investigated. +\blankline +If it may be possible to improve on the current favourite method, the function +will call computational agents to analyse the problem for specific features +and calculate the measure from the results these agents return, +using a variation on the Lucks/Gladwell intensity and compatibility +model if conflict between attributes, as investigated by these computational +agents, may be present. +\blankline +{\bf implementation:} A function which may be one of two distinct kinds. +The first kind uses the interface to the NAG Library to call a particular +routine with the required parameters. Some of the parameters may need to be +calculated from the data provided before the external function call. +\blankline +The other kind will apply a ``high level'' strategy to try to solve the +problem e.g.~a transformation of an expression from one that is difficult to +solve to one which is easier, or a splitting of the problem into several more +easily solvable parts. For example, for a solution of the equation above, +since the integral is semi-infinite we might wish to transform the range by, +say, using the mapping {\it y -> 1/x} on the section {\it 1 +< x < \inputbitmap{\htbmdir{}/infty.xbm}}) and +adding the result to the unmapped section {\it 0 < x < 1}. +\blankline +\end{scroll} +\downlink{ Measure Functions }{UXANNAMeas} +\downlink{ Computational Agents }{UXANNAAgent} +\downlink{ Examples }{UXANNAEx} +\end{page} + +@ +\subsection{Measure Functions} +\label{UXANNAMeas} +\begin{itemize} +\item UXANNAAgent \ref{UXANNAAgent} on page~\pageref{UXANNAAgent} +\item UXANNAEx \ref{UXANNAEx} on page~\pageref{UXANNAEx} +\end{itemize} +\index{pages!UXANNAMeas!annaex.ht} +\index{annaex.ht!pages!UXANNAMeas} +\index{UXANNAMeas!annaex.ht!pages} +<>= +\begin{page}{UXANNAMeas}{Measure Functions} +\begin{scroll} +\blankline +Each measure function will estimate the ability of a particular method to +solve a problem. It will consult whichever agents are needed to perform +analysis on the problem in order to calculate the measure. There is a +parameter which would contain the best compatibility +value found so far. +\blankline +However, the interpretation we give to the results of some tests is not +always clear-cut. If a set of tests give +conflicting advice as to the appropriateness of a particular method, it +becomes important to decide not only {\it whether} certain properties are +present but also their {\it degree}. This gives us a basis for estimating the +compatibility of each property. +\blankline +We have taken for our model the system recommended by Lucks and Gladwell +which uses a system of measurement of compatibility allowing for interaction +and conflict between a number of attributes. All of these processes may not +be required if the choice is clear-cut e.g. we have an integral to calculate +which has a particular singularity structure for which one particular method +has been specifically constructed. However, for more difficult cases a +composite picture should be built up to calculate a true measurement. +\blankline +How the compatibility functions interpret the measurements of various +attributes is up to them and may vary between differing methods. It is this +area that takes as its basis the {\it judgement} of Numerical Analysis +`experts' whether that be from the documentation (which may be deficient in +certain respects) or from alternative sources. However, its assessment of +the suitability or otherwise of a particular method is reflected in a single +normalised value facilitating the direct comparison of the suitability of a +number of possible methods. +\blankline +\end{scroll} +\downlink{ Computational Agents }{UXANNAAgent} +\downlink{ Examples }{UXANNAEx} +\end{page} + +@ +\subsection{Computational Agents} +\label{UXANNAAgent} +See UXANNAEx \ref{UXANNAEx} on page~\pageref{UXANNAEx} +\index{pages!UXANNAAgent!annaex.ht} +\index{annaex.ht!pages!UXANNAAgent} +\index{UXANNAAgent!annaex.ht!pages} +<>= +\begin{page}{UXANNAAgent}{Computational Agents} +\begin{scroll} +\blankline +Computational Agents are those program segments which investigate the +attributes of the input function or functions, such as +{\bf stiffnessAndStabilityOfODEIF} +(the {\em IF} indicates that it is an {\em Intensity Function} i.e. one that +returns a normalised real number or a set of normalised real numbers). They +are usually functions or programs written completely in the Axiom +language and implemented using computer algebra. +\blankline +Some agents will be common to more than one problem domain whereas others +will be specific to a single domain. They also vary greatly in their +complexity. It is a fairly simple task to return details about the range of +a function since this information will have been included in the problem +specification. It is a different order of complexity to return details of +its singularity structure. +\blankline +\xtc{ +As an example, here is a call to the computational agent {\bf +singularitiesOf} to obtain the list of singularities of the function +{\it tan x} which are in the range +{\it 0..12\inputbitmap{\htbmdir{}/pi.xbm}}: +\blankline +}{ +} +\xtc{ +}{ +\spadpaste{s := singularitiesOf(tan x,[x],0..12*\%pi)$ESCONT \free{lib3} } +} +\blankline +Each of these computational agents which may be called by a number of method +domains retain their output in a dynamic hash-table, so speeding the process +and retaining efficiency. +\end{scroll} +\downlink{ Examples }{UXANNAEx} +\end{page} + +@ +\section{array1.ht} +<>= +\newcommand{\OneDimensionalArrayXmpTitle}{OneDimensionalArray} +\newcommand{\OneDimensionalArrayXmpNumber}{9.57} + +@ +\subsection{OneDimensionalArray} +\label{OneDimensionalArrayXmpPage} +\begin{itemize} +\item VectorXmpPage \ref{VectorXmpPage} on +page~pageref{VectorXmpPage} +\item FlexibleArrayXmpPage \ref{FlexibleArrayXmpPage} on +page~pageref{FlexibleArrayXmpPage} +\end{itemize} +\index{pages!OneDimensionalArrayXmpPage!array1.ht} +\index{array1.ht!pages!OneDimensionalArrayXmpPage} +\index{OneDimensionalArrayXmpPage!array1.ht!pages} +<>= +\begin{page}{OneDimensionalArrayXmpPage}{OneDimensionalArray} +\beginscroll +The \spadtype{OneDimensionalArray} domain is used for storing data in a +one-dimensional indexed data structure. +Such an array is a homogeneous data structure in that all the entries of +the array must belong to the same Axiom domain. +Each array has a fixed length specified by the user and arrays are not +extensible. +The indexing of one-dimensional arrays is one-based. +This means that the ``first'' element of an array is given the index +\spad{1}. +See also \downlink{`Vector'}{VectorXmpPage}\ignore{Vector} and +\downlink{`FlexibleArray'}{FlexibleArrayXmpPage}\ignore{FlexibleArray}. +\xtc{ +To create a one-dimensional array, apply the +operation \spadfun{oneDimensionalArray} to a list. +}{ +\spadpaste{oneDimensionalArray [i**2 for i in 1..10]} +} +\xtc{ +Another approach is to first create \spad{a}, a one-dimensional +array of 10 \spad{0}'s. +\spadtype{OneDimensionalArray} has the convenient abbreviation \spadtype{ARRAY1}. +}{ +\spadpaste{a : ARRAY1 INT := new(10,0)\bound{a}} +} +\xtc{ +Set each \spad{i}th element to i, then display the result. +}{ +\spadpaste{for i in 1..10 repeat a.i := i; a\bound{a1}\free{a}} +} +\xtc{ +Square each element by mapping the function +\texht{$i \mapsto i^2$}{i +-> i**2} onto each element. +}{ +\spadpaste{map!(i +-> i ** 2,a); a\bound{a3}\free{a2}} +} +\xtc{ +Reverse the elements in place. +}{ +\spadpaste{reverse! a\bound{a4}\free{a3}} +} +\xtc{ +Swap the \spad{4}th and \spad{5}th element. +}{ +\spadpaste{swap!(a,4,5); a\bound{a5}\free{a4}} +} +\xtc{ +Sort the elements in place. +}{ +\spadpaste{sort! a \bound{a6}\free{a5}} +} +\xtc{ +Create a new one-dimensional array \spad{b} containing the last 5 elements of \spad{a}. +}{ +\spadpaste{b := a(6..10)\bound{b}\free{a6}} +} +\xtc{ +Replace the first 5 elements of \spad{a} with those of \spad{b}. +}{ +\spadpaste{copyInto!(a,b,1)\free{b}} +} + +\endscroll +\autobuttons +\end{page} + +@ +\section{array2.ht} +<>= +\newcommand{\TwoDimensionalArrayXmpTitle}{TwoDimensionalArray} +\newcommand{\TwoDimensionalArrayXmpNumber}{9.82} + +@ +\subsection{TwoDimensionalArray} +\label{TwoDimensionalArrayXmpPage} +\begin{itemize} +\item ugTypesAnyNonePage \ref{ugTypesAnyNonePage} on +page~pageref{ugTypesAnyNonePage} +\item MatrixXmpPage \ref{MatrixXmpPage} on +page~pageref{MatrixXmpPage} +\item OneDimensionalArrayXmpPage \ref{OneDimensionalArrayXmpPage} on +page~pageref{OneDimensionalArrayXmpPage} +\end{itemize} +\index{pages!TwoDimensionalArrayXmpPage!array2.ht} +\index{array2.ht!pages!TwoDimensionalArrayXmpPage} +\index{TwoDimensionalArrayXmpPage!array2.ht!pages} +<>= +\begin{page}{TwoDimensionalArrayXmpPage}{TwoDimensionalArray} +\beginscroll + +The \spadtype{TwoDimensionalArray} domain is used for storing data in a +\twodim{} data structure indexed by row and by column. +Such an array is a homogeneous data structure in that all the entries of +the array must belong to the same Axiom domain (although see +\downlink{``\ugTypesAnyNoneTitle''}{ugTypesAnyNonePage} in +Section \ugTypesAnyNoneNumber\ignore{ugTypesAnyNone}). +Each array has a fixed number of rows and columns specified by the user +and arrays are not extensible. +In Axiom, the indexing of two-dimensional arrays is one-based. +This means that both the ``first'' row of an array and the ``first'' +column of an array are given the index \spad{1}. +Thus, the entry in the upper left corner of an array is in position +\spad{(1,1)}. + + +The operation \spadfunFrom{new}{TwoDimensionalArray} creates +an array with a specified number of rows and columns and fills the components +of that array with a specified entry. +The arguments of this operation specify the number of rows, the number +of columns, and the entry. +\xtc{ +This creates a five-by-four array of integers, all of whose entries are +zero. +}{ +\spadpaste{arr : ARRAY2 INT := new(5,4,0) \bound{arr}} +} +The entries of this array can be set to other integers using +the operation \spadfunFrom{setelt}{TwoDimensionalArray}. + +\xtc{ +Issue this to set the element in the upper left corner of this array to +\spad{17}. +}{ +\spadpaste{setelt(arr,1,1,17) \free{arr}\bound{arr1}} +} +\xtc{ +Now the first element of the array is \spad{17.} +}{ +\spadpaste{arr \free{arr1}} +} +\xtc{ +Likewise, elements of an array are extracted using the operation +\spadfunFrom{elt}{TwoDimensionalArray}. +}{ +\spadpaste{elt(arr,1,1) \free{arr1}} +} +\xtc{ +Another way to use these two operations is as follows. +This sets the element in position \spad{(3,2)} of the array to \spad{15}. +}{ +\spadpaste{arr(3,2) := 15 \free{arr1}\bound{arr2}} +} +\xtc{ +This extracts the element in position \spad{(3,2)} of the array. +}{ +\spadpaste{arr(3,2) \free{arr2}} +} +The operations \spadfunFrom{elt}{TwoDimensionalArray} and +\spadfunFrom{setelt}{TwoDimensionalArray} come equipped with an error +check which verifies that the indices are in the proper ranges. +For example, the above array has five rows and four columns, so if you ask +for the entry in position \spad{(6,2)} with \spad{arr(6,2)} Axiom +displays an error message. +If there is no need for an error check, you can call the operations +\spadfunFrom{qelt}{TwoDimensionalArray} and +\spadfunFromX{qsetelt}{TwoDimensionalArray} which provide the same +functionality but without the error check. +Typically, these operations are called in well-tested programs. + +\xtc{ +The operations \spadfunFrom{row}{TwoDimensionalArray} and +\spadfunFrom{column}{TwoDimensionalArray} extract rows and columns, +respectively, and return objects of \spadtype{OneDimensionalArray} with +the same underlying element type. +}{ +\spadpaste{row(arr,1) \free{arr2}} +} +\xtc{ +}{ +\spadpaste{column(arr,1) \free{arr2}} +} + +\xtc{ +You can determine the dimensions of an array by calling the +operations \spadfunFrom{nrows}{TwoDimensionalArray} and +\spadfunFrom{ncols}{TwoDimensionalArray}, +which return the number of rows and columns, respectively. +}{ +\spadpaste{nrows(arr) \free{arr2}} +} +\xtc{ +}{ +\spadpaste{ncols(arr) \free{arr2}} +} +\xtc{ +To apply an operation to every element of an array, use +\spadfunFrom{map}{TwoDimensionalArray}. +This creates a new array. +This expression negates every element. +}{ +\spadpaste{map(-,arr) \free{arr2}} +} +\xtc{ +This creates an array where all the elements are doubled. +}{ +\spadpaste{map((x +-> x + x),arr) \free{arr2}} +} +\xtc{ +To change the array destructively, use +\spadfunFromX{map}{TwoDimensionalArray} instead of +\spadfunFrom{map}{TwoDimensionalArray}. +If you need to make a copy of any array, use +\spadfunFrom{copy}{TwoDimensionalArray}. +}{ +\spadpaste{arrc := copy(arr) \bound{arrc}\free{arr2}} +} +\xtc{ +}{ +\spadpaste{map!(-,arrc) \free{arrc}} +} +\xtc{ +}{ +\spadpaste{arrc \free{arrc}} +} +\xtc{ +}{ +\spadpaste{arr \free{arr2}} +} + +\xtc{ +Use \spadfunFrom{member?}{TwoDimensionalArray} to see if a given element +is in an array. +}{ +\spadpaste{member?(17,arr) \free{arr2}} +} +\xtc{ +}{ +\spadpaste{member?(10317,arr) \free{arr2}} +} +\xtc{ +To see how many times an element appears in an array, use +\spadfunFrom{count}{TwoDimensionalArray}. +}{ +\spadpaste{count(17,arr) \free{arr2}} +} +\xtc{ +}{ +\spadpaste{count(0,arr) \free{arr2}} +} + +For more information about the operations available for +\spadtype{TwoDimensionalArray}, issue \spadcmd{)show +TwoDimensionalArray}. +For information on related topics, see +\downlink{`Matrix'}{MatrixXmpPage}\ignore{Matrix} and +\downlink{`OneDimensionalArray'}{OneDimensionalArrayXmpPage} +\ignore{OneDimensionalArray}. +\endscroll +\autobuttons +\end{page} +% + +@ +\section{aspex.ht} +\subsection{Asp1 Example Code} +\label{Asp1ExampleCode} +\index{pages!Asp1ExampleCode!aspex.ht} +\index{aspex.ht!pages!Asp1ExampleCode} +\index{Asp1ExampleCode!aspex.ht!pages} +<>= +\begin{page}{Asp1ExampleCode}{Asp1 Example Code} +\begin{verbatim} + DOUBLE PRECISION FUNCTION F(X) + DOUBLE PRECISION X + F=DSIN(X) + RETURN + END +\end{verbatim} +\end{page} + +@ +\subsection{Asp10 Example Code} +\label{Asp10ExampleCode} +\index{pages!Asp10ExampleCode!aspex.ht} +\index{aspex.ht!pages!Asp10ExampleCode} +\index{Asp10ExampleCode!aspex.ht!pages} +<>= +\begin{page}{Asp10ExampleCode}{Asp10 Example Code} +\begin{verbatim} + SUBROUTINE COEFFN(P,Q,DQDL,X,ELAM,JINT) + DOUBLE PRECISION ELAM,P,Q,X,DQDL + INTEGER JINT + P=1.0D0 + Q=((-1.0D0*X**3)+ELAM*X*X-2.0D0)/(X*X) + DQDL=1.0D0 + RETURN + END +\end{verbatim} +\end{page} + + +@ +\subsection{Asp12 Example Code} +\label{Asp12ExampleCode} +\index{pages!Asp12ExampleCode!aspex.ht} +\index{aspex.ht!pages!Asp12ExampleCode} +\index{Asp12ExampleCode!aspex.ht!pages} +<>= +\begin{page}{Asp12ExampleCode}{Asp12 Example Code} +\begin{verbatim} + SUBROUTINE MONIT (MAXIT,IFLAG,ELAM,FINFO) + DOUBLE PRECISION ELAM,FINFO(15) + INTEGER MAXIT,IFLAG + IF(MAXIT.EQ.-1)THEN + PRINT*,"Output from Monit" + ENDIF + PRINT*,MAXIT,IFLAG,ELAM,(FINFO(I),I=1,4) + RETURN + END +\end{verbatim} +\end{page} + +@ +\subsection{Asp19 Example Code} +\label{Asp19ExampleCode} +\index{pages!Asp19ExampleCode!aspex.ht} +\index{aspex.ht!pages!Asp19ExampleCode} +\index{Asp19ExampleCode!aspex.ht!pages} +<>= +\begin{page}{Asp19ExampleCode}{Asp19 Example Code} +\begin{verbatim} + SUBROUTINE LSFUN2(M,N,XC,FVECC,FJACC,LJC) + DOUBLE PRECISION FVECC(M),FJACC(LJC,N),XC(N) + INTEGER M,N,LJC + INTEGER I,J + DO 25003 I=1,LJC + DO 25004 J=1,N + FJACC(I,J)=0.0D0 +25004 CONTINUE +25003 CONTINUE + FVECC(1)=((XC(1)-0.14D0)*XC(3)+(15.0D0*XC(1)-2.1D0)*XC(2)+1.0D0)/( + &XC(3)+15.0D0*XC(2)) + FVECC(2)=((XC(1)-0.18D0)*XC(3)+(7.0D0*XC(1)-1.26D0)*XC(2)+1.0D0)/( + &XC(3)+7.0D0*XC(2)) + FVECC(3)=((XC(1)-0.22D0)*XC(3)+(4.333333333333333D0*XC(1)-0.953333 + &3333333333D0)*XC(2)+1.0D0)/(XC(3)+4.333333333333333D0*XC(2)) + FVECC(4)=((XC(1)-0.25D0)*XC(3)+(3.0D0*XC(1)-0.75D0)*XC(2)+1.0D0)/( + &XC(3)+3.0D0*XC(2)) + FVECC(5)=((XC(1)-0.29D0)*XC(3)+(2.2D0*XC(1)-0.6379999999999999D0)* + &XC(2)+1.0D0)/(XC(3)+2.2D0*XC(2)) + FVECC(6)=((XC(1)-0.32D0)*XC(3)+(1.666666666666667D0*XC(1)-0.533333 + &3333333333D0)*XC(2)+1.0D0)/(XC(3)+1.666666666666667D0*XC(2)) + FVECC(7)=((XC(1)-0.35D0)*XC(3)+(1.285714285714286D0*XC(1)-0.45D0)* + &XC(2)+1.0D0)/(XC(3)+1.285714285714286D0*XC(2)) + FVECC(8)=((XC(1)-0.39D0)*XC(3)+(XC(1)-0.39D0)*XC(2)+1.0D0)/(XC(3)+ + &XC(2)) + FVECC(9)=((XC(1)-0.37D0)*XC(3)+(XC(1)-0.37D0)*XC(2)+1.285714285714 + &286D0)/(XC(3)+XC(2)) + FVECC(10)=((XC(1)-0.58D0)*XC(3)+(XC(1)-0.58D0)*XC(2)+1.66666666666 + &6667D0)/(XC(3)+XC(2)) + FVECC(11)=((XC(1)-0.73D0)*XC(3)+(XC(1)-0.73D0)*XC(2)+2.2D0)/(XC(3) + &+XC(2)) + FVECC(12)=((XC(1)-0.96D0)*XC(3)+(XC(1)-0.96D0)*XC(2)+3.0D0)/(XC(3) + &+XC(2)) + FVECC(13)=((XC(1)-1.34D0)*XC(3)+(XC(1)-1.34D0)*XC(2)+4.33333333333 + &3333D0)/(XC(3)+XC(2)) + FVECC(14)=((XC(1)-2.1D0)*XC(3)+(XC(1)-2.1D0)*XC(2)+7.0D0)/(XC(3)+X + &C(2)) + FVECC(15)=((XC(1)-4.39D0)*XC(3)+(XC(1)-4.39D0)*XC(2)+15.0D0)/(XC(3 + &)+XC(2)) + FJACC(1,1)=1.0D0 + FJACC(1,2)=-15.0D0/(XC(3)**2+30.0D0*XC(2)*XC(3)+225.0D0*XC(2)**2) + FJACC(1,3)=-1.0D0/(XC(3)**2+30.0D0*XC(2)*XC(3)+225.0D0*XC(2)**2) + FJACC(2,1)=1.0D0 + FJACC(2,2)=-7.0D0/(XC(3)**2+14.0D0*XC(2)*XC(3)+49.0D0*XC(2)**2) + FJACC(2,3)=-1.0D0/(XC(3)**2+14.0D0*XC(2)*XC(3)+49.0D0*XC(2)**2) + FJACC(3,1)=1.0D0 + FJACC(3,2)=((-0.1110223024625157D-15*XC(3))-4.333333333333333D0)/( + &XC(3)**2+8.666666666666666D0*XC(2)*XC(3)+18.77777777777778D0*XC(2) + &**2) + FJACC(3,3)=(0.1110223024625157D-15*XC(2)-1.0D0)/(XC(3)**2+8.666666 + &666666666D0*XC(2)*XC(3)+18.77777777777778D0*XC(2)**2) + FJACC(4,1)=1.0D0 + FJACC(4,2)=-3.0D0/(XC(3)**2+6.0D0*XC(2)*XC(3)+9.0D0*XC(2)**2) + FJACC(4,3)=-1.0D0/(XC(3)**2+6.0D0*XC(2)*XC(3)+9.0D0*XC(2)**2) + FJACC(5,1)=1.0D0 + FJACC(5,2)=((-0.1110223024625157D-15*XC(3))-2.2D0)/(XC(3)**2+4.399 + &999999999999D0*XC(2)*XC(3)+4.839999999999998D0*XC(2)**2) + FJACC(5,3)=(0.1110223024625157D-15*XC(2)-1.0D0)/(XC(3)**2+4.399999 + &999999999D0*XC(2)*XC(3)+4.839999999999998D0*XC(2)**2) + FJACC(6,1)=1.0D0 + FJACC(6,2)=((-0.2220446049250313D-15*XC(3))-1.666666666666667D0)/( + &XC(3)**2+3.333333333333333D0*XC(2)*XC(3)+2.777777777777777D0*XC(2) + &**2) + FJACC(6,3)=(0.2220446049250313D-15*XC(2)-1.0D0)/(XC(3)**2+3.333333 + &333333333D0*XC(2)*XC(3)+2.777777777777777D0*XC(2)**2) + FJACC(7,1)=1.0D0 + FJACC(7,2)=((-0.5551115123125783D-16*XC(3))-1.285714285714286D0)/( + &XC(3)**2+2.571428571428571D0*XC(2)*XC(3)+1.653061224489796D0*XC(2) + &**2) + FJACC(7,3)=(0.5551115123125783D-16*XC(2)-1.0D0)/(XC(3)**2+2.571428 + &571428571D0*XC(2)*XC(3)+1.653061224489796D0*XC(2)**2) + FJACC(8,1)=1.0D0 + FJACC(8,2)=-1.0D0/(XC(3)**2+2.0D0*XC(2)*XC(3)+XC(2)**2) + FJACC(8,3)=-1.0D0/(XC(3)**2+2.0D0*XC(2)*XC(3)+XC(2)**2) + FJACC(9,1)=1.0D0 + FJACC(9,2)=-1.285714285714286D0/(XC(3)**2+2.0D0*XC(2)*XC(3)+XC(2)* + &*2) + FJACC(9,3)=-1.285714285714286D0/(XC(3)**2+2.0D0*XC(2)*XC(3)+XC(2)* + &*2) + FJACC(10,1)=1.0D0 + FJACC(10,2)=-1.666666666666667D0/(XC(3)**2+2.0D0*XC(2)*XC(3)+XC(2) + &**2) + FJACC(10,3)=-1.666666666666667D0/(XC(3)**2+2.0D0*XC(2)*XC(3)+XC(2) + &**2) + FJACC(11,1)=1.0D0 + FJACC(11,2)=-2.2D0/(XC(3)**2+2.0D0*XC(2)*XC(3)+XC(2)**2) + FJACC(11,3)=-2.2D0/(XC(3)**2+2.0D0*XC(2)*XC(3)+XC(2)**2) + FJACC(12,1)=1.0D0 + FJACC(12,2)=-3.0D0/(XC(3)**2+2.0D0*XC(2)*XC(3)+XC(2)**2) + FJACC(12,3)=-3.0D0/(XC(3)**2+2.0D0*XC(2)*XC(3)+XC(2)**2) + FJACC(13,1)=1.0D0 + FJACC(13,2)=-4.333333333333333D0/(XC(3)**2+2.0D0*XC(2)*XC(3)+XC(2) + &**2) + FJACC(13,3)=-4.333333333333333D0/(XC(3)**2+2.0D0*XC(2)*XC(3)+XC(2) + &**2) + FJACC(14,1)=1.0D0 + FJACC(14,2)=-7.0D0/(XC(3)**2+2.0D0*XC(2)*XC(3)+XC(2)**2) + FJACC(14,3)=-7.0D0/(XC(3)**2+2.0D0*XC(2)*XC(3)+XC(2)**2) + FJACC(15,1)=1.0D0 + FJACC(15,2)=-15.0D0/(XC(3)**2+2.0D0*XC(2)*XC(3)+XC(2)**2) + FJACC(15,3)=-15.0D0/(XC(3)**2+2.0D0*XC(2)*XC(3)+XC(2)**2) + RETURN + END +\end{verbatim} +\end{page} + +@ +\subsection{Asp20 Example Code} +\label{Asp20ExampleCode} +\index{pages!Asp20ExampleCode!aspex.ht} +\index{aspex.ht!pages!Asp20ExampleCode} +\index{Asp20ExampleCode!aspex.ht!pages} +<>= +\begin{page}{Asp20ExampleCode}{Asp20 Example Code} +\begin{verbatim} + SUBROUTINE QPHESS(N,NROWH,NCOLH,JTHCOL,HESS,X,HX) + DOUBLE PRECISION HX(N),X(N),HESS(NROWH,NCOLH) + INTEGER JTHCOL,N,NROWH,NCOLH + HX(1)=2.0D0*X(1) + HX(2)=2.0D0*X(2) + HX(3)=2.0D0*X(4)+2.0D0*X(3) + HX(4)=2.0D0*X(4)+2.0D0*X(3) + HX(5)=2.0D0*X(5) + HX(6)=(-2.0D0*X(7))+(-2.0D0*X(6)) + HX(7)=(-2.0D0*X(7))+(-2.0D0*X(6)) + RETURN + END +\end{verbatim} +\end{page} + +@ +\subsection{Asp24 Example Code} +\label{Asp24ExampleCode} +\index{pages!Asp24ExampleCode!aspex.ht} +\index{aspex.ht!pages!Asp24ExampleCode} +\index{Asp24ExampleCode!aspex.ht!pages} +<>= +\begin{page}{Asp24ExampleCode}{Asp24 Example Code} +\begin{verbatim} + SUBROUTINE FUNCT1(N,XC,FC) + DOUBLE PRECISION FC,XC(N) + INTEGER N + FC=10.0D0*XC(4)**4+(-40.0D0*XC(1)*XC(4)**3)+(60.0D0*XC(1)**2+5 + &.0D0)*XC(4)**2+((-10.0D0*XC(3))+(-40.0D0*XC(1)**3))*XC(4)+16.0D0*X + &C(3)**4+(-32.0D0*XC(2)*XC(3)**3)+(24.0D0*XC(2)**2+5.0D0)*XC(3)**2+ + &(-8.0D0*XC(2)**3*XC(3))+XC(2)**4+100.0D0*XC(2)**2+20.0D0*XC(1)*XC( + &2)+10.0D0*XC(1)**4+XC(1)**2 + RETURN + END +\end{verbatim} +\end{page} + +@ +\subsection{Asp27 Example Code} +\label{Asp27ExampleCode} +\index{pages!Asp27ExampleCode!aspex.ht} +\index{aspex.ht!pages!Asp27ExampleCode} +\index{Asp27ExampleCode!aspex.ht!pages} +<>= +\begin{page}{Asp27ExampleCode}{Asp27 Example Code} +\begin{verbatim} + FUNCTION DOT(IFLAG,N,Z,W,RWORK,LRWORK,IWORK,LIWORK) + DOUBLE PRECISION W(N),Z(N),RWORK(LRWORK) + INTEGER N,LIWORK,IFLAG,LRWORK,IWORK(LIWORK) + DOT=(W(16)+(-0.5D0*W(15)))*Z(16)+((-0.5D0*W(16))+W(15)+(-0.5D0*W(1 + &4)))*Z(15)+((-0.5D0*W(15))+W(14)+(-0.5D0*W(13)))*Z(14)+((-0.5D0*W( + &14))+W(13)+(-0.5D0*W(12)))*Z(13)+((-0.5D0*W(13))+W(12)+(-0.5D0*W(1 + &1)))*Z(12)+((-0.5D0*W(12))+W(11)+(-0.5D0*W(10)))*Z(11)+((-0.5D0*W( + &11))+W(10)+(-0.5D0*W(9)))*Z(10)+((-0.5D0*W(10))+W(9)+(-0.5D0*W(8)) + &)*Z(9)+((-0.5D0*W(9))+W(8)+(-0.5D0*W(7)))*Z(8)+((-0.5D0*W(8))+W(7) + &+(-0.5D0*W(6)))*Z(7)+((-0.5D0*W(7))+W(6)+(-0.5D0*W(5)))*Z(6)+((-0. + &5D0*W(6))+W(5)+(-0.5D0*W(4)))*Z(5)+((-0.5D0*W(5))+W(4)+(-0.5D0*W(3 + &)))*Z(4)+((-0.5D0*W(4))+W(3)+(-0.5D0*W(2)))*Z(3)+((-0.5D0*W(3))+W( + &2)+(-0.5D0*W(1)))*Z(2)+((-0.5D0*W(2))+W(1))*Z(1) + RETURN + END +\end{verbatim} +\end{page} + +@ +\subsection{Asp28 Example Code} +\label{Asp28ExampleCode} +\index{pages!Asp28ExampleCode!aspex.ht} +\index{aspex.ht!pages!Asp28ExampleCode} +\index{Asp28ExampleCode!aspex.ht!pages} +<>= +\begin{page}{Asp28ExampleCode}{Asp28 Example Code} +\begin{verbatim} + SUBROUTINE IMAGE(IFLAG,N,Z,W,RWORK,LRWORK,IWORK,LIWORK) + DOUBLE PRECISION Z(N),W(N),IWORK(LRWORK),RWORK(LRWORK) + INTEGER N,LIWORK,IFLAG,LRWORK + W(1)=0.01707454969713436D0*Z(16)+0.001747395874954051D0*Z(15)+0.00 + &2106973900813502D0*Z(14)+0.002957434991769087D0*Z(13)+(-0.00700554 + &0882865317D0*Z(12))+(-0.01219194009813166D0*Z(11))+0.0037230647365 + &3087D0*Z(10)+0.04932374658377151D0*Z(9)+(-0.03586220812223305D0*Z( + &8))+(-0.04723268012114625D0*Z(7))+(-0.02434652144032987D0*Z(6))+0. + &2264766947290192D0*Z(5)+(-0.1385343580686922D0*Z(4))+(-0.116530050 + &8238904D0*Z(3))+(-0.2803531651057233D0*Z(2))+1.019463911841327D0*Z + &(1) + W(2)=0.0227345011107737D0*Z(16)+0.008812321197398072D0*Z(15)+0.010 + &94012210519586D0*Z(14)+(-0.01764072463999744D0*Z(13))+(-0.01357136 + &72105995D0*Z(12))+0.00157466157362272D0*Z(11)+0.05258889186338282D + &0*Z(10)+(-0.01981532388243379D0*Z(9))+(-0.06095390688679697D0*Z(8) + &)+(-0.04153119955569051D0*Z(7))+0.2176561076571465D0*Z(6)+(-0.0532 + &5555586632358D0*Z(5))+(-0.1688977368984641D0*Z(4))+(-0.32440166056 + &67343D0*Z(3))+0.9128222941872173D0*Z(2)+(-0.2419652703415429D0*Z(1 + &)) + W(3)=0.03371198197190302D0*Z(16)+0.02021603150122265D0*Z(15)+(-0.0 + &06607305534689702D0*Z(14))+(-0.03032392238968179D0*Z(13))+0.002033 + &305231024948D0*Z(12)+0.05375944956767728D0*Z(11)+(-0.0163213312502 + &9967D0*Z(10))+(-0.05483186562035512D0*Z(9))+(-0.04901428822579872D + &0*Z(8))+0.2091097927887612D0*Z(7)+(-0.05760560341383113D0*Z(6))+(- + &0.1236679206156403D0*Z(5))+(-0.3523683853026259D0*Z(4))+0.88929961 + &32269974D0*Z(3)+(-0.2995429545781457D0*Z(2))+(-0.02986582812574917 + &D0*Z(1)) + W(4)=0.05141563713660119D0*Z(16)+0.005239165960779299D0*Z(15)+(-0. + &01623427735779699D0*Z(14))+(-0.01965809746040371D0*Z(13))+0.054688 + &97337339577D0*Z(12)+(-0.014224695935687D0*Z(11))+(-0.0505181779315 + &6355D0*Z(10))+(-0.04353074206076491D0*Z(9))+0.2012230497530726D0*Z + &(8)+(-0.06630874514535952D0*Z(7))+(-0.1280829963720053D0*Z(6))+(-0 + &.305169742604165D0*Z(5))+0.8600427128450191D0*Z(4)+(-0.32415033802 + &68184D0*Z(3))+(-0.09033531980693314D0*Z(2))+0.09089205517109111D0* + &Z(1) + W(5)=0.04556369767776375D0*Z(16)+(-0.001822737697581869D0*Z(15))+( + &-0.002512226501941856D0*Z(14))+0.02947046460707379D0*Z(13)+(-0.014 + &45079632086177D0*Z(12))+(-0.05034242196614937D0*Z(11))+(-0.0376966 + &3291725935D0*Z(10))+0.2171103102175198D0*Z(9)+(-0.0824949256021352 + &4D0*Z(8))+(-0.1473995209288945D0*Z(7))+(-0.315042193418466D0*Z(6)) + &+0.9591623347824002D0*Z(5)+(-0.3852396953763045D0*Z(4))+(-0.141718 + &5427288274D0*Z(3))+(-0.03423495461011043D0*Z(2))+0.319820917706851 + &6D0*Z(1) + W(6)=0.04015147277405744D0*Z(16)+0.01328585741341559D0*Z(15)+0.048 + &26082005465965D0*Z(14)+(-0.04319641116207706D0*Z(13))+(-0.04931323 + &319055762D0*Z(12))+(-0.03526886317505474D0*Z(11))+0.22295383396730 + &01D0*Z(10)+(-0.07375317649315155D0*Z(9))+(-0.1589391311991561D0*Z( + &8))+(-0.328001910890377D0*Z(7))+0.952576555482747D0*Z(6)+(-0.31583 + &09975786731D0*Z(5))+(-0.1846882042225383D0*Z(4))+(-0.0703762046700 + &4427D0*Z(3))+0.2311852964327382D0*Z(2)+0.04254083491825025D0*Z(1) + W(7)=0.06069778964023718D0*Z(16)+0.06681263884671322D0*Z(15)+(-0.0 + &2113506688615768D0*Z(14))+(-0.083996867458326D0*Z(13))+(-0.0329843 + &8523869648D0*Z(12))+0.2276878326327734D0*Z(11)+(-0.067356038933017 + &95D0*Z(10))+(-0.1559813965382218D0*Z(9))+(-0.3363262957694705D0*Z( + &8))+0.9442791158560948D0*Z(7)+(-0.3199955249404657D0*Z(6))+(-0.136 + &2463839920727D0*Z(5))+(-0.1006185171570586D0*Z(4))+0.2057504515015 + &423D0*Z(3)+(-0.02065879269286707D0*Z(2))+0.03160990266745513D0*Z(1 + &) + W(8)=0.126386868896738D0*Z(16)+0.002563370039476418D0*Z(15)+(-0.05 + &581757739455641D0*Z(14))+(-0.07777893205900685D0*Z(13))+0.23117338 + &45834199D0*Z(12)+(-0.06031581134427592D0*Z(11))+(-0.14805474755869 + &52D0*Z(10))+(-0.3364014128402243D0*Z(9))+0.9364014128402244D0*Z(8) + &+(-0.3269452524413048D0*Z(7))+(-0.1396841886557241D0*Z(6))+(-0.056 + &1733845834199D0*Z(5))+0.1777789320590069D0*Z(4)+(-0.04418242260544 + &359D0*Z(3))+(-0.02756337003947642D0*Z(2))+0.07361313110326199D0*Z( + &1) + W(9)=0.07361313110326199D0*Z(16)+(-0.02756337003947642D0*Z(15))+(- + &0.04418242260544359D0*Z(14))+0.1777789320590069D0*Z(13)+(-0.056173 + &3845834199D0*Z(12))+(-0.1396841886557241D0*Z(11))+(-0.326945252441 + &3048D0*Z(10))+0.9364014128402244D0*Z(9)+(-0.3364014128402243D0*Z(8 + &))+(-0.1480547475586952D0*Z(7))+(-0.06031581134427592D0*Z(6))+0.23 + &11733845834199D0*Z(5)+(-0.07777893205900685D0*Z(4))+(-0.0558175773 + &9455641D0*Z(3))+0.002563370039476418D0*Z(2)+0.126386868896738D0*Z( + &1) + W(10)=0.03160990266745513D0*Z(16)+(-0.02065879269286707D0*Z(15))+0 + &.2057504515015423D0*Z(14)+(-0.1006185171570586D0*Z(13))+(-0.136246 + &3839920727D0*Z(12))+(-0.3199955249404657D0*Z(11))+0.94427911585609 + &48D0*Z(10)+(-0.3363262957694705D0*Z(9))+(-0.1559813965382218D0*Z(8 + &))+(-0.06735603893301795D0*Z(7))+0.2276878326327734D0*Z(6)+(-0.032 + &98438523869648D0*Z(5))+(-0.083996867458326D0*Z(4))+(-0.02113506688 + &615768D0*Z(3))+0.06681263884671322D0*Z(2)+0.06069778964023718D0*Z( + &1) + W(11)=0.04254083491825025D0*Z(16)+0.2311852964327382D0*Z(15)+(-0.0 + &7037620467004427D0*Z(14))+(-0.1846882042225383D0*Z(13))+(-0.315830 + &9975786731D0*Z(12))+0.952576555482747D0*Z(11)+(-0.328001910890377D + &0*Z(10))+(-0.1589391311991561D0*Z(9))+(-0.07375317649315155D0*Z(8) + &)+0.2229538339673001D0*Z(7)+(-0.03526886317505474D0*Z(6))+(-0.0493 + &1323319055762D0*Z(5))+(-0.04319641116207706D0*Z(4))+0.048260820054 + &65965D0*Z(3)+0.01328585741341559D0*Z(2)+0.04015147277405744D0*Z(1) + W(12)=0.3198209177068516D0*Z(16)+(-0.03423495461011043D0*Z(15))+(- + &0.1417185427288274D0*Z(14))+(-0.3852396953763045D0*Z(13))+0.959162 + &3347824002D0*Z(12)+(-0.315042193418466D0*Z(11))+(-0.14739952092889 + &45D0*Z(10))+(-0.08249492560213524D0*Z(9))+0.2171103102175198D0*Z(8 + &)+(-0.03769663291725935D0*Z(7))+(-0.05034242196614937D0*Z(6))+(-0. + &01445079632086177D0*Z(5))+0.02947046460707379D0*Z(4)+(-0.002512226 + &501941856D0*Z(3))+(-0.001822737697581869D0*Z(2))+0.045563697677763 + &75D0*Z(1) + W(13)=0.09089205517109111D0*Z(16)+(-0.09033531980693314D0*Z(15))+( + &-0.3241503380268184D0*Z(14))+0.8600427128450191D0*Z(13)+(-0.305169 + &742604165D0*Z(12))+(-0.1280829963720053D0*Z(11))+(-0.0663087451453 + &5952D0*Z(10))+0.2012230497530726D0*Z(9)+(-0.04353074206076491D0*Z( + &8))+(-0.05051817793156355D0*Z(7))+(-0.014224695935687D0*Z(6))+0.05 + &468897337339577D0*Z(5)+(-0.01965809746040371D0*Z(4))+(-0.016234277 + &35779699D0*Z(3))+0.005239165960779299D0*Z(2)+0.05141563713660119D0 + &*Z(1) + W(14)=(-0.02986582812574917D0*Z(16))+(-0.2995429545781457D0*Z(15)) + &+0.8892996132269974D0*Z(14)+(-0.3523683853026259D0*Z(13))+(-0.1236 + &679206156403D0*Z(12))+(-0.05760560341383113D0*Z(11))+0.20910979278 + &87612D0*Z(10)+(-0.04901428822579872D0*Z(9))+(-0.05483186562035512D + &0*Z(8))+(-0.01632133125029967D0*Z(7))+0.05375944956767728D0*Z(6)+0 + &.002033305231024948D0*Z(5)+(-0.03032392238968179D0*Z(4))+(-0.00660 + &7305534689702D0*Z(3))+0.02021603150122265D0*Z(2)+0.033711981971903 + &02D0*Z(1) + W(15)=(-0.2419652703415429D0*Z(16))+0.9128222941872173D0*Z(15)+(-0 + &.3244016605667343D0*Z(14))+(-0.1688977368984641D0*Z(13))+(-0.05325 + &555586632358D0*Z(12))+0.2176561076571465D0*Z(11)+(-0.0415311995556 + &9051D0*Z(10))+(-0.06095390688679697D0*Z(9))+(-0.01981532388243379D + &0*Z(8))+0.05258889186338282D0*Z(7)+0.00157466157362272D0*Z(6)+(-0. + &0135713672105995D0*Z(5))+(-0.01764072463999744D0*Z(4))+0.010940122 + &10519586D0*Z(3)+0.008812321197398072D0*Z(2)+0.0227345011107737D0*Z + &(1) + W(16)=1.019463911841327D0*Z(16)+(-0.2803531651057233D0*Z(15))+(-0. + &1165300508238904D0*Z(14))+(-0.1385343580686922D0*Z(13))+0.22647669 + &47290192D0*Z(12)+(-0.02434652144032987D0*Z(11))+(-0.04723268012114 + &625D0*Z(10))+(-0.03586220812223305D0*Z(9))+0.04932374658377151D0*Z + &(8)+0.00372306473653087D0*Z(7)+(-0.01219194009813166D0*Z(6))+(-0.0 + &07005540882865317D0*Z(5))+0.002957434991769087D0*Z(4)+0.0021069739 + &00813502D0*Z(3)+0.001747395874954051D0*Z(2)+0.01707454969713436D0* + &Z(1) + RETURN + END +\end{verbatim} +\end{page} + +@ +\subsection{Asp29 Example Code} +\label{Asp29ExampleCode} +\index{pages!Asp29ExampleCode!aspex.ht} +\index{aspex.ht!pages!Asp29ExampleCode} +\index{Asp29ExampleCode!aspex.ht!pages} +<>= +\begin{page}{Asp29ExampleCode}{Asp29 Example Code} +\begin{verbatim} + SUBROUTINE MONIT(ISTATE,NEXTIT,NEVALS,NEVECS,K,F,D) + DOUBLE PRECISION D(K),F(K) + INTEGER K,NEXTIT,NEVALS,NVECS,ISTATE + CALL F02FJZ(ISTATE,NEXTIT,NEVALS,NEVECS,K,F,D) + RETURN + END +\end{verbatim} +\end{page} + +@ +\subsection{Asp30 Example Code} +\label{Asp30ExampleCode} +\index{pages!Asp30ExampleCode!aspex.ht} +\index{aspex.ht!pages!Asp30ExampleCode} +\index{Asp30ExampleCode!aspex.ht!pages} +<>= +\begin{page}{Asp30ExampleCode}{Asp30 Example Code} +\begin{verbatim} + SUBROUTINE APROD(MODE,M,N,X,Y,RWORK,LRWORK,IWORK,LIWORK) + DOUBLE PRECISION X(N),Y(M),RWORK(LRWORK) + INTEGER M,N,LIWORK,IFAIL,LRWORK,IWORK(LIWORK),MODE + DOUBLE PRECISION A(5,5) + EXTERNAL F06PAF + A(1,1)=1.0D0 + A(1,2)=0.0D0 + A(1,3)=0.0D0 + A(1,4)=-1.0D0 + A(1,5)=0.0D0 + A(2,1)=0.0D0 + A(2,2)=1.0D0 + A(2,3)=0.0D0 + A(2,4)=0.0D0 + A(2,5)=-1.0D0 + A(3,1)=0.0D0 + A(3,2)=0.0D0 + A(3,3)=1.0D0 + A(3,4)=-1.0D0 + A(3,5)=0.0D0 + A(4,1)=-1.0D0 + A(4,2)=0.0D0 + A(4,3)=-1.0D0 + A(4,4)=4.0D0 + A(4,5)=-1.0D0 + A(5,1)=0.0D0 + A(5,2)=-1.0D0 + A(5,3)=0.0D0 + A(5,4)=-1.0D0 + A(5,5)=4.0D0 + IF(MODE.EQ.1)THEN + CALL F06PAF('N',M,N,1.0D0,A,M,X,1,1.0D0,Y,1) + ELSEIF(MODE.EQ.2)THEN + CALL F06PAF('T',M,N,1.0D0,A,M,Y,1,1.0D0,X,1) + ENDIF + RETURN + END +\end{verbatim} +\end{page} + +@ +\subsection{Asp31 Example Code} +\label{Asp31ExampleCode} +\index{pages!Asp31ExampleCode!aspex.ht} +\index{aspex.ht!pages!Asp31ExampleCode} +\index{Asp31ExampleCode!aspex.ht!pages} +<>= +\begin{page}{Asp31ExampleCode}{Asp31 Example Code} +\begin{verbatim} + SUBROUTINE PEDERV(X,Y,PW) + DOUBLE PRECISION X,Y(*) + DOUBLE PRECISION PW(3,3) + PW(1,1)=-0.03999999999999999D0 + PW(1,2)=10000.0D0*Y(3) + PW(1,3)=10000.0D0*Y(2) + PW(2,1)=0.03999999999999999D0 + PW(2,2)=(-10000.0D0*Y(3))+(-60000000.0D0*Y(2)) + PW(2,3)=-10000.0D0*Y(2) + PW(3,1)=0.0D0 + PW(3,2)=60000000.0D0*Y(2) + PW(3,3)=0.0D0 + RETURN + END +\end{verbatim} +\end{page} + +@ +\subsection{Asp33 Example Code} +\label{Asp33ExampleCode} +\index{pages!Asp33ExampleCode!aspex.ht} +\index{aspex.ht!pages!Asp33ExampleCode} +\index{Asp33ExampleCode!aspex.ht!pages} +<>= +\begin{page}{Asp33ExampleCode}{Asp33 Example Code} +\begin{verbatim} + SUBROUTINE REPORT(X,V,JINT) + DOUBLE PRECISION V(3),X + INTEGER JINT + RETURN + END +\end{verbatim} +\end{page} + +@ +\subsection{Asp34 Example Code} +\label{Asp34ExampleCode} +\index{pages!Asp34ExampleCode!aspex.ht} +\index{aspex.ht!pages!Asp34ExampleCode} +\index{Asp34ExampleCode!aspex.ht!pages} +<>= +\begin{page}{Asp34ExampleCode}{Asp34 Example Code} +\begin{verbatim} + SUBROUTINE MSOLVE(IFLAG,N,X,Y,RWORK,LRWORK,IWORK,LIWORK) + DOUBLE PRECISION RWORK(LRWORK),X(N),Y(N) + INTEGER I,J,N,LIWORK,IFLAG,LRWORK,IWORK(LIWORK) + DOUBLE PRECISION W1(3),W2(3),MS(3,3) + IFLAG=-1 + MS(1,1)=2.0D0 + MS(1,2)=1.0D0 + MS(1,3)=0.0D0 + MS(2,1)=1.0D0 + MS(2,2)=2.0D0 + MS(2,3)=1.0D0 + MS(3,1)=0.0D0 + MS(3,2)=1.0D0 + MS(3,3)=2.0D0 + CALL F04ASF(MS,N,X,N,Y,W1,W2,IFLAG) + IFLAG=-IFLAG + RETURN + END +\end{verbatim} +\end{page} + +@ +\subsection{Asp35 Example Code} +\label{Asp35ExampleCode} +\index{pages!Asp35ExampleCode!aspex.ht} +\index{aspex.ht!pages!Asp35ExampleCode} +\index{Asp35ExampleCode!aspex.ht!pages} +<>= +\begin{page}{Asp35ExampleCode}{Asp35 Example Code} +\begin{verbatim} + SUBROUTINE FCN(N,X,FVEC,FJAC,LDFJAC,IFLAG) + DOUBLE PRECISION X(N),FVEC(N),FJAC(LDFJAC,N) + INTEGER LDFJAC,N,IFLAG + IF(IFLAG.EQ.1)THEN + FVEC(1)=(-1.0D0*X(2))+X(1) + FVEC(2)=(-1.0D0*X(3))+2.0D0*X(2) + FVEC(3)=3.0D0*X(3) + ELSEIF(IFLAG.EQ.2)THEN + FJAC(1,1)=1.0D0 + FJAC(1,2)=-1.0D0 + FJAC(1,3)=0.0D0 + FJAC(2,1)=0.0D0 + FJAC(2,2)=2.0D0 + FJAC(2,3)=-1.0D0 + FJAC(3,1)=0.0D0 + FJAC(3,2)=0.0D0 + FJAC(3,3)=3.0D0 + ENDIF + END +\end{verbatim} +\end{page} + +@ +\subsection{Asp4 Example Code} +\label{Asp4ExampleCode} +\index{pages!Asp4ExampleCode!aspex.ht} +\index{aspex.ht!pages!Asp4ExampleCode} +\index{Asp4ExampleCode!aspex.ht!pages} +<>= +\begin{page}{Asp4ExampleCode}{Asp4 Example Code} +\begin{verbatim} + DOUBLE PRECISION FUNCTION FUNCTN(NDIM,X) + DOUBLE PRECISION X(NDIM) + INTEGER NDIM + FUNCTN=(4.0D0*X(1)*X(3)**2*DEXP(2.0D0*X(1)*X(3)))/(X(4)**2+(2.0D0* + &X(2)+2.0D0)*X(4)+X(2)**2+2.0D0*X(2)+1.0D0) + RETURN + END +\end{verbatim} +\end{page} + +@ +\subsection{Asp41 Example Code} +\label{Asp41ExampleCode} +\index{pages!Asp41ExampleCode!aspex.ht} +\index{aspex.ht!pages!Asp41ExampleCode} +\index{Asp41ExampleCode!aspex.ht!pages} +<>= +\begin{page}{Asp41ExampleCode}{Asp41 Example Code} +\begin{verbatim} + SUBROUTINE FCN(X,EPS,Y,F,N) + DOUBLE PRECISION EPS,F(N),X,Y(N) + INTEGER N + F(1)=Y(2) + F(2)=Y(3) + F(3)=(-1.0D0*Y(1)*Y(3))+2.0D0*EPS*Y(2)**2+(-2.0D0*EPS) + RETURN + END + SUBROUTINE JACOBF(X,EPS,Y,F,N) + DOUBLE PRECISION EPS,F(N,N),X,Y(N) + INTEGER N + F(1,1)=0.0D0 + F(1,2)=1.0D0 + F(1,3)=0.0D0 + F(2,1)=0.0D0 + F(2,2)=0.0D0 + F(2,3)=1.0D0 + F(3,1)=-1.0D0*Y(3) + F(3,2)=4.0D0*EPS*Y(2) + F(3,3)=-1.0D0*Y(1) + RETURN + END + SUBROUTINE JACEPS(X,EPS,Y,F,N) + DOUBLE PRECISION EPS,F(N),X,Y(N) + INTEGER N + F(1)=0.0D0 + F(2)=0.0D0 + F(3)=2.0D0*Y(2)**2-2.0D0 + RETURN + END +\end{verbatim} +\end{page} + +@ +\subsection{Asp42 Example Code} +\label{Asp42ExampleCode} +\index{pages!Asp42ExampleCode!aspex.ht} +\index{aspex.ht!pages!Asp42ExampleCode} +\index{Asp42ExampleCode!aspex.ht!pages} +<>= +\begin{page}{Asp42ExampleCode}{Asp42 Example Code} +\begin{verbatim} + SUBROUTINE G(EPS,YA,YB,BC,N) + DOUBLE PRECISION EPS,YA(N),YB(N),BC(N) + INTEGER N + BC(1)=YA(1) + BC(2)=YA(2) + BC(3)=YB(2)-1.0D0 + RETURN + END + SUBROUTINE JACOBG(EPS,YA,YB,AJ,BJ,N) + DOUBLE PRECISION EPS,YA(N),AJ(N,N),BJ(N,N),YB(N) + INTEGER N + AJ(1,1)=1.0D0 + AJ(1,2)=0.0D0 + AJ(1,3)=0.0D0 + AJ(2,1)=0.0D0 + AJ(2,2)=1.0D0 + AJ(2,3)=0.0D0 + AJ(3,1)=0.0D0 + AJ(3,2)=0.0D0 + AJ(3,3)=0.0D0 + BJ(1,1)=0.0D0 + BJ(1,2)=0.0D0 + BJ(1,3)=0.0D0 + BJ(2,1)=0.0D0 + BJ(2,2)=0.0D0 + BJ(2,3)=0.0D0 + BJ(3,1)=0.0D0 + BJ(3,2)=1.0D0 + BJ(3,3)=0.0D0 + RETURN + END + SUBROUTINE JACGEP(EPS,YA,YB,BCEP,N) + DOUBLE PRECISION EPS,YA(N),YB(N),BCEP(N) + INTEGER N + BCEP(1)=0.0D0 + BCEP(2)=0.0D0 + BCEP(3)=0.0D0 + RETURN + END +\end{verbatim} +\end{page} + +@ +\subsection{Asp49 Example Code} +\label{Asp49ExampleCode} +\index{pages!Asp49ExampleCode!aspex.ht} +\index{aspex.ht!pages!Asp49ExampleCode} +\index{Asp49ExampleCode!aspex.ht!pages} +<>= +\begin{page}{Asp49ExampleCode}{Asp49 Example Code} +\begin{verbatim} + SUBROUTINE OBJFUN(MODE,N,X,OBJF,OBJGRD,NSTATE,IUSER,USER) + DOUBLE PRECISION X(N),OBJF,OBJGRD(N),USER(*) + INTEGER N,IUSER(*),MODE,NSTATE + OBJF=X(4)*X(9)+((-1.0D0*X(5))+X(3))*X(8)+((-1.0D0*X(3))+X(1))*X(7) + &+(-1.0D0*X(2)*X(6)) + OBJGRD(1)=X(7) + OBJGRD(2)=-1.0D0*X(6) + OBJGRD(3)=X(8)+(-1.0D0*X(7)) + OBJGRD(4)=X(9) + OBJGRD(5)=-1.0D0*X(8) + OBJGRD(6)=-1.0D0*X(2) + OBJGRD(7)=(-1.0D0*X(3))+X(1) + OBJGRD(8)=(-1.0D0*X(5))+X(3) + OBJGRD(9)=X(4) + RETURN + END +\end{verbatim} +\end{page} + +@ +\subsection{Asp50 Example Code} +\label{Asp50ExampleCode} +\index{pages!Asp50ExampleCode!aspex.ht} +\index{aspex.ht!pages!Asp50ExampleCode} +\index{Asp50ExampleCode!aspex.ht!pages} +<>= +\begin{page}{Asp50ExampleCode}{Asp50 Example Code} +\begin{verbatim} + SUBROUTINE LSFUN1(M,N,XC,FVECC) + DOUBLE PRECISION FVECC(M),XC(N) + INTEGER I,M,N + FVECC(1)=((XC(1)-2.4D0)*XC(3)+(15.0D0*XC(1)-36.0D0)*XC(2)+1.0D0)/( + &XC(3)+15.0D0*XC(2)) + FVECC(2)=((XC(1)-2.8D0)*XC(3)+(7.0D0*XC(1)-19.6D0)*XC(2)+1.0D0)/(X + &C(3)+7.0D0*XC(2)) + FVECC(3)=((XC(1)-3.2D0)*XC(3)+(4.333333333333333D0*XC(1)-13.866666 + &66666667D0)*XC(2)+1.0D0)/(XC(3)+4.333333333333333D0*XC(2)) + FVECC(4)=((XC(1)-3.5D0)*XC(3)+(3.0D0*XC(1)-10.5D0)*XC(2)+1.0D0)/(X + &C(3)+3.0D0*XC(2)) + FVECC(5)=((XC(1)-3.9D0)*XC(3)+(2.2D0*XC(1)-8.579999999999998D0)*XC + &(2)+1.0D0)/(XC(3)+2.2D0*XC(2)) + FVECC(6)=((XC(1)-4.199999999999999D0)*XC(3)+(1.666666666666667D0*X + &C(1)-7.0D0)*XC(2)+1.0D0)/(XC(3)+1.666666666666667D0*XC(2)) + FVECC(7)=((XC(1)-4.5D0)*XC(3)+(1.285714285714286D0*XC(1)-5.7857142 + &85714286D0)*XC(2)+1.0D0)/(XC(3)+1.285714285714286D0*XC(2)) + FVECC(8)=((XC(1)-4.899999999999999D0)*XC(3)+(XC(1)-4.8999999999999 + &99D0)*XC(2)+1.0D0)/(XC(3)+XC(2)) + FVECC(9)=((XC(1)-4.699999999999999D0)*XC(3)+(XC(1)-4.6999999999999 + &99D0)*XC(2)+1.285714285714286D0)/(XC(3)+XC(2)) + FVECC(10)=((XC(1)-6.8D0)*XC(3)+(XC(1)-6.8D0)*XC(2)+1.6666666666666 + &67D0)/(XC(3)+XC(2)) + FVECC(11)=((XC(1)-8.299999999999999D0)*XC(3)+(XC(1)-8.299999999999 + &999D0)*XC(2)+2.2D0)/(XC(3)+XC(2)) + FVECC(12)=((XC(1)-10.6D0)*XC(3)+(XC(1)-10.6D0)*XC(2)+3.0D0)/(XC(3) + &+XC(2)) + FVECC(13)=((XC(1)-1.34D0)*XC(3)+(XC(1)-1.34D0)*XC(2)+4.33333333333 + &3333D0)/(XC(3)+XC(2)) + FVECC(14)=((XC(1)-2.1D0)*XC(3)+(XC(1)-2.1D0)*XC(2)+7.0D0)/(XC(3)+X + &C(2)) + FVECC(15)=((XC(1)-4.39D0)*XC(3)+(XC(1)-4.39D0)*XC(2)+15.0D0)/(XC(3 + &)+XC(2)) + END +\end{verbatim} +\end{page} + +@ +\subsection{Asp55 Example Code} +\label{Asp55ExampleCode} +\index{pages!Asp55ExampleCode!aspex.ht} +\index{aspex.ht!pages!Asp55ExampleCode} +\index{Asp55ExampleCode!aspex.ht!pages} +<>= +\begin{page}{Asp55ExampleCode}{Asp55 Example Code} +\begin{verbatim} + SUBROUTINE CONFUN(MODE,NCNLN,N,NROWJ,NEEDC,X,C,CJAC,NSTATE,IUSER + &,USER) + DOUBLE PRECISION C(NCNLN),X(N),CJAC(NROWJ,N),USER(*) + INTEGER N,IUSER(*),NEEDC(NCNLN),NROWJ,MODE,NCNLN,NSTATE + IF(NEEDC(1).GT.0)THEN + C(1)=X(6)**2+X(1)**2 + CJAC(1,1)=2.0D0*X(1) + CJAC(1,2)=0.0D0 + CJAC(1,3)=0.0D0 + CJAC(1,4)=0.0D0 + CJAC(1,5)=0.0D0 + CJAC(1,6)=2.0D0*X(6) + ENDIF + IF(NEEDC(2).GT.0)THEN + C(2)=X(2)**2+(-2.0D0*X(1)*X(2))+X(1)**2 + CJAC(2,1)=(-2.0D0*X(2))+2.0D0*X(1) + CJAC(2,2)=2.0D0*X(2)+(-2.0D0*X(1)) + CJAC(2,3)=0.0D0 + CJAC(2,4)=0.0D0 + CJAC(2,5)=0.0D0 + CJAC(2,6)=0.0D0 + ENDIF + IF(NEEDC(3).GT.0)THEN + C(3)=X(3)**2+(-2.0D0*X(1)*X(3))+X(2)**2+X(1)**2 + CJAC(3,1)=(-2.0D0*X(3))+2.0D0*X(1) + CJAC(3,2)=2.0D0*X(2) + CJAC(3,3)=2.0D0*X(3)+(-2.0D0*X(1)) + CJAC(3,4)=0.0D0 + CJAC(3,5)=0.0D0 + CJAC(3,6)=0.0D0 + ENDIF + RETURN + END +\end{verbatim} +\end{page} + +@ +\subsection{Asp6 Example Code} +\label{Asp6ExampleCode} +\index{pages!Asp6ExampleCode!aspex.ht} +\index{aspex.ht!pages!Asp6ExampleCode} +\index{Asp6ExampleCode!aspex.ht!pages} +<>= +\begin{page}{Asp6ExampleCode}{Asp6 Example Code} +\begin{verbatim} + SUBROUTINE FCN(N,X,FVEC,IFLAG) + DOUBLE PRECISION X(N),FVEC(N) + INTEGER N,IFLAG + FVEC(1)=(-2.0D0*X(2))+(-2.0D0*X(1)**2)+3.0D0*X(1)+1.0D0 + FVEC(2)=(-2.0D0*X(3))+(-2.0D0*X(2)**2)+3.0D0*X(2)+(-1.0D0*X(1))+1. + &0D0 + FVEC(3)=(-2.0D0*X(4))+(-2.0D0*X(3)**2)+3.0D0*X(3)+(-1.0D0*X(2))+1. + &0D0 + FVEC(4)=(-2.0D0*X(5))+(-2.0D0*X(4)**2)+3.0D0*X(4)+(-1.0D0*X(3))+1. + &0D0 + FVEC(5)=(-2.0D0*X(6))+(-2.0D0*X(5)**2)+3.0D0*X(5)+(-1.0D0*X(4))+1. + &0D0 + FVEC(6)=(-2.0D0*X(7))+(-2.0D0*X(6)**2)+3.0D0*X(6)+(-1.0D0*X(5))+1. + &0D0 + FVEC(7)=(-2.0D0*X(8))+(-2.0D0*X(7)**2)+3.0D0*X(7)+(-1.0D0*X(6))+1. + &0D0 + FVEC(8)=(-2.0D0*X(9))+(-2.0D0*X(8)**2)+3.0D0*X(8)+(-1.0D0*X(7))+1. + &0D0 + FVEC(9)=(-2.0D0*X(9)**2)+3.0D0*X(9)+(-1.0D0*X(8))+1.0D0 + RETURN + END +\end{verbatim} +\end{page} + +@ +\subsection{Asp7 Example Code} +\label{Asp7ExampleCode} +\index{pages!Asp7ExampleCode!aspex.ht} +\index{aspex.ht!pages!Asp7ExampleCode} +\index{Asp7ExampleCode!aspex.ht!pages} +<>= +\begin{page}{Asp7ExampleCode}{Asp7 Example Code} +\begin{verbatim} + SUBROUTINE FCN(X,Z,F) + DOUBLE PRECISION F(*),X,Z(*) + F(1)=DTAN(Z(3)) + F(2)=((-0.03199999999999999D0*DCOS(Z(3))*DTAN(Z(3)))+(-0.02D0*Z(2) + &**2))/(Z(2)*DCOS(Z(3))) + F(3)=-0.03199999999999999D0/(X*Z(2)**2) + RETURN + END +\end{verbatim} +\end{page} + +@ +\subsection{Asp73 Example Code} +\label{Asp73ExampleCode} +\index{pages!Asp73ExampleCode!aspex.ht} +\index{aspex.ht!pages!Asp73ExampleCode} +\index{Asp73ExampleCode!aspex.ht!pages} +<>= +\begin{page}{Asp73ExampleCode}{Asp73 Example Code} +\begin{verbatim} + SUBROUTINE PDEF(X,Y,ALPHA,BETA,GAMMA,DELTA,EPSOLN,PHI,PSI) + DOUBLE PRECISION ALPHA,EPSOLN,PHI,X,Y,BETA,DELTA,GAMMA,PSI + ALPHA=DSIN(X) + BETA=Y + GAMMA=X*Y + DELTA=DCOS(X)*DSIN(Y) + EPSOLN=Y+X + PHI=X + PSI=Y + RETURN + END +\end{verbatim} +\end{page} + +@ +\subsection{Asp74 Example Code} +\label{Asp74ExampleCode} +\index{pages!Asp74ExampleCode!aspex.ht} +\index{aspex.ht!pages!Asp74ExampleCode} +\index{Asp74ExampleCode!aspex.ht!pages} +<>= +\begin{page}{Asp74ExampleCode}{Asp74 Example Code} +\begin{verbatim} + SUBROUTINE BNDY(X,Y,A,B,C,IBND) + DOUBLE PRECISION A,B,C,X,Y + INTEGER IBND + IF(IBND.EQ.0)THEN + A=0.0D0 + B=1.0D0 + C=-1.0D0*DSIN(X) + ELSEIF(IBND.EQ.1)THEN + A=1.0D0 + B=0.0D0 + C=DSIN(X)*DSIN(Y) + ELSEIF(IBND.EQ.2)THEN + A=1.0D0 + B=0.0D0 + C=DSIN(X)*DSIN(Y) + ELSEIF(IBND.EQ.3)THEN + A=0.0D0 + B=1.0D0 + C=-1.0D0*DSIN(Y) + ENDIF + END +\end{verbatim} +\end{page} + +@ +\subsection{Asp77 Example Code} +\label{Asp77ExampleCode} +\index{pages!Asp77ExampleCode!aspex.ht} +\index{aspex.ht!pages!Asp77ExampleCode} +\index{Asp77ExampleCode!aspex.ht!pages} +<>= +\begin{page}{Asp77ExampleCode}{Asp77 Example Code} +\begin{verbatim} + SUBROUTINE FCNF(X,F) + DOUBLE PRECISION X + DOUBLE PRECISION F(2,2) + F(1,1)=0.0D0 + F(1,2)=1.0D0 + F(2,1)=0.0D0 + F(2,2)=-10.0D0 + RETURN + END +\end{verbatim} +\end{page} + +@ +\subsection{Asp78 Example Code} +\label{Asp78ExampleCode} +\index{pages!Asp78ExampleCode!aspex.ht} +\index{aspex.ht!pages!Asp78ExampleCode} +\index{Asp78ExampleCode!aspex.ht!pages} +<>= +\begin{page}{Asp78ExampleCode}{Asp78 Example Code} +\begin{verbatim} + SUBROUTINE FCNG(X,G) + DOUBLE PRECISION G(*),X + G(1)=0.0D0 + G(2)=0.0D0 + END +\end{verbatim} +\end{page} + +@ +\subsection{Asp8 Example Code} +\label{Asp8ExampleCode} +\index{pages!Asp8ExampleCode!aspex.ht} +\index{aspex.ht!pages!Asp8ExampleCode} +\index{Asp8ExampleCode!aspex.ht!pages} +<>= +\begin{page}{Asp8ExampleCode}{Asp8 Example Code} +\begin{verbatim} + SUBROUTINE OUTPUT(XSOL,Y,COUNT,M,N,RESULT,FORWRD) + DOUBLE PRECISION Y(N),RESULT(M,N),XSOL + INTEGER M,N,COUNT + LOGICAL FORWRD + DOUBLE PRECISION X02ALF,POINTS(8) + EXTERNAL X02ALF + INTEGER I + POINTS(1)=1.0D0 + POINTS(2)=2.0D0 + POINTS(3)=3.0D0 + POINTS(4)=4.0D0 + POINTS(5)=5.0D0 + POINTS(6)=6.0D0 + POINTS(7)=7.0D0 + POINTS(8)=8.0D0 + COUNT=COUNT+1 + DO 25001 I=1,N + RESULT(COUNT,I)=Y(I) +25001 CONTINUE + IF(COUNT.EQ.M)THEN + IF(FORWRD)THEN + XSOL=X02ALF() + ELSE + XSOL=-X02ALF() + ENDIF + ELSE + XSOL=POINTS(COUNT) + ENDIF + END +\end{verbatim} +\end{page} + +@ +\subsection{Asp80 Example Code} +\label{Asp80ExampleCode} +\index{pages!Asp80ExampleCode!aspex.ht} +\index{aspex.ht!pages!Asp80ExampleCode} +\index{Asp80ExampleCode!aspex.ht!pages} +<>= +\begin{page}{Asp80ExampleCode}{Asp80 Example Code} +\begin{verbatim} + SUBROUTINE BDYVAL(XL,XR,ELAM,YL,YR) + DOUBLE PRECISION ELAM,XL,YL(3),XR,YR(3) + YL(1)=XL + YL(2)=2.0D0 + YR(1)=1.0D0 + YR(2)=-1.0D0*DSQRT(XR+(-1.0D0*ELAM)) + RETURN + END +\end{verbatim} +\end{page} + +@ +\subsection{Asp9 Example Code} +\label{Asp9ExampleCode} +\index{pages!Asp9ExampleCode!aspex.ht} +\index{aspex.ht!pages!Asp9ExampleCode} +\index{Asp9ExampleCode!aspex.ht!pages} +<>= +\begin{page}{Asp9ExampleCode}{Asp9 Example Code} +\begin{verbatim} + DOUBLE PRECISION FUNCTION G(X,Y) + DOUBLE PRECISION X,Y(*) + G=X+Y(1) + RETURN + END +\end{verbatim} +\end{page} + +@ +\section{basic.ht} +\subsection{Basic Commands} +\label{BasicCommand} +See Calculus \ref{Calculus} on page~\pageref{Calculus} +\index{pages!BasicCommand!basic.ht} +\index{basic.ht!pages!BasicCommand} +\index{BasicCommand!basic.ht!pages} +\index{Function!bcMatrix} +\index{bcMatrix Function} +\index{Function!bcExpand} +\index{bcExpand Function} +\index{Function!bcDraw} +\index{bcDraw Function} +\index{Function!bcSeries} +\index{bcSeries Function} +\index{Function!bcSolve} +\index{bcSolve Function} +<>= +\begin{page}{BasicCommand}{Basic Commands} +\beginscroll +\beginmenu +\menumemolink{Calculus}{Calculus}\tab{10} + Compute integrals, derivatives, or limits +\menulispmemolink{Matrix}{(|bcMatrix|)}\tab{10} + Create a matrix +%\menulispmemolink{Operations}{(|bcExpand|)} +% Expand, factor, simplify, substitute, etc. +\menulispmemolink{Draw}{(|bcDraw|)}\tab{10} + Create 2D or 3D plots. +\menulispmemolink{Series}{(|bcSeries|)}\tab{10} + Create a power series +\menulispmemolink{Solve}{(|bcSolve|)}\tab{10} + Solve an equation or system of equations. +\endmenu +\endscroll +\autobuttons +\end{page} + +@ +\subsection{Calculus} +\label{Calculus} +\index{pages!Calculus!basic.ht} +\index{basic.ht!pages!Calculus} +\index{Calculus!basic.ht!pages} +\index{Function!bcDifferentiate} +\index{bcDifferentiate Function} +\index{Function!bcIndefiniteIntegral} +\index{bcIndefiniteIntegral Function} +\index{Function!bcDefiniteIntegral} +\index{bcDefiniteIntegral Function} +\index{Function!bcLimit} +\index{bcLimit Function} +\index{Function!bcSum} +\index{bcSum Function} +\index{Function!bcProduct} +\index{bcProduct Function} +<>= +\begin{page}{Calculus}{Calculus} +\beginscroll +What would you like to do? +\beginmenu +\menulispdownlink{Differentiate}{(|bcDifferentiate|)}\space{} +\menulispdownlink{Do an Indefinite Integral}{(|bcIndefiniteIntegrate|)}\space{} +\menulispdownlink{Do a Definite Integral}{(|bcDefiniteIntegrate|)}\space{} +\menulispdownlink{Find a limit}{(|bcLimit|)}\space{} +\menulispdownlink{Do a summation}{(|bcSum|)}\space{} +%\menulispdownlink{Compute a product}{(|bcProduct|)}\space{} +\endmenu +\endscroll +\autobuttons +\end{page} + +@ +\section{bbtree.ht} +<>= +\newcommand{\BalancedBinaryTreeXmpTitle}{BalancedBinaryTree} +\newcommand{\BalancedBinaryTreeXmpNumber}{9.2} + +@ +\subsection{BalancedBinaryTree} +\label{BalancedBinaryTreeXmpPage} +\index{pages!BalancedBinaryTreeXmpPage!bbtree.ht} +\index{bbtree.ht!pages!BalancedBinaryTreeXmpPage} +\index{BalancedBinaryTreeXmpPage!bbtree.ht!pages} +<>= +\begin{page}{BalancedBinaryTreeXmpPage}{BalancedBinaryTree} +\beginscroll +\spadtype{BalancedBinaryTrees(S)} is the domain +of balanced binary trees with elements of type \spad{S} at the nodes. +A binary tree is either \spadfun{empty} or else +consists of a \spadfun{node} having a \spadfun{value} +and two branches, each branch a binary tree. +A balanced binary tree is one that is balanced with respect its leaves. +One with \texht{$2^k$}{2**k} leaves is +perfectly ``balanced'': the tree has minimum depth, and +the \spadfun{left} and \spadfun{right} +branch of every interior node is identical in shape. + +Balanced binary trees are useful in algebraic computation for +so-called ``divide-and-conquer'' algorithms. +Conceptually, the data for a problem is initially placed at the +root of the tree. +The original data is then split into two subproblems, one for each +subtree. +And so on. +Eventually, the problem is solved at the leaves of the tree. +A solution to the original problem is obtained by some mechanism +that can reassemble the pieces. +In fact, an implementation of the Chinese Remainder Algorithm +using balanced binary trees was first proposed by David Y. Y. +Yun at the IBM T. J. +Watson Research Center in Yorktown Heights, New York, in 1978. +It served as the prototype for polymorphic algorithms in +Axiom. + +In what follows, rather than perform a series of computations with +a single expression, the expression is reduced modulo a number of +integer primes, a computation is done with modular arithmetic for +each prime, and the Chinese Remainder Algorithm is used to obtain +the answer to the original problem. +We illustrate this principle with the computation of \texht{$12^2 += 144$}{12 ** 2 = 144}. + +\xtc{ +A list of moduli. +}{ +\spadpaste{lm := [3,5,7,11]\bound{lm}} +} +\xtc{ +The expression \spad{modTree(n, lm)} +creates a balanced binary tree with leaf values +\spad{n mod m} for each modulus \spad{m} in \spad{lm}. +}{ +\spadpaste{modTree(12,lm)\free{lm}} +} +\xtc{ +Operation \spad{modTree} does this using +operations on balanced binary trees. +We trace its steps. +Create a balanced binary tree \spad{t} of zeros with four leaves. +}{ +\spadpaste{t := balancedBinaryTree(\#lm, 0)\bound{t}\free{lm}} +} +\xtc{ +The leaves of the tree are set to the individual moduli. +}{ +\spadpaste{setleaves!(t,lm)\bound{t1}\free{t}} +} +\xtc{ +Use \spadfunX{mapUp} to do +a bottom-up traversal of \spad{t}, setting each interior node +to the product of the values at the nodes of its children. +}{ +\spadpaste{mapUp!(t,_*)\bound{t2}\free{t1}} +} +\xtc{ +The value at the node of every subtree is the product of the moduli +of the leaves of the subtree. +}{ +\spadpaste{t \bound{t3}\free{t2}} +} +\xtc{ +Operation \spadfunX{mapDown}\spad{(t,a,fn)} replaces the value +\spad{v} at each node of \spad{t} by \spad{fn(a,v)}. +}{ +\spadpaste{mapDown!(t,12,_rem)\bound{t4}\free{t3}} +} +\xtc{ +The operation \spadfun{leaves} returns the leaves of the resulting +tree. +In this case, it returns the list of \spad{12 mod m} for each +modulus \spad{m}. +}{ +\spadpaste{leaves \%\bound{t5}\free{t4}} +} +\xtc{ +Compute the square of the images of \spad{12} modulo each \spad{m}. +}{ +\spadpaste{squares := [x**2 rem m for x in \% for m in lm]\bound{t6}\free{t5}} +} +\xtc{ +Call the Chinese Remainder Algorithm to get the +answer for \texht{$12^2$}{12**2}. +}{ +\spadpaste{chineseRemainder(\%,lm)\free{t6}} +} +\endscroll +\autobuttons +\end{page} + +@ +\section{binary.ht} +% !! DO NOT MODIFY THIS FILE BY HAND !! Created by ht.awk. +<>= +\newcommand{\BinaryExpansionXmpTitle}{BinaryExpansion} +\newcommand{\BinaryExpansionXmpNumber}{9.4} + +@ +\subsection{BinaryExpansion} +\label{BinaryExpansionXmpPage} +\begin{itemize} +\item DecimalExpansionXmpPage \ref{DecimalExpansionXmpPage} on +page~pageref{DecimalExpansionXmpPage} +\item RadixExpansionXmpPage \ref{RadixExpansionXmpPage} on +page~pageref{RadixExpansionXmpPage} +\end{itemize} +\index{pages!BinaryExpansionXmpPage!binary.ht} +\index{binary.ht!pages!BinaryExpansionXmpPage} +\index{BinaryExpansionXmpPage!binary.ht!pages} +<>= +\begin{page}{BinaryExpansionXmpPage}{BinaryExpansion} +\beginscroll + +All rational numbers have repeating binary expansions. +Operations to access the individual bits of a binary expansion can +be obtained by converting the value to \spadtype{RadixExpansion(2)}. +More examples of expansions are available in +\downlink{`DecimalExpansion'}{DecimalExpansionXmpPage}\ignore{DecimalExpansion}, +\downlink{`HexadecimalExpansion'}{HexadecimalExpansionXmpPage}\ignore{HexadecimalExpansion}, and +\downlink{`RadixExpansion'}{RadixExpansionXmpPage}\ignore{RadixExpansion}. + +\xtc{ +The expansion (of type \spadtype{BinaryExpansion}) of a rational number +is returned by the \spadfunFrom{binary}{BinaryExpansion} operation. +}{ +\spadpaste{r := binary(22/7) \bound{r}} +} +\xtc{ +Arithmetic is exact. +}{ +\spadpaste{r + binary(6/7) \free{r}} +} +\xtc{ +The period of the expansion can be short or long \ldots +}{ +\spadpaste{[binary(1/i) for i in 102..106] } +} +\xtc{ +or very long. +}{ +\spadpaste{binary(1/1007) } +} +\xtc{ +These numbers are bona fide algebraic objects. +}{ +\spadpaste{p := binary(1/4)*x**2 + binary(2/3)*x + binary(4/9) \bound{p}} +} +\xtc{ +}{ +\spadpaste{q := D(p, x) \free{p}\bound{q}} +} +\xtc{ +}{ +\spadpaste{g := gcd(p, q) \free{p q}\bound{g}} +} +\endscroll +\autobuttons +\end{page} + +@ +\section{bmcat.ht} +\subsection{Bit Map Catalog} +\label{BitMaps} +\index{pages!BitMaps!bmcat.ht} +\index{bmcat.ht!pages!BitMaps} +\index{BitMaps!bmcat.ht!pages} +<>= +\begin{page}{BitMaps}{Bit Map Catalog} +\beginscroll +%\space{} {\inputbitmap{/usr/include/X11/bitmaps/1x1}} 1x1 +%\space{} {\inputbitmap{/usr/include/X11/bitmaps/2x2}} 2x2 +%\space{} {\inputbitmap{/usr/include/X11/bitmaps/black}} black +%\space{} {\inputbitmap{/usr/include/X11/bitmaps/boxes}} boxes +%\space{} {\inputbitmap{/usr/include/X11/bitmaps/cntr_ptr}} cntr_ptr +%\space{} {\inputbitmap{/usr/include/X11/bitmaps/cntr_ptrmsk}} cntr_ptrmsk +%\space{} {\inputbitmap{/usr/include/X11/bitmaps/cross_weave}} cross_weave +%\space{} {\inputbitmap{/usr/include/X11/bitmaps/dimple1}} dimple1 +%\space{} {\inputbitmap{/usr/include/X11/bitmaps/dimple3}} dimple3 +%\space{} {\inputbitmap{/usr/include/X11/bitmaps/dot}} dot +%\space{} {\inputbitmap{/usr/include/X11/bitmaps/flipped_gray}} flipped_gray +%\space{} {\inputbitmap{/usr/include/X11/bitmaps/gray}} gray +%\space{} {\inputbitmap{/usr/include/X11/bitmaps/gray1}} gray1 +%\space{} {\inputbitmap{/usr/include/X11/bitmaps/gray3}} gray3 +%\space{} {\inputbitmap{/usr/include/X11/bitmaps/icon}} icon +%\space{} {\inputbitmap{/usr/include/X11/bitmaps/left_ptr}} left_ptr +%\space{} {\inputbitmap{/usr/include/X11/bitmaps/left_ptrmsk}} left_ptrmsk +%\space{} {\inputbitmap{/usr/include/X11/bitmaps/light_gray}} light_gray +%\space{} {\inputbitmap{/usr/include/X11/bitmaps/opendot}} opendot +%\space{} {\inputbitmap{/usr/include/X11/bitmaps/opendotMask}} opendotMask +%\space{} {\inputbitmap{/usr/include/X11/bitmaps/right_ptr}} right_ptr +%\space{} {\inputbitmap{/usr/include/X11/bitmaps/right_ptrmsk}} right_ptrmsk +%\space{} {\inputbitmap{/usr/include/X11/bitmaps/root_weave}} root_weave +%\space{} {\inputbitmap{/usr/include/X11/bitmaps/scales}} scales +%\space{} {\inputbitmap{/usr/include/X11/bitmaps/sipb}} sipb +%\space{} {\inputbitmap{/usr/include/X11/bitmaps/star}} star +%\space{} {\inputbitmap{/usr/include/X11/bitmaps/starMask}} starMask +%\space{} {\inputbitmap{/usr/include/X11/bitmaps/stipple}} stipple +%\space{} {\inputbitmap{/usr/include/X11/bitmaps/target}} target +%\space{} {\inputbitmap{/usr/include/X11/bitmaps/tie_fighter}} tie_fighter +%\space{} {\inputbitmap{/usr/include/X11/bitmaps/wide_weave}} wide_weave +%\space{} {\inputbitmap{/usr/include/X11/bitmaps/wierd_size}} wierd_size +%\space{} {\inputbitmap{/usr/include/X11/bitmaps/wingdogs}} wingdogs +%\horizontalline +\space{} {\inputbitmap{\htbmdir{}/xfbox.bitmap}} xfbox \space{4} +\space{} {\inputbitmap{\htbmdir{}/xfcirc.bitmap}} xfcirc \space{4} +\space{} {\inputbitmap{\htbmdir{}/xnobox.bitmap}} Xnobox \space{2} +\space{} {\inputbitmap{\htbmdir{}/xnocirc.bitmap}} xnocirc \space{2} +\newline +\space{} {\inputbitmap{\htbmdir{}/xfullfbox.bitmap}} xfullfbox +\space{} {\inputbitmap{\htbmdir{}/xfullfcirc.bitmap}} xfullfcirc +\space{} {\inputbitmap{\htbmdir{}/xfullbox.bitmap}} xfullbox +\space{} {\inputbitmap{\htbmdir{}/xfullcirc.bitmap}} xfullcirc +\newline +\space{} {\inputbitmap{\htbmdir{}/xgreyfbox.bitmap}} xgreyfbox +\space{} {\inputbitmap{\htbmdir{}/xgreyfcirc.bitmap}} xgreyfcirc +\space{} {\inputbitmap{\htbmdir{}/xgreybox.bitmap}} xgreybox +\space{} {\inputbitmap{\htbmdir{}/xgreycirc.bitmap}} xgreycirc +\newline +\space{} {\inputbitmap{\htbmdir{}/xopenfbox.bitmap}} xopenfbox +\space{} {\inputbitmap{\htbmdir{}/xopenfcirc.bitmap}} xopenfcirc +\space{} {\inputbitmap{\htbmdir{}/xopenbox.bitmap}} xopenbox +\space{} {\inputbitmap{\htbmdir{}/xopencirc.bitmap}} xopencirc +\newline +\space{} {\inputbitmap{\htbmdir{}/xtickfbox.bitmap}} xtickfbox +\space{} {\inputbitmap{\htbmdir{}/xtickfcirc.bitmap}} xtickfcirc +\space{} {\inputbitmap{\htbmdir{}/xtickbox.bitmap}} xtickbox +\space{} {\inputbitmap{\htbmdir{}/xtickcirc.bitmap}} xtickcirc +\newline +\space{} {\inputbitmap{\htbmdir{}/xxfbox.bitmap}} xxfbox \space{3} +\space{} {\inputbitmap{\htbmdir{}/xxfcirc.bitmap}} xxfcirc \space{3} +\space{} {\inputbitmap{\htbmdir{}/xxbox.bitmap}} xxbox \space{3} +\space{} {\inputbitmap{\htbmdir{}/xxcirc.bitmap}} xxcirc \space{3} +\horizontalline +\space{} {\inputbitmap{\htbmdir{}/xnoface.bitmap}} xnoface +\space{} {\inputbitmap{\htbmdir{}/xhappy.bitmap}} xhappy +\space{} {\inputbitmap{\htbmdir{}/xsad.bitmap}} xsad +\space{} {\inputbitmap{\htbmdir{}/xdesp.bitmap}} xdesp +\space{} {\inputbitmap{\htbmdir{}/xperv.bitmap}} xperv +\horizontalline +\space{} {\inputbitmap{\htbmdir{}/exit.bitmap}} exit +\space{} {\inputbitmap{\htbmdir{}/help3.bitmap}} help3 +\newline +\space{} {\inputbitmap{\htbmdir{}/down3.bitmap}} down3 +\space{} {\inputbitmap{\htbmdir{}/up3.bitmap}} up3 +\space{} {\inputbitmap{\htbmdir{}/return3.bitmap}} return3 +\newline +\space{} {\inputbitmap{\htbmdir{}/tear.bitmap}} tear +\space{} {\inputbitmap{\htbmdir{}/eqpage.bitmap}} eqpage +\horizontalline +\space{} {\inputbitmap{\htbmdir{}/sup.bm}} sup.bm +\space{} {\inputbitmap{\htbmdir{}/sup.bitmap}} sup +\space{} {\inputbitmap{\htbmdir{}/sdown.bm}} sdown.bm +\newline +\space{} {\inputbitmap{\htbmdir{}/ht_icon}} ht_icon +\space{} {\inputbitmap{\htbmdir{}/smile.bitmap}} smile +\newline +\space{} {\inputbitmap{\htbmdir{}/drown.bm}} drown.bm +\space{} {\inputbitmap{\htbmdir{}/help2.bitmap}} help2 +\space{} {\inputbitmap{\htbmdir{}/return.bitmap}} return +\space{} {\inputbitmap{\htbmdir{}/back.bitmap}} back +\endscroll +\autobuttons +\end{page} + +@ +\section{bop.ht} +<>= +\newcommand{\BasicOperatorXmpTitle}{BasicOperator} +\newcommand{\BasicOperatorXmpNumber}{9.3} + +@ +\subsection{BasicOperator} +\label{BasicOperatorXmpPage} +\index{pages!BasicOperatorXmpPage!bop.ht} +\index{bop.ht!pages!BasicOperatorXmpPage} +\index{BasicOperatorXmpPage!bop.ht!pages} +<>= +\begin{page}{BasicOperatorXmpPage}{BasicOperator} +\beginscroll + +A basic operator is an object that can be symbolically +applied to a list of arguments from a set, the result being +a kernel over that set or an expression. +In addition to this section, please see \downlink{`Expression'}{ExpressionXmpPage}\ignore{Expression} +and \downlink{`Kernel'}{KernelXmpPage}\ignore{Kernel} for additional information and examples. + +You create an object of type \axiomType{BasicOperator} by using the +\axiomFunFrom{operator}{BasicOperator} operation. +This first form of this operation has one argument and it +must be a symbol. +The symbol should be quoted in case the +name has been used as an identifier to which a value +has been assigned. + +A frequent application of \axiomType{BasicOperator} is the +creation of an operator to represent the unknown function when +solving a differential equation. +\xtc{ +Let \axiom{y} be the unknown function in terms of \axiom{x}. +}{ +\spadpaste{y := operator 'y \bound{y}} +} +% +\xtc{ +This is how you enter +the equation \axiom{y'' + y' + y = 0}. +}{ +\spadpaste{deq := D(y x, x, 2) + D(y x, x) + y x = 0\bound{e1}\free{y}} +} +% +\xtc{ +To solve the above equation, enter this. +}{ +\spadpaste{solve(deq, y, x) \free{e1}\free{y}} +} +See \downlink{``\ugProblemDEQTitle''}{ugProblemDEQPage} in Section \ugProblemDEQNumber\ignore{ugProblemDEQ} +for this kind of use of \axiomType{BasicOperator}. + +Use the single argument form of +\axiomFunFrom{operator}{BasicOperator} (as above) when you intend +to use the operator to create functional expressions with an +arbitrary number of arguments +\xtc{ +{\it Nary} means an arbitrary number of arguments can be used +in the functional expressions. +}{ +\spadpaste{nary? y \free{y}} +} +\xtc{ +}{ +\spadpaste{unary? y \free{y}} +} +Use the two-argument form when you want to restrict the number of +arguments in the functional expressions created with the operator. +\xtc{ +This operator can only be used to create functional expressions +with one argument. +}{ +\spadpaste{opOne := operator('opOne, 1) \bound{opOne}} +} +\xtc{ +}{ +\spadpaste{nary? opOne \free{opOne}} +} +\xtc{ +}{ +\spadpaste{unary? opOne \free{opOne}} +} +\xtc{ +Use \axiomFunFrom{arity}{BasicOperator} to learn the number of +arguments that can be used. +It returns {\tt "false"} if the operator is nary. +}{ +\spadpaste{arity opOne \free{opOne}} +} +\xtc{ +Use \axiomFunFrom{name}{BasicOperator} to learn the name of an +operator. +}{ +\spadpaste{name opOne \free{opOne}} +} +\xtc{ +Use \axiomFunFrom{is?}{BasicOperator} to learn if an operator has a +particular name. +}{ +\spadpaste{is?(opOne, 'z2) \free{opOne}} +} +\xtc{ +You can also use a string as the name to be tested against. +}{ +\spadpaste{is?(opOne, "opOne") \free{opOne}} +} + +You can attached named properties to an operator. +These are rarely used at the top-level of the Axiom +interactive environment but are used with Axiom +library source code. +\xtc{ +By default, an operator has no properties. +}{ +\spadpaste{properties y \free{y}} +} +The interface for setting and getting properties is somewhat awkward +because the property values are stored as values of type +\axiomType{None}. +\xtc{ +Attach a property by using \axiomFunFrom{setProperty}{BasicOperator}. +}{ +\spadpaste{setProperty(y, "use", "unknown function" :: None ) \free{y}\bound{spy}} +} +\xtc{ +}{ +\spadpaste{properties y \free{y spy}} +} +\xtc{ +We {\it know} the property value has type \axiomType{String}. +}{ +\spadpaste{property(y, "use") :: None pretend String \free{y spy}} +} +\xtc{ +Use \axiomFunFrom{deleteProperty!}{BasicOperator} to destructively +remove a property. +}{ +\spadpaste{deleteProperty!(y, "use") \free{y spy}\bound{dpy}} +} +\xtc{ +}{ +\spadpaste{properties y \free{dpy}} +} +\endscroll +\autobuttons +\end{page} +% +@ +\section{bstree.ht} +<>= +\newcommand{\BinarySearchTreeXmpTitle}{BinarySearchTree} +\newcommand{\BinarySearchTreeXmpNumber}{9.5} + +@ +\subsection{BinarySearchTree} +\label{BinarySearchTreeXmpPage} +\index{pages!BinarySearchTreeXmpPage!bstree.ht} +\index{bstree.ht!pages!BinarySearchTreeXmpPage} +\index{BinarySearchTreeXmpPage!bstree.ht!pages} +<>= +\begin{page}{BinarySearchTreeXmpPage}{BinarySearchTree} +\beginscroll +\spadtype{BinarySearchTree(R)} is the domain of binary trees with +elements of type \spad{R}, ordered across the nodes of the tree. +A non-empty binary search tree has a value +of type \spad{R}, and \spadfun{right} and \spadfun{left} +binary search subtrees. +If a subtree is empty, it is displayed as a period (``.''). + +\xtc{ +Define a list of values to be placed across the tree. +The resulting tree has \spad{8} at the root; +all other elements are in the left subtree. +}{ +\spadpaste{lv := [8,3,5,4,6,2,1,5,7]\bound{lv}} +} +\xtc{ +A convenient way to create a binary search tree is to +apply the operation \spadfun{binarySearchTree} to a list of elements. +}{ +\spadpaste{t := binarySearchTree lv\free{lv}\bound{t}} +} +\xtc{ +Another approach is to first create an empty binary search tree of integers. +}{ +\spadpaste{emptybst := empty()\$BSTREE(INT)\bound{e}} +} +\xtc{ +Insert the value \spad{8}. +This establishes \spad{8} as the root of the binary search tree. +Values inserted later that are less than \spad{8} get stored in +the \spadfun{left} subtree, others in the \spadfun{right} subtree. +}{ +\spadpaste{t1 := insert!(8,emptybst)\free{e}\bound{t1}} +} +\xtc{ +Insert the value \spad{3}. This number becomes the root of the +\spadfun{left} subtree of \spad{t1}. +For optimal retrieval, it is thus important to insert the +middle elements first. +}{ +\spadpaste{insert!(3,t1)\free{t1}} +} +\xtc{ +We go back to the original tree \spad{t}. +The leaves of the binary search tree are those which have empty +\spadfun{left} and \spadfun{right} subtrees. +}{ +\spadpaste{leaves t\free{t}} +} +\xtc{ +The operation +\spadfun{split}\spad{(k,t)} returns a \spadgloss{record} containing +the two subtrees: one with all elements ``less'' than \spad{k}, +another with elements ``greater'' than \spad{k}. +}{ +\spadpaste{split(3,t)\free{t}} +} +\xtc{ +Define \userfun{insertRoot} to insert new elements by +creating a new node. +}{ +\spadpaste{insertRoot: (INT,BSTREE INT) -> BSTREE INT\bound{x}} +} +\xtc{ +The new node puts the inserted value between its +``less'' tree and ``greater'' tree. +}{ +\begin{spadsrc}[\bound{x1}\free{x}] +insertRoot(x, t) == + a := split(x, t) + node(a.less, x, a.greater) +\end{spadsrc} +} +\xtc{ +Function \userfun{buildFromRoot} builds +a binary search tree from a list of elements \spad{ls} +and the empty tree \spad{emptybst}. +}{ +\spadpaste{buildFromRoot ls == reduce(insertRoot,ls,emptybst)\bound{x2}\free{x1 e}} +} +\xtc{ +Apply this to the reverse of the list \spad{lv}. +}{ +\spadpaste{rt := buildFromRoot reverse lv\bound{rt}\free{lv x2}} +} +\xtc{ +Have Axiom check that these are equal. +}{ +\spadpaste{(t = rt)@Boolean\free{rt t}} +} +\endscroll +\autobuttons +\end{page} + +@ +\section{card.ht} +<>= +\newcommand{\CardinalNumberXmpTitle}{CardinalNumber} +\newcommand{\CardinalNumberXmpNumber}{9.6} + +@ +\subsection{CardinalNumber} +\label{CardinalNumberXmpPage} +\index{pages!CardinalNumberXmpPage!card.ht} +\index{card.ht!pages!CardinalNumberXmpPage} +\index{CardinalNumberXmpPage!card.ht!pages} +<>= +\begin{page}{CardinalNumberXmpPage}{CardinalNumber} +\beginscroll + +The \spadtype{CardinalNumber} domain can be used for values indicating +the cardinality of sets, both finite and infinite. +For example, the \spadfunFrom{dimension}{VectorSpace} operation in the +category \spadtype{VectorSpace} returns a cardinal number. + +The non-negative integers have a natural construction as cardinals +\begin{verbatim} +0 = #{ }, 1 = {0}, 2 = {0, 1}, ..., n = {i | 0 <= i < n}. +\end{verbatim} +The fact that \spad{0} acts as a zero for the multiplication of cardinals is +equivalent to the axiom of choice. + +\xtc{ +Cardinal numbers can be created by conversion from non-negative integers. +}{ +\spadpaste{c0 := 0 :: CardinalNumber \bound{c0}} +} +\xtc{ +}{ +\spadpaste{c1 := 1 :: CardinalNumber \bound{c1}} +} +\xtc{ +}{ +\spadpaste{c2 := 2 :: CardinalNumber \bound{c2}} +} +\xtc{ +}{ +\spadpaste{c3 := 3 :: CardinalNumber \bound{c3}} +} +\xtc{ +They can also be obtained as the named cardinal \spad{Aleph(n)}. +}{ +\spadpaste{A0 := Aleph 0 \bound{A0}} +} +\xtc{ +}{ +\spadpaste{A1 := Aleph 1 \bound{A1}} +} +\xtc{ +The \spadfunFrom{finite?}{CardinalNumber} operation tests whether a value is a +finite cardinal, that is, a non-negative integer. +}{ +\spadpaste{finite? c2 \free{c2}} +} +\xtc{ +}{ +\spadpaste{finite? A0 \free{A0}} +} +\xtc{ +Similarly, the \spadfunFrom{countable?}{CardinalNumber} +operation determines whether a value is +a countable cardinal, that is, finite or \spad{Aleph(0)}. +}{ +\spadpaste{countable? c2 \free{c2}} +} +\xtc{ +}{ +\spadpaste{countable? A0 \free{A0}} +} +\xtc{ +}{ +\spadpaste{countable? A1 \free{A1}} +} +Arithmetic operations are defined on cardinal numbers as follows: +If \spad{x = \#X} and \spad{y = \#Y} then + +\indent{0} +\spad{x+y = \#(X+Y)} \tab{20} cardinality of the disjoint union \newline +\spad{x-y = \#(X-Y)} \tab{20} cardinality of the relative complement \newline +\spad{x*y = \#(X*Y)} \tab{20} cardinality of the Cartesian product \newline +\spad{x**y = \#(X**Y)}\tab{20} cardinality of the set of maps from \spad{Y} to \spad{X} \newline + +\xtc{ +Here are some arithmetic examples. +}{ +\spadpaste{[c2 + c2, c2 + A1] \free{c2 A1}} +} +\xtc{ +}{ +\spadpaste{[c0*c2, c1*c2, c2*c2, c0*A1, c1*A1, c2*A1, A0*A1] \free{c0,c1,c2,A0,A1}} +} +\xtc{ +}{ +\spadpaste{[c2**c0, c2**c1, c2**c2, A1**c0, A1**c1, A1**c2] \free{c0,c1,c2,A1}} +} +\xtc{ +Subtraction is a partial operation: it is not defined +when subtracting a larger cardinal from a smaller one, nor +when subtracting two equal infinite cardinals. +}{ +\spadpaste{[c2-c1, c2-c2, c2-c3, A1-c2, A1-A0, A1-A1] \free{c1,c2,c3,A0,A1}} +} +The generalized continuum hypothesis asserts that +\begin{verbatim} +2**Aleph i = Aleph(i+1) +\end{verbatim} +and is independent of the axioms of set theory.\footnote{Goedel, +{\it The consistency of the continuum hypothesis,} +Ann. Math. Studies, Princeton Univ. Press, 1940.} +\xtc{ +The \spadtype{CardinalNumber} domain provides an operation to assert +whether the hypothesis is to be assumed. +}{ +\spadpaste{generalizedContinuumHypothesisAssumed true \bound{GCH}} +} +\xtc{ +When the generalized continuum hypothesis +is assumed, exponentiation to a transfinite power is allowed. +}{ +\spadpaste{[c0**A0, c1**A0, c2**A0, A0**A0, A0**A1, A1**A0, A1**A1] \free{c0,c1,c2,A0,A1,GCH}} +} + +Three commonly encountered cardinal numbers are + +\indent{0} +\spad{a = \#}{\bf Z} \tab{20} countable infinity \newline +\spad{c = \#}{\bf R} \tab{20} the continuum \newline +\spad{f = \#\{g| g: [0,1] -> {\bf R}\}} \newline + +\xtc{ +In this domain, these values are obtained under the +generalized continuum hypothesis in this way. +}{ +\spadpaste{a := Aleph 0 \free{GCH}\bound{a}} +} +\xtc{ +}{ +\spadpaste{c := 2**a \free{a} \bound{c}} +} +\xtc{ +}{ +\spadpaste{f := 2**c \free{c} \bound{f}} +} +\endscroll +\autobuttons +\end{page} + +@ +\section{carten.ht} +<>= +\newcommand{\CartesianTensorXmpTitle}{CartesianTensor} +\newcommand{\CartesianTensorXmpNumber}{9.7} + +@ +\subsection{CartesianTensor} +\label{CartesianTensorXmpPage} +\index{pages!CartesianTensorXmpPage!carten.ht} +\index{carten.ht!pages!CartesianTensorXmpPage} +\index{CartesianTensorXmpPage!carten.ht!pages} +<>= +\begin{page}{CartesianTensorXmpPage}{CartesianTensor} +\beginscroll + +\spadtype{CartesianTensor(i0,dim,R)} +provides Cartesian tensors with +components belonging to a commutative ring \spad{R}. +Tensors can be described as a generalization of vectors and matrices. +This gives a concise {\it tensor algebra} for multilinear objects +supported by the \spadtype{CartesianTensor} domain. +You can form the inner or outer product of any two tensors and you can add +or subtract tensors with the same number of components. +Additionally, various forms of traces and transpositions are useful. + +The \spadtype{CartesianTensor} constructor allows you to specify the +minimum index for subscripting. +In what follows we discuss in detail how to manipulate tensors. + +\xtc{ +Here we construct the domain of Cartesian tensors of dimension 2 over the +integers, with indices starting at 1. +}{ +\spadpaste{CT := CARTEN(i0 := 1, 2, Integer) \bound{CT i0}} +} + +\subsubsection{Forming tensors} + +\labelSpace{2pc} +\xtc{ +Scalars can be converted to tensors of rank zero. +}{ +\spadpaste{t0: CT := 8 \free{CT}\bound{t0}} +} +\xtc{ +}{ +\spadpaste{rank t0 \free{t0}} +} +\xtc{ +Vectors (mathematical direct products, rather than one dimensional array +structures) can be converted to tensors of rank one. +}{ +\spadpaste{v: DirectProduct(2, Integer) := directProduct [3,4] \bound{v}} +} +\xtc{ +}{ +\spadpaste{Tv: CT := v \free{v CT}\bound{Tv}} +} +\xtc{ +Matrices can be converted to tensors of rank two. +}{ +\spadpaste{m: SquareMatrix(2, Integer) := matrix [[1,2],[4,5]] \bound{m}} +} +\xtc{ +}{ +\spadpaste{Tm: CT := m \bound{Tm}\free{CT m}} +} +\xtc{ +}{ +\spadpaste{n: SquareMatrix(2, Integer) := matrix [[2,3],[0,1]] \bound{n}} +} +\xtc{ +}{ +\spadpaste{Tn: CT := n \bound{Tn}\free{CT n}} +} +\xtc{ +In general, a tensor of rank \spad{k} can be formed by making a +list of rank \spad{k-1} tensors or, alternatively, a \spad{k}-deep nested +list of lists. +}{ +\spadpaste{t1: CT := [2, 3] \free{CT}\bound{t1}} +} +\xtc{ +}{ +\spadpaste{rank t1 \free{t1}} +} +\xtc{ +}{ +\spadpaste{t2: CT := [t1, t1] \free{CT t1}\bound{t2}} +} +\xtc{ +}{ +\spadpaste{t3: CT := [t2, t2] \free{CT t2}\bound{t3}} +} +\xtc{ +}{ +\spadpaste{tt: CT := [t3, t3]; tt := [tt, tt] \free{CT t3}\bound{tt}} +} +\xtc{ +}{ +\spadpaste{rank tt \free{tt}} +} + +\subsubsection{Multiplication} +%\head{subsection}{Multiplication}{ugxCartenMult} + +Given two tensors of rank \spad{k1} and \spad{k2}, the outer +{CartesianTensor} +\spadfunFrom{product}{CartesianTensor} forms a new tensor of rank +\spad{k1+k2}. +\xtc{ +Here +\texht{$T_{mn}(i,j,k,l) = T_m(i,j) \ T_n(k,l).$}{% +\spad{Tmn(i,j,k,l) = Tm(i,j)*Tn(k,l).}} +}{ +\spadpaste{Tmn := product(Tm, Tn)\free{Tm Tn}\bound{Tmn}} +} + +The inner product (\spadfunFrom{contract}{CartesianTensor}) forms a tensor +{CartesianTensor} +of rank \spad{k1+k2-2}. +This product generalizes the vector dot product and matrix-vector product +by summing component products along two indices. +\xtc{ +Here we sum along the second index of \texht{$T_m$}{\spad{Tm}} and the +first index of \texht{$T_v$}{\spad{Tv}}. +Here +\texht{$T_{mv} = \sum_{j=1}^{\hbox{\tiny\rm dim}} T_m(i,j) \ T_v(j)$}{% +\spad{Tmv(i) = sum Tm(i,j) * Tv(j) for j in 1..dim.}} +}{ +\spadpaste{Tmv := contract(Tm,2,Tv,1)\free{Tm Tv}\bound{Tmv}} +} + +The multiplication operator \spadopFrom{*}{CartesianTensor} is scalar +multiplication or an inner product depending on the ranks of the +arguments. +\xtc{ +If either argument is rank zero it is treated as scalar multiplication. +Otherwise, \spad{a*b} is the inner product summing the last index of +\spad{a} with the first index of \spad{b}. +}{ +\spadpaste{Tm*Tv \free{Tm Tv}} +} +\xtc{ +This definition is consistent with the inner product on matrices +and vectors. +}{ +\spadpaste{Tmv = m * v \free{Tmv m v}} +} + +\subsubsection{Selecting Components} +%\head{subsection}{Selecting Components}{ugxCartenSelect} + +\labelSpace{2pc} +\xtc{ +For tensors of low rank (that is, four or less), components can be selected +by applying the tensor to its indices. +}{ +\spadpaste{t0() \free{t0}} +} +\xtc{ +}{ +\spadpaste{t1(1+1) \free{t1}} +} +\xtc{ +}{ +\spadpaste{t2(2,1) \free{t2}} +} +\xtc{ +}{ +\spadpaste{t3(2,1,2) \free{t3}} +} +\xtc{ +}{ +\spadpaste{Tmn(2,1,2,1) \free{Tmn}} +} +\xtc{ +A general indexing mechanism is provided for a list of indices. +}{ +\spadpaste{t0[] \free{t0}} +} +\xtc{ +}{ +\spadpaste{t1[2] \free{t1}} +} +\xtc{ +}{ +\spadpaste{t2[2,1] \free{t2}} +} +\xtc{ +The general mechanism works for tensors of arbitrary rank, but is +somewhat less efficient since the intermediate index list must be created. +}{ +\spadpaste{t3[2,1,2] \free{t3}} +} +\xtc{ +}{ +\spadpaste{Tmn[2,1,2,1] \free{Tmn}} +} + +\subsubsection{Contraction} +%\head{subsection}{Contraction}{ugxCartenContract} + +A ``contraction'' between two tensors is an inner product, as we have +seen above. +You can also contract a pair of indices of a single tensor. +This corresponds to a ``trace'' in linear algebra. +The expression \spad{contract(t,k1,k2)} forms a new tensor by summing the +diagonal given by indices in position \spad{k1} and \spad{k2}. +\xtc{ +This is the tensor given by +\texht{$xT_{mn} = \sum_{k=1}^{\hbox{\tiny\rm dim}} T_{mn}(k,k,i,j).$}{% +\spad{cTmn(i,j) = sum Tmn(k,k,i,j) for k in 1..dim.}} +}{ +\spadpaste{cTmn := contract(Tmn,1,2) \free{Tmn}\bound{cTmn}} +} +\xtc{ +Since \spad{Tmn} is the outer product of matrix \spad{m} and matrix \spad{n}, +the above is equivalent to this. +}{ +\spadpaste{trace(m) * n \free{m n}} +} +\xtc{ +In this and the next few examples, +we show all possible contractions of \spad{Tmn} and their matrix +algebra equivalents. +}{ +\spadpaste{contract(Tmn,1,2) = trace(m) * n \free{Tmn m n}} +} +\xtc{ +}{ +\spadpaste{contract(Tmn,1,3) = transpose(m) * n \free{Tmn m n}} +} +\xtc{ +}{ +\spadpaste{contract(Tmn,1,4) = transpose(m) * transpose(n) \free{Tmn m n}} +} +\xtc{ +}{ +\spadpaste{contract(Tmn,2,3) = m * n \free{Tmn m n}} +} +\xtc{ +}{ +\spadpaste{contract(Tmn,2,4) = m * transpose(n) \free{Tmn m n}} +} +\xtc{ +}{ +\spadpaste{contract(Tmn,3,4) = trace(n) * m \free{Tmn m n}} +} + +\subsubsection{Transpositions} +%\head{subsection}{Transpositions}{ugxCartenTranspos} + +You can exchange any desired pair of indices using the +\spadfunFrom{transpose}{CartesianTensor} operation. + +\labelSpace{1pc} +\xtc{ +Here the indices in positions one and three are exchanged, that is, +\texht{$tT_{mn}(i,j,k,l) = T_{mn}(k,j,i,l).$}{% +\spad{tTmn(i,j,k,l) = Tmn(k,j,i,l).}} +}{ +\spadpaste{tTmn := transpose(Tmn,1,3) \free{tTmn}} +} +\xtc{ +If no indices are specified, the first and last index are exchanged. +}{ +\spadpaste{transpose Tmn \free{Tmn}} +} +\xtc{ +This is consistent with the matrix transpose. +}{ +\spadpaste{transpose Tm = transpose m \free{Tm m}} +} + +If a more complicated reordering of the indices is required, then the +\spadfunFrom{reindex}{CartesianTensor} operation can be used. +This operation allows the indices to be arbitrarily permuted. +\xtc{ +This defines +\texht{$rT_{mn}(i,j,k,l) = \allowbreak T_{mn}(i,l,j,k).$}{% +\spad{rTmn(i,j,k,l) = Tmn(i,l,j,k).}} +}{ +\spadpaste{rTmn := reindex(Tmn, [1,4,2,3]) \free{Tmn}\bound{rTmn}} +} + +\subsubsection{Arithmetic} +%\head{subsection}{Arithmetic}{ugxCartenArith} + +\labelSpace{2pc} +\xtc{ +Tensors of equal rank can be added or subtracted so arithmetic +expressions can be used to produce new tensors. +}{ +\spadpaste{tt := transpose(Tm)*Tn - Tn*transpose(Tm) \free{Tm Tn}\bound{tt2}} +} +\xtc{ +}{ +\spadpaste{Tv*(tt+Tn) \free{tt2 Tv Tn}} +} +\xtc{ +}{ +\spadpaste{reindex(product(Tn,Tn),[4,3,2,1])+3*Tn*product(Tm,Tm) \free{Tm Tn}} +} + +\subsubsection{Specific Tensors} +%\head{subsection}{Specific Tensors}{ugxCartenSpecific} + +Two specific tensors have properties which depend only on the +dimension. + +\labelSpace{1pc} +\xtc{ +The Kronecker delta satisfies +\texht{}{% +\begin{verbatim} + +- + | 1 if i = j +delta(i,j) = | + | 0 if i ^= j + +- +\end{verbatim} +}% +}{ +\spadpaste{delta: CT := kroneckerDelta() \free{CT}\bound{delta}} +} +\xtc{ +This can be used to reindex via contraction. +}{ +\spadpaste{contract(Tmn, 2, delta, 1) = reindex(Tmn, [1,3,4,2]) \free{Tmn delta}} +} + +\xtc{ +The Levi Civita symbol determines the sign of a permutation of indices. +}{ +\spadpaste{epsilon:CT := leviCivitaSymbol() \free{CT}\bound{epsilon}} +} +Here we have: +\texht{}{% +\begin{verbatim} +epsilon(i1,...,idim) + = +1 if i1,...,idim is an even permutation of i0,...,i0+dim-1 + = -1 if i1,...,idim is an odd permutation of i0,...,i0+dim-1 + = 0 if i1,...,idim is not a permutation of i0,...,i0+dim-1 +\end{verbatim} +} +\xtc{ +This property can be used to form determinants. +}{ +\spadpaste{contract(epsilon*Tm*epsilon, 1,2) = 2 * determinant m \free{epsilon Tm m}} +} + +\subsubsection{Properties of the CartesianTensor domain} +%\head{subsection}{Properties of the CartesianTensor domain}{ugxCartenProp} + +\spadtype{GradedModule(R,E)} denotes ``\spad{E}-graded \spad{R}-module'', +that is, a collection of \spad{R}-modules indexed by an abelian monoid +\spad{E.} +An element \spad{g} of \spad{G[s]} for some specific \spad{s} +in \spad{E} is said to be an element of \spad{G} with +\spadfunFrom{degree}{GradedModule} \spad{s}. +Sums are defined in each module \spad{G[s]} so two elements of +\spad{G} can be added if they have the same degree. +Morphisms can be defined and composed by degree to give the +mathematical category of graded modules. + +\spadtype{GradedAlgebra(R,E)} denotes ``\spad{E}-graded \spad{R}-algebra.'' +{CartesianTensor} +A graded algebra is a graded module together with a degree preserving +\spad{R}-bilinear map, called the \spadfunFrom{product}{GradedAlgebra}. + +\begin{verbatim} +degree(product(a,b))= degree(a) + degree(b) + +product(r*a,b) = product(a,r*b) = r*product(a,b) +product(a1+a2,b) = product(a1,b) + product(a2,b) +product(a,b1+b2) = product(a,b1) + product(a,b2) +product(a,product(b,c)) = product(product(a,b),c) +\end{verbatim} + +The domain \spadtype{CartesianTensor(i0, dim, R)} belongs +to the category \spadtype{GradedAlgebra(R, NonNegativeInteger)}. +The non-negative integer \spadfunFrom{degree}{GradedAlgebra} +is the tensor rank +and the graded algebra \spadfunFrom{product}{GradedAlgebra} +is the tensor outer product. +The graded module addition captures the notion that +only tensors of equal rank can be added. + +If \spad{V} is a vector space of dimension \spad{dim} over \spad{R}, +then the tensor module \spad{T[k](V)} is defined as +\begin{verbatim} +T[0](V) = R +T[k](V) = T[k-1](V) * V +\end{verbatim} +where \spadop{*} denotes the \spad{R}-module tensor +\spadfunFrom{product}{GradedAlgebra}. +\spadtype{CartesianTensor(i0,dim,R)} is the graded algebra in which +the degree \spad{k} module is \spad{T[k](V)}. + +\subsubsection{Tensor Calculus} +%\head{subsection}{Tensor Calculus}{ugxCartenCalculus} + +It should be noted here that often tensors are used in the context of +tensor-valued manifold maps. +This leads to the notion of covariant and contravariant bases with tensor +component functions transforming in specific ways under a change of +coordinates on the manifold. +This is no more directly supported by the \spadtype{CartesianTensor} +domain than it is by the \spadtype{Vector} domain. +However, it is possible to have the components implicitly represent +component maps by choosing a polynomial or expression type for the +components. +In this case, it is up to the user to satisfy any constraints which arise +on the basis of this interpretation. +\endscroll +\autobuttons +\end{page} + +@ +\section{cclass.ht} +<>= +\newcommand{\CharacterClassXmpTitle}{CharacterClass} +\newcommand{\CharacterClassXmpNumber}{9.9} + +@ +\subsection{CharacterClass} +\label{CharacterClassXmpPage} +\index{pages!CharacterClassXmpPage!cclass.ht} +\index{cclass.ht!pages!CharacterClassXmpPage} +\index{CharacterClassXmpPage!cclass.ht!pages} +<>= +\begin{page}{CharacterClassXmpPage}{CharacterClass} +\beginscroll + +The \spadtype{CharacterClass} domain allows classes of characters to be +defined and manipulated efficiently. + +\xtc{ +Character classes can be created by giving either a string or a list +of characters. +}{ +\spadpaste{cl1 := charClass [char "a", char "e", char "i", char "o", char "u", char "y"] \bound{cl1}} +} +\xtc{ +}{ +\spadpaste{cl2 := charClass "bcdfghjklmnpqrstvwxyz" \bound{cl2}} +} +\xtc{ +A number of character classes are predefined for convenience. +}{ +\spadpaste{digit()} +} +\xtc{ +}{ +\spadpaste{hexDigit()} +} +\xtc{ +}{ +\spadpaste{upperCase()} +} +\xtc{ +}{ +\spadpaste{lowerCase()} +} +\xtc{ +}{ +\spadpaste{alphabetic()} +} +\xtc{ +}{ +\spadpaste{alphanumeric()} +} +\xtc{ +You can quickly test whether a character belongs to a class. +}{ +\spadpaste{member?(char "a", cl1) \free{cl1}} +} +\xtc{ +}{ +\spadpaste{member?(char "a", cl2) \free{cl2}} +} +\xtc{ +Classes have the usual set operations because +the \spadtype{CharacterClass} domain belongs to the category +\spadtype{FiniteSetAggregate(Character)}. +}{ +\spadpaste{intersect(cl1, cl2) \free{cl1 cl2}} +} +\xtc{ +}{ +\spadpaste{union(cl1,cl2) \free{cl1 cl2}} +} +\xtc{ +}{ +\spadpaste{difference(cl1,cl2) \free{cl1 cl2}} +} +\xtc{ +}{ +\spadpaste{intersect(complement(cl1),cl2) \free{cl1 cl2}} +} +\xtc{ +You can modify character classes by adding or removing characters. +}{ +\spadpaste{insert!(char "a", cl2) \free{cl2}\bound{cl22}} +} +\xtc{ +}{ +\spadpaste{remove!(char "b", cl2) \free{cl22}\bound{cl23}} +} + +For more information on related topics, see +\downlink{`Character'}{CharacterXmpPage}\ignore{Character} and +\downlink{`String'}{StringXmpPage}\ignore{String}. +% +\showBlurb{CharacterClass} +\endscroll +\autobuttons +\end{page} + +@ +\section{char.ht} +<>= +\newcommand{\CharacterXmpTitle}{Character} +\newcommand{\CharacterXmpNumber}{9.8} + +@ +\subsection{Character} +\label{CharacterXmpPage} +\begin{itemize} +\item CharacterClassXmpPage \ref{CharacterClassXmpPage} on +page~\pageref{CharacterClassXmpPage} +\item StringXmpPage \ref{StringXmpPage} on +page~\pageref{StringXmpPage} +\end{itemize} +\index{pages!CharacterXmpPage!char.ht} +\index{char.ht!pages!CharacterXmpPage} +\index{CharacterXmpPage!char.ht!pages} +<>= +\begin{page}{CharacterXmpPage}{Character} +\beginscroll + +The members of the domain \spadtype{Character} are values +representing letters, numerals and other text elements. +For more information on related topics, see +\downlink{`CharacterClass'}{CharacterClassXmpPage}\ignore{CharacterClass} +and \downlink{`String'}{StringXmpPage}\ignore{String}. + +\xtc{ +Characters can be obtained using \spadtype{String} notation. +}{ +\spadpaste{chars := [char "a", char "A", char "X", char "8", char "+"] \bound{chars}} +} +\xtc{ +Certain characters are available by name. +This is the blank character. +}{ +\spadpaste{space()} +} +\xtc{ +This is the quote that is used in strings. +}{ +\spadpaste{quote()} +} +\xtc{ +This is the escape character that allows quotes and other characters +within strings. +}{ +\spadpaste{escape()} +} +\xtc{ +Characters are represented as integers in a machine-dependent way. +The integer value can be obtained using the \spadfunFrom{ord}{Character} +operation. +It is always true that \spad{char(ord c) = c} and \spad{ord(char i) = i}, +provided that \spad{i} is in the range \spad{0..size()\$Character-1}. +}{ +\spadpaste{[ord c for c in chars] \free{chars}} +} + +\xtc{ +The \spadfunFrom{lowerCase}{Character} operation converts an upper case +letter to the corresponding lower case letter. +If the argument is not an upper case letter, then it is returned +unchanged. +}{ +\spadpaste{[upperCase c for c in chars] \free{chars}} +} +\xtc{ +Likewise, the \spadfunFrom{upperCase}{Character} operation converts lower +case letters to upper case. +}{ +\spadpaste{[lowerCase c for c in chars] \free{chars}} +} + +\xtc{ +A number of tests are available to determine whether characters +belong to certain families. +}{ +\spadpaste{[alphabetic? c for c in chars] \free{chars}} +} +\xtc{ +}{ +\spadpaste{[upperCase? c for c in chars] \free{chars}} +} +\xtc{ +}{ +\spadpaste{[lowerCase? c for c in chars] \free{chars}} +} +\xtc{ +}{ +\spadpaste{[digit? c for c in chars] \free{chars}} +} +\xtc{ +}{ +\spadpaste{[hexDigit? c for c in chars] \free{chars}} +} +\xtc{ +}{ +\spadpaste{[alphanumeric? c for c in chars] \free{chars}} +} +\endscroll +\autobuttons +\end{page} +% + +@ +\section{clif.ht} +\newcommand{\CliffordAlgebraXmpTitle}{CliffordAlgebra} +\newcommand{\CliffordAlgebraXmpNumber}{9.10} + +@ +\subsection{CliffordAlgebra} +\label{CliffordAlgebraXmpPage} +\begin{itemize} +\item ugxCliffordComplexPage \ref{ugxCliffordComplexPage} on +page~pageref{ugxCliffordComplexPage} +\item ugxCliffordQuaternPage \ref{ugxCliffordQuaternPage} on +page~pageref{ugxCliffordQuaternPage} +\item ugxCliffordExteriorPage \ref{ugxCliffordExteriorPage} on +page~pageref{ugxCliffordExteriorPage} +\item ugxCliffordDiracPage \ref{ugxCliffordDiracPage} on +page~pageref{ugxCliffordDiracPage} +\end{itemize} +\index{pages!CliffordAlgebraXmpPage!clif.ht} +\index{clif.ht!pages!CliffordAlgebraXmpPage} +\index{CliffordAlgebraXmpPage!clif.ht!pages} +<>= +\begin{page}{CliffordAlgebraXmpPage}{CliffordAlgebra} +\beginscroll + +\noindent +\spadtype{CliffordAlgebra(n,K,Q)} defines a vector space of dimension +\texht{$2^n$}{\spad{2**n}} over the field \texht{$K$}{\spad{K}} with a +given quadratic form \spad{Q}. +If \texht{$\{e_1, \ldots, e_n\}$}{\spad{\{e(i), 1<=i<=n\}}} +is a basis for \texht{$K^n$}{\spad{K**n}} then +\begin{verbatim} +{ 1, + e(i) 1 <= i <= n, + e(i1)*e(i2) 1 <= i1 < i2 <=n, + ..., + e(1)*e(2)*...*e(n) } +\end{verbatim} +is a basis for the Clifford algebra. +The algebra is defined by the relations +\begin{verbatim} +e(i)*e(i) = Q(e(i)) +e(i)*e(j) = -e(j)*e(i), i ^= j +\end{verbatim} +Examples of Clifford Algebras are +gaussians (complex numbers), quaternions, +exterior algebras and spin algebras. +% + +\beginmenu + \menudownlink{{9.10.1. The Complex Numbers as a Clifford Algebra}} +{ugxCliffordComplexPage} + \menudownlink{{9.10.2. The Quaternion Numbers as a Clifford Algebra}} +{ugxCliffordQuaternPage} + \menudownlink{{9.10.3. The Exterior Algebra on a Three Space}} +{ugxCliffordExteriorPage} + \menudownlink{{9.10.4. The Dirac Spin Algebra}} +{ugxCliffordDiracPage} +\endmenu +\endscroll +\autobuttons +\end{page} + +@ +<>= +\newcommand{\ugxCliffordComplexTitle} +{The Complex Numbers as a Clifford Algebra} +\newcommand{\ugxCliffordComplexNumber}{9.10.1.} + +@ +\subsection{The Complex Numbers as a Clifford Algebra} +\label{ugxCliffordComplexPage} +See ComplexXmpPage \ref{ComplexXmpPage} on page~\pageref{ComplexXmpPage} +\index{pages!ugxCliffordComplexPage!clif.ht} +\index{clif.ht!pages!ugxCliffordComplexPage} +\index{ugxCliffordComplexPage!clif.ht!pages} +<>= +\begin{page}{ugxCliffordComplexPage}{The Complex Numbers as a Clifford Algebra} +\beginscroll + +\labelSpace{5pc} +\xtc{ +This is the field over which we will work, rational functions with +integer coefficients. +}{ +\spadpaste{K := Fraction Polynomial Integer \bound{K}} +} +\xtc{ +We use this matrix for the quadratic form. +}{ +\spadpaste{m := matrix [[-1]] \bound{m}} +} +\xtc{ +We get complex arithmetic by using this domain. +}{ +\spadpaste{C := CliffordAlgebra(1, K, quadraticForm m) \free{K m}\bound{C}} +} +\xtc{ +Here is \spad{i}, the usual square root of \spad{-1.} +}{ +\spadpaste{i: C := e(1) \bound{i}\free{C}} +} +\xtc{ +Here are some examples of the arithmetic. +}{ +\spadpaste{x := a + b * i \bound{x}\free{i}} +} +\xtc{ +}{ +\spadpaste{y := c + d * i \bound{y}\free{i}} +} +\xtc{ +See \downlink{`Complex'}{ComplexXmpPage}\ignore{Complex} +for examples of Axiom's constructor +implementing complex numbers. +}{ +\spadpaste{x * y \free{x y}} +} + +\endscroll +\autobuttons +\end{page} + +@ +<>= +\newcommand{\ugxCliffordQuaternTitle} +{The Quaternion Numbers as a Clifford Algebra} +\newcommand{\ugxCliffordQuaternNumber}{9.10.2.} + +@ +\subsection{The Quaternion Numbers as a Clifford Algebra} +\label{ugxCliffordQuaternPage} +See QuaternionXmpPage \ref{QuaternionXmpPage} on +page~\pageref{QuaternionXmpPage} +\index{pages!ugxCliffordQuaternPage!clif.ht} +\index{clif.ht!pages!ugxCliffordQuaternPage} +\index{ugxCliffordQuaternPage!clif.ht!pages} +<>= +\begin{page}{ugxCliffordQuaternPage} +{The Quaternion Numbers as a Clifford Algebra} +\beginscroll + +\labelSpace{3pc} +\xtc{ +This is the field over which we will work, rational functions with +integer coefficients. +}{ +\spadpaste{K := Fraction Polynomial Integer \bound{K}} +} +\xtc{ +We use this matrix for the quadratic form. +}{ +\spadpaste{m := matrix [[-1,0],[0,-1]] \bound{m}} +} +\xtc{ +The resulting domain is the quaternions. +}{ +\spadpaste{H := CliffordAlgebra(2, K, quadraticForm m) \free{K m}\bound{H}} +} +\xtc{ +We use Hamilton's notation for \spad{i},\spad{j},\spad{k}. +}{ +\spadpaste{i: H := e(1) \free{H}\bound{i}} +} +\xtc{ +}{ +\spadpaste{j: H := e(2) \free{H}\bound{j}} +} +\xtc{ +}{ +\spadpaste{k: H := i * j \free{H,i,j}\bound{k}} +} +\xtc{ +}{ +\spadpaste{x := a + b * i + c * j + d * k \free{i j k}\bound{x}} +} +\xtc{ +}{ +\spadpaste{y := e + f * i + g * j + h * k \free{i j k}\bound{y}} +} +\xtc{ +}{ +\spadpaste{x + y \free{x y}} +} +\xtc{ +}{ +\spadpaste{x * y \free{x y}} +} +\xtc{ +See \downlink{`Quaternion'}{QuaternionXmpPage}\ignore{Quaternion} for examples of Axiom's constructor +implementing quaternions. +}{ +\spadpaste{y * x \free{x y}} +} + +\endscroll +\autobuttons +\end{page} + +@ +<>= +\newcommand{\ugxCliffordExteriorTitle}{The Exterior Algebra on a Three Space} +\newcommand{\ugxCliffordExteriorNumber}{9.10.3.} + +@ +\subsection{The Exterior Algebra on a Three Space} +\label{ugxCliffordExteriorPage} +\index{pages!ugxCliffordExteriorPage!clif.ht} +\index{clif.ht!pages!ugxCliffordExteriorPage} +\index{ugxCliffordExteriorPage!clif.ht!pages} +<>= +\begin{page}{ugxCliffordExteriorPage}{The Exterior Algebra on a Three Space} +\beginscroll + +\labelSpace{4pc} +\xtc{ +This is the field over which we will work, rational functions with +integer coefficients. +}{ +\spadpaste{K := Fraction Polynomial Integer \bound{K}} +} +\xtc{ +If we chose the three by three zero quadratic form, we obtain +the exterior algebra on \spad{e(1),e(2),e(3)}. +}{ +\spadpaste{Ext := CliffordAlgebra(3, K, quadraticForm 0) \bound{Ext}\free{K}} +} +\xtc{ +This is a three dimensional vector algebra. +We define \spad{i}, \spad{j}, \spad{k} as the unit vectors. +}{ +\spadpaste{i: Ext := e(1) \free{Ext}\bound{i}} +} +\xtc{ +}{ +\spadpaste{j: Ext := e(2) \free{Ext}\bound{j}} +} +\xtc{ +}{ +\spadpaste{k: Ext := e(3) \free{Ext}\bound{k}} +} +\xtc{ +Now it is possible to do arithmetic. +}{ +\spadpaste{x := x1*i + x2*j + x3*k \free{i j k}\bound{x}} +} +\xtc{ +}{ +\spadpaste{y := y1*i + y2*j + y3*k \free{i j k}\bound{y}} +} +\xtc{ +}{ +\spadpaste{x + y \free{x y}} +} +\xtc{ +}{ +\spadpaste{x * y + y * x \free{x y}} +} +\xtc{ +On an \spad{n} space, a grade \spad{p} form has a dual \spad{n-p} form. +In particular, in three space the dual of a grade two element identifies +\spad{e1*e2->e3, e2*e3->e1, e3*e1->e2}. +}{ +\spadpaste{dual2 a == coefficient(a,[2,3]) * i + coefficient(a,[3,1]) * j + coefficient(a,[1,2]) * k \free{i j k}\bound{dual2}} +} +\xtc{ +The vector cross product is then given by this. +}{ +\spadpaste{dual2(x*y) \free{x y dual2}} +} + +\endscroll +\autobuttons +\end{page} + +@ +<>= +\newcommand{\ugxCliffordDiracTitle}{The Dirac Spin Algebra} +\newcommand{\ugxCliffordDiracNumber}{9.10.4.} + +@ +\subsection{The Dirac Spin Algebra} +\label{ugxCliffordDiracPage} +\index{pages!ugxCliffordDiracPage!clif.ht} +\index{clif.ht!pages!ugxCliffordDiracPage} +\index{ugxCliffordDiracPage!clif.ht!pages} +<>= +\begin{page}{ugxCliffordDiracPage}{The Dirac Spin Algebra} +\beginscroll + +\labelSpace{4pc} +{The Dirac Spin Algebra} +% +\xtc{ +In this section we will work over the field of rational numbers. +}{ +\spadpaste{K := Fraction Integer \bound{K}} +} +\xtc{ +We define the quadratic form to be the Minkowski space-time metric. +}{ +\spadpaste{g := matrix [[1,0,0,0], [0,-1,0,0], [0,0,-1,0], [0,0,0,-1]] \bound{g}} +} +\xtc{ +We obtain the Dirac spin algebra +used in Relativistic Quantum Field Theory. +{9.10.4.}{The Dirac Spin Algebra} +}{ +\spadpaste{D := CliffordAlgebra(4,K, quadraticForm g) \free{K g}\bound{D}} +} +\xtc{ +The usual notation for the basis is \texht{$\gamma$}{\spad{gamma}} +with a superscript. +For Axiom input we will use \spad{gam(i)}: +}{ +\spadpaste{gam := [e(i)\$D for i in 1..4] \free{D}\bound{gam}} +} +\noindent +There are various contraction identities of the form +\begin{verbatim} +g(l,t)*gam(l)*gam(m)*gam(n)*gam(r)*gam(s)*gam(t) = + 2*(gam(s)gam(m)gam(n)gam(r) + gam(r)*gam(n)*gam(m)*gam(s)) +\end{verbatim} +where a sum over \spad{l} and \spad{t} is implied. +\xtc{ +Verify this identity for particular values of \spad{m,n,r,s}. +}{ +\spadpaste{m := 1; n:= 2; r := 3; s := 4; \bound{m,n,r,s}} +} +\xtc{ +}{ +\spadpaste{lhs := reduce(+, [reduce(+, [ g(l,t)*gam(l)*gam(m)*gam(n)*gam(r)*gam(s)*gam(t) for l in 1..4]) for t in 1..4]) \bound{lhs}\free{g gam m n r s}} +} +\xtc{ +}{ +\spadpaste{rhs := 2*(gam s * gam m*gam n*gam r + gam r*gam n*gam m*gam s) \bound{rhs}\free{lhs g gam m n r s}} +} +\endscroll +\autobuttons +\end{page} + +@ +\section{complex.ht} +<>= +\newcommand{\ComplexXmpTitle}{Complex} +\newcommand{\ComplexXmpNumber}{9.11} + +@ +\subsection{Complex} +\label{ComplexXmpPage} +\begin{itemize} +\item ugProblemNumericPage \ref{ugProblemNumericPage} on +page~pageref{ugProblemNumericPage} +\item ugTypesConvertPage \ref{ugTypesConvertPage} on +page~pageref{ugTypesConvertPage} +\end{itemize} +\index{pages!ComplexXmpPage!complex.ht} +\index{complex.ht!pages!ComplexXmpPage} +\index{ComplexXmpPage!complex.ht!pages} +<>= +\begin{page}{ComplexXmpPage}{Complex} +\beginscroll + +The \spadtype{Complex} constructor implements complex objects over a +commutative ring \spad{R}. +Typically, the ring \spad{R} is \spadtype{Integer}, \spadtype{Fraction +Integer}, \spadtype{Float} or \spadtype{DoubleFloat}. +\spad{R} can also be a symbolic type, like \spadtype{Polynomial Integer}. +For more information about the numerical and graphical aspects of complex +numbers, +see \downlink{``\ugProblemNumericTitle''}{ugProblemNumericPage} in Section +\ugProblemNumericNumber\ignore{ugProblemNumeric}. + +\xtc{ +Complex objects are created by the \spadfunFrom{complex}{Complex} operation. +}{ +\spadpaste{a := complex(4/3,5/2) \bound{a}} +} +\xtc{ +}{ +\spadpaste{b := complex(4/3,-5/2) \bound{b}} +} +\xtc{ +The standard arithmetic operations are available. +}{ +\spadpaste{a + b \free{a b}} +} +\xtc{ +}{ +\spadpaste{a - b \free{a b}} +} +\xtc{ +}{ +\spadpaste{a * b \free{a b}} +} +\xtc{ +If \spad{R} is a field, you can also divide the complex objects. +}{ +\spadpaste{a / b \free{a b}\bound{adb}} +} +\xtc{ +Use a conversion (\downlink{``\ugTypesConvertTitle''}{ugTypesConvertPage} in Section \ugTypesConvertNumber\ignore{ugTypesConvert}) to view the last +object as a fraction of complex integers. +}{ +\spadpaste{\% :: Fraction Complex Integer \free{adb}} +} +\xtc{ +The predefined macro \spad{\%i} is defined to be \spad{complex(0,1)}. +}{ +\spadpaste{3.4 + 6.7 * \%i} +} +\xtc{ +You can also compute the \spadfunFrom{conjugate}{Complex} and +\spadfunFrom{norm}{Complex} of a complex number. +}{ +\spadpaste{conjugate a \free{a}} +} +\xtc{ +}{ +\spadpaste{norm a \free{a}} +} +\xtc{ +The \spadfunFrom{real}{Complex} and \spadfunFrom{imag}{Complex} operations +are provided to extract the real and imaginary parts, respectively. +}{ +\spadpaste{real a \free{a}} +} +\xtc{ +}{ +\spadpaste{imag a \free{a}} +} + +\xtc{ +The domain \spadtype{Complex Integer} is also called the Gaussian +integers. +If \spad{R} is the integers (or, more generally, +a \spadtype{EuclideanDomain}), you can compute greatest common divisors. +}{ +\spadpaste{gcd(13 - 13*\%i,31 + 27*\%i)} +} +\xtc{ +You can also compute least common multiples. +}{ +\spadpaste{lcm(13 - 13*\%i,31 + 27*\%i)} +} +\xtc{ +You can \spadfunFrom{factor}{Complex} Gaussian integers. +}{ +\spadpaste{factor(13 - 13*\%i)} +} +\xtc{ +}{ +\spadpaste{factor complex(2,0)} +} +\endscroll +\autobuttons +\end{page} + +@ +\section{contfrac.ht} +<>= +\newcommand{\ContinuedFractionXmpTitle}{ContinuedFraction} +\newcommand{\ContinuedFractionXmpNumber}{9.12} + +@ +\subsection{ContinuedFraction} +\label{ContinuedFractionXmpPage} +See StreamXmpPage \ref{StreamXmpPage} on page~\pageref{StreamXmpPage} +\index{pages!ContinuedFractionXmpPage!contfrac.ht} +\index{contfrac.ht!pages!ContinuedFractionXmpPage} +\index{ContinuedFractionXmpPage!contfrac.ht!pages} +<>= +\begin{page}{ContinuedFractionXmpPage}{ContinuedFraction} +\beginscroll + +Continued fractions have been a fascinating and useful tool in mathematics +for well over three hundred years. +Axiom implements continued fractions for fractions of any Euclidean +domain. +In practice, this usually means rational numbers. +In this section we demonstrate some of the operations available for +manipulating both finite and infinite continued fractions. +It may be helpful if you review +\downlink{`Stream'}{StreamXmpPage}\ignore{Stream} to remind yourself of some +of the operations with streams. + +The \spadtype{ContinuedFraction} domain is a field and therefore you can +add, subtract, multiply and divide the fractions. +\xtc{ +The \spadfunFrom{continuedFraction}{ContinuedFraction} operation +converts its fractional argument to a continued fraction. +}{ +\spadpaste{c := continuedFraction(314159/100000) \bound{c}} +} +% +This display is a compact form of the bulkier +\texht{\narrowDisplay{% +3 + {\displaystyle 1 \over {\displaystyle +7 + {1 \over {\displaystyle +15 + {1 \over {\displaystyle +1 + {1 \over {\displaystyle +25 + {1 \over {\displaystyle +1 + {1 \over {\displaystyle +7 + {1 \over 4}}}}}}}}}}}}}}% +}{ +\begin{verbatim} + 3 + 1 + ------------------------------- + 7 + 1 + --------------------------- + 15 + 1 + ---------------------- + 1 + 1 + ------------------ + 25 + 1 + ------------- + 1 + 1 + --------- + 7 + 1 + ----- + 4 +\end{verbatim} +} +You can write any rational number in a similar form. +The fraction will be finite and you can always take the ``numerators'' to +be \spad{1}. +That is, any rational number can be written as a simple, finite continued +fraction of the form +\texht{\narrowDisplay{% +a_1 + {\displaystyle 1 \over {\displaystyle +a_2 + {1 \over {\displaystyle +a_3 + {1 \over {\displaystyle \ddots +a_{n-1} + {1 \over a_n}}}}}}}}% +}{ +\begin{verbatim} + a(1) + 1 + ------------------------- + a(2) + 1 + -------------------- + a(3) + + . + . + . + 1 + ------------- + a(n-1) + 1 + ---- + a(n) +\end{verbatim} +} +\xtc{ +The \texht{$a_i$}{\spad{a(i)}} are called partial quotients and the operation +\spadfunFrom{partialQuotients}{ContinuedFraction} creates a stream of them. +}{ +\spadpaste{partialQuotients c \free{c}} +} +\xtc{ +By considering more and more of the fraction, you get the +\spadfunFrom{convergents}{ContinuedFraction}. +For example, the first convergent is \texht{$a_1$}{\spad{a(1)}}, +the second is +\texht{$a_1 + 1/a_2$}{\spad{a(1) + 1/a(2)}} and so on. +}{ +\spadpaste{convergents c \free{c}} +} +% +\xtc{ +Since this is a finite continued fraction, the last convergent is +the original rational number, in reduced form. +The result of \spadfunFrom{approximants}{ContinuedFraction} +is always an infinite stream, though it may just repeat the ``last'' +value. +}{ +\spadpaste{approximants c \free{c}} +} +\xtc{ +Inverting \spad{c} only changes the partial quotients of its fraction +by inserting a \spad{0} at the beginning of the list. +}{ +\spadpaste{pq := partialQuotients(1/c) \free{c}\bound{pq}} +} +\xtc{ +Do this to recover the original continued fraction from this list of +partial quotients. +The three-argument form of the +\spadfunFrom{continuedFraction}{ContinuedFraction} operation takes an +element which is the whole part of the fraction, a stream of elements +which are the numerators of the fraction, and a stream of elements which +are the denominators of the fraction. +}{ +\spadpaste{continuedFraction(first pq,repeating [1],rest pq) \free{pq}} +} +\xtc{ +The streams need not be finite for +\spadfunFrom{continuedFraction}{ContinuedFraction}. +Can you guess which irrational number has the following continued +fraction? +See the end of this section for the answer. +}{ +\spadpaste{z:=continuedFraction(3,repeating [1],repeating [3,6]) \bound{z}} +} +% + +In 1737 Euler discovered the infinite continued fraction expansion +\texht{\narrowDisplay{% +{{e - 1} \over 2} = +{1 \over {\displaystyle +1 + {1 \over {\displaystyle +6 + {1 \over {\displaystyle +10 + {1 \over {\displaystyle +14 + \cdots}}}}}}}}}% +}{ +\begin{verbatim} + e - 1 1 + ----- = --------------------- + 2 1 + 1 + ----------------- + 6 + 1 + ------------- + 10 + 1 + -------- + 14 + ... +\end{verbatim} +} +We use this expansion to compute rational and floating point +approximations of \spad{e}.\footnote{For this and other interesting +expansions, see C. D. Olds, {\it Continued Fractions,} +New Mathematical Library, (New York: Random House, 1963), pp. +134--139.} + +\xtc{ +By looking at the above expansion, we see that the whole part is \spad{0} +and the numerators are all equal to \spad{1}. +This constructs the stream of denominators. +}{ +\spadpaste{dens:Stream Integer := cons(1,generate((x+->x+4),6)) \bound{dens}} +} +\xtc{ +Therefore this is the continued fraction expansion for +\texht{$(e - 1) / 2$}{\spad{(e-1)/2}}. +}{ +\spadpaste{cf := continuedFraction(0,repeating [1],dens) \free{dens}\bound{cf}} +} +\xtc{ +These are the rational number convergents. +}{ +\spadpaste{ccf := convergents cf \free{cf}\bound{ccf}} +} +\xtc{ +You can get rational convergents for \spad{e} by multiplying by \spad{2} and +adding \spad{1}. +}{ +\spadpaste{eConvergents := [2*e + 1 for e in ccf] \bound{ec}\free{ccf}} +} +% +\xtc{ +You can also compute the floating point approximations to these convergents. +}{ +\spadpaste{eConvergents :: Stream Float \free{ec}} +} +\xtc{ +Compare this to the value of \spad{e} computed by the +\spadfunFrom{exp}{Float} operation in \spadtype{Float}. +}{ +\spadpaste{exp 1.0} +} + +In about 1658, Lord Brouncker established the following expansion +for \texht{$4 / \pi$}{\spad{4/pi}}. +\texht{\narrowDisplay{% +1 + {\displaystyle +1 \over {\displaystyle +2 + {9 \over {\displaystyle +2 + {25 \over {\displaystyle +2 + {49 \over {\displaystyle +2 + {81 \over {\displaystyle +2 + \cdots}}}}}}}}}}}% +}{ +\begin{verbatim} + 1 + 1 + ----------------------- + 2 + 9 + ------------------- + 2 + 25 + --------------- + 2 + 49 + ----------- + 2 + 81 + ------- + 2 + ... +\end{verbatim} +} +\xtc{ +Let's use this expansion to compute rational and floating point +approximations for \texht{$\pi$}{\spad{pi}}. +}{ +\spadpaste{ +cf := continuedFraction(1,[(2*i+1)**2 for i in 0..],repeating [2])\bound{cf1}} +} +\xtc{ +}{ +\spadpaste{ccf := convergents cf \free{cf1}\bound{ccf1}} +} +\xtc{ +}{ +\spadpaste{ +piConvergents := [4/p for p in ccf] \bound{piConvergents}\free{ccf1}} +} +\xtc{ +As you can see, the values are converging to +\texht{$\pi$}{\spad{pi}} = 3.14159265358979323846..., +but not very quickly. +}{ +\spadpaste{piConvergents :: Stream Float \free{piConvergents}} +} + +\xtc{ +You need not restrict yourself to continued fractions of integers. +Here is an expansion for a quotient of Gaussian integers. +}{ +\spadpaste{continuedFraction((- 122 + 597*\%i)/(4 - 4*\%i))} +} +\xtc{ +This is an expansion for a quotient of polynomials in one variable +with rational number coefficients. +}{ +\spadpaste{r : Fraction UnivariatePolynomial(x,Fraction Integer) \bound{rdec}} +} +\xtc{ +}{ +\spadpaste{r := ((x - 1) * (x - 2)) / ((x-3) * (x-4)) \free{rdec}\bound{r}} +} +\xtc{ +}{ +\spadpaste{continuedFraction r \free{r}} +} + +To conclude this section, we give you evidence that +\texht{\narrowDisplay{% +z = +{3+\zag{1}{3}+\zag{1}{6}+\zag{1}{3}+\zag{1}{6}+\zag{1}{3}+\zag{1}{6}+ +\zag{1}{3}+\zag{1}{6}+\zag{1}{3}+\zag{1}{6}+...}}% +}{ +\begin{verbatim} + z = 3 + 1 + ----------------------- + 3 + 1 + ------------------- + 6 + 1 + --------------- + 3 + 1 + ----------- + 6 + 1 + ------- + 3 + ... +\end{verbatim} +} +is the expansion of \texht{$\sqrt{11}$}{the square root of \spad{11}}. +% +\xtc{ +}{ +\spadpaste{[i*i for i in convergents(z) :: Stream Float] \free{z}} +} +\endscroll +\autobuttons +\end{page} + +@ +\section{cphelp.ht} +\subsection{Control Panel Bits} +\label{CPHelp} +\index{pages!CPHelp!cphelp.ht} +\index{cphelp.ht!pages!CPHelp} +\index{CPHelp!cphelp.ht!pages} +<>= +\begin{page}{CPHelp}{Control Panel Bits} +\beginscroll + +Here are some stuff from a Three Dimensional Viewport's Control Panel \newline + +Main Control Panel: \newline \newline + +%% +%% These things were generated from the file helpCP3D.dat +%% +%% The help comments are hokey and are not the intended +%% format. +%% + +Rotate: \helpbit{rotate3D} +Zoom: \helpbit{zoom3D} +Translate up/down left/right in the window: \helpbit{translate3D} +Changing the color of the rendered surface: \helpbit{color3D} +Turn the axes on and off: \helpbit{axes3D} +Display surface as a transparent wire mesh: \helpbit{transparent3D} +Display surface with hidden surface removed and the core dumped: \helpbit{opaque3D} +Display rendered surface: \helpbit{render3D} +Show region within which the function is defined: \helpbit{region3D} +Change position of light source: \helpbit{lighting3D} +Change Perspective/Clipping of surface: \helpbit{volume3D} +Reset to original viewpoint: \helpbit{reset3D} +Show grid lines on the rendered surface: \helpbit{outline3D} +Hide the menu: \helpbit{hide3D} +Close the viewport: \helpbit{close3D} + +\newline +\newline +\endscroll +\autobuttons +\end{page} + +@ +\section{cycles.ht} +<>= +\newcommand{\CycleIndicatorsXmpTitle}{CycleIndicators} +\newcommand{\CycleIndicatorsXmpNumber}{9.13} + +@ +\subsection{CycleIndicators} +\label{CycleIndicatorsXmpPage} +\index{pages!CycleIndicatorsXmpPage!cycles.ht} +\index{cycles.ht!pages!CycleIndicatorsXmpPage} +\index{CycleIndicatorsXmpPage!cycles.ht!pages} +<>= +\begin{page}{CycleIndicatorsXmpPage}{CycleIndicators} +\beginscroll +This section is based upon the paper +J. H. Redfield, ``The Theory of Group-Reduced Distributions'', +and is an application of group theory to enumeration problems. +It is a development of the work by P. A. MacMahon on the + +The theory is based upon the power sum symmetric functions +\subscriptIt{s}{i} +which are the sum of the \eth{\it i} powers of the variables. +The cycle index of a permutation is an expression that specifies +the sizes of the cycles of a permutation, and +may be represented as a partition. +A partition of a non-negative integer \spad{n} is a collection +of positive integers called its parts whose sum is \spad{n}. +For example, the partition +\texht{$(3^2 \ 2 \ 1^2)$}{\spad{3**2 2 1**2)}} +will be used to represent +\texht{$s^2_3 s_2 s^2_1$}{\spad{(s_3)**2 s_2 (s_1)**2}} +and will indicate that the permutation has two cycles of length 3, +one of length 2 and two of length 1. +The cycle index of a permutation group is the sum of the cycle indices +of its permutations divided by the number of permutations. +The cycle indices of certain groups are provided. +\xtc{ +We first expose something from the library. +}{ +\spadpaste{)expose EVALCYC} +} +\xtc{ +The operation \spadfun{complete} returns the cycle index of the +symmetric group of order \spad{n} for argument \spad{n}. +Alternatively, it is the \eth{\spad{n}} complete homogeneous symmetric +function expressed in terms of power sum symmetric functions. +}{ +\spadpaste{complete 1} +} +\xtc{ +}{ +\spadpaste{complete 2} +} +\xtc{ +}{ +\spadpaste{complete 3} +} +\xtc{ +}{ +\spadpaste{complete 7} +} +\xtc{ +The operation \spadfun{elementary} computes the \eth{\spad{n}} +elementary symmetric function for argument \spad{n.} +}{ +\spadpaste{elementary 7} +} +\xtc{ +The operation \spadfun{alternating} returns +the cycle index of the alternating group +having an even number of even parts in each cycle partition. +}{ +\spadpaste{alternating 7} +} +\xtc{ +The operation \spadfun{cyclic} returns the cycle index of the cyclic group. +}{ +\spadpaste{cyclic 7} +} +\xtc{ +The operation \spadfun{dihedral} is the cycle index of the +dihedral group. +}{ +\spadpaste{dihedral 7} +} +\xtc{ +The operation \spadfun{graphs} for argument \spad{n} returns +the cycle index of the group of permutations on +the edges of the complete graph with \spad{n} nodes induced by +applying the symmetric group to the nodes. +}{ +\spadpaste{graphs 5} +} + +The cycle index of a direct product of two groups is the product +of the cycle indices of the groups. +Redfield provided two operations on two cycle indices which will +be called ``cup'' and ``cap'' here. +The \spadfun{cup} of two cycle indices is a kind of scalar product +that combines monomials for permutations with the same cycles. +The \spadfun{cap} operation provides the sum of the coefficients +of the result of the \spadfun{cup} operation which will be an +integer that enumerates what Redfield called +group-reduced distributions. + +We can, for example, represent \spad{complete 2 * complete 2} +as the set of objects \spad{a a b b} and +\spad{complete 2 * complete 1 * complete 1} as \spad{c c d e.} + +\xtc{ +This integer +is the number of different sets of four pairs. +}{ +\spadpaste{cap(complete 2**2, complete 2*complete 1**2)} +} +For example, +\begin{verbatim} +a a b b a a b b a a b b a a b b +c c d e c d c e c e c d d e c c +\end{verbatim} + +\xtc{ +This integer +is the number of different sets of four pairs no two pairs being equal. +}{ +\spadpaste{cap(elementary 2**2, complete 2*complete 1**2)} +} +For example, +\begin{verbatim} +a a b b a a b b +c d c e c e c d +\end{verbatim} +In this case the configurations enumerated are easily constructed, +however the theory merely enumerates them providing little help in +actually constructing them. +\xtc{ +Here are the +number of 6-pairs, first from \spad{a a a b b c,} second from +\spad{d d e e f g.} +}{ +\spadpaste{cap(complete 3*complete 2*complete 1,complete 2**2*complete 1**2)} +} +\xtc{ +Here it is again, but with no equal pairs. +}{ +\spadpaste{cap(elementary 3*elementary 2*elementary 1,complete 2**2*complete 1**2)} +} +\xtc{ +}{ +\spadpaste{cap(complete 3*complete 2*complete 1,elementary 2**2*elementary 1**2)} +} +\xtc{ +The number of 6-triples, first from \spad{a a a b b c,} second from +\spad{d d e e f g,} third from \spad{h h i i j j.} +}{ +\spadpaste{eval(cup(complete 3*complete 2*complete 1, cup(complete 2**2*complete 1**2,complete 2**3)))} +} +\xtc{ +The cycle index of vertices of a square is dihedral 4. +}{ +\spadpaste{square:=dihedral 4} +} +\xtc{ +The number of different squares with 2 red vertices and 2 blue vertices. +}{ +\spadpaste{cap(complete 2**2,square)} +} +\xtc{ +The number of necklaces with 3 red beads, 2 blue beads and 2 green beads. +}{ +\spadpaste{cap(complete 3*complete 2**2,dihedral 7)} +} +\xtc{ +The number of graphs with 5 nodes and 7 edges. +}{ +\spadpaste{cap(graphs 5,complete 7*complete 3)} +} +\xtc{ +The cycle index of rotations of vertices of a cube. +}{ +\spadpaste{s(x) == powerSum(x)} +} +\xtc{ +}{ +\spadpaste{cube:=(1/24)*(s 1**8+9*s 2**4 + 8*s 3**2*s 1**2+6*s 4**2)} +} +\xtc{ +The number of cubes with 4 red vertices and 4 blue vertices. +}{ +\spadpaste{cap(complete 4**2,cube)} +} +\xtc{ +The number of labeled graphs with degree sequence \spad{2 2 2 1 1} +with no loops or multiple edges. +}{ +\spadpaste{cap(complete 2**3*complete 1**2,wreath(elementary 4,elementary 2))} +} +\xtc{ +Again, but +with loops allowed but not multiple edges. +}{ +\spadpaste{cap(complete 2**3*complete 1**2,wreath(elementary 4,complete 2))} +} +\xtc{ +Again, but +with multiple edges allowed, but not loops +}{ +\spadpaste{cap(complete 2**3*complete 1**2,wreath(complete 4,elementary 2))} +} +\xtc{ +Again, but +with both multiple edges and loops allowed +}{ +\spadpaste{cap(complete 2**3*complete 1**2,wreath(complete 4,complete 2))} +} + +Having constructed a cycle index for a configuration +we are at liberty to evaluate the +\subscriptIt{s}{i} +components any way we please. +For example we can produce enumerating generating functions. +{CycleIndicators} +This is done by providing a function \spad{f} on an integer \spad{i} to the +value required of \subscriptIt{s}{i}, +and then evaluating \spad{eval(f, cycleindex)}. +\xtc{ +}{ +\spadpaste{x: ULS(FRAC INT,'x,0) := 'x \bound{x}} +} +\xtc{ +}{ +\spadpaste{ZeroOrOne: INT -> ULS(FRAC INT, 'x, 0) \bound{zodec}} +} +\xtc{ +}{ +\spadpaste{Integers: INT -> ULS(FRAC INT, 'x, 0) \bound{idec}} +} +\xtc{ +For the integers 0 and 1, or two colors. +}{ +\spadpaste{ZeroOrOne n == 1+x**n \free{x zodec}\bound{zo}} +} +\xtc{ +}{ +\spadpaste{ZeroOrOne 5 \free{zo}} +} +\xtc{ +For the integers \spad{0, 1, 2, ...} we have this. +}{ +\spadpaste{Integers n == 1/(1-x**n) \free{x idec}\bound{i}} +} +\xtc{ +}{ +\spadpaste{Integers 5 \free{i}} +} + +\xtc{ +The coefficient of \texht{$x^n$}{\spad{x**n}} +is the number of graphs with 5 nodes +and \spad{n} edges. +}{ +\spadpaste{eval(ZeroOrOne, graphs 5) \free{zo}} +} +\xtc{ +The coefficient of \texht{$x^n$}{\spad{x**n}} is the number of necklaces with +\spad{n} red beads and \spad{n-8} green beads. +}{ +\spadpaste{eval(ZeroOrOne,dihedral 8) \free{zo}} +} +\xtc{ +The coefficient of \texht{$x^n$}{\spad{x**n}} is the number of partitions of +\spad{n} into 4 or fewer parts. +}{ +\spadpaste{eval(Integers,complete 4) \free{i}} +} +\xtc{ +The coefficient of \texht{$x^n$}{\spad{x**n}} is the number of +partitions of \spad{n} into 4 +boxes containing ordered distinct parts. +}{ +\spadpaste{eval(Integers,elementary 4) \free{i}} +} +\xtc{ +The coefficient of \texht{$x^n$}{\spad{x**n}} is the number of +different cubes with \spad{n} red vertices and \spad{8-n} green ones. +}{ +\spadpaste{eval(ZeroOrOne,cube) \free{zo}} +} +\xtc{ +The coefficient of \texht{$x^n$}{\spad{x**n}} +is the number of different cubes with integers +on the vertices whose sum is \spad{n.} +}{ +\spadpaste{eval(Integers,cube) \free{i}} +} +\xtc{ +The coefficient of \texht{$x^n$}{\spad{x**n}} is the number of +graphs with 5 nodes and with integers on the edges whose sum is +\spad{n.} +In other words, the enumeration is of multigraphs with 5 nodes and +\spad{n} edges. +}{ +\spadpaste{eval(Integers,graphs 5) \free{i}} +} +\xtc{ +Graphs with 15 nodes enumerated with respect to number of edges. +}{ +\spadpaste{eval(ZeroOrOne ,graphs 15) \free{zo}} +} +\xtc{ +Necklaces with 7 green beads, 8 white beads, 5 yellow beads and 10 +red beads. +}{ +\spadpaste{cap(dihedral 30,complete 7*complete 8*complete 5*complete 10)} +} +The operation \spadfun{SFunction} is the S-function or Schur function +of a partition written +as a descending list of integers expressed in terms of power sum +symmetric functions. +\xtc{ +In this case the argument partition represents a tableau shape. +For example \spad{3,2,2,1} represents a tableau with three boxes in the +first row, two boxes in the second and third rows, and one box in the +fourth row. +\spad{SFunction [3,2,2,1]} +counts the number of different tableaux of shape \spad{3, 2, 2, 1} filled +with objects with an ascending order in the columns and a +non-descending order in the rows. +}{ +\spadpaste{sf3221:= SFunction [3,2,2,1] \bound{sf3221}} +} +\xtc{ +This is the number filled with \spad{a a b b c c d d.} +}{ +\spadpaste{cap(sf3221,complete 2**4) \free{sf3221}} +} +The configurations enumerated above are: +\begin{verbatim} +a a b a a c a a d +b c b b b b +c d c d c c +d d d +\end{verbatim} +\xtc{ +This is the number of tableaux filled with \spad{1..8.} +}{ +\spadpaste{cap(sf3221, powerSum 1**8)\free{sf3221}} +} +\xtc{ +The coefficient of \texht{$x^n$}{\spad{x**n}} is the number +of column strict reverse plane partitions of \spad{n} of shape +\spad{3 2 2 1.} +}{ +\spadpaste{eval(Integers, sf3221)\free{i sf3221}} +} +The smallest is +\begin{verbatim} +0 0 0 +1 1 +2 2 +3 +\end{verbatim} +\endscroll +\autobuttons +\end{page} + +@ +\section{coverex.ht} +\subsection{Examples Of Axiom Commands} +DO NOT EDIT! Created by ex2ht. +\begin{itemize} +\item Menuexdiff \ref{Menuexdiff} on page~\pageref{Menuexdiff} +\item menuexint \ref{menuexint} on page~\pageref{menuexint} +\item Menuexlap \ref{Menuexlap} on page~\pageref{Menuexlap} +\item Menuexlimit \ref{Menuexlimit} on page~\pageref{Menuexlimit} +\item Menuexmatrix \ref{Menuexmatrix} on page~\pageref{Menuexmatrix} +\item Menuexplot2d \ref{Menuexplot2d} on page~\pageref{Menuexplot2d} +\item Menuexplot3d \ref{Menuexplot3d} on page~\pageref{Menuexplot3d} +\item Menuexseries \ref{Menuexseries} on page~\pageref{Menuexseries} +\item Menuexsum \ref{Menuexsum} on page~\pageref{Menuexsum} +\end{itemize} +\label{ExampleCoverPage} +\index{pages!ExampleCoverPage!coverex.ht} +\index{coverex.ht!pages!ExampleCoverPage} +\index{ExampleCoverPage!coverex.ht!pages} +<>= +\begin{page}{ExampleCoverPage}{Examples Of Axiom Commands} +\beginscroll\table{ +{\downlink{Differentiation}{Menuexdiff}} +{\downlink{Integration}{Menuexint}} +{\downlink{Laplace Transforms}{Menuexlap}} +{\downlink{Limits}{Menuexlimit}} +{\downlink{Matrices}{Menuexmatrix}} +{\downlink{2-D Graphics}{Menuexplot2d}} +{\downlink{3-D Graphics}{Menuexplot3d}} +{\downlink{Series}{Menuexseries}} +{\downlink{Summations}{Menuexsum}} +}\endscroll\end{page} + +@ +\subsection{Differentiation} +\label{Menuexdiff} +\begin{itemize} +\item ExDiffBasic \ref{ExDiffBasic} on page~\pageref{ExDiffBasic} +\item ExDiffSeveralVariables \ref{ExDiffSeveralVariables} on +page~\pageref{ExDiffSeveralVariables} +\item ExDiffHigherOrder \ref{ExDiffHigherOrder} on +page~\pageref{ExDiffHigherOrder} +\item ExDiffMultipleI \ref{ExDiffMultipleI} on page~\pageref{ExDiffMultipleI} +\item ExDiffMultipleII \ref{ExDiffMultipleII} on +page~\pageref{ExDiffMultipleII} +\item ExDiffFormalIntegral \ref{ExDiffFormalIntegral} on +page~\pageref{ExDiffFormalIntegral} +\end{itemize} +\index{pages!Menuexdiff!coverex.ht} +\index{coverex.ht!pages!Menuexdiff} +\index{Menuexdiff!coverex.ht!pages} +<>= +\begin{page}{Menuexdiff}{Differentiation} +\beginscroll\beginmenu +\menudownlink{Computing Derivatives}{ExDiffBasic} +\spadpaste{differentiate(sin(x) * exp(x**2),x)} +\menudownlink{Derivatives of Functions of Several Variables}{ExDiffSeveralVariables} +\spadpaste{differentiate(sin(x) * tan(y)/(x**2 + y**2),x)} +\spadpaste{differentiate(sin(x) * tan(y)/(x**2 + y**2),y)} +\menudownlink{Derivatives of Higher Order}{ExDiffHigherOrder} +\spadpaste{differentiate(exp(x**2),x,4)} +\menudownlink{Multiple Derivatives I}{ExDiffMultipleI} +\spadpaste{differentiate(sin(x)/(x**2 + y**2),[x,y])} +\spadpaste{differentiate(sin(x)/(x**2 + y**2),[x,y,y])} +\menudownlink{Multiple Derivatives II}{ExDiffMultipleII} +\spadpaste{differentiate(cos(z)/(x**2 + y**3),[x,y,z],[1,2,3])} +\menudownlink{Derivatives of Functions Involving Formal Integrals}{ExDiffFormalIntegral} +\spadpaste{f := integrate(sqrt(1 + t**3),t) \bound{f}} +\spadpaste{differentiate(f,t) \free{f}} +\spadpaste{differentiate(f * t**2,t) \free{f}} +\endmenu\endscroll\end{page} + +@ +\subsection{Integration} +\label{Menuexint} +\begin{itemize} +\item ExIntRationalFunction \ref{ExIntRationalFunction} on +page~\pageref{ExIntRationalFunction} +\item ExIntRationalWithRealParameter \ref{ExIntRationalWithRealParameter} on +page~\pageref{ExIntRationalWithRealParameter} +\item ExIntRationalWithComplexParameter +\ref{ExIntRationalWithComplexParameter} on +page~\pageref{ExIntRationalWithComplexParameter} +\item ExIntTwoSimilarIntegrands \ref{ExIntTwoSimilarIntegrands} on +page~\pageref{ExIntTwoSimilarIntegrands} +\item ExIntNoSolution \ref{ExIntNoSolution} on page~\pageref{ExIntNoSolution} +\item ExIntTrig \ref{ExIntTrig} on page~\pageref{ExIntTrig} +\item ExIntAlgebraicRelation \ref{ExIntAlgebraicRelation} on +page~\pageref{ExIntAlgebraicRelation} +\item ExIntRadicalOfTranscendental \ref{ExIntRadicalOfTranscendental} on +page~\pageref{ExIntRadicalOfTranscendental} +\item ExIntNonElementary \ref{ExIntNonElementary} on +page~\pageref{ExIntNonElementary} +\end{itemize} +\index{pages!Menuexint!coverex.ht} +\index{coverex.ht!pages!Menuexint} +\index{Menuexint!coverex.ht!pages} +<>= +\begin{page}{Menuexint}{Integration} +\beginscroll\beginmenu +\menudownlink{Integral of a Rational Function}{ExIntRationalFunction} +\spadpaste{integrate((x**2+2*x+1)/((x+1)**6+1),x)} +\spadpaste{integrate(1/(x**3+x+1),x) \bound{i}} +\spadpaste{definingPolynomial(tower(\%).2::EXPR INT) \free{i}} +\menudownlink{Integral of a Rational Function with a Real Parameter}{ExIntRationalWithRealParameter} +\spadpaste{integrate(1/(x**2 + a),x)} +\menudownlink{Integral of a Rational Function with a Complex Parameter}{ExIntRationalWithComplexParameter} +\spadpaste{complexIntegrate(1/(x**2 + a),x)} +\menudownlink{Two Similar Integrands Producing Very Different Results}{ExIntTwoSimilarIntegrands} +\spadpaste{integrate(x**3 / (a+b*x)**(1/3),x)} +\spadpaste{integrate(1 / (x**3 * (a+b*x)**(1/3)),x)} +\menudownlink{An Integral Which Does Not Exist}{ExIntNoSolution} +\spadpaste{integrate(log(1 + sqrt(a*x + b)) / x,x)} +\menudownlink{A Trigonometric Function of a Quadratic}{ExIntTrig} +\spadpaste{integrate((sinh(1+sqrt(x+b))+2*sqrt(x+b))/(sqrt(x+b)*(x+cosh(1+sqrt(x+b)))),x)} +\menudownlink{Integrating a Function with a Hidden Algebraic Relation}{ExIntAlgebraicRelation} +\spadpaste{integrate(tan(atan(x)/3),x)} +\menudownlink{Details for integrating a function wiht a Hidden Algebraic Relation}{ExIntAlgebraicRelationExplain} +\menudownlink{An Integral Involving a Root of a Transcendental Function}{ExIntRadicalOfTranscendental} +\spadpaste{integrate((x + 1) / (x * (x + log x)**(3/2)),x)} +\menudownlink{An Integral of a Non-elementary Function}{ExIntNonElementary} +\spadpaste{integrate(exp(-x**2) * erf(x) / (erf(x)**3 - erf(x)**2 - erf(x) + 1),x)} +\endmenu\endscroll\end{page} + +@ +\subsection{Laplace Transforms} +\label{Menuexlap} +\begin{itemize} +\item ExLapSimplePole \ref{ExLapSimplePole} on page~\pageref{ExLapSimplePole} +\item ExLapTrigTrigh \ref{ExLapTrigTrigh} on page~\pageref{ExLapTrigTrigh} +\item ExLapDefInt \ref{ExLapDefInt} on page~\pageref{ExLapDefInt} +\item ExLapExpExp \ref{ExLapExpExp} on page~\pageref{ExLapExpExp} +\item ExLapSpecial1 \ref{ExLapSpecial1} on page~\pageref{ExLapSpecial1} +\item ExLapSpecial2 \ref{ExLapSpecial2} on page~\pageref{ExLapSpecial2} +\end{itemize} +\index{pages!Menuexlap!coverex.ht} +\index{coverex.ht!pages!Menuexlap} +\index{Menuexlap!coverex.ht!pages} +<>= +\begin{page}{Menuexlap}{Laplace Transforms} +\beginscroll\beginmenu +\menudownlink{Laplace transform with a single pole}{ExLapSimplePole} +\spadpaste{laplace(t**4 * exp(-a*t) / factorial(4), t, s)} +\menudownlink{Laplace transform of a trigonometric function}{ExLapTrigTrigh} +\spadpaste{laplace(sin(a*t) * cosh(a*t) - cos(a*t) * sinh(a*t), t, s)} +\menudownlink{Laplace transform requiring a definite integration}{ExLapDefInt} +\spadpaste{laplace(2/t * (1 - cos(a*t)), t, s)} +\menudownlink{Laplace transform of exponentials}{ExLapExpExp} +\spadpaste{laplace((exp(a*t) - exp(b*t))/t, t, s)} +\menudownlink{Laplace transform of an exponential integral}{ExLapSpecial1} +\spadpaste{laplace(exp(a*t+b)*Ei(c*t), t, s)} +\menudownlink{Laplace transform of special functions}{ExLapSpecial2} +\spadpaste{laplace(a*Ci(b*t) + c*Si(d*t), t, s)} +\endmenu\endscroll\end{page} + +@ +\subsection{Limits} +\label{Menuexlimit} +\begin{itemize} +\item ExLimitBasic \ref{ExLimitBasic} on page~\pageref{ExLimitBasic} +\item ExLimitParameter \ref{ExLimitParameter} on +page~\pageref{ExLimitParameter} +\item ExLimitOneSided \ref{ExLimitOneSided} on page~\pageref{ExLimitOneSided} +\item ExLimitTwoSided \ref{ExLimitTwoSided} on page~\pageref{ExLimitTwoSided} +\item ExLimitInfinite \ref{ExLimitInfinite} on page~\pageref{ExLimitInfinite} +\item ExLimitRealComplex \ref{ExLimitRealComplex} on +page~\pageref{ExLimitRealComplex} +\item ExLimitComplexInfinite \ref{ExLimitComplexInfinite} on +page~\pageref{ExLimitComplexInfinite} +\end{itemize} +\index{pages!Menuexlimit!coverex.ht} +\index{coverex.ht!pages!Menuexlimit} +\index{Menuexlimit!coverex.ht!pages} +<>= +\begin{page}{Menuexlimit}{Limits} +\beginscroll\beginmenu +\menudownlink{Computing Limits}{ExLimitBasic} +\spadpaste{limit((x**2 - 3*x + 2)/(x**2 - 1),x = 1)} +\menudownlink{Limits of Functions with Parameters}{ExLimitParameter} +\spadpaste{limit(sinh(a*x)/tan(b*x),x = 0)} +\menudownlink{One-sided Limits}{ExLimitOneSided} +\spadpaste{limit(x * log(x),x = 0,"right")} +\spadpaste{limit(x * log(x),x = 0)} +\menudownlink{Two-sided Limits}{ExLimitTwoSided} +\spadpaste{limit(sqrt(y**2)/y,y = 0)} +\spadpaste{limit(sqrt(1 - cos(t))/t,t = 0)} +\menudownlink{Limits at Infinity}{ExLimitInfinite} +\spadpaste{limit(sqrt(3*x**2 + 1)/(5*x),x = \%plusInfinity)} +\spadpaste{limit(sqrt(3*x**2 + 1)/(5*x),x = \%minusInfinity)} +\menudownlink{Real Limits vs. Complex Limits}{ExLimitRealComplex} +\spadpaste{limit(z * sin(1/z),z = 0)} +\spadpaste{complexLimit(z * sin(1/z),z = 0)} +\menudownlink{Complex Limits at Infinity}{ExLimitComplexInfinite} +\spadpaste{complexLimit((2 + z)/(1 - z),z = \%infinity)} +\spadpaste{limit(sin(x)/x,x = \%plusInfinity)} +\spadpaste{complexLimit(sin(x)/x,x = \%infinity)} +\endmenu\endscroll\end{page} + +@ +\subsection{Matrices} +\label{Menuexmatrix} +\begin{itemize} +\item ExMatrixBasicFunction \ref{ExMatrixBasicFunction} on +page~\pageref{ExMatrixBasicFunction} +\item ExConstructMatrix \ref{ExConstructMatrix} on +page~\pageref{ExConstructMatrix} +\item ExTraceMatrix \ref{ExTraceMatrix} on page~\pageref{ExTraceMatrix} +\item ExDeterminantMatrix \ref{ExDeterminantMatrix} on +page~\pageref{ExDeterminantMatrix} +\item ExInverseMatrix \ref{ExInverseMatrix} on page~\pageref{ExInverseMatrix} +\item ExRankMatrix \ref{ExRankMatrix} on page~\pageref{ExRankMatrix} +\end{itemize} +\index{pages!Menuexmatrix!coverex.ht} +\index{coverex.ht!pages!Menuexmatrix} +\index{Menuexmatrix!coverex.ht!pages} +<>= +\begin{page}{Menuexmatrix}{Matrices} +\beginscroll\beginmenu +\menudownlink{Basic Arithmetic Operations on Matrices}{ExMatrixBasicFunction} +\spadpaste{m1 := matrix([[1,-2,1],[4,2,-4]]) \bound{m1}} +\spadpaste{m2 := matrix([[1,0,2],[20,30,10],[0,200,100]]) \bound{m2}} +\spadpaste{m3 := matrix([[1,2,3],[2,4,6]]) \bound{m3}} +\spadpaste{m1 + m3 \free{m1} \free{m3}} +\spadpaste{100 * m1 \free{m1}} +\spadpaste{m1 * m2 \free{m1} \free{m2}} +\spadpaste{-m1 + m3 * m2 \free{m1} \free{m2} \free{m3}} +\spadpaste{m3 *vector([1,0,1]) \free{m3}} +\menudownlink{Constructing new Matrices}{ExConstructMatrix} +\spadpaste{diagonalMatrix([1,2,3,2,1])} +\spadpaste{subMatrix(matrix([[0,1,2,3,4],[5,6,7,8,9],[10,11,12,13,14]]), 1,3,2,4)} +\spadpaste{horizConcat(matrix([[1,2,3],[6,7,8]]),matrix([[11,12,13],[55,77,88]])) } +\spadpaste{vertConcat(matrix([[1,2,3],[6,7,8]]),matrix([[11,12,13],[55,77,88]])) } +\spadpaste{b:=matrix([[0,1,2,3,4],[5,6,7,8,9],[10,11,12,13,14]]) \bound{b}} +\spadpaste{setsubMatrix!(b,1,1,transpose(subMatrix(b,1,3,1,3)))\free{b}} +\menudownlink{Trace of a Matrix}{ExTraceMatrix} +\spadpaste{trace( matrix([[1,x,x**2,x**3],[1,y,y**2,y**3],[1,z,z**2,z**3],[1,u,u**2,u**3]]) )} +\menudownlink{Determinant of a Matrix}{ExDeterminantMatrix} +\spadpaste{determinant(matrix([[1,2,3,4],[2,3,2,5],[3,4,5,6],[4,1,6,7]]))} +\menudownlink{Inverse of a Matrix}{ExInverseMatrix} +\spadpaste{inverse(matrix([[1,2,1],[-2,3,4],[-1,5,6]])) } +\menudownlink{Rank of a Matrix}{ExRankMatrix} +\spadpaste{rank(matrix([[0,4,1],[5,3,-7],[-5,5,9]]))} +\endmenu\endscroll\end{page} + +@ +\subsection{2-D Graphics} +\label{Menuexplot2d} +\begin{itemize} +\item ExPlot2DFunctions \ref{ExPlot2DFunctions} on +page~\pageref{ExPlot2DFunctions} +\item ExPlot2DParametric \ref{ExPlot2DParametric} on +page~\pageref{ExPlot2DParametric} +\item ExPlot2DPolar \ref{ExPlot2DPolar} on page~\pageref{ExPlot2DPolar} +\item ExPlot2DAlgebraic \ref{ExPlot2DAlgebraic} on +page~\pageref{ExPlot2DAlgebraic} +\end{itemize} +\index{pages!Menuexplot2d!coverex.ht} +\index{coverex.ht!pages!Menuexplot2d} +\index{Menuexplot2d!coverex.ht!pages} +<>= +\begin{page}{Menuexplot2d}{2-D Graphics} +\beginscroll\beginmenu +\menudownlink{Plotting Functions of One Variable}{ExPlot2DFunctions} +\graphpaste{draw(sin(tan(x)) - tan(sin(x)),x = 0..6)} +\menudownlink{Plotting Parametric Curves}{ExPlot2DParametric} +\graphpaste{draw(curve(9 * sin(3*t/4),8 * sin(t)),t = -4*\%pi..4*\%pi)} +\menudownlink{Plotting Using Polar Coordinates}{ExPlot2DPolar} +\graphpaste{draw(sin(4*t/7),t = 0..14*\%pi,coordinates == polar)} +\menudownlink{Plotting Plane Algebraic Curves}{ExPlot2DAlgebraic} +\graphpaste{draw(y**2 + y - (x**3 - x) = 0, x, y, range == [-2..2,-2..1])} +\endmenu\endscroll\end{page} + +@ +\subsection{3-D Graphics} +\label{Menuexplot3d} +\begin{itemize} +\item ExPlot3DFunctions \ref{ExPlot3DFunctions} on +page~\pageref{ExPlot3DFunctions} +\item ExPlot3DParametricSurface \ref{ExPlot3DParametricSurface} on +page~\pageref{ExPlot3DParametricSurface} +\item ExPlot3DParametricCurve \ref{ExPlot3DParametricCurve} on +page~\pageref{ExPlot3DParametricCurve} +\end{itemize} +\index{pages!Menuexplot3d!coverex.ht} +\index{coverex.ht!pages!Menuexplot3d} +\index{Menuexplot3d!coverex.ht!pages} +<>= +\begin{page}{Menuexplot3d}{3-D Graphics} +\beginscroll\beginmenu +\menudownlink{Plotting Functions of Two Variables}{ExPlot3DFunctions} +\graphpaste{draw(cos(x*y),x = -3..3,y = -3..3)} +\menudownlink{Plotting Parametric Surfaces}{ExPlot3DParametricSurface} +\graphpaste{draw(surface(5*sin(u)*cos(v),4*sin(u)*sin(v),3*cos(u)),u=0..\%pi,v=0..2*\%pi)} +\graphpaste{draw(surface(u*cos(v),u*sin(v),u),u=0..4,v=0..2*\%pi)} +\menudownlink{Plotting Parametric Curves}{ExPlot3DParametricCurve} +\graphpaste{draw(curve(cos(t),sin(t),t),t=0..6)} +\graphpaste{draw(curve(t,t**2,t**3),t=-3..3)} +\endmenu\endscroll\end{page} + +@ +\subsection{Series} +\label{Menuexseries} +\begin{itemize} +\item ExSeriesConvert \ref{ExSeriesConvert} on page~\pageref{ExSeriesConvert} +\item ExSeriesManipulate \ref{ExSeriesManipulate} on +page~\pageref{ExSeriesManipulate} +\item ExSeriesFunctions \ref{ExSeriesFunctions} on +page~\pageref{ExSeriesFunctions} +\item ExSeriesSubstitution \ref{ExSeriesSubstitution} on +page~\pageref{ExSeriesSubstitution} +\end{itemize} +\index{pages!Menuexseries!coverex.ht} +\index{coverex.ht!pages!Menuexseries} +\index{Menuexseries!coverex.ht!pages} +<>= +\begin{page}{Menuexseries}{Series} +\beginscroll\beginmenu +\menudownlink{Converting Expressions to Series}{ExSeriesConvert} +\spadpaste{series(sin(a*x),x = 0)} +\spadpaste{series(sin(a*x),a = \%pi/4)} +\menudownlink{Manipulating Power Series}{ExSeriesManipulate} +\spadpaste{f := series(1/(1-x),x = 0) \bound{f}} +\spadpaste{f ** 2 \free{f}} +\menudownlink{Functions on Power Series}{ExSeriesFunctions} +\spadpaste{f := series(1/(1-x),x = 0) \bound{f1}} +\spadpaste{g := log(f) \free{f1} \bound{g}} +\spadpaste{exp(g) \free{g}} +\menudownlink{Substituting Numerical Values in Power Series}{ExSeriesSubstitution} +\spadpaste{f := taylor(exp(x)) \bound{f2}} +\spadpaste{eval(f,1.0) \free{f2}} +\endmenu\endscroll\end{page} + +@ +\subsection{Summations} +\label{Menuexsum} +\begin{itemize} +\item ExSumListEntriesI \ref{ExSumListEntriesI} on +page~\pageref{ExSumListEntriesI} +\item ExSumListEntriesII \ref{ExSumListEntriesII} on +page~\pageref{ExSumListEntriesII} +\item ExSumApproximateE \ref{ExSumApproximateE} on +page~\pageref{ExSumApproximateE} +\item ExSumClosedForm \ref{ExSumClosedForm} on page~\pageref{ExSumClosedForm} +\item ExSumCubes \ref{ExSumCubes} on page~\pageref{ExSumCubes} +\item ExSumPolynomial \ref{ExSumPolynomial} on page~\pageref{ExSumPolynomial} +\item ExSumGeneralFunction \ref{ExSumGeneralFunction} on +page~\pageref{ExSumGeneralFunction} +\item ExSumInfinite \ref{ExSumInfinite} on page~\pageref{ExSumInfinite} +\end{itemize} +\index{pages!Menuexsum!coverex.ht} +\index{coverex.ht!pages!Menuexsum} +\index{Menuexsum!coverex.ht!pages} +<>= +\begin{page}{Menuexsum}{Summations} +\beginscroll\beginmenu +\menudownlink{Summing the Entries of a List I}{ExSumListEntriesI} +\spadpaste{[i for i in 1..15]} +\spadpaste{reduce(+,[i for i in 1..15])} +\menudownlink{Summing the Entries of a List II}{ExSumListEntriesII} +\spadpaste{[n**2 for n in 5..20]} +\spadpaste{reduce(+,[n**2 for n in 5..20])} +\menudownlink{Approximating e}{ExSumApproximateE} +\spadpaste{reduce(+,[1.0/factorial(n) for n in 0..20])} +\menudownlink{Closed Form Summations}{ExSumClosedForm} +\spadpaste{s := sum(k**2,k = a..b) \bound{s}} +\spadpaste{eval(s,[a,b],[1,25]) \free{s}} +\spadpaste{reduce(+,[i**2 for i in 1..25])} +\menudownlink{Sums of Cubes}{ExSumCubes} +\spadpaste{sum(k**3,k = 1..n)} +\spadpaste{sum(k,k = 1..n) ** 2} +\menudownlink{Sums of Polynomials}{ExSumPolynomial} +\spadpaste{sum(3*k**2/(c**2 + 1) + 12*k/d,k = (3*a)..(4*b))} +\menudownlink{Sums of General Functions}{ExSumGeneralFunction} +\spadpaste{sum(k * x**k,k = 1..n)} +\menudownlink{Infinite Sums}{ExSumInfinite} +\spadpaste{limit( sum(1/(k * (k + 2)),k = 1..n) ,n = \%plusInfinity)} +\endmenu\endscroll\end{page} + +@ +\section{decimal.ht} +<>= +\newcommand{\DecimalExpansionXmpTitle}{DecimalExpansion} +\newcommand{\DecimalExpansionXmpNumber}{9.15} + +@ +\subsection{Decimal Expansion} +\label{DecimalExpansionXmpPage} +\begin{itemize} +\item BinaryExpansionXmpPage \ref{BinaryExpansionXmpPage} on +page~pageref{BinaryExpansionXmpPage} +\item HexadecimalExpansionXmpPage \ref{HexadecimalExpansionXmpPage} on +page~pageref{HexadecimalExpansionXmpPage} +\item RadicExpansionPage \ref{RadicExpansionPage} on +page~pageref{RadicExpansionPage} +\end{itemize} +\index{pages!DecimalExpansionXmpPage!decimal.ht} +\index{decimal.ht!pages!DecimalExpansionXmpPage} +\index{DecimalExpansionXmpPage!decimal.ht!pages} +<>= +\begin{page}{DecimalExpansionXmpPage}{Decimal Expansion} +\beginscroll + + +All rationals have repeating decimal expansions. +Operations to access the individual digits of a decimal expansion can +be obtained by converting the value to \spadtype{RadixExpansion(10)}. +More examples of expansions are available in +\downlink{`BinaryExpansion'}{BinaryExpansionXmpPage}\ignore{BinaryExpansion}, +\downlink{`HexadecimalExpansion'}{HexadecimalExpansionXmpPage}\ignore{HexadecimalExpansion}, and +\downlink{`RadixExpansion'}{RadixExpansionXmpPage}\ignore{RadixExpansion}. +\showBlurb{DecimalExpansion} + +\xtc{ +The operation \spadfunFrom{decimal}{DecimalExpansion} is used to create +this expansion of type \spadtype{DecimalExpansion}. +}{ +\spadpaste{r := decimal(22/7) \bound{r}} +} +\xtc{ +Arithmetic is exact. +}{ +\spadpaste{r + decimal(6/7) \free{r}} +} +\xtc{ +The period of the expansion can be short or long \ldots +}{ +\spadpaste{[decimal(1/i) for i in 350..354] } +} +\xtc{ +or very long. +}{ +\spadpaste{decimal(1/2049) } +} +\xtc{ +These numbers are bona fide algebraic objects. +}{ +\spadpaste{p := decimal(1/4)*x**2 + decimal(2/3)*x + decimal(4/9) \bound{p}} +} +\xtc{ +}{ +\spadpaste{q := differentiate(p, x) \free{p}\bound{q}} +} +\xtc{ +}{ +\spadpaste{g := gcd(p, q) \free{p q} \bound{g}} +} +\endscroll +\autobuttons +\end{page} + +@ +\section{derham.ht} +<>= +\newcommand{\DeRhamComplexXmpTitle}{DeRhamComplex} +\newcommand{\DeRhamComplexXmpNumber}{9.14} +@ +\subsection{DeRhamComplex} +\label{DeRhamComplexXmpPage} +\index{pages!DeRhamComplexXmpPage!derham.ht} +\index{derham.ht!pages!DeRhamComplexXmpPage} +\index{DeRhamComplexXmpPage!derham.ht!pages} +<>= +\begin{page}{DeRhamComplexXmpPage}{DeRhamComplex} +\beginscroll + +The domain constructor \spadtype{DeRhamComplex} creates the +class of differential forms of arbitrary degree over a coefficient ring. +The De Rham complex constructor takes two arguments: a ring, \spad{coefRing,} +and a list of coordinate variables. + +\xtc{ +This is the ring of coefficients. +}{ +\spadpaste{coefRing := Integer \bound{coefRing}} +} +\xtc{ +These are the coordinate variables. +}{ +\spadpaste{lv : List Symbol := [x,y,z] \bound{lv}} +} +\xtc{ +This is the De Rham complex of Euclidean three-space using +coordinates \spad{x, y} and \spad{z.} +}{ +\spadpaste{der := DERHAM(coefRing,lv) \free{coefRing}\free{lv}\bound{der}} +} + +This complex allows us to describe differential forms having +expressions of integers as coefficients. +These coefficients can involve any number of variables, for example, +\spad{f(x,t,r,y,u,z).} +As we've chosen to work with ordinary +Euclidean three-space, expressions involving these forms +are treated as functions of +\spad{x, y} and \spad{z} with the additional arguments +\spad{t, r} and \spad{u} regarded as symbolic constants. +\xtc{ +Here are some examples of coefficients. +}{ +\spadpaste{R := Expression coefRing \free{coefRing}\bound{R}} +} +\xtc{ +}{ +\spadpaste{f : R := x**2*y*z-5*x**3*y**2*z**5 \free{R}\bound{f}} +} +\xtc{ +}{ +\spadpaste{g : R := z**2*y*cos(z)-7*sin(x**3*y**2)*z**2 \free{R}\bound{g}} +} +\xtc{ +}{ +\spadpaste{h : R :=x*y*z-2*x**3*y*z**2 \free{R}\bound{h}} +} +\xtc{ +We now define +the multiplicative basis elements for the exterior algebra over \spad{R}. +}{ +\spadpaste{dx : der := generator(1) \free{der}\bound{dx}} +} +\xtc{ +}{ +\spadpaste{dy : der := generator(2)\free{der}\bound{dy}} +} +\xtc{ +}{ +\spadpaste{dz : der := generator(3)\free{der}\bound{dz}} +} +\xtc{ +This is an alternative way to give the above assignments. +}{ +\spadpaste{[dx,dy,dz] := [generator(i)\$der for i in 1..3] \free{der}\bound{dxyz}} +} +\xtc{ +Now we define some one-forms. +}{ +\spadpaste{alpha : der := f*dx + g*dy + h*dz \bound{alpha}\free{der f g h dxyz}} +} +\xtc{ +}{ +\spadpaste{beta : der := cos(tan(x*y*z)+x*y*z)*dx + x*dy \bound{beta}\free{der f g h dxyz}} +} +\xtc{ +A well-known theorem states that the composition of +\spadfunFrom{exteriorDifferential}{DeRhamComplex} +with itself is the zero map for continuous forms. +Let's verify this theorem for \spad{alpha}. +}{ +\spadpaste{exteriorDifferential alpha; \free{alpha}\bound{ed}} +} +\xtc{ +We suppressed the lengthy output of the last expression, but nevertheless, the +composition is zero. +}{ +\spadpaste{exteriorDifferential \% \free{ed}} +} + +\xtc{ +Now we check that \spadfunFrom{exteriorDifferential}{DeRhamComplex} +is a ``graded derivation'' \spad{D,} that is, \spad{D} satisfies: +\begin{verbatim} +D(a*b) = D(a)*b + (-1)**degree(a)*a*D(b) +\end{verbatim} +}{ +\spadpaste{gamma := alpha * beta \bound{gamma}\free{alpha}\free{beta}} +} +\xtc{ +We try this for the one-forms \spad{alpha} and \spad{beta}. +}{ +\spadpaste{exteriorDifferential(gamma) - (exteriorDifferential(alpha)*beta - alpha * exteriorDifferential(beta)) \free{alpha beta gamma}} +} +\xtc{ +Now we define some ``basic operators'' (see \downlink{`Operator'}{OperatorXmpPage}\ignore{Operator}). +}{ +\spadpaste{a : BOP := operator('a) \bound{ao}} +} +\xtc{ +}{ +\spadpaste{b : BOP := operator('b) \bound{bo}} +} +\xtc{ +}{ +\spadpaste{c : BOP := operator('c) \bound{co}} +} +\xtc{ +We also define +some indeterminate one- and two-forms using these operators. +}{ +\spadpaste{sigma := a(x,y,z) * dx + b(x,y,z) * dy + c(x,y,z) * dz \bound{sigma}\free{ao bo co dxyz}} +} +\xtc{ +}{ +\spadpaste{theta := a(x,y,z) * dx * dy + b(x,y,z) * dx * dz + c(x,y,z) * dy * dz \bound{theta}\free{ao bo co dxyz}} +} + +\xtc{ +This allows us to get formal definitions for the ``gradient'' \ldots +}{ +\spadpaste{totalDifferential(a(x,y,z))\$der \free{ao der}} +} +\xtc{ +the ``curl'' \ldots +}{ +\spadpaste{exteriorDifferential sigma \free{sigma}} +} +\xtc{ +and the ``divergence.'' +}{ +\spadpaste{exteriorDifferential theta \free{theta}} +} + +\xtc{ +Note that the De Rham complex is an algebra with unity. +This element \spad{1} is the basis for elements for zero-forms, that is, +functions in our space. +}{ +\spadpaste{one : der := 1 \bound{one}\free{der}} +} + +\xtc{ +To convert a function to a function lying in the De Rham complex, +multiply the function by ``one.'' +}{ +\spadpaste{g1 : der := a([x,t,y,u,v,z,e]) * one \free{der one ao}\bound{g1}} +} +\xtc{ +A current limitation of Axiom forces you to write +functions with more than four arguments using square brackets in this way. +}{ +\spadpaste{h1 : der := a([x,y,x,t,x,z,y,r,u,x]) * one \free{der one ao}\bound{h1}} +} + +\xtc{ +Now note how the system keeps track of where your coordinate functions +are located in expressions. +}{ +\spadpaste{exteriorDifferential g1 \free{g1}} +} +\xtc{ +}{ +\spadpaste{exteriorDifferential h1 \free{h1}} +} + +\xtc{ +In this example of Euclidean three-space, the basis for the De Rham complex +consists of the eight forms: \spad{1}, \spad{dx}, \spad{dy}, \spad{dz}, +\spad{dx*dy}, \spad{dx*dz}, \spad{dy*dz}, and \spad{dx*dy*dz}. +}{ +\spadpaste{coefficient(gamma, dx*dy) \free{gamma dxyz}} +} +\xtc{ +}{ +\spadpaste{coefficient(gamma, one) \free{gamma one}} +} +\xtc{ +}{ +\spadpaste{coefficient(g1,one) \free{g1 one}} +} +\endscroll +\autobuttons +\end{page} + +@ +\section{dfloat.ht} +<>= +\newcommand{\DoubleFloatXmpTitle}{DoubleFloat} +\newcommand{\DoubleFloatXmpNumber}{9.17} + +@ +\subsection{DoubleFloat} +\label{DoubleFloatXmpPage} +\begin{itemize} +\item ugGraphPage \ref{ugGraphPage} on +page~pageref{ugGraphPage} +\item ugProblemNumericPage \ref{ugProblemNumericPage} on +page~pageref{ugProblemNumericPage} +\item FloatXmpPage \ref{FloatXmpPage} on +page~pageref{FloatXmpPage} +\end{itemize} +\index{pages!DoubleFloatXmpPage!dfloat.ht} +\index{dfloat.ht!pages!DoubleFloatXmpPage} +\index{DoubleFloatXmpPage!dfloat.ht!pages} +<>= +\begin{page}{DoubleFloatXmpPage}{DoubleFloat} +\beginscroll + +Axiom provides two kinds of floating point numbers. +The domain \spadtype{Float} (abbreviation \spadtype{FLOAT}) +implements a model of arbitrary +precision floating point numbers. +The domain \spadtype{DoubleFloat} (abbreviation \spadtype{DFLOAT}) +is intended to make available +hardware floating point arithmetic in Axiom. +The actual model of floating point \spadtype{DoubleFloat} that provides is +system-dependent. +For example, on the IBM system 370 Axiom uses IBM double +precision which has fourteen hexadecimal digits of precision or roughly +sixteen decimal digits. +Arbitrary precision floats allow the user to specify the +precision at which arithmetic operations are computed. +Although this is an attractive facility, it comes at a cost. +Arbitrary-precision floating-point arithmetic typically takes +twenty to two hundred times more time than hardware floating point. + +The usual arithmetic and elementary functions are available for +\spadtype{DoubleFloat}. +Use \spadsys{)show DoubleFloat} to get a list of operations or +the \HyperName{} \Browse{} facility to get more extensive +documentation about \spadtype{DoubleFloat}. + +\xtc{ +By default, floating point numbers that you enter into Axiom +are of type \spadtype{Float}. +}{ +\spadpaste{2.71828} +} +You must therefore tell Axiom that you want to use +\spadtype{DoubleFloat} values and operations. +The following are some conservative guidelines for getting +Axiom to use \spadtype{DoubleFloat}. + +\xtc{ +To get a value of type \spadtype{DoubleFloat}, use a target with +\spadSyntax{@}, \ldots +}{ +\spadpaste{2.71828@DoubleFloat} +} +\xtc{ +a conversion, \ldots +}{ +\spadpaste{2.71828 :: DoubleFloat} +} +\xtc{ +or an assignment to a declared variable. +It is more efficient if you use a target rather than an explicit or +implicit conversion. +}{ +\spadpaste{eApprox : DoubleFloat := 2.71828 \bound{eApprox}} +} + +\xtc{ +You also need to declare functions that work with +\spadtype{DoubleFloat}. +}{ +\spadpaste{avg : List DoubleFloat -> DoubleFloat \bound{avgDec}} +} +\xtc{ +}{ +\begin{spadsrc}[\bound{avg}\free{avgDec}] +avg l == + empty? l => 0 :: DoubleFloat + reduce(_+,l) / #l +\end{spadsrc} +} +\xtc{ +}{ +\spadpaste{avg [] \free{avg}} +} +\xtc{ +}{ +\spadpaste{avg [3.4,9.7,-6.8] \free{avg}} +} +\xtc{ +Use package-calling for operations from \spadtype{DoubleFloat} unless +the arguments themselves are already of type \spadtype{DoubleFloat}. +}{ +\spadpaste{cos(3.1415926)\$DoubleFloat} +} +\xtc{ +}{ +\spadpaste{cos(3.1415926 :: DoubleFloat)} +} + +By far, the most common usage of \spadtype{DoubleFloat} is for functions +to be graphed. +For more information about Axiom's numerical and graphical +facilities, see +\downlink{``\ugGraphTitle''}{ugGraphPage} in Section \ugGraphNumber\ignore{ugGraph}, +\downlink{``\ugProblemNumericTitle''}{ugProblemNumericPage} in Section \ugProblemNumericNumber\ignore{ugProblemNumeric}, and +\downlink{`Float'}{FloatXmpPage}\ignore{Float}. +\endscroll +\autobuttons +\end{page} + +@ +\section{dmp.ht} +<>= +\newcommand{\DistributedMultivariatePolynomialXmpTitle} +{DistributedMultivariatePolynomial} +\newcommand{\DistributedMultivariatePolynomialXmpNumber}{9.16} +@ +\subsection{DistributedMultivariatePolynomial} +\label{DistributedMultivariatePolynomialXmpPage} +\begin{itemize} +\item ugIntroVariablesPage \ref{ugIntroVariablesPage} on +page~pageref{ugIntroVariablesPage} +\item utTypesConvertPage \ref{utTypesConvertPage} on +page~pageref{utTypesConvertPage} +\item PolynomialXmpPage \ref{PolynomialXmpPage} on +page~pageref{PolynomialXmpPage} +\item UnivariatePolynomialXmpPage \ref{UnivariatePolynomialXmpPage} on +page~pageref{UnivariatePolynomialXmpPage} +\item MultivariatePolynomialXmpPage \ref{MultivariatePolynomialXmpPage} on +page~pageref{MultivariatePolynomialXmpPage} +\end{itemize} +\index{pages!DistributedMultivariatePolynomialXmpPage!dmp.ht} +\index{dmp.ht!pages!DistributedMultivariatePolynomialXmpPage} +\index{DistributedMultivariatePolynomialXmpPage!dmp.ht!pages} +<>= +\begin{page}{DistributedMultivariatePolynomialXmpPage} +{DistributedMultivariatePolynomial} +\beginscroll + +\texht{\hyphenation{ +Homo-gen-eous-Dis-tributed-Multi-var-i-ate-Pol-y-nomial +}}{} + +\spadtype{DistributedMultivariatePolynomial} and +\spadtype{HomogeneousDistributedMultivariatePolynomial}, abbreviated +\spadtype{DMP} and \spadtype{HDMP}, respectively, are very similar to +\spadtype{MultivariatePolynomial} except that they are represented and +displayed in a non-recursive manner. +\xtc{ +}{ +\spadpaste{(d1,d2,d3) : DMP([z,y,x],FRAC INT) \bound{d1dec d2dec d3dec}} +} +\xtc{ +The constructor +\spadtype{DMP} orders its monomials lexicographically while +\spadtype{HDMP} orders them by total order refined by reverse +lexicographic order. +}{ +\spadpaste{d1 := -4*z + 4*y**2*x + 16*x**2 + 1 \bound{d1}\free{d1dec}} +} +\xtc{ +}{ +\spadpaste{d2 := 2*z*y**2 + 4*x + 1 \bound{d2}\free{d2dec}} +} +\xtc{ +}{ +\spadpaste{d3 := 2*z*x**2 - 2*y**2 - x \bound{d3}\free{d3dec}} +} +\xtc{ +These constructors are mostly used in \texht{Gr\"{o}bner}{Groebner} +basis calculations. +}{ +\spadpaste{groebner [d1,d2,d3] \free{d1 d2 d3}} +} +\xtc{ +}{ +\spadpaste{(n1,n2,n3) : HDMP([z,y,x],FRAC INT) \bound{ndec}} +} +\xtc{ +}{ +\spadpaste{(n1,n2,n3) := (d1,d2,d3) \free{ndec}\bound{n}\free{d1 d2 d3}} +} +\xtc{ +Note that we get a different +\texht{Gr\"{o}bner}{Groebner} basis +when we use the \spadtype{HDMP} polynomials, as expected. +}{ +\spadpaste{groebner [n1,n2,n3] \free{n}} +} + +\spadtype{GeneralDistributedMultivariatePolynomial} is somewhat +more flexible in the sense that as well as accepting a list of +variables to specify the variable ordering, it also takes a +predicate on exponent vectors to specify the term ordering. +With this polynomial type the user can experiment with the effect +of using completely arbitrary term orderings. +This flexibility is mostly important for algorithms such as +\texht{Gr\"{o}bner}{Groebner} basis calculations which can be very +sensitive to term ordering. + +For more information on related topics, see +\downlink{``\ugIntroVariablesTitle''}{ugIntroVariablesPage} +in Section \ugIntroVariablesNumber\ignore{ugIntroVariables}, +\downlink{``\ugTypesConvertTitle''}{ugTypesConvertPage} +in Section \ugTypesConvertNumber\ignore{ugTypesConvert}, +\downlink{`Polynomial'}{PolynomialXmpPage}\ignore{Polynomial}, +\downlink{`UnivariatePolynomial'}{UnivariatePolynomialXmpPage} +\ignore{UnivariatePolynomial}, and +\downlink{`MultivariatePolynomial'}{MultivariatePolynomialXmpPage} +\ignore{MultivariatePolynomial}. +% +\showBlurb{DistributedMultivariatePolynomial} +\endscroll +\autobuttons +\end{page} +% + +@ +\section{eq.ht} +<>= +\newcommand{\EquationXmpTitle}{Equation} +\newcommand{\EquationXmpNumber}{9.19} + +@ +\subsection{Equation} +\label{EquationXmpPage} +\index{pages!EquationXmpPage!eq.ht} +\index{eq.ht!pages!EquationXmpPage} +\index{EquationXmpPage!eq.ht!pages} +<>= +\begin{page}{EquationXmpPage}{Equation} +\beginscroll + +The \spadtype{Equation} domain provides equations as mathematical objects. +These are used, for example, as the input to various +\spadfunFrom{solve}{TransSolvePackage} operations. + +\xtc{ +Equations are created using the equals symbol, \spadopFrom{=}{Equation}. +}{ +\spadpaste{eq1 := 3*x + 4*y = 5 \bound{eq1}} +} +\xtc{ +}{ +\spadpaste{eq2 := 2*x + 2*y = 3 \bound{eq2}} +} +\xtc{ +The left- and right-hand sides of an equation are accessible using +the operations \spadfunFrom{lhs}{Equation} and \spadfunFrom{rhs}{Equation}. +}{ +\spadpaste{lhs eq1 \free{eq1}} +} +\xtc{ +}{ +\spadpaste{rhs eq1 \free{eq1}} +} + +\xtc{ +Arithmetic operations are supported +and operate on both sides of the equation. +}{ +\spadpaste{eq1 + eq2 \free{eq1 eq2}} +} +\xtc{ +}{ +\spadpaste{eq1 * eq2 \free{eq1 eq2}} +} +\xtc{ +}{ +\spadpaste{2*eq2 - eq1 \free{eq1 eq2}} +} +\xtc{ +Equations may be created for any type so the arithmetic operations +will be defined only when they make sense. For example, +exponentiation is not defined for equations involving non-square matrices. +}{ +\spadpaste{eq1**2 \free{eq1}} +} + +\xtc{ +Note that an equals symbol is also used to {\it test} +for equality of values in certain contexts. +For example, \spad{x+1} and \spad{y} are unequal as polynomials. +}{ +\spadpaste{if x+1 = y then "equal" else "unequal"} +} +\xtc{ +}{ +\spadpaste{eqpol := x+1 = y \bound{eqpol}} +} +\xtc{ +If an equation is used where a \spadtype{Boolean} value +is required, then it is evaluated using the equality +test from the operand type. +}{ +\spadpaste{if eqpol then "equal" else "unequal" \free{eqpol}} +} +\xtc{ +If one wants a \spadtype{Boolean} value rather than an equation, +all one has to do is ask! +}{ +\spadpaste{eqpol::Boolean \free{eqpol}} +} +\endscroll +\autobuttons +\end{page} + +@ +\section{eqtbl.ht} +<>= +\newcommand{\EqTableXmpTitle}{EqTable} +\newcommand{\EqTableXmpNumber}{9.18} +@ +\subsection{EqTable} +\label{EqTableXmpPage} +See TableXmpPage \ref{TableXmpPage} on page~\pageref{TableXmpPage} +\index{pages!EqTableXmpPage!eqtbl.ht} +\index{eqtbl.ht!pages!EqTableXmpPage} +\index{EqTableXmpPage!eqtbl.ht!pages} +<>= +\begin{page}{EqTableXmpPage}{EqTable} +\beginscroll + +The \spadtype{EqTable} domain provides tables where the keys are compared +using \spadfunFrom{eq?}{EqTable}. +Keys are considered equal only if they are the same instance of a +structure. +This is useful if the keys are themselves updatable structures. +Otherwise, all operations are the same as for type \spadtype{Table}. +See \downlink{`Table'}{TableXmpPage}\ignore{Table} +for general information about tables. +\showBlurb{EqTable} + +\xtc{ +The operation \spadfunFrom{table}{EqTable} is here used to create a table +where the keys are lists of integers. +}{ +\spadpaste{e: EqTable(List Integer, Integer) := table() \bound{e}} +} +\xtc{ +These two lists are equal according to \spadopFrom{=}{List}, +but not according to \spadfunFrom{eq?}{List}. +}{ +\spadpaste{l1 := [1,2,3] \bound{l1}} +} +\xtc{ +}{ +\spadpaste{l2 := [1,2,3] \bound{l2}} +} +\xtc{ +Because the two lists are not \spadfunFrom{eq?}{List}, separate values +can be stored under each. +}{ +\spadpaste{e.l1 := 111 \free{e l1} \bound{e1}} +} +\xtc{ +}{ +\spadpaste{e.l2 := 222 \free{e1 l2} \bound{e2}} +} +\xtc{ +}{ +\spadpaste{e.l1 \free{e2 l1}} +} +\endscroll +\autobuttons +\end{page} + +@ +\section{evalex.ht} +\subsection{Example of Standard Evaluation} +\label{PrefixEval} +\index{pages!PrefixEval!evalex.ht} +\index{evalex.ht!pages!PrefixEval} +\index{PrefixEval!evalex.ht!pages} +<>= +\begin{page}{PrefixEval}{Example of Standard Evaluation} +\beginscroll +We illustrate the general evaluation of {\em op a} for some +prefix operator {\em op} and operand {\em a} +by the example: {\em cos(2)}. +The evaluation steps are as follows: +\vspace{1}\newline +1.\tab{3}{\em a} evaluates to a value of some type. +\newline\tab{3}{\em Example:} {\em 2} evaluates to {\em 2} of type +\spadtype{Integer} +\newline +2.\tab{3}Axiom then chooses a function {\em op} based on the +type of {\em a}. +\newline\tab{3}{\em Example:} The function {\em cos:} \spadtype{Float} {\em ->} +\spadtype{Float} is chosen. +\newline +3.\tab{3}If the argument type of the function is different from that +of {\em a}, +then the system coerces +%\downlink{coerces}{Coercion} +the value of {\em a} to the +argument type. +\newline\tab{3}{\em Example:} The integer {\em 2} is coerced to the +float {\em 2.0}. +\newline +4.\tab{3}The function is then applied to the value of {\em a} to +produce the value +for {\em op a}. +\newline\tab{3}{\em Example:} The function {\em cos} is applied to {\em 2.0}. +\vspace{1}\newline +Try it: +\example{cos(2)} +\endscroll +\autobuttons\end{page} + +@ +\subsection{Example of Standard Evaluation} +\label{InfixEval} +\index{pages!InfixEval!evalex.ht} +\index{evalex.ht!pages!InfixEval} +\index{InfixEval!evalex.ht!pages} +<>= +\begin{page}{InfixEval}{Example of Standard Evaluation} +\beginscroll +We illustrate the general evaluation of {\em a op b} for some +infix operator {\em op} with operands {\em a} and {\em b} +by the example: {\em 2 + 3.4}. +The evaluation steps are as follows: +\vspace{1}\newline +1.\tab{3}{\em a} and {\em b} are evaluated, each producing a value and a type. +\newline\tab{3}{\em Example:} {\em 2} evaluates to {\em 2} of type +\spadtype{Integer}; +{\em 3.4} evaluates to {\em 3.4} of type \spadtype{Float}. +\vspace{1}\newline +2.\tab{3}Axiom then chooses a function {\em op} based on the +types of {\em a} and {\em b}. +\newline\tab{3}{\em Example:} The function {\em +: (D,D) -> D} +is chosen requiring a common type {\em D} for both arguments to {\em +}. +An operation called {\em resolve} determines the `smallest common +type' \spadtype{Float}. +\vspace{1}\newline +3.\tab{3}If the argument types for the function are different from +those of {\em a} and {\em b}, +then the system coerces +%\downlink{coerces}{Coercion} +the values to the argument types. +\newline\tab{3}{\em Example:} The integer {\em 2} is coerced to the +float {\em 2.0}. +\vspace{1}\newline +4.\tab{3}The function is then applied to the values of {\em a} and {\em b} +to produce the value for {\em a op b}. +\newline\tab{3}{\em Example:} The function {\em +: (D,D) -> D}, where +{\em D} = \spadtype{Float} is applied to {\em 2.0} and {\em 3.4} to +produce {\em 5.4}. +\vspace{1}\newline +Try it: +\example{2 + 3.4} +\endscroll +\autobuttons\end{page} + +@ +\section{exdiff.ht} +\subsection{Computing Derivatives} +\label{ExDiffBasic} +\index{pages!ExDiffBasic!exdiff.ht} +\index{exdiff.ht!pages!ExDiffBasic} +\index{ExDiffBasic!exdiff.ht!pages} +<>= +\begin{page}{ExDiffBasic}{Computing Derivatives} +\beginscroll +To compute a derivative, you must specify an expression and a variable +of differentiation. +For example, to compute the derivative of {\em sin(x) * exp(x**2)} with respect to the +variable {\em x}, issue the following command: +\spadpaste{differentiate(sin(x) * exp(x**2),x)} +\endscroll +\autobuttons\end{page} + +@ +\subsection{Derivatives of Functions of Several Variables} +\label{ExDiffSeveralVariables} +\index{pages!ExDiffSeveralVariables!exdiff.ht} +\index{exdiff.ht!pages!ExDiffSeveralVariables} +\index{ExDiffSeveralVariables!exdiff.ht!pages} +<>= +\begin{page}{ExDiffSeveralVariables} +{Derivatives of Functions of Several Variables} +\beginscroll +Partial derivatives are computed in the same way as derivatives of functions +of one variable: you specify the function and a variable of differentiation. +For example: +\spadpaste{differentiate(sin(x) * tan(y)/(x**2 + y**2),x)} +\spadpaste{differentiate(sin(x) * tan(y)/(x**2 + y**2),y)} +\endscroll +\autobuttons\end{page} + +@ +\subsection{Derivatives of Higher Order} +\label{ExDiffHigherOrder} +\index{pages!ExDiffHigherOrder!exdiff.ht} +\index{exdiff.ht!pages!ExDiffHigherOrder} +\index{ExDiffHigherOrder!exdiff.ht!pages} +<>= +\begin{page}{ExDiffHigherOrder}{Derivatives of Higher Order} +\beginscroll +To compute a derivative of higher order (e.g. a second or third derivative), +pass the order as the third argument of the function 'differentiate'. +For example, to compute the fourth derivative of {\em exp(x**2)}, issue the +following command: +\spadpaste{differentiate(exp(x**2),x,4)} +\endscroll +\autobuttons\end{page} + +@ +\subsection{Multiple Derivatives I} +\label{ExDiffMultipleI} +\index{pages!ExDiffMultipleI!exdiff.ht} +\index{exdiff.ht!pages!ExDiffMultipleI} +\index{ExDiffMultipleI!exdiff.ht!pages} +<>= +\begin{page}{ExDiffMultipleI}{Multiple Derivatives I} +\beginscroll +When given a function of several variables, you may take derivatives repeatedly +and with respect to different variables. +The following command differentiates the function {\em sin(x)/(x**2 + y**2)} +first with respect to {\em x} and then with respect to {\em y}: +\spadpaste{differentiate(sin(x)/(x**2 + y**2),[x,y])} +As you can see, we first specify the function and then a list of the variables +of differentiation. +Variables may appear on the list more than once. +For example, the following command differentiates the same function with +respect to {\em x} and then twice with respect to {\em y}. +\spadpaste{differentiate(sin(x)/(x**2 + y**2),[x,y,y])} +\endscroll +\autobuttons\end{page} + +@ +\subsection{Multiple Derivatives II} +\label{ExDiffMultipleII} +\index{pages!ExDiffMultipleII!exdiff.ht} +\index{exdiff.ht!pages!ExDiffMultipleII} +\index{ExDiffMultipleII!exdiff.ht!pages} +<>= +\begin{page}{ExDiffMultipleII}{Multiple Derivatives II} +\beginscroll +You may also compute multiple derivatives by specifying a list of variables +together with a list of multiplicities. +For example, to differentiate {\em cos(z)/(x**2 + y**3)} +first with respect to {\em x}, +then twice with respect to {\em y}, then three times with respect to {\em z}, +issue the following command: +\spadpaste{differentiate(cos(z)/(x**2 + y**3),[x,y,z],[1,2,3])} +\endscroll +\autobuttons\end{page} + +@ +\subsection{Derivatives of Functions Involving Formal Integrals} +\label{ExDiffFormalIntegral} +\index{pages!ExDiffFormalIntegral!exdiff.ht} +\index{exdiff.ht!pages!ExDiffFormalIntegral} +\index{ExDiffFormalIntegral!exdiff.ht!pages} +<>= +\begin{page}{ExDiffFormalIntegral} +{Derivatives of Functions Involving Formal Integrals} +\beginscroll +When a function does not have a closed-form antiderivative, Axiom +returns a formal integral. +A typical example is +\spadpaste{f := integrate(sqrt(1 + t**3),t) \bound{f}} +This formal integral may be differentiated, either by itself or in any +combination with other functions: +\spadpaste{differentiate(f,t) \free{f}} +\spadpaste{differentiate(f * t**2,t) \free{f}} +\endscroll +\autobuttons\end{page} + +@ +\section{exit.ht} +\newcommand{\ExitXmpTitle}{Exit} +\newcommand{\ExitXmpNumber}{9.20} + +@ +\subsection{Exit} +\label{ExitXmpPage} +\index{pages!ExitXmpPage!exit.ht} +\index{exit.ht!pages!ExitXmpPage} +\index{ExitXmpPage!exit.ht!pages} +<>= +\begin{page}{ExitXmpPage}{Exit} +% ===================================================================== +\beginscroll + +A function that does not return directly to its caller has +\spadtype{Exit} as its return type. +The operation \spadfun{error} is an example of one which does not return +to its caller. +Instead, it causes a return to top-level. +\xtc{ +}{ +\spadpaste{n := 0 \bound{n}} +} +\xtc{ +The function \userfun{gasp} is given return type \spadtype{Exit} since it is +guaranteed never to return a value to its caller. +}{ +\begin{spadsrc}[\bound{gasp}\free{n}] +gasp(): Exit == + free n + n := n + 1 + error "Oh no!" +\end{spadsrc} +} +\xtc{ +The return type of \userfun{half} is determined by resolving the types of +the two branches of the \spad{if}. +}{ +\begin{spadsrc}[\bound{half}\free{gasp}] +half(k) == + if odd? k then gasp() + else k quo 2 +\end{spadsrc} +} +\xtc{ +Because +\userfun{gasp} has the return type \spadtype{Exit}, +the type of \spad{if} in \userfun{half} is resolved to be \spadtype{Integer}. +}{ +\spadpaste{half 4 \free{half}\bound{app1}} +} +\xtc{ +}{ +\spadpaste{half 3 \free{half app1}\bound{app2}} +} +\xtc{ +}{ +\spadpaste{n \free{app2}} +} + +For functions which return no value at all, use \spadtype{Void}. +See \downlink{``\ugUserTitle''}{ugUserPage} in Section \ugUserNumber\ignore{ugUser} and \downlink{`Void'}{VoidXmpPage}\ignore{Void} for more information. +% +\showBlurb{Exit} +\endscroll +\autobuttons +\end{page} + +@ +\section{exlap.ht} +\subsection{Laplace transform with a single pole} +\label{ExLapSimplePole} +\index{pages!ExLapSimplePole!exlap.ht} +\index{exlap.ht!pages!ExLapSimplePole} +\index{ExLapSimplePole!exlap.ht!pages} +<>= +\begin{page}{ExLapSimplePole}{Laplace transform with a single pole} +\beginscroll +The Laplace transform of t^n e^(a t) has a pole of order n+1 at x = a +and no other pole. We divide by n! to get a monic denominator in the +answer. +\spadpaste{laplace(t**4 * exp(-a*t) / factorial(4), t, s)} +\endscroll +\autobuttons\end{page} + +@ +\subsection{Laplace transform of a trigonometric function} +\label{ExLapTrigTrigh} +\index{pages!ExLapTrigTrigh!exlap.ht} +\index{exlap.ht!pages!ExLapTrigTrigh} +\index{ExLapTrigTrigh!exlap.ht!pages} +<>= +\begin{page}{ExLapTrigTrigh}{Laplace transform of a trigonometric function} +\beginscroll +Rather than looking up into a table, we use the normalizer to rewrite +the trigs and hyperbolic trigs to complex exponentials and +logarithms. +\spadpaste{laplace(sin(a*t) * cosh(a*t) - cos(a*t) * sinh(a*t), t, s)} +\endscroll +\autobuttons\end{page} + +@ +\subsection{Laplace transform requiring a definite integration} +\label{ExLapDefInt} +\index{pages!ExLapDefInt!exlap.ht} +\index{exlap.ht!pages!ExLapDefInt} +\index{ExLapDefInt!exlap.ht!pages} +<>= +\begin{page}{ExLapDefInt}{Laplace transform requiring a definite integration} +\beginscroll +When powers of t appear in the denominator, computing the laplace transform +requires integrating the result of another laplace transform between a +symbol and infinity. We use the full power of Axiom's integrator +in such cases. +\spadpaste{laplace(2/t * (1 - cos(a*t)), t, s)} +\endscroll +\autobuttons\end{page} + +@ +\subsection{Laplace transform of exponentials} +\label{ExLapExpExp} +\index{pages!ExLapExpExp!exlap.ht} +\index{exlap.ht!pages!ExLapExpExp} +\index{ExLapExpExp!exlap.ht!pages} +<>= +\begin{page}{ExLapExpExp}{Laplace transform of exponentials} +\beginscroll +This is another example where it is necessary to +integrate the result of another laplace transform. +\spadpaste{laplace((exp(a*t) - exp(b*t))/t, t, s)} +\endscroll +\autobuttons\end{page} + +@ +\subsection{Laplace transform of an exponential integral} +\label{ExLapSpecial1} +\index{pages!ExLapSpecial1!exlap.ht} +\index{exlap.ht!pages!ExLapSpecial1} +\index{ExLapSpecial1!exlap.ht!pages} +<>= +\begin{page}{ExLapSpecial1}{Laplace transform of an exponential integral} +We can handle some restricted cases of special functions, linear exponential +integrals among them. +\beginscroll +\spadpaste{laplace(exp(a*t+b)*Ei(c*t), t, s)} +\endscroll +\autobuttons\end{page} + +@ +\subsection{Laplace transform of special functions} +\label{ExLapSpecial2} +\index{pages!ExLapSpecial2!exlap.ht} +\index{exlap.ht!pages!ExLapSpecial2} +\index{ExLapSpecial2!exlap.ht!pages} +<>= +\begin{page}{ExLapSpecial2}{Laplace transform of special functions} +\beginscroll +An example with some interesting special functions. +\spadpaste{laplace(a*Ci(b*t) + c*Si(d*t), t, s)} +\endscroll +\autobuttons\end{page} + +@ +\section{exint.ht} +\subsection{Integral of a Rational Function} +\label{ExIntRationalFunction} +\index{pages!ExIntRationalFunction!exint.ht} +\index{exint.ht!pages!ExIntRationalFunction} +\index{ExIntRationalFunction!exint.ht!pages} +<>= +\begin{page}{ExIntRationalFunction}{Integral of a Rational Function} +\beginscroll +The following fraction has a denominator which factors into +a quadratic and a quartic irreducible polynomial. The usual +partial fraction approach used by most other computer algebra +systems either fails or introduces expensive unneeded algebraic +numbers. +We use a factorization-free algorithm. +\spadpaste{integrate((x**2+2*x+1)/((x+1)**6+1),x)} +There are cases where algebraic numbers are absolutely necessary. +In that case the answer to the \spadfun{integrate} command will contain +symbols like \spad{\%\%F0} denoting the algebraic numbers used. +To find out what the definitions for these numbers is use the +\spadfun{definingPolynomial} operation on these numbers. +\spadpaste{integrate(1/(x**3+x+1),x) \bound{i}} +For example, if a symbol like \spad{\%\%F0} appears in the result of this last +integration, then \spad{definingPolynomial \%\%F0} will return the +polynomial that \spad{\%\%F0} is a root of. The next command isolates the +algebraic number from the expression and displays its defining polynomial. +\spadpaste{definingPolynomial(tower(\%).2::EXPR INT) \free{i}} + +\endscroll +\autobuttons\end{page} + +@ +\subsection{Integral of a Rational Function with a Real Parameter} +\label{ExIntRationalWithRealParameter} +\index{pages!ExIntRationalWithRealParameter!exint.ht} +\index{exint.ht!pages!ExIntRationalWithRealParameter} +\index{ExIntRationalWithRealParameter!exint.ht!pages} +<>= +\begin{page}{ExIntRationalWithRealParameter} +{Integral of a Rational Function with a Real Parameter} +\beginscroll +When real parameters are present, the form of the integral can depend on +the signs of some expressions. Rather than query the user or make +sign assumptions, Axiom returns all possible answers. +\spadpaste{integrate(1/(x**2 + a),x)} +The integrate command generally assumes that all parameters are real. +\endscroll +\autobuttons\end{page} + +@ +\subsection{Integral of a Rational Function with a Complex Parameter} +\label{ExIntRationalWithComplexParameter} +\index{pages!ExIntRationalWithComplexParameter!exint.ht} +\index{exint.ht!pages!ExIntRationalWithComplexParameter} +\index{ExIntRationalWithComplexParameter!exint.ht!pages} +<>= +\begin{page}{ExIntRationalWithComplexParameter} +{Integral of a Rational Function with a Complex Parameter} +\beginscroll +If the parameter is complex instead of real, then the notion of +sign is undefined and there is a unique answer. +You can request +this answer by prepending the word `complex' to the command name: +\spadpaste{complexIntegrate(1/(x**2 + a),x)} +\endscroll +\autobuttons\end{page} + +@ +\subsection{Two Similar Integrands Producing Very Different Results} +\label{ExIntTwoSimilarIntegrands} +\index{pages!ExIntTwoSimilarIntegrands!exint.ht} +\index{exint.ht!pages!ExIntTwoSimilarIntegrands} +\index{ExIntTwoSimilarIntegrands!exint.ht!pages} +<>= +\begin{page}{ExIntTwoSimilarIntegrands} +{Two Similar Integrands Producing Very Different Results} +\beginscroll +The following two examples illustrate the limitations of table based +approaches. The two integrands are very similar, but the answer to one +of them requires the addition of two new algebraic numbers. +This one is the easy one: +\spadpaste{integrate(x**3 / (a+b*x)**(1/3),x)} +The next one looks very similar +but the answer is much more complicated. Only an algorithmic approach +is guaranteed to find what new constants must be added in order to +find a solution: +\spadpaste{integrate(1 / (x**3 * (a+b*x)**(1/3)),x)} +\endscroll +\autobuttons\end{page} + +@ +\subsection{An Integral Which Does Not Exist} +\label{ExIntNoSolution} +\index{pages!ExIntNoSolution!exint.ht} +\index{exint.ht!pages!ExIntNoSolution} +\index{ExIntNoSolution!exint.ht!pages} +<>= +\begin{page}{ExIntNoSolution}{An Integral Which Does Not Exist} +\beginscroll +Most computer algebra systems use heuristics or table-driven +approaches to integration. When these systems cannot determine +the answer to an integration problem, they reply "I don't know". +Axiom uses a complete algorithm for integration. +It will conclusively prove that an integral +cannot be expressed in terms of elementary functions. +When Axiom returns an integral sign, it has proved +that no answer exists as an elementary function. +\spadpaste{integrate(log(1 + sqrt(a*x + b)) / x,x)} +\endscroll +\autobuttons\end{page} + +@ +\begin{verbatim} +%% This example is broken +%\begin{page}{ExIntChangeOfVariables}{No Change of Variables is Required} +%\beginscroll +%Unlike computer algebra systems which rely on heuristics and +%table-lookup, the algorithmic integration facility +%of Axiom never requires you to make a change of variables +%in order to integrate a function. +%\spadpaste{integrate(sec(x)**(4/5)*csc(x)**(6/5),x)} +%\endscroll +%\autobuttons\end{page} +\end{verbatim} + +@ +\subsection{A Trigonometric Function of a Quadratic} +\label{ExIntTrig} +\index{pages!ExIntTrig!exint.ht} +\index{exint.ht!pages!ExIntTrig} +\index{ExIntTrig!exint.ht!pages} +<>= +\begin{page}{ExIntTrig}{A Trigonometric Function of a Quadratic} +\beginscroll +Axiom can handle complicated mixed functions way beyond what you can +find in tables: +\spadpaste{integrate((sinh(1+sqrt(x+b))+2*sqrt(x+b))/(sqrt(x+b)*(x+cosh(1+sqrt(x+b)))),x)} +Whenever possible, Axiom tries to express the answer using the functions +present in the integrand. +\endscroll +\autobuttons\end{page} + +@ +\subsection{Integrating a Function with a Hidden Algebraic Relation} +\label{ExIntAlgebraicRelation} +See ExIntAlgebraicRelationExplain \ref{ExIntAlgebraicRelationExplain} +on page~\pageref{ExIntAlgebraicRelationExplain} +\index{pages!ExIntAlgebraicRelation!exint.ht} +\index{exint.ht!pages!ExIntAlgebraicRelation} +\index{ExIntAlgebraicRelation!exint.ht!pages} +<>= +\begin{page}{ExIntAlgebraicRelation} +{Integrating a Function with a Hidden Algebraic Relation} +\beginscroll +A strong structure checking algorithm in Axiom finds hidden algebraic +relationships between functions. +\spadpaste{integrate(tan(atan(x)/3),x)} +The discovery of this algebraic relationship is necessary +for correctly integrating this function. +\downlink{Details.}{ExIntAlgebraicRelationExplain} \space{1} +\endscroll +\autobuttons\end{page} + +@ +\subsection{Details for integrating a function with a Hidden Algebraic Relation} +\label{ExIntAlgebraicRelationExplain} +\index{pages!ExIntAlgebraicRelationExplain!exint.ht} +\index{exint.ht!pages!ExIntAlgebraicRelationExplain} +\index{ExIntAlgebraicRelationExplain!exint.ht!pages} +<>= +\begin{page}{ExIntAlgebraicRelationExplain} +{Details for integrating a function with a Hidden Algebraic Relation} +\beginscroll +Steps taken for integration of: +\centerline{{\em f := tan(atan(x)/3)}} +\beginitems +\item +1. Replace {\em f} by an equivalent algebraic function {\em g} +satisfying the algebraic relation: +\centerline{{\em g**3 - 3*x*g - 3*g + x = 0}} +\item +2. Integrate {\em g} using using this algebraic relation; this produces: +\centerline{{\em (24g**2 - 8)log(3g**2 - 1) + (81x**2 + 24)g**2 + 72xg - 27x**2 - 16}} +\centerline{{\em / (54g**2 - 18)}} +\item +3. Rationalize the denominator, producing: +\centerline{{\em (8log(3g**2-1) - 3g**2 + 18xg + 15)/18}} +\item +4. Replace {\em g} by the initial {\em f} to produce the final result: +\centerline{{\em (8log(3tan(atan(x/3))**2-1) - 3tan(atan(x/3))**2 - 18xtan(atan(x/3) + 16)/18)}} +\enditems +\endscroll +\autobuttons\end{page} + +@ +\subsection{An Integral Involving a Root of a Transcendental Function} +\label{ExIntRadicalOfTranscendental} +\index{pages!ExIntRadicalOfTranscendental!exint.ht} +\index{exint.ht!pages!ExIntRadicalOfTranscendental} +\index{ExIntRadicalOfTranscendental!exint.ht!pages} +<>= +\begin{page}{ExIntRadicalOfTranscendental}{An Integral Involving a Root of a Transcendental Function} +\beginscroll +This is an example of a mixed function where +the algebraic layer is over the transcendental one. +\spadpaste{integrate((x + 1) / (x * (x + log x)**(3/2)),x)} +\endscroll +\autobuttons\end{page} + +@ +\subsection{An Integral of a Non-elementary Function} +\label{ExIntNonElementary} +\index{pages!ExIntNonElementary!exint.ht} +\index{exint.ht!pages!ExIntNonElementary} +\index{ExIntNonElementary!exint.ht!pages} +<>= +\begin{page}{ExIntNonElementary}{An Integral of a Non-elementary Function} +\beginscroll +While incomplete for non-elementary functions, Axiom can +handle some of them: +\spadpaste{integrate(exp(-x**2) * erf(x) / (erf(x)**3 - erf(x)**2 - erf(x) + 1),x)} +\endscroll +\autobuttons\end{page} + +@ +\section{exlimit.ht} +\subsection{Computing Limits} +\label{ExLimitBasic} +\begin{itemize} +\item ExLimitTwoSided \ref{ExLimitTwoSided} on page~\pageref{ExLimitTwoSided} +\item ExLimitOneSided \ref{ExLimitOneSided} on page~\pageref{ExLimitOneSided} +\end{itemize} +\index{pages!ExLimitBasic!exlimit.ht} +\index{exlimit.ht!pages!ExLimitBasic} +\index{ExLimitBasic!exlimit.ht!pages} +<>= +\begin{page}{ExLimitBasic}{Computing Limits} +\beginscroll +To compute a limit, you must specify a functional expression, +a variable, and a limiting value for that variable. +For example, to compute the limit of (x**2 - 3*x + 2)/(x**2 - 1) +as x approaches 1, issue the following command: +\spadpaste{limit((x**2 - 3*x + 2)/(x**2 - 1),x = 1)} +% answer := -1/2 +Since you have not specified a direction, Axiom will attempt +to compute a two-sided limit. Sometimes the limit when approached from +the left is different from the limit from the right. +\downlink{Example}{ExLimitTwoSided}. In this case, you may +wish to ask for a one-sided limit. +\downlink{How. }{ExLimitOneSided} +\endscroll +\autobuttons\end{page} + +@ +\subsection{Limits of Functions with Parameters} +\label{ExLimitParameter} +\index{pages!ExLimitParameter!exlimit.ht} +\index{exlimit.ht!pages!ExLimitParameter} +\index{ExLimitParameter!exlimit.ht!pages} +<>= +\begin{page}{ExLimitParameter}{Limits of Functions with Parameters} +\beginscroll +You may also take limits of functions with parameters. The limit +will be expressed in terms of those parameters. +Here's an example: +\spadpaste{limit(sinh(a*x)/tan(b*x),x = 0)} +% answer := a/b +\endscroll +\autobuttons\end{page} + +@ +\subsection{One-sided Limits} +\label{ExLimitOneSided} +\index{pages!ExLimitOneSided!exlimit.ht} +\index{exlimit.ht!pages!ExLimitOneSided} +\index{ExLimitOneSided!exlimit.ht!pages} +<>= +\begin{page}{ExLimitOneSided}{One-sided Limits} +\beginscroll +If you have a function which is only defined on one side of a particular value, +you may wish to compute a one-sided limit. +For instance, the function \spad{log(x)} is only defined to the right of zero, +i.e. for \spad{x > 0}. +Thus, when computing limits of functions involving \spad{log(x)}, you probably +will want a 'right-hand' limit. +Here's an example: +\spadpaste{limit(x * log(x),x = 0,"right")} +% answer := 0 +When you do not specify \spad{right} or \spad{left} as an optional fourth +argument, the function \spadfun{limit} will try to compute a two-sided limit. +In the above case, the limit from the left does not exist, as Axiom +will indicate when you try to take a two-sided limit: +\spadpaste{limit(x * log(x),x = 0)} +% answer := [left = "failed",right = 0] +\endscroll +\autobuttons\end{page} + +@ +\subsection{Two-sided Limits} +\label{ExLimitTwoSided} +\index{pages!ExLimitTwoSided!exlimit.ht} +\index{exlimit.ht!pages!ExLimitTwoSided} +\index{ExLimitTwoSided!exlimit.ht!pages} +<>= +\begin{page}{ExLimitTwoSided}{Two-sided Limits} +\beginscroll +A function may be defined on both sides of a particular value, but will +tend to different limits as its variable tends to that value from the +left and from the right. +We can construct an example of this as follows: +Since { \em sqrt(y**2)} is simply the absolute value of \spad{y}, +the function \spad{sqrt(y**2)/y} +is simply the sign (+1 or -1) of the real number \spad{y}. +Therefore, \spad{sqrt(y**2)/y = -1} for \spad{y < 0} and +\spad{sqrt(y**2)/y = +1} for \spad{y > 0}. +Watch what happens when we take the limit at \spad{y = 0}. +\spadpaste{limit(sqrt(y**2)/y,y = 0)} +% answer := [left = -1,right = 1] +The answer returned by Axiom gives both a 'left-hand' and a 'right-hand' +limit. +Here's another example, this time using a more complicated function: +\spadpaste{limit(sqrt(1 - cos(t))/t,t = 0)} +% answer := [left = -sqrt(1/2),right = sqrt(1/2)] +\endscroll +\autobuttons\end{page} + +@ +\subsection{Limits at Infinity} +\label{ExLimitInfinite} +\index{pages!ExLimitInfinite!exlimit.ht} +\index{exlimit.ht!pages!ExLimitInfinite} +\index{ExLimitInfinite!exlimit.ht!pages} +<>= +\begin{page}{ExLimitInfinite}{Limits at Infinity} +\beginscroll +You can compute limits at infinity by passing either 'plus infinity' +or 'minus infinity' as the third argument of the function \spadfun{limit}. +To do this, use the constants \spad{\%plusInfinity} and \spad{\%minusInfinity}. +Here are two examples: +\spadpaste{limit(sqrt(3*x**2 + 1)/(5*x),x = \%plusInfinity)} +\spadpaste{limit(sqrt(3*x**2 + 1)/(5*x),x = \%minusInfinity)} +\endscroll +\autobuttons\end{page} + +@ +\subsection{Real Limits vs. Complex Limits} +\label{ExLimitRealComplex} +\index{pages!ExLimitRealComplex!exlimit.ht} +\index{exlimit.ht!pages!ExLimitRealComplex} +\index{ExLimitRealComplex!exlimit.ht!pages} +<>= +\begin{page}{ExLimitRealComplex}{Real Limits vs. Complex Limits} +\beginscroll +When you use the function \spadfun{limit}, you will be taking the limit of a real +function of a real variable. +For example, you can compute +\spadpaste{limit(z * sin(1/z),z = 0)} +Axiom returns \spad{0} because as a function of a real variable +\spad{sin(1/z)} is always between \spad{-1} and \spad{1}, so \spad{z * sin(1/z)} +tends to \spad{0} as \spad{z} tends to \spad{0}. +However, as a function of a complex variable, \spad{sin(1/z)} is badly +behaved around \spad{0} +(one says that \spad{sin(1/z)} has an 'essential singularity' at \spad{z = 0}). +When viewed as a function of a complex variable, \spad{z * sin(1/z)} +does not approach any limit as \spad{z} tends to \spad{0} in the complex plane. +Axiom indicates this when we call the function \spadfun{complexLimit}: +\spadpaste{complexLimit(z * sin(1/z),z = 0)} +\endscroll +\autobuttons\end{page} + +@ +\subsection{Complex Limits at Infinity} +\label{ExLimitComplexInfinite} +\index{pages!ExLimitComplexInfinite!exlimit.ht} +\index{exlimit.ht!pages!ExLimitComplexInfinite} +\index{ExLimitComplexInfinite!exlimit.ht!pages} +<>= +\begin{page}{ExLimitComplexInfinite}{Complex Limits at Infinity} +\beginscroll +You may also take complex limits at infinity, i.e. limits of a function of +\spad{z} as \spad{z} approaches infinity on the Riemann sphere. +Use the symbol \spad{\%infinity} to denote `complex infinity'. +Also, to compute complex limits rather than real limits, use the +function \spadfun{complexLimit}. +Here is an example: +\spadpaste{complexLimit((2 + z)/(1 - z),z = \%infinity)} +In many cases, a limit of a real function of a real variable will exist +when the corresponding complex limit does not. +For example: +\spadpaste{limit(sin(x)/x,x = \%plusInfinity)} +\spadpaste{complexLimit(sin(x)/x,x = \%infinity)} +\endscroll +\autobuttons\end{page} + +@ +\section{exmatrix.ht} +\subsection{Basic Arithmetic Operations on Matrices} +\label{ExMatrixBasicFunction} +\index{pages!ExMatrixBasicFunction!exmatrix.ht} +\index{exmatrix.ht!pages!ExMatrixBasicFunction} +\index{ExMatrixBasicFunction!exmatrix.ht!pages} +<>= +\begin{page}{ExMatrixBasicFunction}{Basic Arithmetic Operations on Matrices} +\beginscroll +You can create a matrix using the function \spadfun{matrix}. +The function takes a list of lists of elements of the ring and produces a +matrix whose \spad{i}th row contains the elements of the \spad{i}th list. +For example: +\spadpaste{m1 := matrix([[1,-2,1],[4,2,-4]]) \bound{m1}} +\spadpaste{m2 := matrix([[1,0,2],[20,30,10],[0,200,100]]) \bound{m2}} +\spadpaste{m3 := matrix([[1,2,3],[2,4,6]]) \bound{m3}} +Some of the basic arithmetic operations on matrices are: +\spadpaste{m1 + m3 \free{m1} \free{m3}} +\spadpaste{100 * m1 \free{m1}} +\spadpaste{m1 * m2 \free{m1} \free{m2}} +\spadpaste{-m1 + m3 * m2 \free{m1} \free{m2} \free{m3}} +You can also multiply a matrix and a vector provided +that the matrix and vector have compatible dimensions. +\spadpaste{m3 *vector([1,0,1]) \free{m3}} +However, the dimensions of the matrices must be compatible in order for +Axiom to perform an operation - otherwise an error message will occur. +\endscroll +\autobuttons\end{page} + +@ +\subsection{Constructing new Matrices} +\label{ExConstructMatrix} +\index{pages!ExConstructMatrix!exmatrix.ht} +\index{exmatrix.ht!pages!ExConstructMatrix} +\index{ExConstructMatrix!exmatrix.ht!pages} +<>= +\begin{page}{ExConstructMatrix}{Constructing new Matrices} +\beginscroll +A number of functions exist for constructing new matrices from existing ones. + +If you want to create a matrix whose entries are 0 except on the main +diagonal you can use \spadfun{diagonalMatrix}. +This function takes a list of ring elements as an argument and returns a +square matrix which has these elements on the main diagonal. +Consider the following example: +\spadpaste{diagonalMatrix([1,2,3,2,1])} + +The function \spadfun{subMatrix}(\spad{a},\spad{i},\spad{j},\spad{k},\spad{l}) +constructs a new matrix +consisting of rows \spad{i} through \spad{j} and columns \spad{k} through +\spad{l} of \spad{a} , inclusive. +\spadpaste{subMatrix(matrix([[0,1,2,3,4],[5,6,7,8,9],[10,11,12,13,14]]), 1,3,2,4)} + + +The functions \spadfun{horizConcat} and \spadfun{vertConcat} +concatenate matrices +horizontally and vertically, respectively. +\spadpaste{horizConcat(matrix([[1,2,3],[6,7,8]]),matrix([[11,12,13],[55,77,88]])) } +\spadpaste{vertConcat(matrix([[1,2,3],[6,7,8]]),matrix([[11,12,13],[55,77,88]])) } + + +The function \spadfunX{setsubMatrix}(\spad{a},\spad{i},\spad{k},\spad{b}) +replaces the submatrix of \spad{a} +starting at row \spad{i} and column \spad{k} with the elements of the matrix i\spad{b}. +\spadpaste{b:=matrix([[0,1,2,3,4],[5,6,7,8,9],[10,11,12,13,14]]) \bound{b}} +\spadpaste{setsubMatrix!(b,1,1,transpose(subMatrix(b,1,3,1,3)))\free{b}} +changes the submatrix of \spad{b} consisting of the first 3 rows and columns +to its transpose. +\endscroll +\autobuttons\end{page} + +@ +\subsection{Trace of a Matrix} +\label{ExTraceMatrix} +\index{pages!ExTraceMatrix!exmatrix.ht} +\index{exmatrix.ht!pages!ExTraceMatrix} +\index{ExTraceMatrix!exmatrix.ht!pages} +<>= +\begin{page}{ExTraceMatrix}{Trace of a Matrix} +\beginscroll +If you have a square matrix, then you can compute its `trace'. +The function \spadfun{trace} computes the sum of all elements on the diagonal +of a matrix. +For example `trace' for a four by four Vandermonde matrix. +\spadpaste{trace( matrix([[1,x,x**2,x**3],[1,y,y**2,y**3],[1,z,z**2,z**3],[1,u,u**2,u**3]]) )} +\endscroll +\autobuttons\end{page} + +@ +\subsection{Determinant of a Matrix} +\label{ExDeterminantMatrix} +\index{pages!ExDeterminantMatrix!exmatrix.ht} +\index{exmatrix.ht!pages!ExDeterminantMatrix} +\index{ExDeterminantMatrix!exmatrix.ht!pages} +<>= +\begin{page}{ExDeterminantMatrix}{Determinant of a Matrix} +\beginscroll +The function \spadfun{determinant} computes the determinant of a matrix over a +commutative ring, that is a ring whose multiplication is commutative. +\spadpaste{determinant(matrix([[1,2,3,4],[2,3,2,5],[3,4,5,6],[4,1,6,7]]))} +\endscroll +\autobuttons\end{page} + +@ +\subsection{Inverse of a Matrix} +\label{ExInverseMatrix} +\index{pages!ExInverseMatrix!exmatrix.ht} +\index{exmatrix.ht!pages!ExInverseMatrix} +\index{ExInverseMatrix!exmatrix.ht!pages} +<>= +\begin{page}{ExInverseMatrix}{Inverse of a Matrix} +\beginscroll +The function \spadfun{inverse} computes the inverse of a square matrix. +\spadpaste{inverse(matrix([[1,2,1],[-2,3,4],[-1,5,6]])) } +(If the inverse doesn't exist, then Axiom returns `failed'.) +\endscroll +\autobuttons\end{page} + +@ +\subsection{Rank of a Matrix} +\label{ExRankMatrix} +\index{pages!ExRankMatrix!exmatrix.ht} +\index{exmatrix.ht!pages!ExRankMatrix} +\index{ExRankMatrix!exmatrix.ht!pages} +<>= +\begin{page}{ExRankMatrix}{Rank of a Matrix} +\beginscroll +The function \spadfun{rank} gives you the rank of a matrix: +\spadpaste{rank(matrix([[0,4,1],[5,3,-7],[-5,5,9]]))} +\endscroll +\autobuttons\end{page} + +@ +\section{expr.ht} +<>= +\newcommand{\ExpressionXmpTitle}{Expression} +\newcommand{\ExpressionXmpNumber}{9.21} + +@ +\subsection{Expression} +\label{ExpressionXmpPage} +\begin{itemize} +\item KernelXmpPage \ref{KernelXmpPage} on +page~pageref{KernelXmpPage} +\item ugIntroCalcDerivPage \ref{ugIntroCalcDerivPage} on +page~pageref{ugIntroCalcDerivPage} +\item ugIntroCalcLimitsPage \ref{ugIntroCalcLimitsPage} on +page~pageref{ugIntroCalcLimitsPage} +\item ugIntroSeriesPage \ref{ugIntroSeriesPage} on +page~pageref{ugIntroSeriesPage} +\item ugProblemDEQPage \ref{ugProblemDEQPage} on +page~pageref{ugProblemDEQPage} +\item ugProblemIntegrationPage \ref{ugProblemIntegrationPage} on +page~pageref{ugProblemIntegrationPage} +\item ugUserRulesPage \ref{ugUserRulesPage} on +page~pageref{ugUserRulesPage} +\end{itemize} +\index{pages!ExpressionXmpPage!expr.ht} +\index{expr.ht!pages!ExpressionXmpPage} +\index{ExpressionXmpPage!expr.ht!pages} +<>= +\begin{page}{ExpressionXmpPage}{Expression} +\beginscroll + +\axiomType{Expression} is a constructor that creates domains whose +objects can have very general symbolic forms. +Here are some examples: +\xtc{ +This is an object of type \axiomType{Expression Integer}. +}{ +\spadpaste{sin(x) + 3*cos(x)**2} +} +\xtc{ +This is an object of type \axiomType{Expression Float}. +}{ +\spadpaste{tan(x) - 3.45*x} +} +\xtc{ +This object contains symbolic function applications, sums, +products, square roots, and a quotient. +}{ +\spadpaste{(tan sqrt 7 - sin sqrt 11)**2 / (4 - cos(x - y))} +} +As you can see, \axiomType{Expression} actually takes an argument +domain. +The {\it coefficients} of the terms within the expression belong +to the argument domain. +\axiomType{Integer} and \axiomType{Float}, along with +\axiomType{Complex Integer} and \axiomType{Complex Float} +are the most common coefficient domains. +\xtc{ +The choice of whether +to use a \axiomType{Complex} coefficient domain or not is +important since Axiom can perform some simplifications +on real-valued objects +}{ +\spadpaste{log(exp x)@Expression(Integer)} +} +\xtc{ +... which are not valid on complex ones. +}{ +\spadpaste{log(exp x)@Expression(Complex Integer)} +} +\xtc{ +Many potential coefficient domains, such as +\axiomType{AlgebraicNumber}, are not usually used because +\axiomType{Expression} can subsume them. +}{ +\spadpaste{sqrt 3 + sqrt(2 + sqrt(-5)) \bound{algnum1}} +} +\xtc{ +}{ +\spadpaste{\% :: Expression Integer \free{algnum1}} +} +Note that we sometimes talk about ``an object of type +\axiomType{Expression}.'' This is not really correct because we +should say, for example, ``an object of type \axiomType{Expression +Integer}'' or ``an object of type \axiomType{Expression Float}.'' +By a similar abuse of language, when we refer to an ``expression'' +in this section we will mean an object of type +\axiomType{Expression R} for some domain {\bf R}. + +The Axiom documentation contains many examples of the use +of \axiomType{Expression}. +For the rest of this section, we'll give you some pointers to those +examples plus give you some idea of how to manipulate expressions. + +It is important for you to know that \axiomType{Expression} +creates domains that have category \axiomType{Field}. +Thus you can invert any non-zero expression and you shouldn't +expect an operation like \axiomFun{factor} to give you much +information. +You can imagine expressions as being represented as quotients of +``multivariate'' polynomials where the ``variables'' are kernels +(see \downlink{`Kernel'}{KernelXmpPage}\ignore{Kernel}). +A kernel can either be a symbol such as \axiom{x} or a symbolic +function application like \axiom{sin(x + 4)}. +The second example is actually a nested kernel since the argument +to \axiomFun{sin} contains the kernel \axiom{x}. +\xtc{ +}{ +\spadpaste{height mainKernel sin(x + 4)} +} +Actually, the argument to \axiomFun{sin} is an expression, and so +the structure of \axiomType{Expression} is recursive. +\downlink{`Kernel'}{KernelXmpPage}\ignore{Kernel} +demonstrates how to extract the kernels in an +expression. + +Use the \HyperName{} Browse facility to see what operations are +applicable to expression. +At the time of this writing, there were 262 operations with 147 +distinct name in \axiomType{Expression Integer}. +For example, \axiomFunFrom{numer}{Expression} and +\axiomFunFrom{denom}{Expression} extract the numerator and +denominator of an expression. +\xtc{ +}{ +\spadpaste{e := (sin(x) - 4)**2 / ( 1 - 2*y*sqrt(- y) ) \bound{e}} +} +\xtc{ +}{ +\spadpaste{numer e \free{e}} +} +\xtc{ +}{ +\spadpaste{denom e \free{e}} +} +\xtc{ +Use \axiomFunFrom{D}{Expression} to compute partial derivatives. +}{ +\spadpaste{D(e, x) \free{e}} +} +\xtc{ +See \downlink{``\ugIntroCalcDerivTitle''}{ugIntroCalcDerivPage} in Section +\ugIntroCalcDerivNumber\ignore{ugIntroCalcDeriv} +for more examples of expressions and derivatives. +}{ +\spadpaste{D(e, [x, y], [1, 2]) \free{e}} +} +See \downlink{``\ugIntroCalcLimitsTitle''}{ugIntroCalcLimitsPage} +in Section \ugIntroCalcLimitsNumber\ignore{ugIntroCalcLimits} and +\downlink{``\ugIntroSeriesTitle''}{ugIntroSeriesPage} in Section +\ugIntroSeriesNumber\ignore{ugIntroSeries} +for more examples of expressions and calculus. +Differential equations involving expressions are discussed in +\downlink{``\ugProblemDEQTitle''}{ugProblemDEQPage} in Section +\ugProblemDEQNumber\ignore{ugProblemDEQ}. +Chapter 8 has many advanced examples: see +\downlink{``\ugProblemIntegrationTitle''}{ugProblemIntegrationPage} +in Section \ugProblemIntegrationNumber\ignore{ugProblemIntegration} +for a discussion of Axiom's integration facilities. + +When an expression involves no ``symbol kernels'' (for example, +\axiom{x}), it may be possible to numerically evaluate the +expression. +\xtc{ +If you suspect the evaluation will create a complex number, +use \axiomFun{complexNumeric}. +}{ +\spadpaste{complexNumeric(cos(2 - 3*\%i))} +} +\xtc{ +If you know it will be real, use \axiomFun{numeric}. +}{ +\spadpaste{numeric(tan 3.8)} +} +The \axiomFun{numeric} operation will display an error message if +the evaluation yields a calue with an non-zero imaginary part. +Both of these operations have an optional second argument +\axiom{n} which specifies that the accuracy of the approximation +be up to \axiom{n} decimal places. + +When an expression involves no ``symbolic application'' kernels, +it may be possible to convert it a polynomial or rational +function in the variables that are present. +\xtc{ +}{ +\spadpaste{e2 := cos(x**2 - y + 3) \bound{e2}} +} +\xtc{ +}{ +\spadpaste{e3 := asin(e2) - \%pi/2 \free{e2}\bound{e3}} +} +\xtc{ +}{ +\spadpaste{e3 :: Polynomial Integer \free{e3}} +} +\xtc{ +This also works for the polynomial types where specific variables +and their ordering are given. +}{ +\spadpaste{e3 :: DMP([x, y], Integer) \free{e3}} +} + +Finally, a certain amount of simplication takes place as +expressions are constructed. +\xtc{ +}{ +\spadpaste{sin \%pi} +} +\xtc{ +}{ +\spadpaste{cos(\%pi / 4)} +} +\xtc{ +For simplications that involve multiple terms of the expression, +use \axiomFun{simplify}. +}{ +\spadpaste{tan(x)**6 + 3*tan(x)**4 + 3*tan(x)**2 + 1 \bound{tan6}} +} +\xtc{ +}{ +\spadpaste{simplify \% \free{tan6}} +} +See \downlink{``\ugUserRulesTitle''}{ugUserRulesPage} in Section \ugUserRulesNumber\ignore{ugUserRules} +for examples of how to write your own rewrite rules for +expressions. + + +\endscroll +\autobuttons +\end{page} + +@ +\section{explot2d.ht} +\subsection{Plotting Functions of One Variable} +\label{ExPlot2DFunctions} +\index{pages!ExPlot2DFunctions!explot2d.ht} +\index{explot2d.ht!pages!ExPlot2DFunctions} +\index{ExPlot2DFunctions!explot2d.ht!pages} +<>= +\begin{page}{ExPlot2DFunctions}{Plotting Functions of One Variable} +\beginscroll +To plot a function {\em y = f(x)}, you need only specify the function and the +interval on which it is to be plotted. +\graphpaste{draw(sin(tan(x)) - tan(sin(x)),x = 0..6)} +\endscroll +\autobuttons\end{page} + +@ +\subsection{Plotting Parametric Curves} +\label{ExPlot2DParametric} +\index{pages!ExPlot2DParametric!explot2d.ht} +\index{explot2d.ht!pages!ExPlot2DParametric} +\index{ExPlot2DParametric!explot2d.ht!pages} +<>= +\begin{page}{ExPlot2DParametric}{Plotting Parametric Curves} +\beginscroll +To plot a parametric curve defined by {\em x = f(t)}, +{\em y = g(t)}, specify the functions +{\em f(t)} and {\em g(t)} +as arguments of the function `curve', then give +the interval over which {\em t} is to range. +\graphpaste{draw(curve(9 * sin(3*t/4),8 * sin(t)),t = -4*\%pi..4*\%pi)} +\endscroll +\autobuttons\end{page} + +@ +\subsection{Plotting Using Polar Coordinates} +\label{ExPlot2DPolar} +\index{pages!ExPlot2DPolar!explot2d.ht} +\index{explot2d.ht!pages!ExPlot2DPolar} +\index{ExPlot2DPolar!explot2d.ht!pages} +<>= +\begin{page}{ExPlot2DPolar}{Plotting Using Polar Coordinates} +\beginscroll +To plot the function {\em r = f(theta)} in polar coordinates, +use the option {\em coordinates == polar}. +As usual, +call the function 'draw' and specify the function {\em f(theta)} and the +interval over which {\em theta} is to range. +\graphpaste{draw(sin(4*t/7),t = 0..14*\%pi,coordinates == polar)} +\endscroll +\autobuttons\end{page} + +@ +\subsection{Plotting Plane Algebraic Curves} +\label{ExPlot2DAlgebraic} +\index{pages!ExPlot2DAlgebraic!explot2d.ht} +\index{explot2d.ht!pages!ExPlot2DAlgebraic} +\index{ExPlot2DAlgebraic!explot2d.ht!pages} +<>= +\begin{page}{ExPlot2DAlgebraic}{Plotting Plane Algebraic Curves} +\beginscroll +Axiom can also plot plane algebraic curves (i.e. curves defined by +an equation {\em f(x,y) = 0}) provided that the curve is non-singular in the +region to be sketched. +Here's an example: +\graphpaste{draw(y**2 + y - (x**3 - x) = 0, x, y, range == [-2..2,-2..1])} +Here the region of the sketch is {\em -2 <= x <= 2, -2 <= y <= 1}. +\endscroll +\autobuttons\end{page} + +@ +\section{explot3d.ht} +\subsection{Plotting Functions of Two Variables} +\label{ExPlot3DFunctions} +\index{pages!ExPlot3DFunctions!explot3d.ht} +\index{explot3d.ht!pages!ExPlot3DFunctions} +\index{ExPlot3DFunctions!explot3d.ht!pages} +<>= +\begin{page}{ExPlot3DFunctions}{Plotting Functions of Two Variables} +\beginscroll +To plot a function {\em z = f(x,y)}, you need only specify the function and the +intervals over which the dependent variables will range. +For example, here's how you plot the function {\em z = cos(x*y)} as the +variables {\em x} and {\em y} both range between -3 and 3: +\graphpaste{draw(cos(x*y),x = -3..3,y = -3..3)} +\endscroll +\autobuttons\end{page} + +@ +\subsection{Plotting Parametric Surfaces} +\label{ExPlot3DParametricSurface} +\index{pages!ExPlot3DParametricSurface!explot3d.ht} +\index{explot3d.ht!pages!ExPlot3DParametricSurface} +\index{ExPlot3DParametricSurface!explot3d.ht!pages} +<>= +\begin{page}{ExPlot3DParametricSurface}{Plotting Parametric Surfaces} +\beginscroll +To plot a parametric surface defined by {\em x = f(u,v)}, +{\em y = g(u,v)}, {\em z = h(u,v)}, specify the functions +{\em f(u,v)}, {\em g(u,v)}, and {\em h(u,v)} +as arguments of the function `surface', then give +the intervals over which {\em u} and {\em v} are to range. +With parametric surfaces, we can create some interesting graphs. +Here's an egg: +\graphpaste{draw(surface(5*sin(u)*cos(v),4*sin(u)*sin(v),3*cos(u)),u=0..\%pi,v=0..2*\%pi)} +Here's a cone: +\graphpaste{draw(surface(u*cos(v),u*sin(v),u),u=0..4,v=0..2*\%pi)} +\endscroll +\autobuttons\end{page} + +@ +\subsection{Plotting Parametric Curves} +\label{ExPlot3DParametricCurve} +\index{pages!ExPlot3DParametricCurve!explot3d.ht} +\index{explot3d.ht!pages!ExPlot3DParametricCurve} +\index{ExPlot3DParametricCurve!explot3d.ht!pages} +<>= +\begin{page}{ExPlot3DParametricCurve}{Plotting Parametric Curves} +\beginscroll +To plot a parametric curve defined by {\em x = f(t)}, +{\em y = g(t)}, {\em z = h(t)}, specify the functions +{\em f(t)}, {\em g(t)}, and {\em h(t)} +as arguments of the function `curve', then give +the interval over which {\em t} is to range. +Here is a spiral: +\graphpaste{draw(curve(cos(t),sin(t),t),t=0..6)} +Here is the {\em twisted cubic curve}: +\graphpaste{draw(curve(t,t**2,t**3),t=-3..3)} +\endscroll +\autobuttons\end{page} + +@ +\section{expose.ht} +\subsection{TITLE} +\label{TPD} +\index{pages!TPD!expose.ht} +\index{expose.ht!pages!TPD} +\index{TPD!expose.ht!pages} +<>= +\begin{page}{helpExpose}{Exposure} +\beginscroll +Exposure determines what part of the Axiom library +is available to you when using Axiom. +At any time during your interactive Axiom session, +each constructor is either {\em exposed} or {\em unexposed}. +If a constructor is exposed, its operations are available to +the interpreter and therefore to you. +If a constructor is unexposed, the operations are not seen +by the interpreter and thus not available to you. +\par +If you are a beginner, you may only want +basic parts of the library exposed. +If you are an expert, you may want to have all parts of the +library exposed. +If you have an application that requires +only a segment of the library, you may want to arrange to expose +only that segment for your use. +\par +\endscroll +Additional Information: +\beginmenu +\menulink{What}{ExposureDef}\tab{8}What is an exposure group? +\menulink{System}{ExposureSystem}\tab{8}What exposure groups are system defined? +% \menulink{User}{ExposureUser}\tab{8}How can I define my own? +\menulink{Details}{ExposureDetails}\tab{8}Some details on exposure +\endmenu +\end{page} + +@ +\subsection{System Defined Exposure Groups} +\label{ExposureSystem} +\index{pages!ExposureSystem!expose.ht} +\index{expose.ht!pages!ExposureSystem} +\index{ExposureSystem!expose.ht!pages} +<>= +\begin{page}{ExposureSystem}{System Defined Exposure Groups} +\beginscroll +Exposure is defined by {\em groups}. +Groups have names. +Seven exposure groups are system-defined:\beginmenu +\item\tab{3}{\em current}\tab{12}The currently active exposure group +\item\tab{3}{\em basic}\tab{12}The default value of {\em current} +\item\tab{3}{\em category}\tab{13}Category constructors not in {\em basic} +\item\tab{3}{\em domain}\tab{13}Domain constructors not in {\em basic} +\item\tab{3}{\em package}\tab{13}Package constructors not in {\em basic} +\item\tab{3}{\em default}\tab{13}Default constructors not in {\em basic} +\item\tab{3}{\em hidden}\tab{13}All constructors not in {\em basic} +\item\tab{3}{\em naglink}\tab{13}All constructors used in the AXIOM NAG Link +\endmenu +\par +When you first use Axiom, the {\em current} exposure group is +set to {\em basic} and {\em naglink}. Using \HyperName{} or the system command +{\em expose}, you may +change the current exposure group by +adding or dropping constructors or by setting {\em current} +to an exposure group you have created. +\endscroll +Additional Information: +\beginmenu +\menulink{What}{ExposureDef}\tab{10}What is an exposure group? +%\menulink{User}{ExposureUser}\tab{10}How you can define your own exposure group +\endmenu +\end{page} + +@ +\subsection{What is an Exposure Group?} +\label{ExposureDef} +\index{pages!ExposureDef!expose.ht} +\index{expose.ht!pages!ExposureDef} +\index{ExposureDef!expose.ht!pages} +<>= +\begin{page}{ExposureDef}{What is an Exposure Group?} +\beginscroll +\par +An exposure group is a list of constructors to be exposed. +Those constructors on the list are exposed; +those not on the list are not exposed. +The library contains 4 kinds of constructors intuitively described as follows: +\beginmenu +\item\menuitemstyle{}{\em domain}\tab{10}Describes computational objects and functions defined on these objects +\item\menuitemstyle{}{\em package}\tab{10}Describes functions which will work over a variety of domains +\item\menuitemstyle{}{\em category}\tab{10}Names a set of operations +\item\menuitemstyle{}{\em default}\tab{10}Provides default functions for a cateogry +\endmenu + +An exposure group is defined by three lists: +\beginmenu +\item\menuitemstyle{}{\em groups}\tab{13}A list of other (more basic) groups +\item\menuitemstyle{}{\em additions}\tab{13}A list of explicit constructors to be included +\item\menuitemstyle{}{\em subtractions}\tab{13}A list of explicit constructors to be dropped +\endmenu +You can define your own exposure groups: give them names and define the +three above lists to be anything you like. +Using \HyperName{}, you can conveniently edit your exposure groups, +install them as the {\em current} exposure, and so on. +\endscroll +\end{page} + +@ +\subsection{Details on Exposure} +\label{ExposureDetails} +\index{pages!ExposureDetails!expose.ht} +\index{expose.ht!pages!ExposureDetails} +\index{ExposureDetails!expose.ht!pages} +<>= +\begin{page}{ExposureDetails}{Details on Exposure} +\beginscroll +Exposure is generally defined by the set of domain and package constructors +you want to have available. +Category and default constructors are generally implied. +A category constructor is exposed if mentioned by {\em any} other constructor +(including another category). +A default constructor is exposed if its corresponding category constructor is exposed. +\par +If you explicitly add a domain or package +constructor, its name will be put in an {\em Additions} list. +The system will also add automatically to the {\em Additions} list +any category explicity exported by that domain or package. +If that category has a corresponding default constructor, that +default constructor will also be added as well. +\par +If you like, you can explicitly drop a constructor. +Any such name is added to the {\em Subtractions} list. +The system will drop this name from the {\em Additions} list if it appears. +\par +If the package or domain takes arguments from an unexported +domain or declares that its arguments can come from a domain +which is a member of an unexported category, these constructors +will {\em not} be added. +\endscroll +\end{page} + +@ +\section{exseries.ht} +\subsection{Converting Expressions to Series} +\label{ExSeriesConvert} +\index{pages!ExSeriesConvert!exseries.ht} +\index{exseries.ht!pages!ExSeriesConvert} +\index{ExSeriesConvert!exseries.ht!pages} +<>= +\begin{page}{ExSeriesConvert}{Converting Expressions to Series} +\beginscroll +You can convert a functional expression to a power series by using the +function 'series'. +Here's an example: +\spadpaste{series(sin(a*x),x = 0)} +This causes {\em sin(a*x)} to be expanded in powers of {\em (x - 0)}, that is, in powers +of {\em x}. +You can have {\em sin(a*x)} expanded in powers of {\em (a - \%pi/4)} by +issuing the following command: +\spadpaste{series(sin(a*x),a = \%pi/4)} +\endscroll +\autobuttons\end{page} + +@ +\subsection{Manipulating Power Series} +\label{ExSeriesManipulate} +\index{pages!ExSeriesManipulate!exseries.ht} +\index{exseries.ht!pages!ExSeriesManipulate} +\index{ExSeriesManipulate!exseries.ht!pages} +<>= +\begin{page}{ExSeriesManipulate}{Manipulating Power Series} +\beginscroll +Once you have created a power series, you can perform arithmetic operations +on that series. +First compute the Taylor expansion of {\em 1/(1-x)}: +\spadpaste{f := series(1/(1-x),x = 0) \bound{f}} +Now compute the square of that series: +\spadpaste{f ** 2 \free{f}} +It's as easy as 1, 2, 3,... +\endscroll +\autobuttons\end{page} + +@ +\subsection{Functions on Power Series} +\label{ExSeriesFunctions} +\index{pages!ExSeriesFunctions!exseries.ht} +\index{exseries.ht!pages!ExSeriesFunctions} +\index{ExSeriesFunctions!exseries.ht!pages} +<>= +\begin{page}{ExSeriesFunctions}{Functions on Power Series} +\beginscroll +The usual elementary functions ({\em log}, {\em exp}, +trigonometric functions, etc.) +are defined for power series. +You can create a power series: +% Warning: currently there are (interpretor) problems with converting +% rational functions and polynomials to power series. +\spadpaste{f := series(1/(1-x),x = 0) \bound{f1}} +and then apply these functions to the series: +\spadpaste{g := log(f) \free{f1} \bound{g}} +\spadpaste{exp(g) \free{g}} +\endscroll +\autobuttons\end{page} + +@ +\subsection{Substituting Numerical Values in Power Series} +\label{ExSeriesSubstitution} +\index{pages!ExSeriesSubstitution!exseries.ht} +\index{exseries.ht!pages!ExSeriesSubstitution} +\index{ExSeriesSubstitution!exseries.ht!pages} +<>= +\begin{page}{ExSeriesSubstitution} +{Substituting Numerical Values in Power Series} +\beginscroll +Here's a way to obtain numerical approximations of +{\em e} from the Taylor series +expansion of {\em exp(x)}. +First you create the desired Taylor expansion: +\spadpaste{f := taylor(exp(x)) \bound{f2}} +Now you evaluate the series at the value {\em 1.0}: +% Warning: syntax for evaluating power series may change. +\spadpaste{eval(f,1.0) \free{f2}} +You get a sequence of partial sums. +\endscroll +\autobuttons\end{page} + +@ +\section{exsum.ht} +\subsection{Summing the Entries of a List I} +\label{ExSumListEntriesI} +\index{pages!ExSumListEntriesI!exsum.ht} +\index{exsum.ht!pages!ExSumListEntriesI} +\index{ExSumListEntriesI!exsum.ht!pages} +<>= +\begin{page}{ExSumListEntriesI}{Summing the Entries of a List I} +\beginscroll +In Axiom, you can create lists of consecutive integers by giving the +first and last entries of the list. +Here's how you create a list of the integers between {\em 1} and {\em 15}: +\spadpaste{[i for i in 1..15]} +To sum the entries of a list, simply put {\em +/} in front of the list. +For example, the following command will sum the integers from 1 to 15: +\spadpaste{reduce(+,[i for i in 1..15])} +\endscroll +\autobuttons\end{page} + +@ +\subsection{Summing the Entries of a List II} +\label{ExSumListEntriesII} +\index{pages!ExSumListEntriesII!exsum.ht} +\index{exsum.ht!pages!ExSumListEntriesII} +\index{ExSumListEntriesII!exsum.ht!pages} +<>= +\begin{page}{ExSumListEntriesII}{Summing the Entries of a List II} +\beginscroll +In Axiom, you can also create lists whose elements are some expression +{\em f(n)} as the parameter n ranges between two integers. +For example, the following command will create a list of the squares of +the integers between {\em 5} and {\em 20}: +\spadpaste{[n**2 for n in 5..20]} +You can also compute the sum of the entries of this list: +\spadpaste{reduce(+,[n**2 for n in 5..20])} +\endscroll +\autobuttons\end{page} + +@ +\subsection{Approximating $e$} +\label{ExSumApproximateE} +\index{pages!ExSumApproximateE!exsum.ht} +\index{exsum.ht!pages!ExSumApproximateE} +\index{ExSumApproximateE!exsum.ht!pages} +<>= +\begin{page}{ExSumApproximateE}{Approximating e} +\beginscroll +You can obtain a numerical approximation of the number {\em e} by summing the +entries of the following list: +\spadpaste{reduce(+,[1.0/factorial(n) for n in 0..20])} +\endscroll +\autobuttons\end{page} + +@ +\subsection{Closed Form Summations} +\label{ExSumClosedForm} +\index{pages!ExSumClosedForm!exsum.ht} +\index{exsum.ht!pages!ExSumClosedForm} +\index{ExSumClosedForm!exsum.ht!pages} +<>= +\begin{page}{ExSumClosedForm}{Closed Form Summations} +\beginscroll +In a previous example, we found the sum of the squares of the integers +between {\em 5} and {\em 20}. +We can also use Axiom to find a formula for the sum of the squares of +the integers between {\em a} and {\em b}, where {\em a} and {\em b} +are integers which will remain +unspecified: +\spadpaste{s := sum(k**2,k = a..b) \bound{s}} +{\em sum(k**2,k = a..b)} returns the sum of {\em k**2} as the index {\em k} +runs from {\em a} to {\em b}. +Let's check our answer in one particular case by substituting specific values +for {\em a} and {\em b} in our formula: +% Warning: syntax for polynomial evaluation will probably change. +\spadpaste{eval(s,[a,b],[1,25]) \free{s}} +\spadpaste{reduce(+,[i**2 for i in 1..25])} +\endscroll +\autobuttons\end{page} + +@ +\subsection{Sums of Cubes} +\label{ExSumCubes} +\index{pages!ExSumCubes!exsum.ht} +\index{exsum.ht!pages!ExSumCubes} +\index{ExSumCubes!exsum.ht!pages} +<>= +\begin{page}{ExSumCubes}{Sums of Cubes} +\beginscroll +Here's a cute example. +First compute the sum of the cubes from {\em 1} to {\em n}: +\spadpaste{sum(k**3,k = 1..n)} +Then compute the square of the sum of the integers from {\em 1} to {\em n}: +\spadpaste{sum(k,k = 1..n) ** 2} +The answers are the same. +\endscroll +\autobuttons\end{page} + +@ +\subsection{Sums of Polynomials} +\label{ExSumPolynomial} +\index{pages!ExSumPolynomial!exsum.ht} +\index{exsum.ht!pages!ExSumPolynomial} +\index{ExSumPolynomial!exsum.ht!pages} +<>= +\begin{page}{ExSumPolynomial}{Sums of Polynomials} +\beginscroll +Axiom can compute {\em sum(f,k = a..b)} +when {\em f} is any polynomial in {\em k}, even +one with parameters. +\spadpaste{sum(3*k**2/(c**2 + 1) + 12*k/d,k = (3*a)..(4*b))} +\endscroll +\autobuttons\end{page} + +@ +\begin{verbatim} +%\begin{page}{ExSumRationalFunction}{Sums of Rational Functions} +%\beginscroll +%Axiom can compute {\em sum(f,k = a..b)} for some rational functions (quotients +%of polynomials) in {\em k}. +%\spadpaste{sum(1/(k * (k + 2)),k = 1..n)} +%However, the method used (Gosper's method) does not guarantee an answer +%in every case. +%Here's an example of a sum that the method cannot compute: +%\spadpaste{sum(1/(k**2 + 1),k = 1..n)} +%\endscroll +%\autobuttons\end{page} +\end{verbatim} + +\subsection{Sums of General Functions} +\label{ExSumGeneralFunction} +\index{pages!ExSumGeneralFunction!exsum.ht} +\index{exsum.ht!pages!ExSumGeneralFunction} +\index{ExSumGeneralFunction!exsum.ht!pages} +<>= +\begin{page}{ExSumGeneralFunction}{Sums of General Functions} +\beginscroll +Gosper's method can also be used to compute {\em sum(f,k = a..b)} +for some functions +f which are not rational functions in {\em k}. +Here's an example: +\spadpaste{sum(k * x**k,k = 1..n)} +\endscroll +\autobuttons\end{page} + +@ +\subsection{Infinite Sums} +\label{ExSumInfinite} +Provide a package for infinite sums +\index{pages!ExSumInfinite!exsum.ht} +\index{exsum.ht!pages!ExSumInfinite} +\index{ExSumInfinite!exsum.ht!pages} +<>= +\begin{page}{ExSumInfinite}{Infinite Sums} +\beginscroll +In a few cases, we can compute infinite sums by taking limits of finite +sums. +For instance, you can compute the sum of {\em 1/(k * (k + 2))} as {\em k} +ranges from +{\em 1} to {\em infinity}. +Use {\em \%plusInfinity} to denote `plus infinity'. +\spadpaste{limit( sum(1/(k * (k + 2)),k = 1..n) ,n = \%plusInfinity)} +\endscroll +\autobuttons\end{page} + +@ +\section{farray.ht} +<>= +\newcommand{\FlexibleArrayXmpTitle}{FlexibleArray} +\newcommand{\FlexibleArrayXmpNumber}{9.26} + +@ +\subsection{FlexibleArray} +\label{FlexibleArrayXmpPage} +\begin{itemize} +\item OneDimensionalArrayXmpPage \ref{OneDimensionalArrayXmpPage} on +page~pageref{OneDimensionalArrayXmpPage} +\item VectorXmpPage \ref{VectorXmpPage} on +page~pageref{VectorXmpPage} +\end{itemize} +\index{pages!FlexibleArrayXmpPage!farray.ht} +\index{farray.ht!pages!FlexibleArrayXmpPage} +\index{FlexibleArrayXmpPage!farray.ht!pages} +<>= +\begin{page}{FlexibleArrayXmpPage}{FlexibleArray} +\beginscroll +The \spadtype{FlexibleArray} domain constructor creates +one-dimensional arrays of elements of the same type. +Flexible arrays are an attempt to provide a data type that has the +best features of both one-dimensional arrays (fast, random access +to elements) and lists (flexibility). +They are implemented by a fixed block of storage. +When necessary for expansion, a new, larger block of storage is +allocated and the elements from the old storage area are copied +into the new block. + +Flexible arrays have available most of the operations provided by +\spadtype{OneDimensionalArray} +(see \downlink{`OneDimensionalArray'}{OneDimensionalArrayXmpPage} +\ignore{OneDimensionalArray} +and \downlink{`Vector'}{VectorXmpPage}\ignore{Vector}). +Since flexible arrays are also of category +\spadtype{ExtensibleLinearAggregate}, they have operations +\spadfunX{concat}, \spadfunX{delete}, \spadfunX{insert}, +\spadfunX{merge}, \spadfunX{remove}, \spadfunX{removeDuplicates}, +and \spadfunX{select}. +In addition, the operations \spadfun{physicalLength} and +\spadfunX{physicalLength} provide user-control over expansion and +contraction. + +\xtc{ +A convenient way to create a flexible array is to apply +the operation \spadfun{flexibleArray} to a list of values. +}{ +\spadpaste{flexibleArray [i for i in 1..6]} +} +\xtc{ +Create a flexible array of six zeroes. +}{ +\spadpaste{f : FARRAY INT := new(6,0)\bound{f}} +} +\xtc{ +For \texht{$i=1\ldots 6$}{i = 1..6}, +set the \eth{\smath{i}} element to \smath{i}. +Display \spad{f}. +}{ +\spadpaste{for i in 1..6 repeat f.i := i; f\bound{f1}\free{f}} +} +\xtc{ +Initially, the physical length is the same as the number of elements. +}{ +\spadpaste{physicalLength f\free{f1}} +} +\xtc{ +Add an element to the end of \spad{f}. +}{ +\spadpaste{concat!(f,11)\bound{f2}\free{f1}} +} +\xtc{ +See that its physical length has grown. +}{ +\spadpaste{physicalLength f\free{f2}} +} +\xtc{ +Make \spad{f} grow to have room for \spad{15} elements. +}{ +\spadpaste{physicalLength!(f,15)\bound{f3}\free{f2}} +} +\xtc{ +Concatenate the elements of \spad{f} to itself. +The physical length allows room for three more values at the end. +}{ +\spadpaste{concat!(f,f)\bound{f4}\free{f3}} +} +\xtc{ +Use \spadfunX{insert} to add an element to the front of +a flexible array. +}{ +\spadpaste{insert!(22,f,1)\bound{f5}\free{f4}} +} +\xtc{ +Create a second flexible array from \spad{f} consisting of the +elements from index 10 forward. +}{ +\spadpaste{g := f(10..)\bound{g}\free{f5}} +} +\xtc{ +Insert this array at the front of \spad{f}. +}{ +\spadpaste{insert!(g,f,1)\bound{g1}\free{g f5}} +} +\xtc{ +Merge the flexible array \spad{f} into \spad{g} after sorting each in place. +}{ +\spadpaste{merge!(sort! f, sort! g)\bound{f6}\free{g f5}} +} +\xtc{ +Remove duplicates in place. +}{ +\spadpaste{removeDuplicates! f\bound{f7}\free{f6}} +} +\xtc{ +Remove all odd integers. +}{ +\spadpaste{select!(i +-> even? i,f)\bound{f8}\free{f7}} +} +\xtc{ +All these operations have shrunk the physical length of \spad{f}. +}{ +\spadpaste{physicalLength f\free{b8}} +} +\xtc{ +To force Axiom not to shrink flexible arrays call the +\spadfun{shrinkable} operation with the argument \axiom{false}. +You must package call this operation. +The previous value is returned. +}{ +\spadpaste{shrinkable(false)\$FlexibleArray(Integer)} +} + +\endscroll +\autobuttons +\end{page} +% + +@ +\section{file.ht} +<>= +\newcommand{\FileXmpTitle}{File} +\newcommand{\FileXmpNumber}{9.24} + +@ +\subsection{File} +\label{FileXmpPage} +\begin{itemize} +\item TextFileXmpPage \ref{TextFileXmpPage} on +page~pageref{TextFileXmpPage} +\item KeyedAccessFileXmpPage \ref{KeyedAccessFileXmpPage} on +page~pageref{KeyedAccessFileXmpPage} +\item LibraryXmpPage \ref{LibraryXmpPage} on +page~pageref{LibraryXmpPage} +\item FileNameXmpPage \ref{FileNameXmpPage} on +page~pageref{FileNameXmpPage} +\end{itemize} +\index{pages!FileXmpPage!file.ht} +\index{file.ht!pages!FileXmpPage} +\index{FileXmpPage!file.ht!pages} +<>= +\begin{page}{FileXmpPage}{File} +\beginscroll + +The \spadtype{File(S)} domain provides a basic interface to read and +write values of type \spad{S} in files. +\xtc{ +Before working with a file, it must be made accessible to Axiom with +the \spadfunFrom{open}{File} operation. +}{ +\spadpaste{ifile:File List Integer:=open("/tmp/jazz1","output") \bound{ifile}} +} +The \spadfunFrom{open}{File} function arguments are a \spadtype{FileName} +and a \spadtype{String} specifying the mode. +If a full pathname is not specified, the current default directory is +assumed. +The mode must be one of \spad{"input"} or \spad{"output"}. +If it is not specified, \spad{"input"} is assumed. +Once the file has been opened, you can read or write data. +\xtc{ +The operations \spadfunFromX{read}{File} and \spadfunFromX{write}{File} are +provided. +}{ +\spadpaste{write!(ifile, [-1,2,3]) \free{ifile}\bound{ifile1}} +} +\xtc{ +}{ +\spadpaste{write!(ifile, [10,-10,0,111]) \free{ifile1}\bound{ifile2}} +} +\xtc{ +}{ +\spadpaste{write!(ifile, [7]) \free{ifile2}\bound{ifile3}} +} +\xtc{ +You can change from writing to reading (or vice versa) +by reopening a file. +}{ +\spadpaste{reopen!(ifile, "input") \free{ifile3}\bound{ifile4}} +} +\xtc{ +}{ +\spadpaste{read! ifile \free{ifile4}\bound{ifile5}} +} +\xtc{ +}{ +\spadpaste{read! ifile \free{ifile5}\bound{ifile6}} +} +\xtc{ +The \spadfunFromX{read}{File} operation can cause an error if one tries to +read more data than is in the file. +To guard against this possibility the \spadfunFromX{readIfCan}{File} +operation should be used. +}{ +\spadpaste{readIfCan! ifile \free{ifile6}\bound{ifile7}} +} +\xtc{ +}{ +\spadpaste{readIfCan! ifile \free{ifile7}\bound{ifile8}} +} +\xtc{ +You can find the current mode of the file, and the file's name. +}{ +\spadpaste{iomode ifile \free{ifile}} +} +\xtc{ +}{ +\spadpaste{name ifile \free{ifile}} +} +\xtc{ +When you are finished with a file, you should close it. +}{ +\spadpaste{close! ifile \free{ifile}\bound{ifileA}} +} +\noOutputXtc{ +}{ +\spadpaste{)system rm /tmp/jazz1 \free{ifileA}} +} +%\xtc{ +%}{ +%\spadcommand{)clear all \free{}\bound{}} +%} + +A limitation of the underlying LISP system is that not all values can be +represented in a file. +In particular, delayed values containing compiled functions cannot be +saved. + +For more information on related topics, see +\downlink{`TextFile'}{TextFileXmpPage}\ignore{TextFile}, +\downlink{`KeyedAccessFile'}{KeyedAccessFileXmpPage}\ignore{KeyedAccessFile}, +\downlink{`Library'}{LibraryXmpPage}\ignore{Library}, and +\downlink{`FileName'}{FileNameXmpPage}\ignore{FileName}. +\showBlurb{File} +\endscroll +\autobuttons +\end{page} + +@ +\section{float.ht} +<>= +\newcommand{\FloatXmpTitle}{Float} +\newcommand{\FloatXmpNumber}{9.27} + +@ +\subsection{Float} +\label{FloatXmpPage} +\begin{itemize} +\item ugGraphPage \ref{ugGraphPage} on +page~pageref{ugGraphPage} +\item ugProblemNumericPage \ref{ugProblemNumericPage} on +page~pageref{ugProblemNumericPage} +\item DoubleFloatXmpPage \ref{DoubleFloatXmpPage} on +page~pageref{DoubleFloatXmpPage} +\item ugFloatIntroPage \ref{ugFloatIntroPage} on +page~pageref{ugFloatIntroPage} +\item ugFloatConvertPage \ref{ugFloatConvertPage} on +page~pageref{ugFloatConvertPage} +\item ugxFloatOutputPage \ref{ugxFloatOutputPage} on +page~pageref{ugxFloatOutputPage} +\item ugxFloatHilbertPage \ref{ugxFloatHilbertPage} on +page~pageref{ugxFloatHilbertPage} +\end{itemize} +\index{pages!FloatXmpPage!float.ht} +\index{float.ht!pages!FloatXmpPage} +\index{FloatXmpPage!float.ht!pages} +<>= +\begin{page}{FloatXmpPage}{Float} +\beginscroll + + Axiom provides two kinds of floating point numbers. +The domain \spadtype{Float} (abbreviation \spadtype{FLOAT}) +implements a model of arbitrary +precision floating point numbers. +The domain \spadtype{DoubleFloat} (abbreviation \spadtype{DFLOAT}) +is intended to make available +hardware floating point arithmetic in Axiom. +The actual model of floating point that \spadtype{DoubleFloat} provides is +system-dependent. +For example, on the IBM system 370 Axiom uses IBM double +precision which has fourteen hexadecimal digits of precision or roughly +sixteen decimal digits. +Arbitrary precision floats allow the user to specify the +precision at which arithmetic operations are computed. +Although this is an attractive facility, it comes at a cost. +Arbitrary-precision floating-point arithmetic typically takes +twenty to two hundred times more time than hardware floating point. + +For more information about Axiom's numeric and graphic +facilities, see +\downlink{``\ugGraphTitle''}{ugGraphPage} in +Section \ugGraphNumber\ignore{ugGraph}, +\downlink{``\ugProblemNumericTitle''}{ugProblemNumericPage} in +Section \ugProblemNumericNumber\ignore{ugProblemNumeric}, and +\downlink{`DoubleFloat'}{DoubleFloatXmpPage}\ignore{DoubleFloat}. + +\beginmenu + \menudownlink{{9.27.1. Introduction to Float}}{ugxFloatIntroPage} + \menudownlink{{9.27.2. Conversion Functions}}{ugxFloatConvertPage} + \menudownlink{{9.27.3. Output Functions}}{ugxFloatOutputPage} + \menudownlink{{9.27.4. An Example: Determinant of a Hilbert Matrix}} +{ugxFloatHilbertPage} +\endmenu +\endscroll +\autobuttons +\end{page} + +@ +<>= +\newcommand{\ugxFloatIntroTitle}{Introduction to Float} +\newcommand{\ugxFloatIntroNumber}{9.27.1.} + +@ +\subsection{Introduction to Float} +\label{ugxFloatIntroPage} +\index{pages!ugxFloatIntroPage!float.ht} +\index{float.ht!pages!ugxFloatIntroPage} +\index{ugxFloatIntroPage!float.ht!pages} +<>= +\begin{page}{ugxFloatIntroPage}{Introduction to Float} +\beginscroll + +Scientific notation is supported for input and output +of floating point numbers. +A floating point number is written as a string of digits containing a +decimal point optionally followed by the letter ``{\tt E}'', and then +the exponent. +\xtc{ +We begin by doing some calculations using arbitrary precision floats. +The default precision is twenty decimal digits. +}{ +\spadpaste{1.234} +} +\xtc{ +A decimal base for the exponent is assumed, so the number +\spad{1.234E2} denotes +\texht{$1.234 \cdot 10^2$}{\spad{1.234 * 10**2}}. +}{ +\spadpaste{1.234E2} +} +\xtc{ +The normal arithmetic operations are available for floating point +numbers. +}{ +\spadpaste{sqrt(1.2 + 2.3 / 3.4 ** 4.5)} +} + +\endscroll +\autobuttons +\end{page} + +@ +<>= +\newcommand{\ugxFloatConvertTitle}{Conversion Functions} +\newcommand{\ugxFloatConvertNumber}{9.27.2.} + +@ +\subsection{Conversion Functions} +\label{ugxFloatConvertPage} +See ugTypesConvertPage \ref{ugTypesConvertPage} on +page~/pageref{ugTypesConvertPage} +\index{pages!ugxFloatConvertPage!float.ht} +\index{float.ht!pages!ugxFloatConvertPage} +\index{ugxFloatConvertPage!float.ht!pages} +<>= +\begin{page}{ugxFloatConvertPage}{Conversion Functions} +\beginscroll + +\labelSpace{3pc} +\xtc{ +You can use conversion +(\downlink{``\ugTypesConvertTitle''}{ugTypesConvertPage} +in Section \ugTypesConvertNumber\ignore{ugTypesConvert}) +to go back and forth between \spadtype{Integer}, \spadtype{Fraction Integer} +and \spadtype{Float}, as appropriate. +}{ +\spadpaste{i := 3 :: Float \bound{i}} +} +\xtc{ +}{ +\spadpaste{i :: Integer \free{i}} +} +\xtc{ +}{ +\spadpaste{i :: Fraction Integer \free{i}} +} +\xtc{ +Since you are explicitly asking for a conversion, you must take +responsibility for any loss of exactness. +}{ +\spadpaste{r := 3/7 :: Float \bound{r}} +} +\xtc{ +}{ +\spadpaste{r :: Fraction Integer \free{r}} +} +\xtc{ +This conversion cannot be performed: use +\spadfunFrom{truncate}{Float} or \spadfunFrom{round}{Float} if that +is what you intend. +}{ +\spadpaste{r :: Integer \free{r}} +} + +\xtc{ +The operations \spadfunFrom{truncate}{Float} and \spadfunFrom{round}{Float} +truncate \ldots +}{ +\spadpaste{truncate 3.6} +} +\xtc{ +and round +to the nearest integral \spadtype{Float} respectively. +}{ +\spadpaste{round 3.6} +} +\xtc{ +}{ +\spadpaste{truncate(-3.6)} +} +\xtc{ +}{ +\spadpaste{round(-3.6)} +} +\xtc{ +The operation \spadfunFrom{fractionPart}{Float} computes the fractional part of +\spad{x}, that is, \spad{x - truncate x}. +}{ +\spadpaste{fractionPart 3.6} +} +\xtc{ +The operation \spadfunFrom{digits}{Float} allows the user to set the +precision. +It returns the previous value it was using. +}{ +\spadpaste{digits 40 \bound{d40}} +} +\xtc{ +}{ +\spadpaste{sqrt 0.2} +} +\xtc{ +}{ +\spadpaste{pi()\$Float \free{d40}} +} +\xtc{ +The precision is only limited by the computer memory available. +Calculations at 500 or more digits of precision are not difficult. +}{ +\spadpaste{digits 500 \bound{d1000}} +} +\xtc{ +}{ +\spadpaste{pi()\$Float \free{d1000}} +} +\xtc{ +Reset \spadfunFrom{digits}{Float} to its default value. +}{ +\spadpaste{digits 20} +} +Numbers of type \spadtype{Float} are represented as a record of two +integers, namely, the mantissa and the exponent where the base of the +exponent is binary. +That is, the floating point number \spad{(m,e)} represents the number +\texht{$m \cdot 2^e$}{\spad{m * 2**e}}. +A consequence of using a binary base is that decimal numbers can not, in +general, be represented exactly. + +\endscroll +\autobuttons +\end{page} + +<>= +\newcommand{\ugxFloatOutputTitle}{Output Functions} +\newcommand{\ugxFloatOutputNumber}{9.27.3.} + +@ +\subsection{Output Functions} +\label{ugxFloatOutputPage} +\index{pages!ugxFloatOutputPage!float.ht} +\index{float.ht!pages!ugxFloatOutputPage} +\index{ugxFloatOutputPage!float.ht!pages} +<>= +\begin{page}{ugxFloatOutputPage}{Output Functions} +\beginscroll + +A number of operations exist for specifying how +numbers of type \spadtype{Float} are to be +displayed. +By default, spaces are inserted every ten digits in the +output for readability.\footnote{Note that you cannot include spaces +in the input form of a floating point number, though you can use +underscores.} + + +\xtc{ +Output spacing can be modified with the \spadfunFrom{outputSpacing}{Float} +operation. +This inserts no spaces and then displays the value of \spad{x}. +}{ +\spadpaste{outputSpacing 0; x := sqrt 0.2 \bound{x}\bound{os0}} +} +\xtc{ +Issue this to have the spaces inserted every \spad{5} digits. +}{ +\spadpaste{outputSpacing 5; x \bound{os5}\free{x}} +} +\xtc{ +By default, the system displays floats in either fixed format +or scientific format, depending on the magnitude of the number. +}{ +\spadpaste{y := x/10**10 \bound{y}\free{x os5}} +} +\xtc{ +A particular format may be requested with the operations +\spadfunFrom{outputFloating}{Float} and \spadfunFrom{outputFixed}{Float}. +}{ +\spadpaste{outputFloating(); x \bound{of} \free{os5 x}} +} +\xtc{ +}{ +\spadpaste{outputFixed(); y \bound{ox} \free{os5 y}} +} +\xtc{ +Additionally, you can ask for \spad{n} digits to be displayed after the +decimal point. +}{ +\spadpaste{outputFloating 2; y \bound{of2} \free{os5 y}} +} +\xtc{ +}{ +\spadpaste{outputFixed 2; x \bound{ox2} \free{os5 x}} +} +\xtc{ +This resets the output printing to the default behavior. +}{ +\spadpaste{outputGeneral()} +} +% + +\endscroll +\autobuttons +\end{page} + +@ +<>= +\newcommand{\ugxFloatHilbertTitle}{An Example: Determinant of a Hilbert Matrix} +\newcommand{\ugxFloatHilbertNumber}{9.27.4.} + +@ +\subsection{An Example: Determinant of a Hilbert Matrix} +\label{ugxFloatHilbertPage} +\index{pages!ugxFloatHilbertPage!float.ht} +\index{float.ht!pages!ugxFloatHilbertPage} +\index{ugxFloatHilbertPage!float.ht!pages} +<>= +\begin{page}{ugxFloatHilbertPage}{An Example: Determinant of a Hilbert Matrix} +\beginscroll + +Consider the problem of computing the determinant of a \spad{10} by \spad{10} +Hilbert matrix. +The \eth{\spad{(i,j)}} entry of a Hilbert matrix is given by +\spad{1/(i+j+1)}. + +\labelSpace{2pc} +\xtc{ +First do the computation using rational numbers to obtain the +exact result. +}{ +\spadpaste{a: Matrix Fraction Integer := matrix [[1/(i+j+1) for j in 0..9] for i in 0..9] \bound{a}} +} +\xtc{ +This version of \spadfunFrom{determinant}{Matrix} uses Gaussian +elimination. +}{ +\spadpaste{d:= determinant a \free{a}\bound{d}} +} +\xtc{ +}{ +\spadpaste{d :: Float \free{d}} +} +\xtc{ +Now use hardware floats. Note that a semicolon (;) is used +to prevent the display of the matrix. +}{ +\spadpaste{b: Matrix DoubleFloat := matrix [[1/(i+j+1\$DoubleFloat) for j in 0..9] for i in 0..9]; \bound{b}} +} +\xtc{ +The result given by hardware floats is correct only to four significant +digits of precision. +In the jargon of numerical analysis, the Hilbert matrix is said to be +``ill-conditioned.'' +}{ +\spadpaste{determinant b \free{b}} +} +\xtc{ +Now repeat the computation at a higher precision using \spadtype{Float}. +}{ +\spadpaste{digits 40 \bound{d40}} +} +\xtc{ +}{ +\spadpaste{c: Matrix Float := matrix [[1/(i+j+1\$Float) for j in 0..9] for i in 0..9]; \free{d40} \bound{c}} +} +\xtc{ +}{ +\spadpaste{determinant c \free{c}} +} +\xtc{ +Reset \spadfunFrom{digits}{Float} to its default value. +}{ +\spadpaste{digits 20} +} +\endscroll +\autobuttons +\end{page} + +@ +\section{fname.ht} +<>= +\newcommand{\FileNameXmpTitle}{FileName} +\newcommand{\FileNameXmpNumber}{9.25} + +@ +\subsection{FileName} +\label{FileNameXmpPage} +\index{pages!FileNameXmpPage!fname.ht} +\index{fname.ht!pages!FileNameXmpPage} +\index{FileNameXmpPage!fname.ht!pages} +<>= +\begin{page}{FileNameXmpPage}{FileName} +\beginscroll + +The \spadtype{FileName} domain provides an interface to the computer's +file system. +Functions are provided to manipulate file names and to test properties of +files. + +The simplest way to use file names in the Axiom interpreter is to +rely on conversion to and from strings. +The syntax of these strings depends on the operating system. +\xtc{ +}{ +\spadpaste{fn: FileName \bound{fndecl}} +} +\xtc{ +On AIX, this is a proper file syntax: +}{ +\spadpaste{fn := "/spad/src/input/fname.input" \free{fndecl}\bound{fn}} +} + +Although it is very convenient to be able to use string notation +for file names in the interpreter, it is desirable to have a portable +way of creating and manipulating file names from within programs. +\xtc{ +A measure of portability is obtained by considering a file name +to consist of three parts: the {\it directory}, the {\it name}, +and the {\it extension}. +}{ +\spadpaste{directory fn \free{fn}} +} +\xtc{ +}{ +\spadpaste{name fn \free{fn}} +} +\xtc{ +}{ +\spadpaste{extension fn \free{fn}} +} +The meaning of these three parts depends on the operating system. +For example, on CMS the file \spad{"SPADPROF INPUT M"} +would have directory \spad{"M"}, name \spad{"SPADPROF"} and +extension \spad{"INPUT"}. + +\xtc{ +It is possible to create a filename from its parts. +}{ +\spadpaste{fn := filename("/u/smwatt/work", "fname", "input") \free{fndecl}\bound{fn1}} +} +\xtc{ +When writing programs, it is helpful to refer to directories via +variables. +}{ +\spadpaste{objdir := "/tmp" \bound{objdir}} +} +\xtc{ +}{ +\spadpaste{fn := filename(objdir, "table", "spad") \free{fndecl,objdir}\bound{fn2}} +} +\xtc{ +If the directory or the extension is given as an empty string, then +a default is used. On AIX, the defaults are the current directory +and no extension. +}{ +\spadpaste{fn := filename("", "letter", "") \free{fndecl}\bound{fn3}} +} + +Three tests provide information about names in the file system. +\xtc{ +The \spadfunFrom{exists?}{FileName} operation tests whether +the named file exists. +}{ +\spadpaste{exists? "/etc/passwd"} +} +\xtc{ +The operation \spadfunFrom{readable?}{FileName} tells whether the named file +can be read. If the file does not exist, then it cannot be read. +}{ +\spadpaste{readable? "/etc/passwd"} +} +\xtc{ +}{ +\spadpaste{readable? "/etc/security/passwd"} +} +\xtc{ +}{ +\spadpaste{readable? "/ect/passwd"} +} +\xtc{ +Likewise, the operation \spadfunFrom{writable?}{FileName} +tells whether the named file +can be written. +If the file does not exist, the test is determined +by the properties of the directory. +}{ +\spadpaste{writable? "/etc/passwd"} +} +\xtc{ +}{ +\spadpaste{writable? "/dev/null"} +} +\xtc{ +}{ +\spadpaste{writable? "/etc/DoesNotExist"} +} +\xtc{ +}{ +\spadpaste{writable? "/tmp/DoesNotExist"} +} + +The \spadfunFrom{new}{FileName} operation constructs the name of a new +writable file. +The argument sequence is the same as for \spadfunFrom{filename}{FileName}, +except that the name part is actually a prefix for a constructed +unique name. +\xtc{ +The resulting file is in the specified directory +with the given extension, and the same defaults are used. +}{ +\spadpaste{fn := new(objdir, "xxx", "yy") \free{objdir,fndecl}\bound{fn4}} +} +\endscroll +\autobuttons +\end{page} + +@ +\section{fr.ht} +<>= +\newcommand{\FactoredXmpTitle}{Factored} +\newcommand{\FactoredXmpNumber}{9.22} + +@ +\subsection{Factored} +\label{FactoredXmpPage} +\begin{itemize} +\item ugxFactoredDecompPage \ref{ugxFactoredDecompPage} on +page~pageref{ugxFactoredDecompPage} +\item ugxFactoredExpandPage \ref{ugxFactoredExpandPage} on +page~pageref{ugxFactoredExpandPage} +\item ugxFactoredArithPage \ref{ugxFactoredArithPage} on +page~pageref{ugxFactoredArithPage} +\item ugxFactoredNewPage \ref{ugxFactoredNewPage} on +page~pageref{ugxFactoredNewPage} +\item ugxFactoredVarPage \ref{ugxFactoredVarPage} on +page~pageref{ugxFactoredVarPage} +\end{itemize} +\index{pages!FactoredXmpPage!fr.ht} +\index{fr.ht!pages!FactoredXmpPage} +\index{FactoredXmpPage!fr.ht!pages} +<>= +\begin{page}{FactoredXmpPage}{Factored} +\beginscroll + +\spadtype{Factored} creates a domain whose objects are kept in factored +form as long as possible. +Thus certain operations like \spadopFrom{*}{Factored} +(multiplication) and \spadfunFrom{gcd}{Factored} are relatively +easy to do. +Others, such as addition, require somewhat more work, and +the result may not be completely factored +unless the argument domain \spad{R} provides a +\spadfunFrom{factor}{Factored} operation. +Each object consists of a unit and a list of factors, where each +factor consists of a member of \spad{R} (the {\em base}), an +exponent, and a flag indicating what is known about the base. +A flag may be one of \spad{"nil"}, \spad{"sqfr"}, \spad{"irred"} +or \spad{"prime"}, which mean that nothing is known about the +base, it is square-free, it is irreducible, or it is prime, +respectively. +The current restriction to factored objects of integral domains +allows simplification to be performed without worrying about +multiplication order. + +\beginmenu + \menudownlink{{9.22.1. Decomposing Factored Objects}} +{ugxFactoredDecompPage} + \menudownlink{{9.22.2. Expanding Factored Objects}} +{ugxFactoredExpandPage} + \menudownlink{{9.22.3. Arithmetic with Factored Objects}} +{ugxFactoredArithPage} + \menudownlink{{9.22.4. Creating New Factored Objects}} +{ugxFactoredNewPage} + \menudownlink{{9.22.5. Factored Objects with Variables}} +{ugxFactoredVarPage} +\endmenu +\endscroll +\autobuttons +\end{page} + +@ +<>= +\newcommand{\ugxFactoredDecompTitle}{Decomposing Factored Objects} +\newcommand{\ugxFactoredDecompNumber}{9.22.1.} + +@ +\subsection{Decomposing Factored Objects} +\label{ugxFactoredDecompPage} +\index{pages!ugxFactoredDecompPage!fr.ht} +\index{fr.ht!pages!ugxFactoredDecompPage} +\index{ugxFactoredDecompPage!fr.ht!pages} +<>= +\begin{page}{ugxFactoredDecompPage}{Decomposing Factored Objects} +\beginscroll + +\labelSpace{4pc} +\xtc{ +In this section we will work with a factored integer. +}{ +\spadpaste{g := factor(4312) \bound{g}} +} +\xtc{ +Let's begin by decomposing \spad{g} into pieces. +The only possible units for integers are \spad{1} and \spad{-1}. +}{ +\spadpaste{unit(g) \free{g}} +} +\xtc{ +There are three factors. +}{ +\spadpaste{numberOfFactors(g) \free{g}} +} +\xtc{ +We can make a list of the bases, \ldots +}{ +\spadpaste{[nthFactor(g,i) for i in 1..numberOfFactors(g)] \free{g}} +} +\xtc{ +and the exponents, \ldots +}{ +\spadpaste{[nthExponent(g,i) for i in 1..numberOfFactors(g)] \free{g}} +} +\xtc{ +and the flags. +You can see that all the bases (factors) are prime. +}{ +\spadpaste{[nthFlag(g,i) for i in 1..numberOfFactors(g)] \free{g}} +} +\xtc{ +A useful operation for pulling apart a factored object into a list +of records of the components is \spadfunFrom{factorList}{Factored}. +}{ +\spadpaste{factorList(g) \free{g}} +} +\xtc{ +If you don't care about the flags, use \spadfunFrom{factors}{Factored}. +}{ +\spadpaste{factors(g) \free{g}\bound{prev1}} +} +\xtc{ +Neither of these operations returns the unit. +}{ +\spadpaste{first(\%).factor \free{prev1}} +} + +\endscroll +\autobuttons +\end{page} + +@ +<>= +\newcommand{\ugxFactoredExpandTitle}{Expanding Factored Objects} +\newcommand{\ugxFactoredExpandNumber}{9.22.2.} + +@ +@ +\subsection{Expanding Factored Objects} +\label{ugxFactoredExpandPage} +\index{pages!ugxFactoredExpandPage!fr.ht} +\index{fr.ht!pages!ugxFactoredExpandPage} +\index{ugxFactoredExpandPage!fr.ht!pages} +<>= +\begin{page}{ugxFactoredExpandPage}{Expanding Factored Objects} +\beginscroll + +\labelSpace{4pc} +\xtc{ +Recall that we are working with this factored integer. +}{ +\spadpaste{g := factor(4312) \bound{g}} +} +\xtc{ +To multiply out the factors with their multiplicities, use +\spadfunFrom{expand}{Factored}. +}{ +\spadpaste{expand(g) \free{g}} +} +\xtc{ +If you would like, say, the distinct factors multiplied together +but with multiplicity one, you could do it this way. +}{ +\spadpaste{reduce(*,[t.factor for t in factors(g)]) \free{g}} +} + +\endscroll +\autobuttons +\end{page} + +@ +<>= +\newcommand{\ugxFactoredArithTitle}{Arithmetic with Factored Objects} +\newcommand{\ugxFactoredArithNumber}{9.22.3.} + +@ +\subsection{Arithmetic with Factored Objects} +\label{ugxFactoredArithPage} +\index{pages!ugxFactoredArithPage!fr.ht} +\index{fr.ht!pages!ugxFactoredArithPage} +\index{ugxFactoredArithPage!fr.ht!pages} +<>= +\begin{page}{ugxFactoredArithPage}{Arithmetic with Factored Objects} +\beginscroll + +\labelSpace{4pc} +\xtc{ +We're still working with this factored integer. +}{ +\spadpaste{g := factor(4312) \bound{g}} +} +\xtc{ +We'll also define this factored integer. +}{ +\spadpaste{f := factor(246960) \bound{f}} +} +\xtc{ +Operations involving multiplication and division are particularly +easy with factored objects. +}{ +\spadpaste{f * g \free{f g}} +} +\xtc{ +}{ +\spadpaste{f**500 \free{f}} +} +\xtc{ +}{ +\spadpaste{gcd(f,g) \free{f g}} +} +\xtc{ +}{ +\spadpaste{lcm(f,g) \free{f g}} +} +\xtc{ +If we use addition and subtraction things can slow down because +we may need to compute greatest common divisors. +}{ +\spadpaste{f + g \free{f g}} +} +\xtc{ +}{ +\spadpaste{f - g \free{f g}} +} +\xtc{ +Test for equality with \spad{0} and \spad{1} by using +\spadfunFrom{zero?}{Factored} and \spadfunFrom{one?}{Factored}, +respectively. +}{ +\spadpaste{zero?(factor(0))} +} +\xtc{ +}{ +\spadpaste{zero?(g) \free{g}} +} +\xtc{ +}{ +\spadpaste{one?(factor(1))} +} +\xtc{ +}{ +\spadpaste{one?(f) \free{f}} +} +\xtc{ +Another way to get the zero and one factored objects is to use +package calling +(see \downlink{``\ugTypesPkgCallTitle''}{ugTypesPkgCallPage} +in Section \ugTypesPkgCallNumber\ignore{ugTypesPkgCall}). +}{ +\spadpaste{0\$Factored(Integer)} +} +\xtc{ +}{ +\spadpaste{1\$Factored(Integer)} +} + +\endscroll +\autobuttons +\end{page} + +@ +<>= +\newcommand{\ugxFactoredNewTitle}{Creating New Factored Objects} +\newcommand{\ugxFactoredNewNumber}{9.22.4.} + +@ +\subsection{Creating New Factored Objects} +\label{ugxFactoredNewPage} +See FactoredFunctionsTwoXmpPage \ref{FactoredFunctionsTwoXmpPage} on +page~\pageref{FactoredFunctionsTwoXmpPage} +\index{pages!ugxFactoredNewPage!fr.ht} +\index{fr.ht!pages!ugxFactoredNewPage} +\index{ugxFactoredNewPage!fr.ht!pages} +<>= +\begin{page}{ugxFactoredNewPage}{Creating New Factored Objects} +\beginscroll + +The \spadfunFrom{map}{Factored} operation is used to iterate across the +unit and bases of a factored object. +See \downlink{`FactoredFunctions2'}{FactoredFunctionsTwoXmpPage} +\ignore{FactoredFunctions2} for a discussion of +\spadfunFrom{map}{Factored}. + +\labelSpace{1pc} +\xtc{ +The following four operations take a base and an exponent and create +a factored object. +They differ in handling the flag component. +}{ +\spadpaste{nilFactor(24,2) \bound{prev2}} +} +\xtc{ +This factor has no associated information. +}{ +\spadpaste{nthFlag(\%,1) \free{prev2}} +} +\xtc{ +This factor is asserted to be square-free. +}{ +\spadpaste{sqfrFactor(30,2) \bound{prev3}} +} +\xtc{ +This factor is asserted to be irreducible. +}{ +\spadpaste{irreducibleFactor(13,10) \bound{prev4}} +} +\xtc{ +This factor is asserted to be prime. +}{ +\spadpaste{primeFactor(11,5) \bound{prev5}} +} + +\xtc{ +A partial inverse to \spadfunFrom{factorList}{Factored} is +\spadfunFrom{makeFR}{Factored}. +}{ +\spadpaste{h := factor(-720) \bound{h}} +} +\xtc{ +The first argument is the unit and the second is a list of records as +returned by \spadfunFrom{factorList}{Factored}. +}{ +\spadpaste{h - makeFR(unit(h),factorList(h)) \free{h}} +} + +\endscroll +\autobuttons +\end{page} + +@ +<>= +\newcommand{\ugxFactoredVarTitle}{Factored Objects with Variables} +\newcommand{\ugxFactoredVarNumber}{9.22.5.} + +@ +\subsection{Factored Objects with Variables} +\label{ugxFactoredVarPage} +\index{pages!ugxFactoredVarPage!fr.ht} +\index{fr.ht!pages!ugxFactoredVarPage} +\index{ugxFactoredVarPage!fr.ht!pages} +<>= +\begin{page}{ugxFactoredVarPage}{Factored Objects with Variables} +% ===================================================================== +\beginscroll + +\labelSpace{4pc} +\xtc{ +Some of the operations available for polynomials are also available +for factored polynomials. +}{ +\spadpaste{p := (4*x*x-12*x+9)*y*y + (4*x*x-12*x+9)*y + 28*x*x - 84*x + 63 \bound{p}} +} +\xtc{ +}{ +\spadpaste{fp := factor(p) \free{p}\bound{fp}} +} +\xtc{ +You can differentiate with respect to a variable. +}{ +\spadpaste{D(p,x) \free{p}} +} +\xtc{ +}{ +\spadpaste{D(fp,x) \free{fp}\bound{prev1}} +} +\xtc{ +}{ +\spadpaste{numberOfFactors(\%) \free{prev1}} +} +\endscroll +\autobuttons +\end{page} + +@ +\section{fr2.ht} +<>= +\newcommand{\FactoredFunctionsTwoXmpTitle}{FactoredFunctions2} +\newcommand{\FactoredFunctionsTwoXmpNumber}{9.23} + +@ +\subsection{FactoredFunctions2} +\label{FactoredFunctionsTwoXmpPage} +\begin{itemize} +\item FactoredXmpPage \ref{FactoredXmpPage} on +page~pageref{FactoredXmpPage} +\item ugProblemGaloisPage \ref{ugProblemGaloisPage} on +page~pageref{ugProblemGaloisPage} +\end{itemize} +\index{pages!FactoredFunctionsTwoXmpPage!fr2.ht} +\index{fr2.ht!pages!FactoredFunctionsTwoXmpPage} +\index{FactoredFunctionsTwoXmpPage!fr2.ht!pages} +<>= +\begin{page}{FactoredFunctionsTwoXmpPage}{FactoredFunctions2} +\beginscroll + +The \spadtype{FactoredFunctions2} package implements one operation, +\spadfunFrom{map}{FactoredFunctions2}, for applying an operation to every +base in a factored object and to the unit. +\xtc{ +}{ +\spadpaste{double(x) == x + x \bound{double}} +} +\xtc{ +}{ +\spadpaste{f := factor(720) \bound{f}} +} +\xtc{ +Actually, the \spadfunFrom{map}{FactoredFunctions2} operation used +in this example comes from \spadtype{Factored} itself, since +\userfun{double} takes an integer argument and returns an integer +result. +}{ +\spadpaste{map(double,f) \free{f}\free{double}} +} +\xtc{ +If we want to use an operation that returns an object that has a type +different from the operation's argument, +the \spadfunFrom{map}{FactoredFunctions2} in \spadtype{Factored} +cannot be used and we use the one in \spadtype{FactoredFunctions2}. +}{ +\spadpaste{makePoly(b) == x + b \bound{makePoly}} +} +\xtc{ +In fact, the ``2'' in the name of the package means that we might +be using factored objects of two different types. +}{ +\spadpaste{g := map(makePoly,f) \free{f}\free{makePoly}\bound{g}} +} +It is important to note that both versions of +\spadfunFrom{map}{FactoredFunctions2} +destroy any information known about the bases (the fact that they are prime, +for instance). +\xtc{ +The flags for each base are set to ``nil'' in the object returned +by \spadfunFrom{map}{FactoredFunctions2}. +}{ +\spadpaste{nthFlag(g,1) \free{g}} +} + +For more information about factored objects and their use, see +\downlink{`Factored'}{FactoredXmpPage}\ignore{Factored} and +\downlink{``\ugProblemGaloisTitle''}{ugProblemGaloisPage} in +Section \ugProblemGaloisNumber\ignore{ugProblemGalois}. +\endscroll +\autobuttons +\end{page} + +@ +\section{frac.ht} +<>= +\newcommand{\FractionXmpTitle}{Fraction} +\newcommand{\FractionXmpNumber}{9.28} + +@ +\subsection{Fraction} +\label{FractionXmpPage} +\begin{itemize} +\item ContinuedFractionXmpPage \ref{ContinuedFractionXmpPage} on +page~pageref{ContinuedFractionXmpPage} +\item PartialFractionXmpPage \ref{PartialFractionXmpPage} on +page~pageref{PartialFractionXmpPage} +\end{itemize} +\index{pages!FractionXmpPage!frac.ht} +\index{frac.ht!pages!FractionXmpPage} +\index{FractionXmpPage!frac.ht!pages} +<>= +\begin{page}{FractionXmpPage}{Fraction} +\beginscroll + +The \spadtype{Fraction} domain implements quotients. +The elements must belong to a domain of category \spadtype{IntegralDomain}: +multiplication must be commutative and the product of two non-zero +elements must not be zero. +This allows you to make fractions of most things you would think of, +but don't expect to create a fraction of two matrices! +The abbreviation for \spadtype{Fraction} is \spadtype{FRAC}. + +\xtc{ +Use \spadopFrom{/}{Fraction} to create a fraction. +}{ +\spadpaste{a := 11/12 \bound{a}} +} +\xtc{ +}{ +\spadpaste{b := 23/24 \bound{b}} +} +\xtc{ +The standard arithmetic operations are available. +}{ +\spadpaste{3 - a*b**2 + a + b/a \free{a}\free{b}} +} +\xtc{ +Extract the numerator and denominator by using +\spadfunFrom{numer}{Fraction} and \spadfunFrom{denom}{Fraction}, +respectively. +}{ +\spadpaste{numer(a) \free{a}} +} +\xtc{ +}{ +\spadpaste{denom(b) \free{b}} +} +Operations like \spadfunFrom{max}{Fraction}, \spadfunFrom{min}{Fraction}, +\spadfunFrom{negative?}{Fraction}, \spadfunFrom{positive?}{Fraction} and +\spadfunFrom{zero?}{Fraction} are all available if they are provided for +the numerators and denominators. +See \downlink{`Integer'}{IntegerXmpPage}\ignore{Integer} for examples. + +Don't expect a useful answer from \spadfunFrom{factor}{Fraction}, +\spadfunFrom{gcd}{Fraction} or \spadfunFrom{lcm}{Fraction} if you apply +them to fractions. +\xtc{ +}{ +\spadpaste{r := (x**2 + 2*x + 1)/(x**2 - 2*x + 1) \bound{r}} +} +\xtc{ +Since all non-zero fractions are invertible, these operations have trivial +definitions. +}{ +\spadpaste{factor(r) \free{r}} +} +\xtc{ +Use \spadfunFrom{map}{Fraction} to apply \spadfunFrom{factor}{Fraction} to +the numerator and denominator, which is probably what you mean. +}{ +\spadpaste{map(factor,r) \free{r}} +} + +\xtc{ +Other forms of fractions are available. +Use \spadfun{continuedFraction} to create a continued fraction. +}{ +\spadpaste{continuedFraction(7/12)} +} +\xtc{ +Use \spadfun{partialFraction} to create a partial fraction. +See \downlink{`ContinuedFraction'}{ContinuedFractionXmpPage} +\ignore{ContinuedFraction} and +\downlink{`PartialFraction'}{PartialFractionXmpPage} +\ignore{PartialFraction} for +additional information and examples. +}{ +\spadpaste{partialFraction(7,12)} +} + +\xtc{ +Use conversion to create alternative views of fractions with objects +moved in and out of the numerator and denominator. +}{ +\spadpaste{g := 2/3 + 4/5*\%i \bound{g}} +} +\xtc{ +Conversion is discussed in detail in +\downlink{``\ugTypesConvertTitle''}{ugTypesConvertPage} +in Section \ugTypesConvertNumber\ignore{ugTypesConvert}. +}{ +\spadpaste{g :: FRAC COMPLEX INT \free{g}} +} +\endscroll +\autobuttons +\end{page} + +@ +\section{fparfrac.ht} +<>= +\newcommand{\FullPartialFractionExpansionXmpTitle} +{FullPartialFractionExpansion} +\newcommand{\FullPartialFractionExpansionXmpNumber}{9.29} + +@ +\subsection{FullPartialFractionExpansion} +\label{FullPartialFractionExpansionXmpPage} +See PartialFractionXmpPage \ref{PartialFractionXmpPage} on +page~\pageref{PartialFractionXmpPage} +\index{pages!FullPartialFractionExpansionXmpPage!fparfrac.ht} +\index{fparfrac.ht!pages!FullPartialFractionExpansionXmpPage} +\index{FullPartialFractionExpansionXmpPage!fparfrac.ht!pages} +<>= +\begin{page}{FullPartialFractionExpansionXmpPage}{FullPartialFractionExpansion} +\beginscroll + +The domain \spadtype{FullPartialFractionExpansion} implements +factor-free conversion of quotients to full partial fractions. + +\xtc{ +Our examples will all involve quotients of univariate polynomials +with rational number coefficients. +}{ +\spadpaste{Fx := FRAC UP(x, FRAC INT) \bound{Fx}} +} +\xtc{ +Here is a simple-looking rational function. +}{ +\spadpaste{f : Fx := 36 / (x**5-2*x**4-2*x**3+4*x**2+x-2) \bound{f}\free{Fx}} +} +\xtc{ +We use \spadfunFrom{fullPartialFraction}{FullPartialFractionExpansion} +to convert it to an object of type +\spadtype{FullPartialFractionExpansion}. +}{ +\spadpaste{g := fullPartialFraction f \bound{g}\free{f}} +} +\xtc{ +Use a coercion to change it back into a quotient. +}{ +\spadpaste{g :: Fx \free{g}} +} +\xtc{ +Full partial fractions differentiate faster than rational +functions. +}{ +\spadpaste{g5 := D(g, 5) \free{g}\bound{g5}} +} +\xtc{ +}{ +\spadpaste{f5 := D(f, 5) \free{f}\bound{f5}} +} +\xtc{ +We can check that the two forms represent the same function. +}{ +\spadpaste{g5::Fx - f5 \free{Fx g5 f5}} +} +\xtc{ +Here are some examples that are more complicated. +}{ +\spadpaste{f : Fx := (x**5 * (x-1)) / ((x**2 + x + 1)**2 * (x-2)**3) \free{Fx}\bound{f2}} +} +\xtc{ +}{ +\spadpaste{g := fullPartialFraction f \free{f2}\bound{g2}} +} +\xtc{ +}{ +\spadpaste{g :: Fx - f \free{f2 g2 Fx}} +} +\xtc{ +}{ +\spadpaste{f : Fx := (2*x**7-7*x**5+26*x**3+8*x) / (x**8-5*x**6+6*x**4+4*x**2-8) \free{Fx}\bound{f3}} +} +\xtc{ +}{ +\spadpaste{g := fullPartialFraction f \free{f3}\bound{g3}} +} +\xtc{ +}{ +\spadpaste{g :: Fx - f \free{f3 g3 Fx}} +} +\xtc{ +}{ +\spadpaste{f:Fx := x**3 / (x**21 + 2*x**20 + 4*x**19 + 7*x**18 + 10*x**17 + 17*x**16 + 22*x**15 + 30*x**14 + 36*x**13 + 40*x**12 + 47*x**11 + 46*x**10 + 49*x**9 + 43*x**8 + 38*x**7 + 32*x**6 + 23*x**5 + 19*x**4 + 10*x**3 + 7*x**2 + 2*x + 1)\free{Fx}\bound{f4}} +} +\xtc{ +}{ +\spadpaste{g := fullPartialFraction f \free{f4}\bound{g4}} +} +\xtc{ +This verification takes much longer than the conversion to +partial fractions. +}{ +\spadpaste{g :: Fx - f \free{f4 g4 Fx}} +} + +For more information, see the paper: +Bronstein, M and Salvy, B. +``Full Partial Fraction Decomposition of Rational Functions,'' +{\it Proceedings of ISSAC'93, Kiev}, ACM Press. +All see \downlink{`PartialFraction'}{PartialFractionXmpPage} +\ignore{PartialFraction} for standard partial fraction +decompositions. +\endscroll +\autobuttons +\end{page} + +@ +\section{function.ht} +\subsection{Functions in Axiom} +\label{FunctionPage} +\begin{itemize} +\item RationalFunctionPage \ref{RationalFunctionPage} on +page~\pageref{RationalFunctionPage} +\item AlgebraicFunctionPage \ref{AlgebraicFunctionPage} on +page~\pageref{AlgebraicFunctionPage} +\item ElementaryFunctionPage \ref{ElementaryFunctionPage} on +page~\pageref{ElementaryFunctionPage} +\item FunctionSimplificationPage \ref{FunctionSimplificationPage} on +page~\pageref{FunctionSimplificationPage} +\end{itemize} +\index{pages!FunctionPage!function.ht} +\index{function.ht!pages!FunctionPage} +\index{FunctionPage!function.ht!pages} +<>= +\begin{page}{FunctionPage}{Functions in Axiom} +% +In Axiom, a function is an expression in one or more variables. +(Think of it as a function of those variables). +You can also define a function by rules or use a built-in function +Axiom lets you convert expressions to compiled functions. +\beginscroll +\beginmenu +\menulink{Rational Functions}{RationatFunctionPage} \tab{22} +Quotients of polynomials. + +\menulink{Algebraic Functions}{AlgebraicFunctionPage} \tab{22} +Those defined by polynomial equations. + +\menulink{Elementary Functions}{ElementaryFunctionPage} \tab{22} +The elementary functions of calculus. + +\menulink{Simplification}{FunctionSimplificationPage} \tab{22} +How to simplify expressions. + +\menulink{Pattern Matching}{ugUserRulesPage} \tab{22} +How to use the pattern matcher. +\endmenu +\endscroll +\newline +Additional Topics: +\beginmenu + +\menulink{Operator Algebra}{OperatorXmpPage}\tab{22} +The operator algebra facility. + +\endmenu +\autobuttons \end{page} + +@ +\subsection{Rational Functions} +\label{RationatFunctionPage} +\index{pages!RationatFunctionPage!function.ht} +\index{function.ht!pages!RationatFunctionPage} +\index{RationatFunctionPage!function.ht!pages} +<>= +\begin{page}{RationatFunctionPage}{Rational Functions} +\beginscroll +To create a rational function, just compute the +quotient of two polynomials: +\spadpaste{f := (x - y) / (x + y)\bound{f}} +Use the functions \spadfun{numer} and \spadfun{denom}: +to recover the numerator and denominator of a fraction: +% +\spadpaste{numer f\free{f}} +\spadpaste{denom f\free{f}} +% +Since these are polynomials, you can apply all the +\downlink{polynomial operations}{PolynomialPage} +to them. +You can substitute values or +other rational functions for the variables using +the function \spadfun{eval}. The syntax for \spadfun{eval} is +similar to the one for polynomials: +\spadpaste{eval(f, x = 1/x)\free{f}} +\spadpaste{eval(f, [x = y, y = x])\free{f}} +\endscroll +\autobuttons +\end{page} + +@ +\subsection{Algebraic Functions} +\label{AlgebraicFunctionPage} +\index{pages!AlgebraicFunctionPage!function.ht} +\index{function.ht!pages!AlgebraicFunctionPage} +\index{AlgebraicFunctionPage!function.ht!pages} +<>= +\begin{page}{AlgebraicFunctionPage}{Algebraic Functions} +\beginscroll +Algebraic functions are functions defined by algebraic equations. There +are two ways of constructing them: using rational powers, or implicitly. +For rational powers, use \spadopFrom{**}{RadicalCategory} +(or the system functions \spadfun{sqrt} and +\spadfun{nthRoot} for square and nth roots): +\spadpaste{f := sqrt(1 + x ** (1/3))\bound{f}} +To define an algebraic function implicitly +use \spadfun{rootOf}. The following +line defines a function \spad{y} of \spad{x} satisfying the equation +\spad{y**3 = x*y - y**2 - x**3 + 1}: +\spadpaste{y := rootOf(y**3 + y**2 - x*y + x**3 - 1, y)\bound{y}} +You can manipulate, differentiate or integrate an implicitly defined +algebraic function like any other Axiom function: +\spadpaste{differentiate(y, x)\free{y}} +Higher powers of algebraic functions are automatically reduced during +calculations: +\spadpaste{(y + 1) ** 3\free{y}} +But denominators, are not automatically rationalized: +\spadpaste{g := inv f\bound{g}\free{y}} +Use \spadfun{ratDenom} to remove the algebraic quantities from the denominator: +\spadpaste{ratDenom g\free{g}} +\endscroll +\autobuttons \end{page} + +@ +\subsection{Elementary Functions} +\label{ElementaryFunctionPage} +\index{pages!ElementaryFunctionPage!function.ht} +\index{function.ht!pages!ElementaryFunctionPage} +\index{ElementaryFunctionPage!function.ht!pages} +<>= +\begin{page}{ElementaryFunctionPage}{Elementary Functions} +\beginscroll +Axiom has most of the usual functions from calculus built-in: +\spadpaste{f := x * log y * sin(1/(x+y))\bound{f}} +You can substitute values or another elementary functions for the variables +with the function \spadfun{eval}: +\spadpaste{eval(f, [x = y, y = x])\free{f}} +As you can see, the substitutions are made 'in parallel' as in the case +of polynomials. It's also possible to substitute expressions for kernels +instead of variables: +\spadpaste{eval(f, log y = acosh(x + sqrt y))\free{f}} +\endscroll +\autobuttons \end{page} + +@ +\subsection{Simplification} +\label{FunctionSimplificationPage} +See ugUserRulesPage \ref{ugUserRulesPage} on page~\pageref{ugUserRulesPage} +\index{pages!FunctionSimplificationPage!function.ht} +\index{function.ht!pages!FunctionSimplificationPage} +\index{FunctionSimplificationPage!function.ht!pages} +<>= +\begin{page}{FunctionSimplificationPage}{Simplification} +\beginscroll +Simplifying an expression often means different things at +different times, so Axiom offers a large number of +`simplification' functions. +The most common one, which performs the usual trigonometric +simplifications is \spadfun{simplify}: +\spadpaste{f := cos(x)/sec(x) * log(sin(x)**2/(cos(x)**2+sin(x)**2)) \bound{f}} +\spadpaste{g := simplify f\bound{g}\free{f}} +If the result of \spadfun{simplify} is not satisfactory, specific +transformations are available. +For example, to rewrite \spad{g} in terms of secants and +cosecants instead of sines and cosines, issue: +% +\spadpaste{h := sin2csc cos2sec g\bound{h}\free{g}} +% +To apply the logarithm simplification rules to \spad{h}, issue: +\spadpaste{expandLog h\free{h}} +Since the square root of \spad{x**2} is the absolute value of +\spad{x} and not \spad{x} itself, algebraic radicals are not +automatically simplified, but you can specifically request it by +calling \spadfun{rootSimp}: +% +\spadpaste{f1 := sqrt((x+1)**3)\bound{f1}} +\spadpaste{rootSimp f1\free{f1}} +% +There are other transformations which are sometimes useful. +Use the functions \spadfun{complexElementary} and \spadfun{trigs} +to go back and forth between the complex exponential and +trigonometric forms of an elementary function: +% +\spadpaste{g1 := sin(x + cos x)\bound{g1}} +\spadpaste{g2 := complexElementary g1\bound{g2}\free{g1}} +\spadpaste{trigs g2\free{g2}} +% +Similarly, the functions \spadfun{realElementary} and +\spadfun{htrigs} convert hyperbolic functions in and out of their +exponential form: +% +\spadpaste{h1 := sinh(x + cosh x)\bound{h1}} +\spadpaste{h2 := realElementary h1\bound{h2}\free{h1}} +\spadpaste{htrigs h2\free{h2}} +% +Axiom has other transformations, most of which +are in the packages +\spadtype{ElementaryFunctionStructurePackage}, +\spadtype{TrigonometricManipulations}, +\spadtype{AlgebraicManipulations}, +and \spadtype{TranscendentalManipulations}. +If you need to apply a simplification rule not built into the +system, you can use Axiom's \downlink{pattern +matcher}{ugUserRulesPage}. +\endscroll +\autobuttons +\end{page} + +@ +\section{gbf.ht} +<>= +\newcommand{\GroebnerFactorizationPackageXmpTitle}{GroebnerFactorizationPackage} +\newcommand{\GroebnerFactorizationPackageXmpNumber}{9.31} + +@ +\subsection{GroebnerFactorizationPackage} +\label{GroebnerFactorizationPackageXmpPage} +\index{pages!GroebnerFactorizationPackageXmpPage!gbf.ht} +\index{gbf.ht!pages!GroebnerFactorizationPackageXmpPage} +\index{GroebnerFactorizationPackageXmpPage!gbf.ht!pages} +<>= +\begin{page}{GroebnerFactorizationPackageXmpPage}{GroebnerFactorizationPackage} +% ===================================================================== +\beginscroll +%% J. Grabmeier 26 04 91 +Solving systems of polynomial equations with the +\texht{Gr\"{o}bner}{Groebner} +basis algorithm can often be very time consuming because, in general, +the algorithm has exponential run-time. +These systems, which often come from concrete applications, +frequently have symmetries which are not taken advantage of by the +algorithm. +However, it often happens in this case +that the polynomials which occur during the +\texht{Gr\"{o}bner}{Groebner} calculations are reducible. +Since Axiom has an excellent polynomial factorization algorithm, it is +very natural to combine the +\texht{Gr\"{o}bner}{Groebner} and factorization algorithms. + +\spadtype{GroebnerFactorizationPackage} exports the +\axiomFunFrom{groebnerFactorize}{GroebnerFactorizationPackage} +operation which implements a modified +\texht{Gr\"{o}bner}{Groebner} basis algorithm. +In this algorithm, each polynomial that is to be put into the +partial list of the basis is first factored. +The remaining calculation is split into as many parts as there are +irreducible factors. +Call these factors \texht{$p_1, \ldots,p_n.$}{\spad{p1, ..., pn.}} +In the branches corresponding to \texht{$p_2, +\ldots,p_n,$}{\spad{p2, ..., pn,}} the factor +\texht{$p_1$}{\spad{p1}} can be divided out, and so on. +This package also contains operations that allow you to specify the +polynomials that are not zero on the common roots of the final +\texht{Gr\"{o}bner}{Groebner} basis. + +Here is an example from chemistry. +In a theoretical model of the cyclohexan +\texht{${\rm C}_6{\rm H}_{12}$}{C6H12}, +the six carbon atoms each sit in the center of gravity of a +tetrahedron that has two hydrogen atoms and two carbon atoms at its +corners. +We first normalize and set the length of each edge to 1. +Hence, the distances of one fixed carbon atom to each of its +immediate neighbours is 1. +We will denote the distances to the other three carbon atoms by +\smath{x}, \smath{y} and \smath{z}. + +\texht{A.~Dress}{A. Dress} developed a theory to decide whether a set of points +%% reference? +and distances between them can be realized in an \smath{n}-dimensional space. +Here, of course, we have \smath{n = 3}. +\xtc{ +}{ +\spadpaste{mfzn : SQMATRIX(6,DMP([x,y,z],Fraction INT)) := [[0,1,1,1,1,1], [1,0,1,8/3,x,8/3], [1,1,0,1,8/3,y], [1,8/3,1,0,1,8/3], [1,x,8/3,1,0,1], [1,8/3,y,8/3,1,0]] \bound{mfzn}} +} +\xtc{ +For the cyclohexan, the distances have to satisfy this equation. +}{ +\spadpaste{eq := determinant mfzn \free{mfzn}\bound{eq}} +} +\xtc{ +They also must satisfy the equations +given by cyclic shifts of the indeterminates. +}{ +\spadpaste{groebnerFactorize [eq, eval(eq, [x,y,z], [y,z,x]), eval(eq, [x,y,z], [z,x,y])] \free{eq}} +} + +The union of the solutions of this list is the +solution of our original problem. +If we impose positivity conditions, we get two relevant ideals. +One ideal is +zero-dimensional, namely \smath{x = y = z = 11/3}, and this +determines the ``boat'' form of the cyclohexan. +The other ideal is one-dimensional, +which means that we have a solution space given by one parameter. +This gives the ``chair'' form of the cyclohexan. +The parameter describes the angle of the ``back of the chair.'' + +\axiomFunFrom{groebnerFactorize}{GroebnerFactorizationPackage} +has an optional \axiomType{Boolean}-valued second argument. +When it is \spad{true} partial results are displayed, +since it may happen that the +calculation does not terminate in a reasonable time. +See the source code for \spadtype{GroebnerFactorizationPackage} +in {\bf groebf\spadFileExt{}} for more details about the algorithms +used. +\endscroll +\autobuttons +\end{page} + +@ +\section{gloss.ht} +\subsection{Glossary} +\label{GlossaryPage} +\index{pages!GlossaryPage!gloss.ht} +\index{gloss.ht!pages!GlossaryPage} +\index{GlossaryPage!gloss.ht!pages} +<>= +\begin{page}{GlossaryPage}{G l o s s a r y} +\beginscroll\beginmenu\item\newline{\em \menuitemstyle{}}{\em !}\space{} +{\em (syntax)} Suffix character for \spadgloss{destructive operations}. +\item\newline{\em \menuitemstyle{}}{\em ,}\space{} +{\em (syntax)} a separator for items in a \spadgloss{tuple},{} +\spadignore{e.g.} to separate arguments of a function \spad{f(x,{}y)}. +\item\newline{\em \menuitemstyle{}}{\em =>}\space{} +{\em (syntax)} the expression \spad{a => b} is equivalent to +\spad{if a then} \spadgloss{exit} \spad{b}. +\item\newline{\em \menuitemstyle{}}{\em ?}\space{} +1. {\em (syntax)} a suffix character for Boolean-valued \spadfun{function} +names,{} \spadignore{e.g.} \spadfun{odd?}. 2. Suffix character for pattern +variables. 3. The special type \spad{?} means {\em don't care}. For +example,{} the declaration \newline\center{\spad{x : Polynomial ?}}\newline, +{} means that values assigned to \spad{x} must be polynomials over an +arbitrary \spadgloss{underlying domain}. +\item\newline{\em \menuitemstyle{}}{\em abstract datatype}\space{} +a programming language principle used in Axiom where a datatype +is defined in two parts: (1) a {\em public} part describing a set of +\spadgloss{exports},{} principally operations that apply to objects of +that type,{} and (2) a {\em private} part describing the implementation +of the datatype usually in terms of a \spadgloss{representation} for +objects of the type. Programs which create and otherwise manipulate +objects of the type may only do so through its exports. The representation +and other implementation information is specifically hidden. +\item\newline{\em \menuitemstyle{}}{\em abstraction}\space{} +described functionally or conceptually without regard to implementation +\item\newline{\em \menuitemstyle{}}{\em accuracy}\space{} +the degree of exactness of an approximation or measurement. In computer +algebra systems,{} computations are typically carried out with complete +accuracy using integers or rational numbers of indefinite size. Domain +\spadtype{Float} provides a function \spadfunFrom{precision}{Float} to +change the precision for floating point computations. Computations using +\spadtype{DoubleFloat} have a fixed precision but uncertain accuracy. +\item\newline{\em \menuitemstyle{}}{\em add-chain}\space{} +a hierarchy formed by \spadgloss{domain extensions}. If domain \spad{A} +extends domain \spad{B} and domain \spad{B} extends domain \spad{C},{} +then \spad{A} has {\em add-chain} \spad{B}-\spad{C}. +\item\newline{\em \menuitemstyle{}}{\em aggregate}\space{} +a data structure designed to hold multiple values. Examples of aggregates +are \spadtype{List},{} \spadtype{Set},{} \spadtype{Matrix} and \spadtype{Bits}. +\item\newline{\em \menuitemstyle{}}{\em AKCL}\space{} +Austin Kyoto Common LISP,{} a version of \spadgloss{\spad{KCL}} produced +by William Schelter,{} Austin,{} Texas. +\item\newline{\em \menuitemstyle{}}{\em algorithm}\space{} +a step-by-step procedure for a solution of a problem; a program +\item\newline{\em \menuitemstyle{}}{\em ancestor}\space{} +(of a domain) a category which is a \spadgloss{parent} of the domain,{} +or a \spadgloss{parent} of a \spadgloss{parent},{} and so on. +\item\newline{\em \menuitemstyle{}}{\em application}\space{} +{\em (syntax)} an expression denoting "application" of a function to a set +of \spadgloss{argument} parameters. Applications are written as a +\spadgloss{parameterized form}. For example,{} the form \spad{f(x,{}y)} +indicates the "application of the function \spad{f} to the tuple of +arguments \spad{x} and \spad{y}". See also \spadgloss{evaluation} and +\spadgloss{invocation}. +\item\newline{\em \menuitemstyle{}}{\em apply}\space{} +See \spadgloss{application}. +\item\newline{\em \menuitemstyle{}}{\em argument}\space{} +1. (actual argument) a value passed to a function at the time of a +\spadglossSee{function call}{application}; also called an {\em actual +parameter}. 2. (formal argument) a variable used in the definition of +a function to denote the actual argument passed when the function is called. +\item\newline{\em \menuitemstyle{}}{\em arity}\space{} +1. (function) the number of arguments. 2. (operator or operation) +corresponds to the arity of a function implementing the operator or operation. +\item\newline{\em \menuitemstyle{}}{\em assignment}\space{} +{\em (syntax)} an expression of the form \spad{x := e},{} meaning +"assign the value of \spad{e} to \spad{x"}. After \spadgloss{evaluation},{} +the \spadgloss{variable} \spad{x} \spadglossSee{points}{pointer} to an +object obtained by evaluating the expression \spad{e}. If \spad{x} has +a \spadgloss{type} as a result of a previous \spadgloss{declaration},{} +the object assigned to \spad{x} must have that type. An interpreter must +often \spadglossSee{coerce}{coercion} the value of \spad{e} to make that +happen. For example,{} in the interpreter,{} \center{\spad{x : Float := 11}} +first \spadglossSee{declares}{declaration} \spad{x} to be a float. This +declaration causes the interpreter to coerce 11 to 11.0 in order to assign +a floating point value to \spad{x}. +\item\newline{\em \menuitemstyle{}}{\em attribute}\space{} +a name or functional form denoting {\em any} useful computational property. +For example,{} \spadatt{commutative(\spad{"*"})} asserts that "\spadfun{*} +is commutative". Also,{} \spadatt{finiteAggregate} is used to assert that +an aggregate has a finite number of immediate components. +\item\newline{\em \menuitemstyle{}}{\em basis}\space{} +{\em (algebra)} \spad{S} is a basis of a module \spad{M} over a +\spadgloss{ring} if \spad{S} generates \spad{M},{} and \spad{S} is +linearly independent +\item\newline{\em \menuitemstyle{}}{\em benefactor}\space{} +(of a given domain) a domain or package that the given domain explicitly +references (for example,{} calls functions from) in its implementation +\item\newline{\em \menuitemstyle{}}{\em binary}\space{} +operation or function with \spadgloss{arity} 2 +\item\newline{\em \menuitemstyle{}}{\em binding}\space{} +the association of a variable with properties such as \spadgloss{value} +and \spadgloss{type}. The top-level \spadgloss{environment} in the +interpreter consists of bindings for all user variables and functions. +Every \spadgloss{function} has an associated set of bindings,{} one for +each formal \spadgloss{argument} and \spadgloss{local variable}. +\item\newline{\em \menuitemstyle{}}{\em block}\space{} +{\em (syntax)} a control structure where expressions are sequentially +\spadglossSee{evaluated}{evaluation}. +\item\newline{\em \menuitemstyle{}}{\em body}\space{} +a \spadgloss{function body} or \spadgloss{loop body}. +\item\newline{\em \menuitemstyle{}}{\em boolean}\space{} +objects denoted by the \spadgloss{literals} \spad{true} and \spad{false}; +elements of domain \spadtype{Boolean}. See also \spadtype{Bits}. +\item\newline{\em \menuitemstyle{}}{\em built-in function}\space{} +a \spadgloss{function} in the standard Axiom library. Contrast +\spadgloss{user function}. +\item\newline{\em \menuitemstyle{}}{\em cache}\space{} +1. (noun) a mechanism for immediate retrieval of previously computed data. +For example,{} a function which does a lengthy computation might store its +values in a \spadgloss{hash table} using argument as a key. The hash table +then serves as a cache for the function (see also \spadsyscom{)set function +cache}). Also,{} when \spadgloss{recurrence relations} which depend upon +\spad{n} previous values are compiled,{} the previous \spad{n} values are +normally cached (use \spadsyscom{)set functions recurrence} to change this). +2. (verb) to save values in a cache. +\item\newline{\em \menuitemstyle{}}{\em capsule}\space{} +the part of the \spadglossSee{body}{function body} of a +\spadgloss{domain constructor} that defines the functions implemented by +the constructor. +\item\newline{\em \menuitemstyle{}}{\em case}\space{} +{\em (syntax)} an operator used to conditionally evaluate code based on +the branch of a \spadgloss{Union}. For example,{} if value \spad{u} is +\spad{Union(Integer,{}"failed")},{} the conditional expression \spad{if u +case Integer then A else B} evaluate \spad{A} if \spad{u} is an integer +and \spad{B} otherwise. +\item\newline{\em \menuitemstyle{}}{\em Category}\space{} +the distinguished object denoting the type of a category; the class of +all categories. +\item\newline{\em \menuitemstyle{}}{\em category}\space{} +{\em (basic concept)} second-order types which serve to define useful +"classification worlds" for domains,{} such as algebraic constructs +(\spadignore{e.g.} groups,{} rings,{} fields),{} and data structures +(\spadignore{e.g.} homogeneous aggregates,{} collections,{} dictionaries). +Examples of categories are \spadtype{Ring} ("the class of all rings") and +\spadtype{Aggregate} ("the class of all aggregates"). The categories of a +given world are arranged in a hierarchy (formally,{} a directed acyclic +graph). Each category inherits the properties of all its ancestors. Thus,{} +for example,{} the category of ordered rings (\spadtype{OrderedRing}) +inherits the properties of the category of rings (\spadtype{Ring}) and +those of the ordered sets (\spadtype{OrderedSet}). Categories provide a +database of algebraic knowledge and ensure mathematical correctness,{} +\spadignore{e.g.} that "matrices of polynomials" is correct but "polynomials +of hash tables" is not,{} that the multiply operation for "polynomials of +continued fractions" is commutative,{} but that for "matrices of power +series" is not. optionally provide "default definitions" for operations +they export. Categories are defined in Axiom by functions called +\spadgloss{category constructors}. Technically,{} a category designates +a class of domains with common \spadgloss{operations} and +\spadgloss{attributes} but usually with different \spadgloss{functions} +and \spadgloss{representations} for its constituent \spadgloss{objects}. +Categories are always defined using the Axiom library language +(see also \spadgloss{category extension}). See also file {\em catdef.spad} +for definitions of basic algebraic categories in Axiom. +\item\newline{\em \menuitemstyle{}}{\em category constructor}\space{} +a function that creates categories,{} described by an abstract datatype in +the Axiom programming language. For example,{} the category +constructor \spadtype{Module} is a function which takes a domain parameter +\spad{R} and creates the category "modules over \spad{R}". +\item\newline{\em \menuitemstyle{}}{\em category extension}\space{} +created by a category definition,{} an expression usually of the form +\spad{A == B with ...}. In English,{} this means "category A is a \spad{B} +with the new operations and attributes as given by ... . See,{} for +example,{} file {\em catdef.spad} for a definitions of the algebra +categories in Axiom,{} {\em aggcat.spad} for data structure categories. +\item\newline{\em \menuitemstyle{}}{\em category hierarchy}\space{} +hierarchy formed by category extensions. The root category is +\spadtype{Object}. A category can be defined as a \spadgloss{Join} of two +or more categories so as to have multiple \spadgloss{parents}. Categories +may also have parameterized so as to allow conditional inheritance. +\item\newline{\em \menuitemstyle{}}{\em character}\space{} +1. an element of a character set,{} as represented by a keyboard key. 2. +a component of a string. For example,{} the 0th element of the string +\spad{"hello there"} is the character {\em h}. +\item\newline{\em \menuitemstyle{}}{\em client}\space{} +(of a given domain) any domain or package that explicitly calls functions +from the given domain +\item\newline{\em \menuitemstyle{}}{\em coercion}\space{} +an automatic transformation of an object of one \spadgloss{type} to an +object of a similar or desired target type. In the interpreter,{} coercions +and \spadgloss{retractions} are done automatically by the interpreter when +a type mismatch occurs. Compare \spadgloss{conversion}. +\item\newline{\em \menuitemstyle{}}{\em comment}\space{} +textual remarks imbedded in code. Comments are preceded by a double dash +({\em --}). For Axiom library code,{} stylized comments for on-line +documentation are preceded by a two plus signs ({\em ++}). +\item\newline{\em \menuitemstyle{}}{\em Common LISP}\space{} +A version of \spadgloss{LISP} adopted as an informal standard by major +users and suppliers of LISP +\item\newline{\em \menuitemstyle{}}{\em compile-time}\space{} +the time when category or domain constructors are compiled. Contrast +\spadgloss{run-time}. +\item\newline{\em \menuitemstyle{}}{\em compiler}\space{} +a program that generates low-level code from a higher-level source +language. Axiom has three compilers. A {\em graphics compiler} +converts graphical formulas to a compiled subroutine so that points can +be rapidly produced for graphics commands. An {\em interpreter compiler} +optionally compiles \spadgloss{user functions} when first +\spadglossSee{invoked}{invocation} (use \spadsyscom{)set functions compile} +to turn this feature on). A {\em library compiler} compiles all constructors +(not available in Release 1) +\item\newline{\em \menuitemstyle{}}{\em computational object}\space{} +In Axiom,{} domains are objects. This term is used to distinquish +the objects which are members of domains rather than domains themselves. +\item\newline{\em \menuitemstyle{}}{\em conditional}\space{} +a \spadgloss{control structure} of the form \spad{if A then B else C}; +The \spadgloss{evaluation} of \spad{A} produces \spad{true} or \spad{false}. +If \spad{true},{} \spad{B} evaluates to produce a value; otherwise \spad{C} +evaluates to produce a value. When the value is not used,{} \spad{else C} +part can be omitted. +\item\newline{\em \menuitemstyle{}}{\em constant}\space{} +{\em (syntax)} a reserved word used in \spadgloss{signatures} in Axiom +programming language to signify that mark an operation always returns the +same value. For example,{} the signature \spad{0: constant -> \$} in the +source code of \spadtype{AbelianMonoid} tells the Axiom compiler that +\spad{0} is a constant so that suitable optimizations might be performed. +\item\newline{\em \menuitemstyle{}}{\em constructor}\space{} +a \spadgloss{function} which creates a \spadgloss{category},{} +\spadgloss{domain},{} or \spadgloss{package}. +\item\newline{\em \menuitemstyle{}}{\em continuation}\space{} +when a line of a program is so long that it must be broken into several +lines,{} then all but the first line are called {\em continuation lines}. +If such a line is given interactively,{} then each incomplete line must +end with an underscore. +\item\newline{\em \menuitemstyle{}}{\em control structure}\space{} +program structures which can specify a departure from normal sequential +execution. Axiom has four kinds of control structures: +\spadgloss{blocks},{} \spadgloss{case} statements,{} +\spadgloss{conditionals},{} and \spadgloss{loops}. +\item\newline{\em \menuitemstyle{}}{\em conversion}\space{} +the transformation of an object on one \spadgloss{type} to one of another +type. Conversions performed automatically are called \spadgloss{coercions}. +These happen when the interpreter has a type mismatch and a similar or +declared target type is needed. In general,{} the user must use the +infix operation \spadSyntax{\spad{::}} to cause this transformation. +\item\newline{\em \menuitemstyle{}}{\em copying semantics}\space{} +the programming language semantics used in Pascal but {\em not} in +Axiom. See also \spadgloss{pointer semantics} for details. +\item\newline{\em \menuitemstyle{}}{\em data structure}\space{} +a structure for storing data in the computer. Examples are +\spadgloss{lists} and \spadgloss{hash tables}. +\item\newline{\em \menuitemstyle{}}{\em datatype}\space{} +equivalent to \spadgloss{domain} in Axiom. +\item\newline{\em \menuitemstyle{}}{\em declaration}\space{} +{\em (syntax)} an expression of the form \spad{x : T} where \spad{T} is +some \spad{type}. A declaration forces all values +\spadglossSee{assigned}{assignment} to \spad{T} to be of that type. If a +value is of a different type,{} the interpreter will try to +\spadglossSee{coerce}{coercion} the value to type \spad{T}. Declarations +are necessary in case of ambiguity or when a user wants to introduce an +an \spadglossSee{unexposed}{expose} domain. +\item\newline{\em \menuitemstyle{}}{\em default definition}\space{} +a function defined by a \spadgloss{category}. Such definitions appear +category definitions of the form \spad{C: Category == T add I} in an +optional implmentation part \spad{I} to the right of the keyword \spad{add}. +\item\newline{\em \menuitemstyle{}}{\em default package}\space{} +a optional \spadgloss{package} of \spadgloss{functions} associated with +a category. Such functions are necessarily defined in terms over other +functions exported by the category. +\item\newline{\em \menuitemstyle{}}{\em definition}\space{} +{\em (syntax)} 1. An expression of the form \spad{f(a) == b} defining +function \spad{f} with \spadgloss{formal arguments} \spad{a} and +\spadgloss{body} \spad{b}; equivalent to the statement +\spad{f == (a) +-> b}. 2. An expression of the form \spad{a == b} +where \spad{a} is a \spadgloss{symbol},{} equivalent to \spad{a() == b}. +See also \spadgloss{macro} where a similar substitution is done at +\spadgloss{parse} time. +\item\newline{\em \menuitemstyle{}}{\em delimiter}\space{} +a \spadgloss{character} which marks the beginning or end of some +syntactically correct unit in the language,{} \spadignore{e.g.} +\spadSyntax{"} for strings,{} blanks for identifiers. +\item\newline{\em \menuitemstyle{}}{\em destructive operation}\space{} +An operation which changes a component or structure of a value. In +Axiom,{} all destructive operations have names which end with an +exclamation mark ({\em !}). For example,{} domain \spadtype{List} has +two operations to reverse the elements of a list,{} one named +\spadfunFrom{reverse}{List} which returns a copy of the original list with +the elements reversed,{} another named \spadfunFrom{reverse!}{List} which +reverses the elements {\em in place} thus destructively changing the +original list. +\item\newline{\em \menuitemstyle{}}{\em documentation}\space{} +1. on-line or hard copy descriptions of Axiom; 2. text in library +code preceded by {\em ++} comments as opposed to general comments +preceded by {\em --}. +\item\newline{\em \menuitemstyle{}}{\em domain}\space{} +{\em (basic concept)} a domain corresponds to the usual notion of +abstract datatypes: that of a set of values and a set of "exported +operations" for the creation and manipulation of these values. Datatypes +are parameterized,{} dynamically constructed,{} and can combine with others +in any meaningful way,{} \spadignore{e.g.} "lists of floats" +(\spadtype{List Float}),{} "fractions of polynomials with integer +coefficients" (\spadtype{Fraction Polynomial Integer}),{} "matrices +of infinite \spadgloss{streams} of cardinal numbers" +(\spadtype{Matrix Stream CardinalNumber}). The term {\em domain} is +actually abbreviates {\em domain of computation}. Technically,{} a +domain denotes a class of objects,{} a class of \spadgloss{operations} +for creating and other manipulating these objects,{} and a class of +\spadgloss{attributes} describing computationally useful properties. +Domains also provide \spadgloss{functions} for each operation often in +terms of some \spadgloss{representation} for the objects. A domain +itself is an \spadgloss{object} created by a \spadgloss{function} +called a \spadgloss{domain constructor}. +\item\newline{\em \menuitemstyle{}}{\em domain constructor}\space{} +a function that creates domains,{} described by an abstract datatype +in the Axiom programming language. Simple domains like +\spadtype{Integer} and \spadtype{Boolean} are created by domain +constructors with no arguments. Most domain constructors take one or +more parameters,{} one usually denoting an \spadgloss{underlying domain}. +For example,{} the domain \spadtype{Matrix(R)} denotes "matrices over +\spad{R"}. Domains {\em Mapping},{} {\em Record},{} and {\em Union} are +primitive domains. All other domains are written in the Axiom +programming language and can be modified by users with access to the +library source code. +\item\newline{\em \menuitemstyle{}}{\em domain extension}\space{} +a domain constructor \spad{A} is said to {\em extend} a domain +constructor \spad{B} if \spad{A}\spad{'s} definition has the form +\spad{A == B add ...}. This intuitively means "functions not defined +by \spad{A} are assumed to come from \spad{B}". Successive domain +extensions form \spadgloss{add-chains} affecting the the +\spadglossSee{search order}{lineage} for functions not implemented +directly by the domain during \spadgloss{dynamic lookup}. +\item\newline{\em \menuitemstyle{}}{\em dot notation}\space{} +using an infix dot ({\em .}) for function application. If \spad{u} is +the list \spad{[7,{}4,{}-11]} then both \spad{u(2)} and \spad{u.2} return +4. Dot notation nests to left. Thus \spad{f . g . h} is equivalent to +\spad{(f . g) . h}. +\item\newline{\em \menuitemstyle{}}{\em dynamic}\space{} +that which is done at \spadgloss{run-time} as opposed to +\spadgloss{compile-time}. For example,{} the interpreter will build the +domain "matrices over integers" dynamically in response to user input. +However,{} the compilation of all functions for matrices and integers is +done during \spadgloss{compile-time}. Constrast \spadgloss{static}. +\item\newline{\em \menuitemstyle{}}{\em dynamic lookup}\space{} +In Axiom,{} a \spadgloss{domain} may or may not explicitly provide +\spadgloss{function} definitions for all of its exported +\spadgloss{operations}. These definitions may instead come from domains +in the \spadgloss{add-chain} or from \spadgloss{default packages}. When +a \spadglossSee{function call}{application} is made for an operation in +the domain,{} up to five steps are carried out.\begin{items} \item (1) +If the domain itself implements a function for the operation,{} that +function is returned. \item (2) Each of the domains in the +\spadgloss{add-chain} are searched for one which implements the function; +if found,{} the function is returned. \item (3) Each of the +\spadgloss{default packages} for the domain are searched in order of +the \spadgloss{lineage}. If any of the default packages implements the +function,{} the first one found is returned. \item (4) Each of the +\spadgloss{default packages} for each of the domains in the +\spadgloss{add-chain} are searched in the order of their +\spadgloss{lineage}. If any of the default packages implements the +function,{} the first one found is returned. \item (5) If all of the +above steps fail,{} an error message is reported. \end{items} +\item\newline{\em \menuitemstyle{}}{\em empty}\space{} +the unique value of objects with type \spadtype{Void}. +\item\newline{\em \menuitemstyle{}}{\em environment}\space{} +a set of \spadgloss{bindings}. +\item\newline{\em \menuitemstyle{}}{\em evaluation}\space{} +a systematic process which transforms an \spadgloss{expression} into an +object called the \spadgloss{value} of the expression. Evaluation may +produce \spadgloss{side effects}. +\item\newline{\em \menuitemstyle{}}{\em exit}\space{} +{\em (reserved word)} an \spadgloss{operator} which forces an exit from +the current block. For example,{} the \spadgloss{block} \spad{(a := 1; +if i > 0 then exit a; a := 2)} will prematurely exit at the second +statement with value 1 if the value of \spad{i} is greater than 0. +See \spadgloss{\spad{=>}} for an alternate syntax. +\item\newline{\em \menuitemstyle{}}{\em explicit export}\space{} +1. (of a domain \spad{D}) any \spadgloss{attribute},{} +\spadgloss{operation},{} or \spadgloss{category} explicitly mentioned +in the \spadgloss{type} specification part \spad{T} for the domain +constructor definition \spad{D: T == I} 2. (of a category \spad{C}) +any \spadgloss{attribute},{} \spadgloss{operation},{} or +\spadgloss{category} explicitly mentioned in the \spadgloss{type} +specification part \spad{T} for the domain constructor definition +\spad{C: \spadgloss{Category} == T} +\item\newline{\em \menuitemstyle{}}{\em export}\space{} +\spadgloss{explicit export} or \spadgloss{implicit export} of a domain +or category +\item\newline{\em \menuitemstyle{}}{\em expose}\space{} +some constructors are {\em exposed},{} others {\em unexposed}. Exposed +domains and packages are recognized by the interpreter. Use \spadsys{)set +expose} to control change what is exposed. To see both exposed and +unexposed constructors,{} use \Browse{} with give the system command +\spadsyscom{)set hyperdoc browse exposure on}. Unexposed constructors +will now appear prefixed by star(\spad{*}). +\item\newline{\em \menuitemstyle{}}{\em expression}\space{} +1. any syntactically correct program fragment. 2. an element of domain +\spadtype{Expression} +\item\newline{\em \menuitemstyle{}}{\em extend}\space{} +see \spadgloss{category extension} or \spadgloss{domain extension} +\item\newline{\em \menuitemstyle{}}{\em field}\space{} +{\em (algebra)} a \spadgloss{domain} which is \spadgloss{ring} where +every non-zero element is invertible and where \spad{xy=yx}; a member +of category \spadtype{Field}. For a complete list of fields,{} click +on {\em Domains} under {\em Cross Reference} for \spadtype{Field}. +\item\newline{\em \menuitemstyle{}}{\em file}\space{} +a program or collection of data stored on disk,{} tape or other medium. +\item\newline{\em \menuitemstyle{}}{\em float}\space{} +a floating-point number with user-specified precision; an element of +domain \spadtype{Float}. Floats are \spadgloss{literals} which are written +two ways: without an exponent (\spadignore{e.g.} \spad{3.1416}),{} or with +an exponent (\spadignore{e.g.} \spad{3.12E-12}). Use function +\spadgloss{precision} to change the precision of the mantissage +(20 digits by default). See also \spadgloss{small float}. +\item\newline{\em \menuitemstyle{}}{\em formal parameter}\space{} +(of a function) an identifier \spadglossSee{bound}{binding} to the value +of an actual \spadgloss{argument} on \spadgloss{invocation}. In the +function definition \spad{f(x,{}y) == u},{} for example,{} \spad{x} +and \spad{y} are the formal parameter. +\item\newline{\em \menuitemstyle{}}{\em frame}\space{} +the basic unit of an interactive session; each frame has its own +\spadgloss{step number},{} \spadgloss{environment},{} and +\spadgloss{history}. In one interactive session,{} users can create +and drop frames,{} and have several active frames simultaneously. +\item\newline{\em \menuitemstyle{}}{\em free}\space{} +{\em (syntax)} A keyword used in user-defined functions to declare +that a variable is a \spadgloss{free variable} of that function. For +example,{} the statement \spad{free x} declares the variable \spad{x} +within the body of a function \spad{f} to be a free variable in \spad{f}. +Without such a declaration,{} any variable \spad{x} which appears on the +left hand side of an assignment is regarded as a \spadgloss{local variable} +of that function. If the intention of the assignment is to give an value +to a \spadgloss{global variable} \spad{x},{} the body of that function +must contain the statement \spad{free x}. +\item\newline{\em \menuitemstyle{}}{\em free variable}\space{} +(of a function) a variable which appears in a body of a function but is +not \spadglossSee{bound}{binding} by that function. See \spadgloss{local +variable} by default. +\item\newline{\em \menuitemstyle{}}{\em function}\space{} +implementation of \spadgloss{operation}; it takes zero or more +\spadgloss{argument} parameters and produces zero or more values. +Functions are objects which can be passed as parameters to functions +and can be returned as values of functions. Functions can also create +other functions (see also \spadtype{InputForm}). See also +\spadgloss{application} and \spadgloss{invocation}. The terms +{\em operation} and {\em function} are distinct notions in Axiom. +An operation is an abstraction of a function,{} described by declaring a +\spadgloss{signature}. A function is created by providing an implementation +of that operation by some piece of Axiom code. Consider the example +of defining a user-function \spad{fact} to compute the \spadfun{factorial} +of a nonnegative integer. The Axiom statement +\spad{fact: Integer -> Integer} describes the operation,{} whereas the +statement \spad{fact(n) = reduce(*,{}[1..n])} defines the functions. See +also \spadgloss{generic function}. +\item\newline{\em \menuitemstyle{}}{\em function body}\space{} +the part of a \spadgloss{function}\spad{'s} definition which is evaluated +when the function is called at \spadgloss{run-time}; the part of the +function definition to the right of the \spadSyntax{\spad{==}}. +\item\newline{\em \menuitemstyle{}}{\em garbage collection}\space{} +a system function that automatically recycles memory cells from the +\spadgloss{heap}. Axiom is built upon \spadgloss{Common LISP} +which provides this facility. +\item\newline{\em \menuitemstyle{}}{\em garbage collector}\space{} +a mechanism for reclaiming storage in the \spadgloss{heap}. +\item\newline{\em \menuitemstyle{}}{\em Gaussian}\space{} +a complex-valued expression,{} \spadignore{e.g.} one with both a real +and imaginary part; a member of a \spadtype{Complex} domain. +\item\newline{\em \menuitemstyle{}}{\em generic function}\space{} +the use of one function to operate on objects of different types; One +might regard Axiom as supporting generic \spadgloss{operations} +but not generic functions. One operation \spad{+: (D,{}D) -> D} exists +for adding elements in a ring; each ring however provides its own +type-specific function for implementing this operation. +\item\newline{\em \menuitemstyle{}}{\em global variable}\space{} +A variable which can be referenced freely by functions. In Axiom,{} +all top-level user-defined variables defined during an interactive user +session are global variables. Axiom does not allow {\em fluid +variables},{} that is,{} variables \spadglossSee{bound}{binding} by +functions which can be referenced by functions those functions call. +\item\newline{\em \menuitemstyle{}}{\em Groebner basis}\space{} +{\em (algebra)} a special basis for a polynomial ideal that allows a +simple test for membership. It is useful in solving systems of polynomial +equations. +\item\newline{\em \menuitemstyle{}}{\em group}\space{} +{\em (algebra)} a monoid where every element has a multiplicative inverse. +\item\newline{\em \menuitemstyle{}}{\em hash table}\space{} +A data structure that efficiency maps a given object to another. A hash +table consists of a set of {\em entries},{} each of which associates a +{\em key} with a {\em value}. Finding the object stored under a key can +be very fast even if there are a large number of entries since keys are +{\em hashed} into numerical codes for fast lookup. +\item\newline{\em \menuitemstyle{}}{\em heap}\space{} +an area of storage used by data in programs. For example,{} AXIOM will +use the heap to hold the partial results of symbolic computations. When +cancellations occur,{} these results remain in the heap until +\spadglossSee{garbage collected}{garbage collector}. +\item\newline{\em \menuitemstyle{}}{\em history}\space{} +a mechanism which records the results for an interactive computation. +Using the history facility,{} users can save computations,{} review +previous steps of a computation,{} and restore a previous interactive +session at some later time. For details,{} issue the system command +{\em )history ?} to the interpreter. See also \spadgloss{frame}. +\item\newline{\em \menuitemstyle{}}{\em ideal}\space{} +{\em (algebra)} a subset of a ring that is closed under addition and +multiplication by arbitrary ring elements,{} \spadignore{i.e.} +it\spad{'s} a module over the ring. +\item\newline{\em \menuitemstyle{}}{\em identifier}\space{} +{\em (syntax)} an Axiom name; a \spadgloss{literal} of type +\spadtype{Symbol}. An identifier begins with an alphabetical character or +\% and may be followed by alphabetic characters,{} digits,{} ? or !. +Certain distinquished \spadgloss{reserved words} are not allowed as +identifiers but have special meaning in the Axiom. +\item\newline{\em \menuitemstyle{}}{\em immutable}\space{} +an object is immutable if it cannot be changed by an \spadgloss{operation}; +not a \spadglossSee{mutable object}{mutable}. Algebraic objects generally +immutable: changing an algebraic expression involves copying parts of the +original object. One exception is a matrix object of type \spadtype{Matrix}. +Examples of mutable objects are data structures such as those of type +\spadtype{List}. See also \spadgloss{pointer semantics}. +\item\newline{\em \menuitemstyle{}}{\em implicit export}\space{} +(of a domain or category) any \spadgloss{attribute} or +\spadgloss{operation} which is either an explicit export or else an +explicit export of some category which an explicit category export +\spadglossSee{extends}{category extension}. +\item\newline{\em \menuitemstyle{}}{\em index}\space{} +1. a variable that counts the number of times a \spadgloss{loop} is +repeated. 2. the "address" of an element in a data structure (see also +category \spadtype{LinearAggregate}). +\item\newline{\em \menuitemstyle{}}{\em infix}\space{} +{\em (syntax)} an \spadgloss{operator} placed between two +\spadgloss{operands}; also called a {\em binary operator},{} +\spadignore{e.g.} \spad{a + b}. An infix operator may also be used as a +\spadgloss{prefix},{} \spadignore{e.g.} \spad{+(a,{}b)} is also permissable +in the Axiom language. Infix operators have a relative +\spadgloss{precedence}. +\item\newline{\em \menuitemstyle{}}{\em input area}\space{} +a rectangular area on a \HyperName{} screen into which users can enter text. +\item\newline{\em \menuitemstyle{}}{\em instantiate}\space{} +to build a \spadgloss{category},{} \spadgloss{domain},{} or +\spadgloss{package} at run-time +\item\newline{\em \menuitemstyle{}}{\em integer}\space{} +a \spadgloss{literal} object of domain \spadtype{Integer},{} the class +of integers with an unbounded number of digits. Integer literals consist +of one or more consecutive digits (0-9) with no embedded blanks. +Underscores can be used to separate digits in long integers if desirable. +\item\newline{\em \menuitemstyle{}}{\em interactive}\space{} +a system where the user interacts with the computer step-by-step +\item\newline{\em \menuitemstyle{}}{\em interpreter}\space{} +the subsysystem of Axiom responsible for handling user input during +an interactive session. The following somewhat simplified description of +the typical action of the interpreter. The interpreter parsers the +user\spad{'s} input expression to create an expression tree then does a +bottom-up traversal of the tree. Each subtree encountered which is not a +value consists of a root node denoting an operation name and one or more +leaf nodes denoting \spadgloss{operands}. The interpreter resolves type +mismatches and uses type-inferencing and a library database to determine +appropriate types of the operands and the result,{} and an operation to +be performed. The interpreter then builds a domain to perform the +indicated operation,{} then invokes a function from the domain to compute +a value. The subtree is then replaced by that value and the process +continues. Once the entire tree has been processed,{} the value replacing +the top node of the tree is displayed back to the user as the value of the +expression. +\item\newline{\em \menuitemstyle{}}{\em invocation}\space{} +(of a function) the run-time process involved in +\spadglossSee{evaluating}{evaluation} a a \spadgloss{function} +\spadgloss{application}. This process has two steps. First,{} a +local \spadgloss{environment} is created where \spadgloss{formal arguments} +are locally \spadglossSee{bound}{binding} by \spadgloss{assignment} to +their respective actual \spadgloss{argument}. Second,{} the +\spadgloss{function body} is evaluated in that local environment. +The evaluation of a function is terminated either by completely +evaluating the function body or by the evaluation of a +\spadSyntax{return} expression. +\item\newline{\em \menuitemstyle{}}{\em iteration}\space{} +repeated evaluation of an expression or a sequence of expressions. +Iterations use the reserved words \spadSyntax{for},{} +\spadSyntax{while},{} and \spadSyntax{repeat}. +\item\newline{\em \menuitemstyle{}}{\em Join}\space{} +a primitive Axiom function taking two or more categories as +arguments and producing a category containing all of the operations +and attributes from the respective categories. +\item\newline{\em \menuitemstyle{}}{\em KCL}\space{} +Kyoto Common LISP,{} a version of \spadgloss{Common LISP} which +features compilation of the compilation of LISP into the \spad{C} +Programming Language +\item\newline{\em \menuitemstyle{}}{\em library}\space{} +In Axiom,{} a collection of compiled modules respresenting +the \spadgloss{category} or \spadgloss{domain} constructor. +\item\newline{\em \menuitemstyle{}}{\em lineage}\space{} +the sequence of \spadgloss{default packages} for a given domain to be +searched during \spadgloss{dynamic lookup}. This sequence is computed +first by ordering the category \spadgloss{ancestors} of the domain +according to their {\em level number},{} an integer equal to to the +minimum distance of the domain from the category. Parents have level +1,{} parents of parents have level 2,{} and so on. Among categories +with equal level numbers,{} ones which appear in the left-most branches +of {\em Join}\spad{s} in the source code come first. See also +\spadgloss{dynamic lookup}. +\item\newline{\em \menuitemstyle{}}{\em LISP}\space{} +acronymn for List Processing Language,{} a language designed for the +manipulation of nonnumerical data. The Axiom library is +translated into LISP then compiled into machine code by an underlying LISP. +\item\newline{\em \menuitemstyle{}}{\em list}\space{} +an object of a \spadtype{List} domain. +\item\newline{\em \menuitemstyle{}}{\em literal}\space{} +an object with a special syntax in the language. In Axiom,{} +there are five types of literals: \spadgloss{booleans},{} +\spadgloss{integers},{} \spadgloss{floats},{} \spadgloss{strings},{} +and \spadgloss{symbols}. +\item\newline{\em \menuitemstyle{}}{\em local}\space{} +{\em (syntax)} A keyword used in user-defined functions to declare that +a variable is a \spadgloss{local variable} of that function. Because of +default assumptions on variables,{} such a declaration is not necessary +but is available to the user for clarity when appropriate. +\item\newline{\em \menuitemstyle{}}{\em local variable}\space{} +(of a function) a variable \spadglossSee{bound}{binding} by that function +and such that its binding is invisible to any function that function calls. +Also called a {\em lexical} variable. By default in the +interpreter:\begin{items} \item 1. any variable \spad{x} which appears +on the left hand side of an assignment is regarded a local variable of +that function. If the intention of an assignment is to change the value +of a \spadgloss{global variable} \spad{x},{} the body of the function +must then contain the statement \spad{free x}. \item 2. any other variable +is regarded as a \spadgloss{free variable}. \end{items} An optional +declaration \spad{local x} is available to explicitly declare a variable +to be a local variable. All \spadgloss{formal parameters} to the function +can be regarded as local variables to the function. +\item\newline{\em \menuitemstyle{}}{\em loop}\space{} +1. an expression containing a \spadSyntax{repeat} 2. a collection expression +having a \spadSyntax{for} or a \spadSyntax{while},{} \spadignore{e.g.} +\spad{[f(i) for i in S]}. +\item\newline{\em \menuitemstyle{}}{\em loop body}\space{} +the part of a loop following the \spadSyntax{repeat} that tells what to +do each iteration. For example,{} the body of the loop \spad{for x in S +repeat B} is \spad{B}. For a collection expression,{} the body of the loop +precedes the initial \spadSyntax{for} or \spadSyntax{while}. +\item\newline{\em \menuitemstyle{}}{\em macro}\space{} +{\em (syntax)} 1. An expression of the form \spad{macro a == b} where +\spad{a} is a \spadgloss{symbol} causes \spad{a} to be textually replaced +by the expression \spad{b} at \spadgloss{parse} time. 2. An expression of +the form \spad{macro f(a) == b} defines a parameterized macro expansion for +a parameterized form \spad{f} This macro causes a form \spad{f}(\spad{x}) +to be textually replaced by the expression \spad{c} at parse time,{} where +\spad{c} is the expression obtained by replacing \spad{a} by \spad{x} +everywhere in \spad{b}. See also \spadgloss{definition} where a similar +substitution is done during \spadgloss{evaluation}. +\item\newline{\em \menuitemstyle{}}{\em mode}\space{} +a type expression containing a question-mark ({\em ?}). For example,{} +the mode {\em P ?} designates {\em the class of all polynomials over an +arbitrary ring}. +\item\newline{\em \menuitemstyle{}}{\em mutable}\space{} +objects which contain \spadgloss{pointers} to other objects and which +have operations defined on them which alter these pointers. Contrast +\spadgloss{immutable}. Axiom uses \spadgloss{pointer semantics} +as does \spadgloss{LISP} in contrast with many other languages such as +Pascal which use \spadgloss{copying semantics}. See \spadgloss{pointer +semantics} for details. +\item\newline{\em \menuitemstyle{}}{\em name}\space{} +1. a \spadgloss{symbol} denoting a \spadgloss{variable},{} +\spadignore{i.e.} the variable \spad{x}. 2. a \spadgloss{symbol} +denoting an \spadgloss{operation},{} \spadignore{i.e.} the operation +\spad{divide: (Integer,{}Integer) -> Integer}. +\item\newline{\em \menuitemstyle{}}{\em nullary}\space{} +a function with no arguments,{} \spadignore{e.g.} \spadfun{characteristic}. +\item\newline{\em \menuitemstyle{}}{\em nullary}\space{} +operation or function with \spadgloss{arity} 0 +\item\newline{\em \menuitemstyle{}}{\em Object}\space{} +a category with no operations or attributes,{} from which most categories +in Axiom are \spadglossSee{extensions}{category extension}. +\item\newline{\em \menuitemstyle{}}{\em object}\space{} +a data entity created or manipulated by programs. Elements of domains,{} +functions,{} and domains themselves are objects. Whereas categories are +created by functions,{} they cannot be dynamically manipulated in the +current system and are thus not considered as objects. The most basic +objects are \spadgloss{literals}; all other objects must be created +\spadgloss{functions}. Objects can refer to other objects using +\spadgloss{pointers}. Axiom language uses +\spadgloss{pointer semantics} when dealing with \spadgloss{mutable} objects. +\item\newline{\em \menuitemstyle{}}{\em object code}\space{} +code which can be directly executed by hardware; also known as +{\em machine language}. +\item\newline{\em \menuitemstyle{}}{\em operand}\space{} +an argument of an \spadgloss{operator} (regarding an operator as a +\spadgloss{function}). +\item\newline{\em \menuitemstyle{}}{\em operation}\space{} +an abstraction of a \spadgloss{function},{} described by a +\spadgloss{signature}. For example,{} +\center{\spad{fact: NonNegativeInteger -> NonNegativeInteger}} describes +an operation for "the factorial of a (non-negative) integer". +\item\newline{\em \menuitemstyle{}}{\em operator}\space{} +special reserved words in the language such as \spadop{+} and \spadop{*}; +operators can be either \spadgloss{prefix} or \spadgloss{infix} and have a +relative \spadgloss{precedence}. +\item\newline{\em \menuitemstyle{}}{\em overloading}\space{} +the use of the same name to denote distinct functions; a function is +identified by a \spadgloss{signature} identifying its name,{} the number +and types of its arguments,{} and its return types. If two functions can +have identical signatures,{} a \spadgloss{package call} must be made to +distinquish the two. +\item\newline{\em \menuitemstyle{}}{\em package}\space{} +a domain whose exported operations depend solely on the parameters and +other explicit domains,{} \spadignore{e.g.} a package for solving systems +of equations of polynomials over any field,{} \spadignore{e.g.} floats,{} +rational numbers,{} complex rational functions,{} or power series. +Facilities for integration,{} differential equations,{} solution of +linear or polynomial equations,{} and group theory are provided by +"packages". Technically,{} a package is a domain which has no +\spadgloss{signature} containing the symbol \$. While domains intuitively +provide computational objects you can compute with,{} packages intuitively +provide functions (\spadgloss{polymorphic} functions) which will work over +a variety of datatypes. +\item\newline{\em \menuitemstyle{}}{\em package call}\space{} +{\em (syntax)} an expression of the form \spad{e \$ D} where \spad{e} is +an \spadgloss{application} and \spad{D} denotes some \spadgloss{package} +(or \spadgloss{domain}). +\item\newline{\em \menuitemstyle{}}{\em package call}\space{} +{\em (syntax)} an expression of the form \spad{f(x,{}y)\$D} used to identify +that the function \spad{f} is to be one from \spad{D}. +\item\newline{\em \menuitemstyle{}}{\em package constructor}\space{} +same as \spadgloss{domain constructor}. +\item\newline{\em \menuitemstyle{}}{\em parameter}\space{} +see \spadgloss{argument} +\item\newline{\em \menuitemstyle{}}{\em parameterized datatype}\space{} +a domain that is built on another,{} for example,{} polynomials with integer +coefficients. +\item\newline{\em \menuitemstyle{}}{\em parameterized form}\space{} +a expression of the form \spad{f(x,{}y)},{} an \spadgloss{application} of a +function. +\item\newline{\em \menuitemstyle{}}{\em parent}\space{} +(of a domain) a category which is explicitly declared in the source code +definition for the domain to be an \spadgloss{export} of the domain. +\item\newline{\em \menuitemstyle{}}{\em parse}\space{} +1. (verb) to produce an internal representation of a user input string; +the resultant internal representation is then "interpreted" by Axiom +to perform some indicated action. +\item\newline{\em \menuitemstyle{}}{\em parse}\space{} +the transformation of a user input string representing a valid Axiom +expression into an internal representation as a tree-structure. +\item\newline{\em \menuitemstyle{}}{\em partially ordered set}\space{} +a set with a reflexive,{} transitive and antisymetric \spadgloss{binary} +operation. +\item\newline{\em \menuitemstyle{}}{\em pattern match}\space{} +1. (on expressions) Given a expression called a "subject" \spad{u},{} the +attempt to rewrite \spad{u} using a set of "rewrite rules". Each rule has +the form \spad{A == B} where \spad{A} indicates a expression called a +"pattern" and \spad{B} denotes a "replacement". The meaning of this rule +is "replace \spad{A} by \spad{B"}. If a given pattern \spad{A} matches a +subexpression of \spad{u},{} that subexpression is replaced by \spad{B}. +Once rewritten,{} pattern matching continues until no further changes occur. +2. (on strings) the attempt to match a string indicating a "pattern" to +another string called a "subject",{} for example,{} for the purpose of +identifying a list of names. In \Browse{},{} users may enter +\spadgloss{search strings} for the purpose of identifying constructors,{} +operations,{} and attributes. +\item\newline{\em \menuitemstyle{}}{\em pile}\space{} +alternate syntax for a block,{} using indentation and column alignment +(see also \spadgloss{block}). +\item\newline{\em \menuitemstyle{}}{\em pointer}\space{} +a reference implemented by a link directed from one object to another in +the computer memory. An object is said to {\em refer} to another if it +has a pointer to that other object. Objects can also refer to themselves +(cyclic references are legal). Also more than one object can refer to the +same object. See also \spadgloss{pointer semantics}. +\item\newline{\em \menuitemstyle{}}{\em pointer semantics}\space{} +the programming language semantics used in languages such as LISP which +allow objects to be \spadgloss{mutable}. Consider the following sequence +of Axiom statements:\begin{items} \item \spad{x : Vector Integer := +[1,{}4,{}7]} \item \spad{y := x} \item \spad{swap!(x,{}2,{}3)} \end{items} +The function \spadfunFrom{swap!}{Vector} is used to interchange the 2nd and +3rd value in the list \spad{x} producing the value \spad{[1,{}7,{}4]}. +What value does \spad{y} have after evaluation of the third statement? +The answer is different in Axiom than it is in a language with +\spadgloss{copying semantics}. In Axiom,{} first the vector +[1,{}2,{}3] is created and the variable \spad{x} set to +\spadglossSee{point}{pointer} to this object. Let\spad{'s} call this +object \spad{V}. Now \spad{V} refers to its \spadgloss{immutable} +components 1,{}2,{} and 3. Next,{} the variable \spad{y} is made to +point to \spad{V} just as \spad{x} does. Now the third statement +interchanges the last 2 elements of \spad{V} (the {\em !} at the end +of the name \spadfunFrom{swap!}{Vector} tells you that this operation +is destructive,{} that is,{} it changes the elements {\em in place}). +Both \spad{x} and \spad{y} perceive this change to \spad{V}. Thus both +\spad{x} and \spad{y} then have the value \spad{[1,{}7,{}4]}. In Pascal,{} +the second statement causes a copy of \spad{V} to be stored under \spad{y}. +Thus the change to \spad{V} made by the third statement does not affect +\spad{y}. +\item\newline{\em \menuitemstyle{}}{\em polymorphic}\space{} +a \spadgloss{function} parameterized by one or more \spadgloss{domains}; +a \spadgloss{algorithm} defined \spadglossSee{categorically}{category}. +Every function defined in a domain or package constructor with a +domain-valued parameter is polymorphic. For example,{} the same matrix +\spadSyntax{*} function is used to multiply "matrices over integers" as +"matrices over matrices over integers" Likewise,{} various +\ignore{\spad{???}} in \ignore{\spad{???}} solves polynomial equations +over any \spadgloss{field}. +\item\newline{\em \menuitemstyle{}}{\em postfix}\space{} +an \spadgloss{operator} that follows its single \spadgloss{operand}. +Postfix operators are not available in Axiom. +\item\newline{\em \menuitemstyle{}}{\em precedence}\space{} +{\em (syntax)} refers to the so-called {\em binding power} of an operator. +For example,{} \spad{*} has higher binding power than \spad{+} so that the +expression \spad{a + b * c} is equivalent to \spad{a + (b * c)}. +\item\newline{\em \menuitemstyle{}}{\em precision}\space{} +the number of digits in the specification of a number,{} \spadignore{e.g.} +as set by \spadfunFrom{precision}{Float}. +\item\newline{\em \menuitemstyle{}}{\em predicate}\space{} +1. a Boolean valued function,{} \spadignore{e.g.} +\spad{odd: Integer -> Boolean}. 2. an Boolean valued expression +\item\newline{\em \menuitemstyle{}}{\em prefix}\space{} +{\em (syntax)} an \spadgloss{operator} such as \spad{-} and \spad{not} +that is written {\em before} its single \spadgloss{operand}. Every +function of one argument can be used as a prefix operator. For example,{} +all of the following have equivalent meaning in Axiom: \spad{f(x)},{} +\spad{f x},{} and \spad{f.x}. See also \spadgloss{dot notation}. +\item\newline{\em \menuitemstyle{}}{\em quote}\space{} +the prefix \spadgloss{operator} \spadSyntax{'} meaning {\em do not evaluate}. +\item\newline{\em \menuitemstyle{}}{\em Record}\space{} +(basic domain constructor) a domain constructor used to create a +inhomogeneous aggregate composed of pairs of "selectors" and +\spadgloss{values}. A Record domain is written in the form +\spad{Record(a1:D1,{}...,{}an:Dn)} (\spad{n} > 0) where +\spad{a1},{}...,{}\spad{an} are identifiers called the +{\em selectors} of the record,{} and \spad{D1},{}...,{}\spad{Dn} +are domains indicating the type of the component stored under selector +\spad{an}. +\item\newline{\em \menuitemstyle{}}{\em recurrence relation}\space{} +A relation which can be expressed as a function \spad{f} with some +argument \spad{n} which depends on the value of \spad{f} at \spad{k} +previous values. In many cases,{} Axiom will rewrite a recurrence +relation on compilation so as to \spadgloss{cache} its previous \spad{k} +values and therefore make the computation significantly more efficient. +\item\newline{\em \menuitemstyle{}}{\em recursion}\space{} +use of a self-reference within the body of a function. Indirect recursion +is when a function uses a function below it in the call chain. +\item\newline{\em \menuitemstyle{}}{\em recursive}\space{} +1. A function that calls itself,{} either directly or indirectly through +another function. 2. self-referential. See also \spadgloss{recursive}. +\item\newline{\em \menuitemstyle{}}{\em reference}\space{} +see \spadgloss{pointer} +\item\newline{\em \menuitemstyle{}}{\em Rep}\space{} +a special identifier used as \spadgloss{local variable} of a domain +constructor body to denote the representation domain for objects of a domain. +\item\newline{\em \menuitemstyle{}}{\em representation}\space{} +a \spadgloss{domain} providing a data structure for elements of a +domain; generally denoted by the special identifier \spadgloss{Rep} +in the Axiom programming language. As domains are +\spadgloss{abstract datatypes},{} this representation is not available +to users of the domain,{} only to functions defined in the +\spadgloss{function body} for a domain constructor. Any domain can be used +as a representation. +\item\newline{\em \menuitemstyle{}}{\em reserved word}\space{} +a special sequence of non-blank characters with special meaning in the +Axiom language. Examples of reserved words are names such as +\spadSyntax{for},{} \spadSyntax{if},{} and \spadSyntax{free},{} operator +names such as \spadSyntax{+} and \spad{mod},{} special character strings +such as spadSyntax{\spad{==}} and spadSyntax{\spad{:=}}. +\item\newline{\em \menuitemstyle{}}{\em retraction}\space{} +to move an object in a parameterized domain back to the underlying +domain,{} for example to move the object \spad{7} from a "fraction of +integers" (domain \spadtype{Fraction Integer}) to "the integers" +(domain \spadtype{Integer}). +\item\newline{\em \menuitemstyle{}}{\em return}\space{} +when leaving a function,{} the value of the expression following +\spadSyntax{return} becomes the value of the function. +\item\newline{\em \menuitemstyle{}}{\em ring}\space{} +a set with a commutative addition,{} associative multiplication,{} +a unit element,{} and multiplication distributes over addition and +subtraction. +\item\newline{\em \menuitemstyle{}}{\em rule}\space{} +{\em (syntax)} 1. An expression of the form \spad{rule A == B} +indicating a "rewrite rule". 2. An expression of the form +\spad{rule (R1;...;Rn)} indicating a set of "rewrite rules" +\spad{R1},{}...,{}\spad{Rn}. See \spadgloss{pattern matching} for details. +\item\newline{\em \menuitemstyle{}}{\em run-time}\space{} +the time of doing a computation. Contrast \spadgloss{compile-time}. + rather than prior to it; \spadgloss{dynamic} as opposed to +\spadgloss{static}. For example,{} the decision of the intepreter +to build a structure such as "matrices with power series entries" +in response to user input is made at run-time. +\item\newline{\em \menuitemstyle{}}{\em run-time check}\space{} +an error-checking which can be done only when the program receives +user input; for example,{} confirming that a value is in the proper +range for a computation. +\item\newline{\em \menuitemstyle{}}{\em search string}\space{} +a string entered into an \spadgloss{input area} on a \HyperName{} screen +\item\newline{\em \menuitemstyle{}}{\em selector}\space{} +an identifier used to address a component value of a gloss{Record} datatype. +\item\newline{\em \menuitemstyle{}}{\em semantics}\space{} +the relationships between symbols and their meanings. The rules for +obtaining the {\em meaning} of any syntactically valid expression. +\item\newline{\em \menuitemstyle{}}{\em semigroup}\space{} +{\em (algebra)} a \spadgloss{monoid} which need not have an identity; +it is closed and associative. +\item\newline{\em \menuitemstyle{}}{\em side effect}\space{} +action which changes a component or structure of a value. See +\spadgloss{destructive operation} for details. +\item\newline{\em \menuitemstyle{}}{\em signature}\space{} +{\em (syntax)} an expression describing an \spadgloss{operation}. +A signature has the form as \spad{name : source -> target},{} where +\spad{source} gives the type of the arguments of the operation,{} and +\spad{target} gives the type of the result. +\item\newline{\em \menuitemstyle{}}{\em small float}\space{} +the domain for hardware floating point arithmetic as provided by the +computer hardware. +\item\newline{\em \menuitemstyle{}}{\em small integer}\space{} +the domain for hardware integer arithmetic. as provided by the computer +hardware. +\item\newline{\em \menuitemstyle{}}{\em source}\space{} +the \spadgloss{type} of the argument of a \spadgloss{function}; the type +expression before the \spad{->} in a \spadgloss{signature}. For example,{} +the source of \spad{f : (Integer,{}Integer) -> Integer} is +\spad{(Integer,{}Integer)}. +\item\newline{\em \menuitemstyle{}}{\em sparse}\space{} +data structure whose elements are mostly identical (a sparse matrix is +one filled with mostly zeroes). +\item\newline{\em \menuitemstyle{}}{\em static}\space{} +that computation done before run-time,{} such as compilation. Contrast +\spadgloss{dynamic}. +\item\newline{\em \menuitemstyle{}}{\em step number}\space{} +the number which precedes user input lines in an interactive session; +the output of user results is also labeled by this number. +\item\newline{\em \menuitemstyle{}}{\em stream}\space{} +an object of \spadtype{Stream(R)},{} a generalization of a +\spadgloss{list} to allow an infinite number of elements. Elements of +a stream are computed "on demand". Strings are used to implement various +forms of power series (\ignore{\spad{???}}). +\item\newline{\em \menuitemstyle{}}{\em string}\space{} +an object of domain \spadtype{String}. Strings are \spadgloss{literals} +consisting of an arbitrary sequence of \spadgloss{characters} surrounded +by double-quotes (\spadSyntax{"}),{} \spadignore{e.g.} \spad{"Look here!"}. +\item\newline{\em \menuitemstyle{}}{\em subdomain}\space{} +{\em (basic concept)} a \spadgloss{domain} together with a +\spadgloss{predicate} characterizing which members of the domain belong +to the subdomain. The exports of a subdomain are usually distinct from the +domain itself. A fundamental assumption however is that values in the +subdomain are automatically \spadglossSee{coerceable}{coercion} to values +in the domain. For example,{} if \spad{n} and \spad{m} are declared to be +members of a subdomain of the integers,{} then {\em any} \spadgloss{binary} +operation from \spadtype{Integer} is available on \spad{n} and \spad{m}. +On the other hand,{} if the result of that operation is to be assigned +to,{} say,{} \spad{k},{} also declared to be of that subdomain,{} a +\spadgloss{run-time} check is generally necessary to ensure that the +result belongs to the subdomain. +\item\newline{\em \menuitemstyle{}}{\em such that clause}\space{} +the use of \spadSyntax{|} followed by an expression to filter an iteration. +\item\newline{\em \menuitemstyle{}}{\em suffix}\space{} +{\em (syntax)} an \spadgloss{operator} which placed after its operand. +Suffix operators are not allowed in the Axiom language. +\item\newline{\em \menuitemstyle{}}{\em symbol}\space{} +objects denoted by \spadgloss{identifier} \spadgloss{literals}; an +element of domain \spadtype{Symbol}. The interpreter defaultly converts +a symbol \spad{x} into \spadtype{Variable(x)}. +\item\newline{\em \menuitemstyle{}}{\em syntax}\space{} +rules of grammar,{} punctuation etc. for forming correct expressions. +\item\newline{\em \menuitemstyle{}}{\em system commands}\space{} +top-level Axiom statements that begin with {\em )}. System +commands allow users to query the database,{} read files,{} trace +functions,{} and so on. +\item\newline{\em \menuitemstyle{}}{\em tag}\space{} +an identifier used to discriminate a branch of a \spadgloss{Union} type. +\item\newline{\em \menuitemstyle{}}{\em target}\space{} +the \spadgloss{type} of the result of a \spadgloss{function}; the +type expression following the \spad{->} in a \spadgloss{signature}. +\item\newline{\em \menuitemstyle{}}{\em top-level}\space{} +refers to direct user interactions with the Axiom interpreter. +\item\newline{\em \menuitemstyle{}}{\em totally ordered set}\space{} +{\em (algebra)} a partially ordered set where any two elements are +comparable. +\item\newline{\em \menuitemstyle{}}{\em trace}\space{} +use of system function \spadsys{)trace} to track the arguments passed +to a function and the values returned. +\item\newline{\em \menuitemstyle{}}{\em tuple}\space{} +an expression of two or more other expressions separated by commas,{} +\spadignore{e.g.} \spad{4,{}7,{}11}. Tuples are also used for multiple +arguments both for \spadgloss{applications} (\spadignore{e.g.} +\spad{f(x,{}y)}) and in \spadgloss{signatures} (\spadignore{e.g.} +\spad{(Integer,{}Integer) -> Integer}). A tuple is not a data structure,{} +rather a syntax mechanism for grouping expressions. +\item\newline{\em \menuitemstyle{}}{\em type}\space{} +The type of any \spadgloss{subdomain} is the unique symbol {\em Category}. +The type of a \spadgloss{domain} is any \spadgloss{category} that domain +belongs to. The type of any other object is either the (unique) domain +that object belongs to or any \spadgloss{subdomain} of that domain. The +type of objects is in general not unique. +\item\newline{\em \menuitemstyle{}}{\em type checking}\space{} +a system function which determines whether the datatype of an object is +appropriate for a given operation. +\item\newline{\em \menuitemstyle{}}{\em type constructor}\space{} +a \spadgloss{domain constructor} or \spadgloss{category constructor}. +\item\newline{\em \menuitemstyle{}}{\em type inference}\space{} +when the interpreter chooses the type for an object based on context. +For example,{} if the user interactively issues the definition +\center{\spad{f(x) == (x + \%i)**2}} then issues \spad{f(2)},{} +the interpreter will infer the type of \spad{f} to be +\spad{Integer -> Complex Integer}. +\item\newline{\em \menuitemstyle{}}{\em unary}\space{} +operation or function with \spadgloss{arity} 1 +\item\newline{\em \menuitemstyle{}}{\em underlying domain}\space{} +for a \spadgloss{domain} that has a single domain-valued parameter,{} +the {\em underlying domain} refers to that parameter. For example,{} +the domain "matrices of integers" (\spadtype{Matrix Integer}) has +underlying domain \spadtype{Integer}. +\item\newline{\em \menuitemstyle{}}{\em Union}\space{} +(basic domain constructor) a domain constructor used to combine any +set of domains into a single domain. A Union domain is written in the +form \spad{Union(a1:D1,{}...,{}an:Dn)} (\spad{n} > 0) where +\spad{a1},{}...,{}\spad{an} are identifiers called the {\em tags} of +the union,{} and \spad{D1},{}...,{}\spad{Dn} are domains called the +{\em branches} of the union. The tags \spad{\spad{ai}} are optional,{} +but required when two of the \spad{\spad{Di}} are equal,{} +\spadignore{e.g.} \spad{Union(inches:Integer,{}centimeters:Integer)}. +In the interpreter,{} values of union domains are automatically coerced +to values in the branches and vice-versa as appropriate. See also +\spadgloss{case}. +\item\newline{\em \menuitemstyle{}}{\em unit}\space{} +{\em (algebra)} an invertible element. +\item\newline{\em \menuitemstyle{}}{\em user function}\space{} +a function defined by a user during an interactive session. Contrast +\spadgloss{built-in function}. +\item\newline{\em \menuitemstyle{}}{\em user variable}\space{} +a variable created by the user at top-level during an interactive session +\item\newline{\em \menuitemstyle{}}{\em value}\space{} +1. the result of \spadglossSee{evaluating}{evaluation} an expression. +2. a property associated with a \spadgloss{variable} in a \spadgloss{binding} +in an \spadgloss{environment}. +\item\newline{\em \menuitemstyle{}}{\em variable}\space{} +a means of referring to an object but itself {\spad{\bf} not} an object. A +variable has a name and an associated \spadgloss{binding} created by +\spadgloss{evaluation} of Axiom expressions such as +\spadgloss{declarations},{} \spadgloss{assignments},{} and +\spadgloss{definitions}. In the top-level \spadgloss{environment} of the +interpreter,{} variables are \spadgloss{global variables}. Such variables +can be freely referenced in user-defined functions although a +\spadgloss{free} declaration is needed to assign values to them. +See \spadgloss{local variable} for details. +\item\newline{\em \menuitemstyle{}}{\em Void}\space{} +the type given when the \spadgloss{value} and \spadgloss{type} of an +expression are not needed. Also used when there is no guarantee at +run-time that a value and predictable mode will result. +\item\newline{\em \menuitemstyle{}}{\em wild card}\space{} +a symbol which matches any substring including the empty string; for +example,{} the search string {\em *an*} matches an word containing the +consecutive letters {\em a} and {\em n} +\item\newline{\em \menuitemstyle{}}{\em workspace}\space{} +an interactive record of the user input and output held in an interactive +history file. Each user input and corresponding output expression in the +workspace has a corresponding \spadgloss{step number}. The current output +expression in the workspace is referred to as \spad{\%}. The output +expression associated with step number \spad{n} is referred to by +\spad{\%\%(n)}. The \spad{k}-th previous output expression relative +to the current step number \spad{n} is referred to by \spad{\%\%(- k)}. +Each interactive \spadgloss{frame} has its own workspace. +\endmenu\endscroll\lispdownlink{Search}{(|htGloss| "\stringvalue{pattern}")} +for glossary entry matching \inputstring{pattern}{24}{*}\end{page} + +@ +\section{graphics.ht} +\subsection{Graphics} +\begin{itemize} +\item GraphicsExamplePage \ref{GraphicsExamplePage} on +page~\pageref{GraphicsExamplePage} +\item TwoDimensionalGraphicsPage \ref{TwoDimensionalGraphicsPage} on +page~\pageref{TwoDimensionalGraphicsPage} +\item ThreeDimensionalGraphicsPage \ref{ThreeDimensionalGraphicsPage} on +page~\pageref{ThreeDimensionalGraphicsPage} +\item ViewportPage \ref{ViewportPage} on page~\pageref{ViewportPage} +\end{itemize} +\label{GraphicsPage} +\index{pages!GraphicsPage!graphics.ht} +\index{graphics.ht!pages!GraphicsPage} +\index{GraphicsPage!graphics.ht!pages} +<>= +\begin{page}{GraphicsPage}{Graphics} +Axiom can plot curves and surfaces of +various types, as well as lists of points in the plane. +% sequences, and complex functions. +\beginscroll +\beginmenu +\menulink{Examples}{GraphicsExamplePage} \tab{13} +See examples of Axiom graphics +\menulink{2D Graphics}{TwoDimensionalGraphicsPage} \tab{13} +Graphics in the real and complex plane +\menulink{3D Graphics}{ThreeDimensionalGraphicsPage} \tab{13} +Plot surfaces, curves, or tubes around curves +\menulink{Viewports}{ViewportPage} \tab{13} +Customize graphics using Viewports +\endmenu +\endscroll +\autobuttons \end{page} + +@ +\subsection{Graphics Examples} +\label{GraphicsExamplePage} +\begin{itemize} +\item AssortedGraphicsExamplePage \ref{AssortedGraphicsExamplePage} on +page~\pageref{AssortedGraphicsExamplePage} +\item ThreeDimensionalGraphicsExamplePage +\ref{ThreeDimensionalGraphicsExamplePage} on +page~\pageref{ThreeDimensionalGraphicsExamplePage} +\item OneVariableGraphicsExamplePage +\ref{OneVariableGraphicsExamplePage} on +page~\pageref{OneVariableGraphicsExamplePage} +\item PolarGraphicsExamplePage \ref{PolarGraphicsExamplePage} on +page~\pageref{PolarGraphicsExamplePage} +\item ParametricCurveGraphicsExamplePage +\ref{ParametricCurveGraphicsExamplePage} on +page~\ref{ParametricCurveGraphicsExamplePage} +\item ImplicitCurveGraphicsExamplePage \ref{ImplicitCurveGraphicsExamplePage} +on page~\pageref{ImplicitCurveGraphicsExamplePage} +\item ListPointsGraphicsExamplePage \ref{ListPointsGraphicsExamplePage} +on page~\pageref{ListPointsGraphicsExamplePage} +\end{itemize} +\index{pages!GraphicsExamplePage!graphics.ht} +\index{graphics.ht!pages!GraphicsExamplePage} +\index{GraphicsExamplePage!graphics.ht!pages} +<>= +\begin{page}{GraphicsExamplePage}{Graphics Examples} +\beginscroll +Here are some examples of Axiom graphics. +Choose a specific type of graph or choose Assorted Examples. +\beginmenu +\menulink{Assorted Examples}{AssortedGraphicsExamplePage} \newline +Examples of each type of Axiom graphics. +\menulink{Three Dimensional Graphics}{ThreeDimensionalGraphicsExamplePage} \newline +Plot parametrically defined surfaces of three functions. +\menulink{Functions of One Variable}{OneVariableGraphicsExamplePage} \newline +Plot curves defined by an equation y = f(x). +\menulink{Parametric Curves}{ParametricCurveGraphicsExamplePage} \newline +Plot curves defined by parametric equations x = f(t), y = g(t). +\menulink{Polar Coordinates}{PolarGraphicsExamplePage} \newline +Plot curves given in polar form by an equation r = f(theta). +\menulink{Implicit Curves}{ImplicitCurveGraphicsExamplePage} \newline +Plot non-singular curves defined by a polynomial equation +\menulink{Lists of Points}{ListPointsGraphicsExamplePage} \newline +Plot lists of points in the (x,y)-plane. +% \menulink{Sequences}{SequenceGraphicsExamplePage} +% Plot a sequence a1, a2, a3,... +% \menulink{Complex Functions}{ComplexFunctionGraphicsExamplePage} +% Plot complex functions of a complex variable by means of grid plots. +\endmenu +\endscroll +\autobuttons \end{page} + +@ +\subsection{Assorted Graphics Examples} +\label{AssortedGraphicsExamplePage} +\index{pages!AssortedGraphicsExamplePage!graphics.ht} +\index{graphics.ht!pages!AssortedGraphicsExamplePage} +\index{AssortedGraphicsExamplePage!graphics.ht!pages} +<>= +\begin{page}{AssortedGraphicsExamplePage}{Assorted Graphics Examples} +\beginscroll +Pick a specific example or choose 'All' to see all the examples.\newline +Function of two variables: z = f(x,y). +\graphpaste{draw(sin(x * y), x = -2.5..2.5, y = -2.5..2.5) \bound{example1}} +Function of one variable: y = f(x). +\graphpaste{draw(sin tan x - tan sin x,x = 0..6) \bound{example2} } +Plane parametric curve: x = f(t), y = g(t). +\graphpaste{draw(curve(sin(t)*sin(2*t), sin(3*t)*sin(4*t)), t = 0..2*\%pi) \bound{example3}} +Space parametric curve: x = f(t), y = g(t), z = h(t). +\graphpaste{draw(curve(sin(t)*sin(2*t), sin(3*t)*sin(4*t), sin(5*t)*sin(6*t)), t = 0..2*\%pi) \bound{example4}} +Polar coordinates: r = f(theta). +\graphpaste{draw(sin(17*t), t = 0..2*\%pi, coordinates == polar) \bound{example5}} +Implicit curves: p(x,y) = 0. +\graphpaste{draw(y**2 + y = x**3 - x, x, y,range == \[-2..2,-2..1\]) \bound{example6}} +Run all examples. +\spadpaste{All \free{example1 example2 example3 example4 example5 example6}} +\endscroll +\autobuttons \end{page} + +@ +\subsection{Three Dimensional Graphics} +\label{ThreeDimensionalGraphicsExamplePage} +\index{pages!ThreeDimensionalGraphicsExamplePage!graphics.ht} +\index{graphics.ht!pages!ThreeDimensionalGraphicsExamplePage} +\index{ThreeDimensionalGraphicsExamplePage!graphics.ht!pages} +<>= +\begin{page}{ThreeDimensionalGraphicsExamplePage}{Three Dimensional Graphics} +Plots of parametric surfaces defined by functions f(u,v), g(u,v), and h(u,v). +Choose a particular example or choose 'All' to see all the examples. +\beginscroll +Pear Surface. +\graphpaste{draw(surface((1+exp(-100*u*u))*sin(\%pi*u)*sin(\%pi*v), (1+exp(-100*u*u))*sin(\%pi*u)*cos(\%pi*v), (1+exp(-100*u*u))*cos(\%pi*u)), u=0..1, v=0..2, title=="Pear") \bound{example1}} +Trigonometric Screw. +\graphpaste{draw(surface(x*cos(y),x*sin(y),y*cos(x)), x=-4..4, y=0..2*\%pi, var1Steps==40, var2Steps==40, title=="Trig Screw") \bound{example2}} +Etruscan Venus. \newline +(click on the draw button to execute this example) +\spadpaste{a := 1.3 * cos(2*x) * cos(y) + sin(y) * cos(x)\bound{a}} +\newline +\spadpaste{b := 1.3 * sin(2*x) * cos(y) - sin(y) * sin(x)\bound{b}} +\newline +\spadpaste{c := 2.5 * cos(y) \bound{c}} +\newline +\graphpaste{draw(surface(a,b,c), x=0..\%pi, y=-\%pi..\%pi, var1Steps==40, var2Steps==40, title=="Etruscan Venus") \free{a b c} \bound{example3}} +Banchoff Klein Bottle. \newline +(click on the draw button to execute this example) +\spadpaste{f:=cos(x)*(cos(x/2)*(sqrt(2) + cos(y))+(sin(x/2)*sin(y)*cos(y)))\bound{f}} +\newline +\spadpaste{g:=sin(x)*(cos(x/2)*(sqrt(2) + cos(y))+(sin(x/2)*sin(y)*cos(y)))\bound{g}} +\newline +\spadpaste{h:=-sin(x/2)*(sqrt(2)+cos(y)) + cos(x/2)*sin(y)*cos(y) \bound{h}} +\newline +\graphpaste{draw(surface(f,g,h), x=0..4*\%pi, y=0..2*\%pi, var1Steps==50, var2Steps==50, title=="Banchoff Klein Bottle") \free{f g h} \bound{example4}} +\newline +\spadpaste{All \free{example1 example2 example3 example4}} +\endscroll +\autobuttons \end{page} + +@ +\subsection{Functions of One Variable} +\label{OneVariableGraphicsExamplePage} +\index{pages!OneVariableGraphicsExamplePage!graphics.ht} +\index{graphics.ht!pages!OneVariableGraphicsExamplePage} +\index{OneVariableGraphicsExamplePage!graphics.ht!pages} +<>= +\begin{page}{OneVariableGraphicsExamplePage}{Functions of One Variable} +\beginscroll +Plots of functions y = f(x). +Choose a particular example or choose 'All' to see all the examples. +\graphpaste{draw(sin tan x - tan sin x, x = 0..6) \bound{example1}} +\newline +\graphpaste{draw(sin x + cos x, x = 0..2*\%pi) \bound{example2}} +\newline +\graphpaste{draw(sin(1/x), x = -1..1) \bound{example3}} +\newline +\graphpaste{draw(x * sin(1/x), x = -1..1) \bound{example4}} +\newline +\spadpaste{All \free{example1 example2 example3 example4}} +\endscroll +\autobuttons \end{page} + +@ +\subsection{Parametric Curves} +\label{ParametricCurveGraphicsExamplePage} +\index{pages!ParametricCurveGraphicsExamplePage!graphics.ht} +\index{graphics.ht!pages!ParametricCurveGraphicsExamplePage} +\index{ParametricCurveGraphicsExamplePage!graphics.ht!pages} +<>= +\begin{page}{ParametricCurveGraphicsExamplePage}{Parametric Curves} +Plots of parametric curves x = f(t), y = g(t). +Pick a particular example or choose 'All' to see all the examples. +\beginscroll +The Lemniscate of Bernoulli. +\graphpaste{draw(curve(cos(t/(1+sin(t)**2)), sin(t)*cos(t)/(1+sin(t)**2)), t = -\%pi..\%pi) \bound{example1}} +Lissajous curve. +\graphpaste{draw(curve(9*sin(3*t/4), 8*sin(t)), t = -4*\%pi..4*\%pi) \bound{example2}} +A gnarly closed curve. +\graphpaste{draw(curve(sin(t)*sin(2*t)*sin(3*t), sin(4*t)*sin(5*t)*sin(6*t)),t = 0..2*\%pi) + \bound{example3}} +Another closed curve. +\graphpaste{draw(curve(cos(4*t)*cos(7*t), cos(4*t)*sin(7*t)), t = 0..2*\%pi) \bound{example4}} +Run all examples on this page. +\spadpaste{All \free{example1 example2 example3 example4}} +\endscroll +\autobuttons \end{page} + +@ +\subsection{Polar Coordinates} +\label{PolarGraphicsExamplePage} +\index{pages!PolarGraphicsExamplePage!graphics.ht} +\index{graphics.ht!pages!PolarGraphicsExamplePage} +\index{PolarGraphicsExamplePage!graphics.ht!pages} +<>= +\begin{page}{PolarGraphicsExamplePage}{Polar Coordinates} +Plots of curves given by an equation in polar coordinates, r = f(theta). +Pick a particular example or choose 'All' to see all the examples. +\beginscroll +A Circle. +\graphpaste{draw(1,t = 0..2*\%pi, coordinates == polar) \bound{example1} } +A Spiral. +\graphpaste{draw(t,t = 0..100, coordinates == polar) \bound{example2} } +A Petal Curve. +\graphpaste{draw(sin(4*t), t = 0..2*\%pi, coordinates == polar) \bound{example3} } +A Limacon. +\graphpaste{draw(2 + 3 * sin t, t = 0..2*\%pi, coordinates == polar) \bound{example4} } +Run all examples on this page. +\spadpaste{All \free{ +%example1 +example2 example3 example4}} +\endscroll +\autobuttons \end{page} + +@ +\subsection{Implicit Curves} +\label{ImplicitCurveGraphicsExamplePage} +\index{pages!ImplicitCurveGraphicsExamplePage!graphics.ht} +\index{graphics.ht!pages!ImplicitCurveGraphicsExamplePage} +\index{ImplicitCurveGraphicsExamplePage!graphics.ht!pages} +<>= +\begin{page}{ImplicitCurveGraphicsExamplePage}{Implicit Curves} +Non-singular curves defined by a polynomial equation p(x,y) = 0 +in a rectangular region in the plane. +Pick a particular example or choose 'All' to see all the examples. +\beginscroll +A Conic Section (Hyperbola). +\graphpaste{draw(x * y = 1, x, y, range == \[-3..3, -3..3\]) \bound{example1} } +An Elliptic Curve. +\graphpaste{draw(y**2 + y = x**3 - x, x, y, range == \[-2..2, -2..1\]) \bound{example2} } +Cartesian Ovals. +\spadpaste{p := ((x**2 + y**2 + 1) - 8*x)**2 - (8*(x**2 + y**2 + 1) - 4*x - 1) \bound{p} } +\graphpaste{draw(p = 0, x, y, range == \[-1..11, -7..7\], title == "Cartesian Ovals") \free{p} \bound{example3} } +Cassinian Ovals: two loops. +\spadpaste{q := (x**2 + y**2 + 7**2)**2 - (6**4 + 4*7**2*x**2) \bound{q} } +\graphpaste{draw(q = 0, x, y, range == \[-10..10, -4..4\], title == "Cassinian oval with two loops") \free{q} \bound{example4} } +Run all examples on this page. +\spadpaste{All \free{example1 example2 example3 example4}} +\endscroll +\autobuttons \end{page} + +@ +\subsection{Lists of Points} +\label{ListPointsGraphicsExamplePage} +\index{pages!ListPointsGraphicsExamplePage!graphics.ht} +\index{graphics.ht!pages!ListPointsGraphicsExamplePage} +\index{ListPointsGraphicsExamplePage!graphics.ht!pages} +<>= +\begin{page}{ListPointsGraphicsExamplePage}{Lists of Points} +Axiom has the ability to create lists of points in a two dimensional +graphics viewport. This is done by utilizing the \spadtype{GraphImage} and +\spadtype{TwoDimensionalViewport} domain facilities. +\beginscroll +\indent{5}\newline +{\em NOTE: It is only necessary to click on the makeViewport2D command button to plot this curve example}. +\indent{0}\newline +\spadpaste{p1 := point [1::SF,1::SF]\$(Point SF) \bound{p1}} +\newline +\spadpaste{p2 := point [0::SF,1::SF]\$(Point SF) \bound{p2}} +\newline +\spadpaste{p3 := point [0::SF,0::SF]\$(Point SF) \bound{p3}} +\newline +\spadpaste{p4 := point [1::SF,0::SF]\$(Point SF) \bound{p4}} +\newline +\spadpaste{p5 := point [1::SF,.5::SF]\$(Point SF) \bound{p5}} +\newline +\spadpaste{p6 := point [.5::SF,0::SF]\$(Point SF) \bound{p6}} +\newline +\spadpaste{p7 := point [0::SF,0.5::SF]\$(Point SF) \bound{p7}} +\newline +\spadpaste{p8 := point [.5::SF,1::SF]\$(Point SF) \bound{p8}} +\newline +\spadpaste{p9 := point [.25::SF,.25::SF]\$(Point SF) \bound{p9}} +\newline +\spadpaste{p10 := point [.25::SF,.75::SF]\$(Point SF) \bound{p10}} +\newline +\spadpaste{p11 := point [.75::SF,.75::SF]\$(Point SF) \bound{p11}} +\newline +\spadpaste{p12 := point [.75::SF,.25::SF]\$(Point SF) \bound{p12}} +\newline +\spadpaste{llp := [[p1,p2],[p2,p3],[p3,p4],[p4,p1],[p5,p6],[p6,p7],[p7,p8],[p8,p5],[p9,p10],[p10,p11],[p11,p12],[p12,p9]] \bound{llp} \free{p1 p2 p3 p4 p5 p6 p7 p8 p9 p10 p11 p12}} +\newline +\spadpaste{size1 := 6::PositiveInteger \bound{size1}} +\newline +\spadpaste{size2 := 8::PositiveInteger \bound{size2}} +\newline +\spadpaste{size3 := 10::PositiveInteger \bound{size3}} +\newline +\spadpaste{lsize := [size1, size1, size1, size1, size2, size2, size2, size2, size3, size3, size3, size3] \bound{lsize} \free{size1 size2 size3}} +\newline +\spadpaste{pc1 := pastel red() \bound{pc1}} +\newline +\spadpaste{pc2 := dim green() \bound{pc2}} +\newline +\spadpaste{pc3 := pastel yellow() \bound{pc3}} +\newline +\spadpaste{lpc := [pc1, pc1, pc1, pc1, pc2, pc2, pc2, pc2, pc3, pc3, pc3, pc3] \bound{lpc} \free{pc1 pc2 pc3}} +\newline +\spadpaste{lc := [pastel blue(), light yellow(), dim green(), bright red(), light green(), dim yellow(), bright blue(), dark red(), pastel red(), light blue(), dim green(), light yellow()] \bound{lc}} +\newline +\spadpaste{g := makeGraphImage(llp,lpc,lc,lsize)\$GRIMAGE \bound{g} \free{llp lpc lc lsize}} +\newline +\graphpaste{makeViewport2D(g,[title("Lines")])\$VIEW2D \free{g}} +The \spadfun{makeViewport2D} command takes a list of options as a parameter +in this example. The string "Lines" is designated as the viewport's title. +\endscroll +\autobuttons \end{page} + +@ +\subsection{Three Dimensional Graphing} +\label{ThreeDimensionalGraphicsPage} +\begin{itemize} +\item TwoVariableGraphicsPage \ref{TwoVariableGraphicsPage} on +page~\pageref{TwoVariableGraphicsPage} +\item SpaceCurveGraphicsPage \ref{SpaceCurveGraphicsPage} on +page~\pageref{SpaceCurveGraphicsPage} +\item ParametricTubeGraphicsPage \ref{ParametricTubeGraphicsPage} on +page~\pageref{ParametricTubeGraphicsPage} +\item ParametricSurfaceGraphicsPage \ref{ParametricSurfaceGraphicsPage} on +page~\pageref{ParametricSurfaceGraphicsPage} +\item ugGraphThreeDBuildPage \ref{ugGraphThreeDBuildPage} on +page~\pageref{ugGraphThreeDBuildPage} +\end{itemize} +\index{pages!ThreeDimensionalGraphicsPage!graphics.ht} +\index{graphics.ht!pages!ThreeDimensionalGraphicsPage} +\index{ThreeDimensionalGraphicsPage!graphics.ht!pages} +<>= +\begin{page}{ThreeDimensionalGraphicsPage}{Three Dimensional Graphing} +\beginscroll +\beginmenu +\menulink{Functions of Two Variables}{TwoVariableGraphicsPage} \newline +Plot surfaces defined by an equation z = f(x,y). +\menulink{Parametric Curves}{SpaceCurveGraphicsPage} \newline +Plot curves defined by equations x = f(t), y = g(t), z = g(t). +\menulink{Parametric Tube Plots}{ParametricTubeGraphicsPage} \newline +Plot a tube around a parametric space curve. +\menulink{Parametric Surfaces}{ParametricSurfaceGraphicsPage} \newline +Plot surfaces defined by x = f(u,v), y = g(u,v), z = h(u,v). +\menulink{Building Objects}{ugGraphThreeDBuildPage} \newline +Create objects constructed from geometric primitives. +\endmenu +\endscroll +\autobuttons \end{page} + +@ +\subsection{Functions of Two Variables} +\label{TwoVariableGraphicsPage} +\index{pages!TwoVariableGraphicsPage!graphics.ht} +\index{graphics.ht!pages!TwoVariableGraphicsPage} +\index{TwoVariableGraphicsPage!graphics.ht!pages} +<>= +\begin{page}{TwoVariableGraphicsPage}{Functions of Two Variables} +\beginscroll +This page describes the plotting of surfaces defined by an equation +of two variables, z = f(x,y), for which the ranges of x and y are explicitly +defined. The basic draw command for this function utilizes either the +uncompiled function or compiled function format. The general format for an +uncompiled function is: +\indent{5}\newline +{\em draw(f(x,y), x = a..b, y = c..d)} +\indent{0}\newline +where a..b and c..d are segments defining the intervals [a,b] and [c,d] over +which the variables x and y span. In this case the function is not compiled +until the draw command is executed. Here is an example: +\graphpaste{draw(cos(x*y),x=-3..3,y=-3..3)} +In the case of a compiled function, the function is named and compiled +independently. This is useful if you intend to use a function often, or +if the function is long and complex. The following line shows a function +whose parameters are of the type Small Float. The function is compiled and +stored by Axiom when it is entered. +\indent{5}\newline +{\em NOTE: It is only necessary to click on the draw command button to plot +this example}. +\indent{0}\newline +\spadpaste{f(x:SF,y:SF):SF == sin(x)*cos(y) \bound{f}} +\newline +Once the function is compiled the draw command only needs the name of the +function to execute. Here is a compiled function example: +\graphpaste{draw(f,-\%pi..\%pi,-\%pi..\%pi) \free{f}} +Note that the parameter ranges do not take the variable names as in the +case of uncompiled functions. The variables are entered in the order in +which they are defined in the function specification. In this case the +first range specifies the x-variable and the second range specifies the +y-variable. +\endscroll +\autobuttons \end{page} + +@ +\subsection{Parametric Space Curves} +\label{SpaceCurveGraphicsPage} +\index{pages!SpaceCurveGraphicsPage!graphics.ht} +\index{graphics.ht!pages!SpaceCurveGraphicsPage} +\index{SpaceCurveGraphicsPage!graphics.ht!pages} +<>= +\begin{page}{SpaceCurveGraphicsPage}{Parametric Space Curves} +\beginscroll +This page describes the plotting in three dimensional space of a curve +defined by the parametric equations x = f(t), y = g(t), z = h(t), where +f, g, and h are functions of the parameter t which ranges over a specified +interval. The basic draw command for this function utilizes either the +uncompiled functions or compiled functions format and uses the \spadfun{curve} +command to specify the three functions for the x, y, and z components of +the curve. The general format for uncompiled functions is: +\indent{5}\newline +{\em draw(curve(f(t),g(t),h(t)), t = a..b)} +\indent{0}\newline +where a..b is the segment defining the interval [a,b] over which the +parameter t ranges. In this case the functions are not compiled until the +draw command is executed. Here is an example: +\graphpaste{draw(curve(cos(t),sin(t),t), t=-12..12)} +In the case of compiled functions, the functions are named and compiled +independently. This is useful if you intend to use the functions often, or +if the functions are long and complex. The following lines show functions +whose parameters are of the type Small Float. The functions are compiled and +stored by Axiom when entered. +\indent{5}\newline +{\em NOTE: It is only necessary to click on the draw command button to plot +this example}. +\indent{0}\newline +\spadpaste{i1(t:SF):SF == sin(t)*cos(3*t/5) \bound{i1}} +\spadpaste{i2(t:SF):SF == cos(t)*cos(3*t/5) \bound{i2}} +\spadpaste{i3(t:SF):SF == cos(t)*sin(3*t/5) \bound{i3}} +Once the functions are compiled the draw command only needs the names of +the functions to execute. Here is a compiled functions example: +\graphpaste{draw(curve(i1,i2,i3),0..15*\%pi) \free{i1 i2 i3}} +Note that the parameter range does not take the variable name as in the +case of uncompiled functions. It is understood that the indicated range +applies to the parameter of the functions, which in this case is t. +\endscroll +\autobuttons \end{page} + +@ +\subsection{Parametric Tube Plots} +\label{ParametricTubeGraphicsPage} +\index{pages!ParametricTubeGraphicsPage!graphics.ht} +\index{graphics.ht!pages!ParametricTubeGraphicsPage} +\index{ParametricTubeGraphicsPage!graphics.ht!pages} +<>= +\begin{page}{ParametricTubeGraphicsPage}{Parametric Tube Plots} +\beginscroll +This page describes the plotting in three dimensional space of a tube +around a parametric space curve defined by the parametric equations +x = f(t), y = g(t), z = h(t), where f, g, and h are functions of the +parameter t which ranges over a specified interval. The basic draw command +for this function utilizes either the uncompiled functions or compiled +functions format and uses the \spadfun{curve} command to specify the three +functions for the x, y, and z components of the curve. This uses the same +format as that for space curves except that it requires a specification for +the radius of the tube. If the radius of the tube is 0, then the result is +the space curve itself. The general format for uncompiled functions is: +\indent{5}\newline +{\em draw(curve(f(t),g(t),h(t)), t = a..b, tubeRadius == r)} +\indent{0}\newline +where a..b is the segment defining the interval [a,b] over which the +parameter t ranges, and the tubeRadius is indicated by the variable r. +In this case the functions are not compiled until the draw command is +executed. Here is an example: +\graphpaste{draw(curve(sin(t)*cos(3*t/5), cos(t)*cos(3*t/5), cos(t)*sin(3*t/5)), t=0..15*\%pi,tubeRadius == .15)} +In the case of compiled functions, the functions are named and compiled +independently. This is useful if you intend to use the functions often, or +if the functions are long and complex. The following lines show functions +whose parameters are of the type Small Float. The functions are compiled and +stored by Axiom when entered. +\indent{5}\newline +{\em NOTE: It is only necessary to click on the draw command button to plot +this example}. +\indent{0}\newline +\spadpaste{t1(t:SF):SF == 4/(2-sin(3*t))*cos(2*t) \bound{t1}} +\newline +\spadpaste{t2(t:SF):SF == 4/(2-sin(3*t))*sin(2*t) \bound{t2}} +\newline +\spadpaste{t3(t:SF):SF == 4/(2-sin(3*t))*cos(3*t) \bound{t3}} +\newline +Once the function is compiled the draw command only needs the names of the +functions to execute. Here is a compiled functions example of a trefoil knot: +\graphpaste{draw(curve(t1,t2,t3),0..2*\%pi,tubeRadius == .2) \free{t1 t2 t3}} +Note that the parameter range does not take the variable name as in the +case of uncompiled functions. It is understood that the indicated range +applies to the parameter of the functions, which in this case is t. +Typically, the radius of the tube should be set between 0 and 1. A radius +of less than 0 results in it's positive counterpart and a radius of greater +than one causes self intersection. +\endscroll +\autobuttons \end{page} + +@ +\subsection{Parametric Surfaces} +\label{ParametricSurfaceGraphicsPage} +\index{pages!ParametricSurfaceGraphicsPage!graphics.ht} +\index{graphics.ht!pages!ParametricSurfaceGraphicsPage} +\index{ParametricSurfaceGraphicsPage!graphics.ht!pages} +<>= +\begin{page}{ParametricSurfaceGraphicsPage}{Parametric Surfaces} +\beginscroll +Graphing a surface defined by x = f(u,v), y = g(u,v), z = h(u,v). \newline +This page describes the plotting of surfaces defined by the parametric +equations of two variables, x = f(u,v), y = g(u,v), and z = h(u,v), +for which the ranges of u and v are explicitly defined. The basic draw +command for this function utilizes either the uncompiled function or +compiled function format and uses the \spadfun{surface} command to specify the +three functions for the x, y and z components of the surface. The general +format for uncompiled functions is: +\indent{5}\newline +{\em draw(surface(f(u,v),g(u,v),h(u,v)), u = a..b, v = c..d)} +\indent{0}\newline +where a..b and c..d are segments defining the intervals [a,b] and [c,d] over +which the parameters u and v span. In this case the functions are not +compiled until the draw command is executed. Here is an example of a +surface plotted using the parabolic cylindrical coordinate system option: +\graphpaste{draw(surface(u*cos(v), u*sin(v),v*cos(u)),u=-4..4,v=0..2*\%pi, + coordinates== parabolicCylindrical)} +In the case of compiled functions, the functions are named and compiled +independently. This is useful if you intend to use the functions often, or +if the functions are long and complex. The following lines show functions +whose parameters are of the type Small Float. The functions are compiled and +stored by Axiom when entered. +\indent{5}\newline +{\em NOTE: It is only necessary to click on the draw command button to plot +this example}. +\indent{0}\newline +\spadpaste{n1(u:SF,v:SF):SF == u*cos(v) \bound{n1}} +\newline +\spadpaste{n2(u:SF,v:SF):SF == u*sin(v) \bound{n2}} +\newline +\spadpaste{n3(u:SF,v:SF):SF == u \bound{n3}} +Once the function is compiled the draw command only needs the names of the +functions to execute. Here is a compiled functions example plotted using +the toroidal coordinate system option: \newline +\graphpaste{draw(surface(n1,n2,n3), 1.0..4.0, 1.0..4*\%pi, + coordinates == toroidal(1\$SF)) \free{n1 n2 n3}} +Note that the parameter ranges do not take the variable names as in the +case of uncompiled functions. The variables are entered in the order in +which they are defined in the function specification. In this case the +first range specifies the u-variable and the second range specifies the +v-variable. +\endscroll +\autobuttons \end{page} + +@ +\subsection{Building 3D Objects} +\label{3DObjectGraphicsPage} +\index{pages!3DObjectGraphicsPage!graphics.ht} +\index{graphics.ht!pages!3DObjectGraphicsPage} +\index{3DObjectGraphicsPage!graphics.ht!pages} +<>= +\begin{page}{3DObjectGraphicsPage}{Building 3D Objects} +\beginscroll +This page describes the Axiom facilities for creating three dimensional +objects constructed from geometric primitives. The Axiom operation +\spadfun{create3Space()} creates a space to which points, curves, and +polygons can be added using the operations from the \spadtype{ThreeSpace} +domain. The contents of this space can then be displayed in a viewport +using the \spadfun{makeViewport3D()} command. It will be necessary to +have these operations exposed in order to use them. \indent{5}\newline +{\em NOTE: It is only necessary to click on the makeViewport3D command button +to plot this curve example}. +\indent{0}\newline +Initially, the space which will hold the objects must be defined and +compiled, as in the following example: +\spadpaste{space := create3Space()\$(ThreeSpace SF) \bound{space}} +Now objects can be sent to this {\em space} as per the operations allowed by +the \spadtype{ThreeSpace} domain. The following examples place curves into +{\em space}. +\spadpaste{curve(space,[[0,20,20],[0,20,30],[0,30,30],[0,30,100], [0,20,100],[0,20,110],[0,50,110],[0,50,100],[0,40,100], [0,40,30],[0,50,30],[0,50,20],[0,20,20]]) \bound{curveI}} +\newline +\spadpaste{curve(space,[[0,80,20],[0,70,20],[0,70,110],[0,110,110], [0,120,100],[0,120,70],[0,115,65],[0,120,60],[0,120,30], [0,110,20],[0,80,20],[0,80,30],[0,105,30],[0,110,35]]) \bound{curveB1}} +\newline +\spadpaste{curve(space,[[0,110,35],[0,110,55],[0,105,60],[0,80,60],[0,80,70], [0,105,70],[0,110,75],[0,110,95],[0,105,100],[0,80,100], [0,80,30]]) \bound{curveB2}} +\newline +\spadpaste{closedCurve(space,[[0,140,20],[0,140,110],[0,150,110],[0,170,50], [0,190,110],[0,200,110],[0,200,20],[0,190,20],[0,190,75], [0,175,35],[0,165,35],[0,150,75],[0,150,20]]) \bound{curveM}} +\spadpaste{closedCurve(space,[[200,0,20], [200,0,110], [185,0,110], [160,0,45], [160,0,110], [150,0,110], [150,0,20], [165,0,20], [190,0,85], [190,0,20]]) \bound{curveN}} +\spadpaste{closedCurve(space,[[140,0,20], [120,0,110], [110,0,110], [90,0,20], [100,0,20], [108,0,50], [123,0,50], [121,0,60], [110,0,60], [115,0,90], [130,0,20]]) \bound{curveA}} +\spadpaste{closedCurve(space,[[80,0,30], [80,0,100], [70,0,110], [40,0,110], [30,0,100], [30,0,90], [40,0,90], [40,0,95], [45,0,100], [65,0,100], [70,0,95], [70,0,35], [65,0,30], [45,0,30], [40,0,35], [40,0,60], [50,0,60], [50,0,70], [30,0,70], [30,0,30], [40,0,20], [70,0,20]]) \bound{curveG}} +Once {\em space} contains the desired elements a viewport is created and +displayed with the following command: +\graphpaste{makeViewport3D(space,[title("Curves")])\$VIEW3D \free{space curveI curveB1 curveB2 curveM curveN curveA curveG}} +The parameters for \spadfun{makeViewport3D()} in this example are {\em space}, +which is the name of the three dimensional space that was defined, and a +string, "curve", which is the title for the viewport. The tailing string +{\em \$VIEW3D} exposes the command \spadfun{makeViewport3D()} from the domain +\spadtype{ThreeDimensionalViewport} if these commands are unexposed. +\endscroll +\autobuttons \end{page} + +@ +\subsection{Two Dimensional Graphics} +\label{TwoDimensionalGraphicsPage} +\begin{itemize} +\item OneVariableGraphicsPage \ref{OneVariableGraphicsPage} on +page~\pageref{OneVariableGraphicsPage} +\item ParametricCurveGraphicsPage \ref{ParametricCurveGraphicsPage} on +page~\pageref{ParametricCurveGraphicsPage} +\item PolarGraphicsPage \ref{PolarGraphicsPage} on +page~\pageref{PolarGraphicsPage} +\item ImplicitCurveGraphicsPage \ref{ImplicitCurveGraphicsPage} on +page~\pageref{ImplicitCurveGraphicsPage} +\item ListPoinstsGraphicsPage \ref{ListPoinstsGraphicsPage} on +page~\pageref{ListPoinstsGraphicsPage} +\end{itemize} +\index{pages!TwoDimensionalGraphicsPage!graphics.ht} +\index{graphics.ht!pages!TwoDimensionalGraphicsPage} +\index{TwoDimensionalGraphicsPage!graphics.ht!pages} +<>= +\begin{page}{TwoDimensionalGraphicsPage}{Two Dimensional Graphics} +\beginscroll +\beginmenu +\menulink{Functions of One Variable}{OneVariableGraphicsPage} \newline +Plot curves defined by an equation y = f(x). +\menulink{Parametric Curves}{ParametricCurveGraphicsPage} \newline +Plot curves defined by parametric equations x = f(t), y = g(t). +\menulink{Polar Coordinates}{PolarGraphicsPage} \newline +Plot curves given in polar form by an equation r = f(theta). +\menulink{Implicit Curves}{ImplicitCurveGraphicsPage} \newline +Plot non-singular curves defined by a polynomial equation +\menulink{Lists of Points}{ListPointsGraphicsPage} \newline +Plot lists of points in the (x,y)-plane. +% \menulink{Sequences}{SeqGraphicsPage} \newline +% Plot a sequence a1, a2, a3,... +% \menulink{Complex Functions}{CxFuncGraphicsPage} \newline +% Plot functions of a complex variable using grid plots. +\endmenu +\endscroll +\autobuttons \end{page} + +@ +\subsection{Functions of One Variable} +\label{OneVariableGraphicsPage} +\index{pages!OneVariableGraphicsPage!graphics.ht} +\index{graphics.ht!pages!OneVariableGraphicsPage} +\index{OneVariableGraphicsPage!graphics.ht!pages} +<>= +\begin{page}{OneVariableGraphicsPage}{Functions of One Variable} +\beginscroll +Here we wish to plot a function y = f(x) on an interval [a,b]. +As an example, let's take the function y = sin(tan(x)) - tan(sin(x)) +on the interval [0,6]. +Here is the simplest command that will do this: +\graphpaste{draw(sin(tan(x)) - tan(sin(x)),x = 0..6)} +Notice that Axiom compiled a function before the graph was put +on the screen. +The expression sin(tan(x)) - tan(sin(x)) was converted to a compiled +function so that it's value for various values of x could be computed +quickly and efficiently. +Let's graph the same function on a different interval and this time +we'll give the graph a title. +The title is a String, which is an optional argument of the command 'draw'. +\graphpaste{draw(sin(tan(x)) - tan(sin(x)),x = 10..16,title == "y = sin tan x - tan sin x")} +Once again the expression sin(tan(x)) - tan(sin(x)) was converted to a +compiled function before any points were computed. +If you want to graph the same function on a number of intervals, it's +a good idea to write down a function definition so that the function +only has to be compiled once. +Here's an example: +\spadpaste{f(x) == (x-1)*(x-2)*(x-3) \bound{f}} +\newline +\graphpaste{draw(f, 0..2, title == "y = f(x) on \[0,2\]") \free{f}} +\newline +\graphpaste{draw(f, 0..4,title == "y = f(x) on \[0,4\]") \free{f}} +Notice that our titles can be whatever we want, as long as they are +enclosed by double quotes. However, a title which is too long to fit +within the viewport title window will be clipped. +\endscroll +\autobuttons \end{page} + +@ +\subsection{Parametric Curves} +\label{ParametricCurveGraphicsPage} +\index{pages!ParametricCurveGraphicsPage!graphics.ht} +\index{graphics.ht!pages!ParametricCurveGraphicsPage} +\index{ParametricCurveGraphicsPage!graphics.ht!pages} +<>= +\begin{page}{ParametricCurveGraphicsPage}{Parametric Curves} +\beginscroll +One way of producing interesting curves is by using parametric equations. +Let x = f(t) and y = g(t) for two functions f and g as the parameter +t ranges over an interval \[a,b\]. +Here's an example: +\graphpaste{draw(curve(sin(t)*sin(2*t)*sin(3*t), sin(4*t)*sin(5*t)*sin(6*t)), t = 0..2*\%pi)} +Here 0..2*\%pi represents the interval over which the variable t ranges. +In the case of parametric curves, Axiom will compile two functions, +one for each of the functions f and g. +You may also put a title on a graph. +The title may be an arbitrary string and is an optional argument +to the command 'draw'. +For example: +\graphpaste{draw(curve(cos(t), sin(t)), t = 0..2*\%pi, title == "The Unit Circle")} +If you plan on plotting x = f(t), y = g(t) as t ranges over several intervals, +you may want to define functions f and g, so that they need not be +recompiled every time you create a new graph. +Here's an example: +\spadpaste{f(t:SF):SF == sin(3*t/4) \bound{f}} +\newline +\spadpaste{g(t:SF):SF == sin(t) \bound{g}} +\newline +\graphpaste{draw(curve(f,g), 0..\%pi) \free{f g}} +\newline +\graphpaste{draw(curve(f,g) ,\%pi..2*\%pi) \free{f g}} +\newline +\graphpaste{draw(curve(f,g), -4*\%pi..4*\%pi) \free{f g}} +These examples show how the curve changes as the range of parameter t varies. +\endscroll +\autobuttons \end{page} + +@ +\subsection{Polar Coordinates} +\label{PolarGraphicsPage} +\index{pages!PolarGraphicsPage!graphics.ht} +\index{graphics.ht!pages!PolarGraphicsPage} +\index{PolarGraphicsPage!graphics.ht!pages} +<>= +\begin{page}{PolarGraphicsPage}{Polar Coordinates} +\beginscroll +Graphs in polar coordinates are given by an equation r = f(theta) as +theta ranges over an interval. +This is equivalent to the parametric curve x = f(theta) * cos(theta), +y = f(theta) * sin(theta) as theta ranges over the same interval. +You may create such curves using the command 'draw', with the optional +argument 'coordinates == polar'. +Here are some examples: +\graphpaste{draw(1,t = 0..2*\%pi,coordinates == polar, title == "The Unit Circle")} +\newline +\graphpaste{draw(sin(17*t), t = 0..2*\%pi, coordinates == polar, title == "A Petal Curve")} +%When you don't specify an interval, Axiom will assume that you +%mean 0..2*\%pi. +You may also define your own functions, when you plan on plotting the +same curve as theta varies over several intervals. +\spadpaste{f(t) == cos(4*t/7) \bound{f}} +\newline +\graphpaste{draw(f, 0..2*\%pi, coordinates == polar) \free{f}} +\newline +\graphpaste{draw(f, 0..14*\%pi, coordinates == polar) \free{f}} +For information on plotting graphs in other coordinate systems see the +pages for the \spadtype{CoordinateSystems} domain. +\endscroll +\autobuttons \end{page} + +@ +\subsection{Implicit Curves} +\label{ImplicitCurveGraphicsPage} +\index{pages!ImplicitCurveGraphicsPage!graphics.ht} +\index{graphics.ht!pages!ImplicitCurveGraphicsPage} +\index{ImplicitCurveGraphicsPage!graphics.ht!pages} +<>= +\begin{page}{ImplicitCurveGraphicsPage}{Implicit Curves} +\beginscroll +Axiom has facilities for graphing a non-singular algebraic curve +in a rectangular region of the plane. +An algebraic curve is a curve defined by a polynomial equation +p(x,y) = 0. +Non-singular means that the curve is "smooth" in that it does not +cross itself or come to a point (cusp). +Algebraically, this means that for any point (a,b) on the curve +(i.e. a point such that p(a,b) = 0), the partial derivatives +dp/dx(a,b) and dp/dy(a,b) are not both zero. +We require that the polynomial have rational or integral coefficients. +Here is a Cartesian ovals algebraic curve example: +(click on the draw button to execute this example) +\spadpaste{p := ((x**2 + y**2 + 1) - 8*x)**2 - (8*(x**2 + y**2 + 1) - 4*x - 1) \bound{p} } +\graphpaste{draw(p = 0, x, y, range == \[-1..11, -7..7\], title == "Cartesian Ovals") \free{p}} +{\em A range must be declared for each variable specified in the algebraic +curve equation}. +\endscroll +\autobuttons \end{page} + +@ +\subsection{Lists of Points} +\label{ListPointsGraphicsPage} +\index{pages!ListPointsGraphicsPage!graphics.ht} +\index{graphics.ht!pages!ListPointsGraphicsPage} +\index{ListPointsGraphicsPage!graphics.ht!pages} +<>= +\begin{page}{ListPointsGraphicsPage}{Lists of Points} +\beginscroll +Axiom has the ability to create lists of points in a two dimensional +graphics viewport. This is done by utilizing the \spadtype{GraphImage} and +\spadtype{TwoDimensionalViewport} domain facilities. +\indent{5}\newline +{\em NOTE: It is only necessary to click on the makeViewport2D command button +to plot this curve example}. +\indent{0}\newline +In this example, the \spadfun{makeGraphImage} command takes a list of lists of +points parameter, a list of colors for each point in the graph, a list of +colors for each line in the graph, and a list of numbers which indicate the +size of each point in the graph. The following lines create list of lists of +points which can be read be made into two dimensional graph images. +\spadpaste{p1 := point [1::SF,1::SF]\$(Point SF) \bound{p1}} +\newline +\spadpaste{p2 := point [0::SF,1::SF]\$(Point SF) \bound{p2}} +\newline +\spadpaste{p3 := point [0::SF,0::SF]\$(Point SF) \bound{p3}} +\newline +\spadpaste{p4 := point [1::SF,0::SF]\$(Point SF) \bound{p4}} +\newline +\spadpaste{p5 := point [1::SF,.5::SF]\$(Point SF) \bound{p5}} +\newline +\spadpaste{p6 := point [.5::SF,0::SF]\$(Point SF) \bound{p6}} +\newline +\spadpaste{p7 := point [0::SF,0.5::SF]\$(Point SF) \bound{p7}} +\newline +\spadpaste{p8 := point [.5::SF,1::SF]\$(Point SF) \bound{p8}} +\newline +\spadpaste{p9 := point [.25::SF,.25::SF]\$(Point SF) \bound{p9}} +\newline +\spadpaste{p10 := point [.25::SF,.75::SF]\$(Point SF) \bound{p10}} +\newline +\spadpaste{p11 := point [.75::SF,.75::SF]\$(Point SF) \bound{p11}} +\newline +\spadpaste{p12 := point [.75::SF,.25::SF]\$(Point SF) \bound{p12}} +\newline +\spadpaste{llp := [[p1,p2],[p2,p3],[p3,p4],[p4,p1],[p5,p6],[p6,p7],[p7,p8],[p8,p5],[p9,p10],[p10,p11],[p11,p12],[p12,p9]] \bound{llp} \free{p1 p2 p3 p4 p5 p6 p7 p8 p9 p10 p11 p12}} +\newline +These lines set the point color and size, and the line color for all components +of the graph. +\spadpaste{size1 := 6::PositiveInteger \bound{size1}} +\newline +\spadpaste{size2 := 8::PositiveInteger \bound{size2}} +\newline +\spadpaste{size3 := 10::PositiveInteger \bound{size3}} +\newline +\spadpaste{lsize := [size1, size1, size1, size1, size2, size2, size2, size2, size3, size3, size3, size3] \bound{lsize} \free{size1 size2 size3}} +\newline +\spadpaste{pc1 := pastel red() \bound{pc1}} +\newline +\spadpaste{pc2 := dim green() \bound{pc2}} +\newline +\spadpaste{pc3 := pastel yellow() \bound{pc3}} +\newline +\spadpaste{lpc := [pc1, pc1, pc1, pc1, pc2, pc2, pc2, pc2, pc3, pc3, pc3, pc3] \bound{lpc} \free{pc1 pc2 pc3}} +\newline +\spadpaste{lc := [pastel blue(), light yellow(), dim green(), bright red(), light green(), dim yellow(), bright blue(), dark red(), pastel red(), light blue(), dim green(), light yellow()] \bound{lc}} +\newline +Now the graph image is created and named according to the component +specifications indicated above. The \spadfun{makeViewport2D} command then +creates a two dimensional viewport for this graph according to the list of +options specified within the brackets. +\spadpaste{g := makeGraphImage(llp,lpc,lc,lsize)\$GRIMAGE \bound{g} \free{llp lpc lc lsize}} +\newline +\graphpaste{makeViewport2D(g,[title("Lines")])\$VIEW2D \free{g}} +The \spadfun{makeViewport2D} command takes a list of options as a parameter. +In this example the string "Lines" is designated as the viewport's title. +\endscroll +\autobuttons \end{page} + +@ + +\subsection{Stand-alone Viewport} +\label{ViewportPage} +\index{pages!ViewportPage!graphics.ht} +\index{graphics.ht!pages!ViewportPage} +\index{ViewportPage!graphics.ht!pages} +<>= +\begin{page}{ViewportPage}{Stand-alone Viewport} +\beginscroll +To get a viewport on a \HyperName{} page, you first need to +create one in Axiom and write it out to a +file that \HyperName{} can call up. \newline +For example, here we draw a saddle function and assign +the result to the variable \spad{v}: +\newline +\graphpaste{v := draw(x*x-y*y,x=-1..1,y=-1..1) \bound{v}} +\newline +Now that we've created the viewport, we want to write +the data out to a file. \newline +To do this, we use the \spadfunFrom{write}{ThreeDimensionalViewport} command which takes +the following arguments: the viewport to write out, +the title of the file to write it out to, and an optional +argument telling the write command what type (or types) of +data you want to write in additional to the one Axiom will +always write out. The optional argument could be +a string, like "pixmap", or a list of strings, like \["postscript","pixmap"\]. +\HyperName{} needs a "pixmap" data type to include a graph in a page +so in this case, we write the viewport and tell it to +also write a "pixmap" file, as well: +\newline +\spadpaste{write(v,"saddle","pixmap") \free{v}} +\newline +Now we want to put this viewport into a \HyperName{} page. +Say you've created a viewport and written it out +to a file called "/tmp/mobius". (Axiom actually +tags a ".view" at the end of a viewport data file to +make it easier to spot when you're rummaging through +your file system, but you needn't worry about that here +since Axiom will always automatically add on a +".view" for you.) \newline + +{\bf Including Viewports} \newline +To put a viewport in a +\HyperName{} page, include the following line in your \HyperName{} +source code: \newline +\space{5}\\viewport\{/tmp/mobius\} \newline +You will get this on your page: \newline +%%%\space{4}\viewport{/tmp/mobius}\newline +%\space{4}\spadviewport{mobius}\newline +\centerline{\spadviewport{mobius}}\newline + +{\bf Creating Viewport Buttons} \newline +To make an active button that would make this viewport +come to life, include the following: \newline +\space{5}\\viewportbutton\{ViewButton\}\{/tmp/mobius\} \newline +this creates this button...\newline +%%%\centerline{\viewportbutton{ViewButton}{/tmp/mobius}}\newline +\centerline{\spadviewportbutton{ViewButton}{mobius}}\newline + +{\bf Creating Active Viewports} \newline +To merge the two things descibed above, namely, getting a picture of a +viewport and creating a button to invoke a live viewport, you can do +the following: \newline + +%\space{5}\\viewportasbutton\{/tmp/mobius\} \newline +\centerline{\\viewportasbutton\{/tmp/mobius\}}\newline + +This would create a picture of a viewport that is an active +button as well. Try it: + +%%%\space{5}\viewportasbutton{/tmp/mobius} \newline +%\space{5}\spadviewportasbutton{mobius} \newline +\centerline{\spadviewportasbutton{mobius}}\newline + +{\bf Including Viewports Distributed with Axiom} \newline +All the above commands have counterparts that allow you to +access viewports that are already packaged with Axiom. +To include those viewports, just add on an "axiom" prefix +to the above commands: \newline + +\centerline{\\axiomviewport\{vA\} for \\viewport\{vA\}} +\centerline{\\axiomviewportbutton\{vB\} for \\viewportbutton\{vB\}} +\centerline{\\axiomviewportasbutton\{vC\} for \\viewportasbutton\{vC\}} \newline + +All these macros really do is include some path that +indicates where Axiom stores the viewports. +\newline +\endscroll +\autobuttons \end{page} + +@ +\section{grpthry.ht} +\subsection{Group Theory} +\label{GroupTheoryPage} +\begin{itemize} +\item InfoGroupTheoryPage \ref{InfoGroupTheoryPage} on +page~\pageref{InfoGroupTheoryPage} +\item InfoRepTheoryPage \ref{InfoRepTheoryPage} on +page~\pageref{InfoRepTheoryPage} +\item RepA6Page \ref{RepA6Page} on +page~\pageref{RepA6Page} +\end{itemize} +\index{pages!GroupTheoryPage!grpthry.ht} +\index{grpthry.ht!pages!GroupTheoryPage} +\index{GroupTheoryPage!grpthry.ht!pages} +<>= +\begin{page}{GroupTheoryPage}{Group Theory} +% authors: H. Gollan, J. Grabmeier, August 1989 +\beginscroll +Axiom can work with individual permutations, permutation +groups and do representation theory. +\horizontalline +\newline +\beginmenu +\menulink{Info on Group Theory}{InfoGroupTheoryPage} + + +%\menulink{Permutation}{PermutationXmpPage} +%Calculating within symmetric groups. + +%\menulink{Permutation Groups}{PermutationGroupXmpPage} +%Working with subgroups of a symmetric group. + +%\menulink{Permutation Group Examples}{PermutationGroupExampleXmpPage} +%Working with permutation groups, predefined in the system as Rubik's group. + +\menulink{Info on Representation Theory}{InfoRepTheoryPage} + +%\menulink{Irreducible Representations of Symmetric Groups}{IrrRepSymNatXmpPage} +%Alfred Young's natural form for these representations. + +%\menulink{Representations of Higher Degree}{RepresentationPackage1XmpPage} +%Constructing new representations by symmetric and antisymmetric +%tensors. + +%\menulink{Decomposing Representations}{RepresentationPackage2XmpPage} +%Parker's `Meat-Axe', working in prime characteristics. + +\menulink{Representations of \texht{$A_6$}{A6}}{RepA6Page} +The irreducible representations of the alternating group \texht{$A_6$}{A6} over fields +of characteristic 2. +\endmenu +\endscroll +\autobuttons \end{page} + +@ +\subsection{Representations of $A_6$ A6} +\label{RepA6Page} +\index{pages!RepA6Page!grpthry.ht} +\index{grpthry.ht!pages!RepA6Page} +\index{RepA6Page!grpthry.ht!pages} +<>= +\begin{page}{RepA6Page}{Representations of \texht{$A_6$}{A6}} +% author: J. Grabmeier, 08/08/89 +\beginscroll +In what follows you'll see how to use Axiom to get all the irreducible +representations of the alternating group \texht{$A_6$}{A6} over the field with two +elements (GF 2). +First, we generate \texht{$A_6$}{A6} by a three-cycle: x = (1,2,3) +and a 5-cycle: y = (2,3,4,5,6). Next we have Axiom calculate +the permutation representation over the integers and over GF 2: +\spadpaste{genA6 : LIST PERM INT := [cycle [1,2,3],cycle [2,3,4,5,6]] \bound{genA6}} +\spadpaste{pRA6 := permutationRepresentation (genA6, 6) \bound{pRA6} \free{genA6} +} +Now we apply Parker's 'Meat-Axe' and split it: +\spadpaste{sp0 := meatAxe (pRA6::(LIST MATRIX PF 2)) \free{pRA6} \bound{sp0}} +We have found the trivial module as a quotient module +and a 5-dimensional sub-module. +Try to split again: +\spadpaste{sp1 := meatAxe sp0.1 \bound{sp1}} +and we find a 4-dimensional sub-module and the trivial one again. +Now we can test if this representaton is absolutely irreducible: +\spadpaste{isAbsolutelyIrreducible? sp1.2 } +and we see that this 4-dimensional representation is absolutely irreducible. +So, we have found a second irreducible representation. +Now, we construct a representation by reducing an irreducible one +of the symmetric group S_6 over the integers mod 2. +We take the one labelled by the partition [2,2,1,1] and +restrict it to \texht{$A_6$}{A6}: +\spadpaste{d2211 := irreducibleRepresentation ([2,2,1,1],genA6) \bound{d2211} } +Now split it: +\spadpaste{d2211m2 := d2211:: (LIST MATRIX PF 2); sp2 := meatAxe d2211m2 \free{d2211} +\bound{sp2}} +This gave both a five and a four dimensional representation. +Now we take the 4-dimensional one +and we shall see that it is absolutely irreducible: +\spadpaste{isAbsolutelyIrreducible? sp2.1} +The two 4-dimensional representations are not equivalent: +\spadpaste{areEquivalent? (sp1.2, sp2.1)} +So we have found a third irreducible representation. +Now we construct a new representation using the tensor product +and try to split it: +\spadpaste{dA6d16 := tensorProduct(sp1.2,sp2.1); meatAxe dA6d16 \bound{dA6d16}} +The representation is irreducible, but it may be not absolutely irreducible. +\spadpaste{isAbsolutelyIrreducible? dA6d16} +So let's try the same procedure over the field with 4 elements: +\spadpaste{sp3 := meatAxe (dA6d16 :: (LIST MATRIX FF(2,2))) \bound{sp3}} +Now we find two 8-dimensional representations, dA6d8a and dA6d8b. +Both are absolutely irreducible... +\spadpaste{isAbsolutelyIrreducible? sp3.1} +\spadpaste{isAbsolutelyIrreducible? sp3.2} +and they are not equivalent: +\spadpaste{areEquivalent? (sp3.1,sp3.2)} +So we have found five absolutely irreducible representations of \texht{$A_6$}{A6} +in characteristic 2. +General theory now tells us that there are no more irreducible ones. +Here, for future reference are all the absolutely irreducible 2-modular +representations of \texht{$A_6$}{A6} +\spadpaste{sp0.2 \free{sp0}} +\spadpaste{sp1.2 \free{sp1}} +\spadpaste{sp2.1 \free{sp2}} +\spadpaste{sp3.1 \free{sp3}} +\spadpaste{sp3.2 \free{sp3}} +And here again is the irreducible, but not absolutely irreducible +representations of \texht{$A_6$}{A6} over GF 2 +\spadpaste{dA6d16 \free{dA6d16}} +\endscroll +\autobuttons \end{page} + +@ +\subsection{Representation Theory} +\label{InfoRepTheoryPage} +\index{pages!InfoRepTheoryPage!grpthry.ht} +\index{grpthry.ht!pages!InfoRepTheoryPage} +\index{InfoRepTheoryPage!grpthry.ht!pages} +<>= +\begin{page}{InfoRepTheoryPage}{Representation Theory} +\beginscroll +\horizontalline +Representation theory for finite groups studies finite groups by +embedding them in a general linear group over a field or an +integral domain. +Hence, we are representing each element of the group by +an invertible matrix. +Two matrix representations of a given group are equivalent, if, by changing the +basis of the underlying +space, you can go from one to the other. When you change bases, you +transform the matrices that are the images of elements by +conjugating them by an invertible matrix. +\newline +\newline +If we can find a subspace which is fixed under the image +of the group, then there exists a `base change' after which all the representing + matrices +are in upper triangular block form. The block matrices on +the main diagonal give a new representation of the group of lower degree. +Such a representation is said to be `reducible'. +\newline +\beginmenu +%\menulink{Irreducible Representations of Symmetric Groups}{IrrRepSymNatXmpPage} + +%Alfred Young's natural form for these representations. + +%\menulink{Representations of Higher Degree}{RepresentationPackage1XmpPage} +%Constructing new representations by symmetric and antisymmetric +%tensors. + +%\menulink{Decomposing Representations}{RepresentationPackage2XmpPage} +%Parker's `Meat-Axe', working in prime characteristics. + +\menulink{Representations of \texht{$A_6$}{A6}}{RepA6Page} +The irreducible representations of the alternating group \texht{$A_6$}{A6} over fields +of characteristic 2. +\endmenu +\endscroll +\autobuttons \end{page} + +@ +\subsection{Group Theory} +\label{InfoGroupTheoryPage} +\index{pages!InfoGroupTheoryPage!grpthry.ht} +\index{grpthry.ht!pages!InfoGroupTheoryPage} +\index{InfoGroupTheoryPage!grpthry.ht!pages} +<>= +\begin{page}{InfoGroupTheoryPage}{Group Theory} +%% +%% Johannes Grabmeier 03/02/90 +%% +\beginscroll +A {\it group} is a set G together with an associative operation +* satisfying the axioms of existence +of a unit element and an inverse of every element of the group. +The Axiom category \spadtype{Group} represents this setting. +Many data structures in Axiom are groups and therefore there +is a large variety of examples as fields and polynomials, +although the main interest there is not the group structure. + +To work with and in groups in a concrete manner some way of +representing groups has to be chosen. A group can be given +as a list of generators and a set of relations. If there +are no relations, then we have a {\it free group}, realized +in the domain \spadtype{FreeMonoid} which won't be discussed here. +We consider {\it permutation groups}, where a group +is realized as a subgroup of the symmetric group of a set, i.e. +the group of all bijections of a set, the operation being the +composition of maps. +Indeed, every group can be realized this way, although +this may not be practical. + +Furthermore group elements can be given as invertible matrices. +The group operation is reflected by matrix multiplication. +More precise in representation theory group homomophisms +from a group to general linear groups are contructed. +Some algorithms are implemented in Axiom. +\newline +%\beginmenu +%\menulink{Permutation}{PermutationXmpPage} +%Calculating within symmetric groups. + +%\menulink{Permutation Groups}{PermutationGroupXmpPage} +%Working with subgroups of a symmetric group. + +%\menulink{Permutation Group Examples}{PermutationGroupExampleXmpPage} +%Working with permutation groups, predefined in the system as Rubik's group. +%\endmenu +\endscroll +\autobuttons \end{page} + + +@ +\section{gstbl.ht} +<>= +\newcommand{\GeneralSparseTableXmpTitle}{GeneralSparseTable} +\newcommand{\GeneralSparseTableXmpNumber}{9.30} + +@ +\subsection{GeneralSparseTable} +\label{GeneralSparseTableXmpPage} +See TableXmpPage \ref{TableXmpPage} on page~\pageref{TableXmpPage} +\index{pages!GeneralSparseTableXmpPage!gstbl.ht} +\index{gstbl.ht!pages!GeneralSparseTableXmpPage} +\index{GeneralSparseTableXmpPage!gstbl.ht!pages} +<>= +\begin{page}{GeneralSparseTableXmpPage}{GeneralSparseTable} +\beginscroll + +Sometimes when working with tables there is a natural value to use +as the entry in all but a few cases. +The \spadtype{GeneralSparseTable} constructor can be used to provide any +table type with a default value for entries. +See \downlink{`Table'}{TableXmpPage}\ignore{Table} +for general information about tables. +\showBlurb{GeneralSparseTable} + +Suppose we launched a fund-raising campaign to raise fifty thousand dollars. +To record the contributions, we want a table with strings as keys +(for the names) and integer entries (for the amount). +In a data base of cash contributions, unless someone +has been explicitly entered, it is reasonable to assume they have made +a zero dollar contribution. +\xtc{ +This creates a keyed access file with default entry \spad{0}. +}{ +\spadpaste{patrons: GeneralSparseTable(String, Integer, KeyedAccessFile(Integer), 0) := table() ; \bound{patrons}} +} +\xtc{ +Now \spad{patrons} can be used just as any other table. +Here we record two gifts. +}{ +\spadpaste{patrons."Smith" := 10500 \free{patrons}\bound{smith}} +} +\xtc{ +}{ +\spadpaste{patrons."Jones" := 22000 \free{smith}\bound{jones}} +} +\xtc{ +Now let us look up the size of the contributions from Jones and Stingy. +}{ +\spadpaste{patrons."Jones" \free{jones}} +} +\xtc{ +}{ +\spadpaste{patrons."Stingy" \free{jones}} +} +\xtc{ +Have we met our seventy thousand dollar goal? +}{ +\spadpaste{reduce(+, entries patrons) \free{jones}} +} +\noOutputXtc{ +So the project is cancelled and we can delete the data base: +}{ +\spadpaste{)system rm -r kaf*.sdata \free{patrons}} +} +\endscroll +\autobuttons +\end{page} + +@ +\section{heap.ht} +<>= +\newcommand{\HeapXmpTitle}{Heap} +\newcommand{\HeapXmpNumber}{9.32} + +@ +\subsection{Heap} +\label{HeapXmpPage} +See FlexibleArrayXmpPage \ref{FlexibleArrayXmpPage} +on page~\pageref{FlexibleArrayXmpPage} +\index{pages!HeapXmpPage!heap.ht} +\index{heap.ht!pages!HeapXmpPage} +\index{HeapXmpPage!heap.ht!pages} +<>= +\begin{page}{HeapXmpPage}{Heap} +\beginscroll +The domain \spadtype{Heap(S)} implements a priority queue of +objects of type \spad{S} such that +the operation \spadfunX{extract} removes and returns +the maximum element. +The implementation represents heaps as flexible arrays +(see \downlink{`FlexibleArray'}{FlexibleArrayXmpPage}\ignore{FlexibleArray}). +The representation and algorithms give complexity +of \texht{$O(\log(n))$}{O(log n)} for insertion and extractions, +and \texht{$O(n)$}{O(n)} for construction. + +\xtc{ +Create a heap of six elements. +}{ +\spadpaste{h := heap [-4,9,11,2,7,-7]\bound{h}} +} +\xtc{ +Use \spadfunX{insert} to add an element. +}{ +\spadpaste{insert!(3,h)\bound{h1}\free{h}} +} +\xtc{ +The operation \spadfunX{extract} removes and returns +the maximum element. +}{ +\spadpaste{extract! h\bound{h2}\free{h1}} +} +\xtc{ +The internal structure of \spad{h} has been +appropriately adjusted. +}{ +\spadpaste{h\free{h2}} +} +\xtc{ +Now \spadfunX{extract} elements repeatedly +until none are left, collecting the elements in a list. +}{ +\spadpaste{[extract!(h) while not empty?(h)]\bound{h2}} +} +\xtc{ +Another way to produce the same result is by defining +a \userfun{heapsort} function. +}{ +\spadpaste{heapsort(x) == (empty? x => []; cons(extract!(x),heapsort x))\bound{f}} +} +\xtc{ +Create another sample heap. +}{ +\spadpaste{h1 := heap [17,-4,9,-11,2,7,-7]\bound{h1}} +} +\xtc{ +Apply \spadfun{heapsort} to present elements in order. +}{ +\spadpaste{heapsort h1\free{f}} +} +\endscroll +\autobuttons +\end{page} + +@ +\section{hexadec.ht} +<>= +\newcommand{\HexadecimalExpansionXmpTitle}{HexadecimalExpansion} +\newcommand{\HexadecimalExpansionXmpNumber}{9.33} + +@ +\subsection{HexadecimalExpansion} +\label{HexadecimalExpansionXmpPage} +\begin{itemize} +\item DecimalExpansion \ref{DecimalExpansion} on +page~pageref{DecimalExpansion} +\item BinaryExpansion \ref{BinaryExpansion} on +page~pageref{BinaryExpansion} +\item RadixExpansion \ref{RadixExpansion} on +page~pageref{RadixExpansion} +\end{itemize} +\index{pages!HexadecimalExpansionXmpPage!hexadec.ht} +\index{hexadec.ht!pages!HexadecimalExpansionXmpPage} +\index{HexadecimalExpansionXmpPage!hexadec.ht!pages} +<>= +\begin{page}{HexadecimalExpansionXmpPage}{HexadecimalExpansion} +\beginscroll + +All rationals have repeating hexadecimal expansions. +The operation \spadfunFrom{hex}{HexadecimalExpansion} returns these +expansions of type \spadtype{HexadecimalExpansion}. +Operations to access the individual numerals of a hexadecimal expansion can +be obtained by converting the value to \spadtype{RadixExpansion(16)}. +More examples of expansions are available in the +\downlink{`DecimalExpansion'}{DecimalExpansionXmpPage}\ignore{DecimalExpansion}, +\downlink{`BinaryExpansion'}{BinaryExpansionXmpPage}\ignore{BinaryExpansion}, and +\downlink{`RadixExpansion'}{RadixExpansionXmpPage}\ignore{RadixExpansion}. + +\showBlurb{HexadecimalExpansion} + +\xtc{ +This is a hexadecimal expansion of a rational number. +}{ +\spadpaste{r := hex(22/7) \bound{r}} +} +\xtc{ +Arithmetic is exact. +}{ +\spadpaste{r + hex(6/7) \free{r}} +} +\xtc{ +The period of the expansion can be short or long \ldots +}{ +\spadpaste{[hex(1/i) for i in 350..354] } +} +\xtc{ +or very long! +}{ +\spadpaste{hex(1/1007) } +} +\xtc{ +These numbers are bona fide algebraic objects. +}{ +\spadpaste{p := hex(1/4)*x**2 + hex(2/3)*x + hex(4/9) \bound{p}} +} +\xtc{ +}{ +\spadpaste{q := D(p, x) \free{p}\bound{q}} +} +\xtc{ +}{ +\spadpaste{g := gcd(p, q) \free{p}\free{q}} +} +\endscroll +\autobuttons +\end{page} + +@ +\section{htxadvpage1.ht} +\subsection{Input Areas} +\label{HTXAdvPage1} +See HTXAdvPage2 \ref{HTXAdvPage2} on page~\pageref{HTXAdvPage2} +\index{pages!HTXAdvPage1!htxadvpage1.ht} +\index{htxadvpage1.ht!pages!HTXAdvPage1} +\index{HTXAdvPage1!htxadvpage1.ht!pages} +<>= +\begin{page}{HTXAdvPage1}{Input areas} +\centerline{\fbox{{\tt \thispage}}}\newline +\begin{scroll} + +You have probably seen input areas in other \HyperName{} +pages. They provide {\it dynamic link} capabilities. +Instead of having a choice between certain actions, +they allow you to specify an action on--the--fly. +To use them, you need the following commands: +\beginImportant +\newline +{\tt \\inputstring\{{\it label}\}\{{\it length}\}\{{\it default value}\}} +\newline +{\tt \\stringvalue\{{\it label}\}} +\endImportant + +The first command puts up an input area of the {\it length} +specified. The {\it default value} is placed in it. +The first argument, {\it label} gives a name to the +contents of the input area. +You can refer to those contents by using +the second command. Never place a {\tt \\stringvalue} command +in an "exposed" part of the page. It is only meant +to be used as an argument to an {\it action}. +Here are some examples. + + + + + +\beginImportant +\begin{paste}{HTXAdvPage1xPaste1}{HTXAdvPage1xPatch1} +\pastebutton{HTXAdvPage1xPaste1}{Interpret} +\newline +{\tt Page name \\tab\{16\} } +{\tt \\inputstring\{pagetogo\}\{30\}\{RootPage\}}\newline +{\tt \\newline}\newline +{\tt \\downlink\{GO!\}\{\\stringvalue\{pagetogo\}\}}\newline +\end{paste} +\endImportant + +\beginImportant +\begin{paste}{HTXAdvPage1xPaste2}{HTXAdvPage1xPatch2} +\pastebutton{HTXAdvPage1xPaste2}{Interpret} +\newline +{\tt File to edit \\tab\{16\}}\newline +{\tt \\inputstring\{filetoedit\}\{30\}\{/etc/passwd\}}\newline +{\tt \\newline}\newline +{\tt \\unixcommand\{Ready!\}\{xterm -e vi \\stringvalue\{filetoedit\}\}} +\end{paste} +\endImportant + + +\end{scroll} +\beginmenu +\menulink{Next Page --- Radio boxes}{HTXAdvPage2} +\endmenu + +\end{page} + +@ +\subsection{HTXAdvPage1xPatch1 patch} +\label{HTXAdvPage1xPatch1} +\index{patch!HTXAdvPage1xPatch1!htxadvpage1.ht} +\index{htxadvpage1.ht!patch!HTXAdvPage1xPatch1} +\index{HTXAdvPage1xPatch1!htxadvpage1.ht!patch} +<>= +\begin{patch}{HTXAdvPage1xPatch1} +\begin{paste}{HTXAdvPage1xPaste1A}{HTXAdvPage1xPatch1A} +\pastebutton{HTXAdvPage1xPaste1A}{Source} +\newline +Page name \tab{16} +\inputstring{pagetogo}{30}{RootPage} +\newline +\downlink{GO!}{\stringvalue{pagetogo}} +\end{paste} +\end{patch} + +@ +\subsection{HTXAdvPage1xPatch1A patch} +\label{HTXAdvPage1xPatch1A} +\index{patch!HTXAdvPage1xPatch1A!htxadvpage1.ht} +\index{htxadvpage1.ht!patch!HTXAdvPage1xPatch1A} +\index{HTXAdvPage1xPatch1A!htxadvpage1.ht!patch} +<>= +\begin{patch}{HTXAdvPage1xPatch1A} +\begin{paste}{HTXAdvPage1xPaste1B}{HTXAdvPage1xPatch1} +\pastebutton{HTXAdvPage1xPaste1B}{Interpret} +\newline +{\tt Page name \\tab\{16\} } +{\tt \\inputstring\{pagetogo\}\{30\}\{RootPage\}}\newline +{\tt \\newline}\newline +{\tt \\downlink\{GO!\}\{\\stringvalue\{pagetogo\}\}}\newline +\end{paste} +\end{patch} + +@ +\subsection{HTXAdvPage1xPatch2 patch} +\label{HTXAdvPage1xPatch2} +\index{patch!HTXAdvPage1xPatch2!htxadvpage1.ht} +\index{htxadvpage1.ht!patch!HTXAdvPage1xPatch2} +\index{HTXAdvPage1xPatch2!htxadvpage1.ht!patch} +<>= +\begin{patch}{HTXAdvPage1xPatch2} +\begin{paste}{HTXAdvPage1xPaste2A}{HTXAdvPage1xPatch2A} +\pastebutton{HTXAdvPage1xPaste2A}{Source} +\newline +File to edit \tab{16} +\inputstring{filetoedit}{30}{/etc/passwd} +\newline +\unixcommand{Ready!}{xterm -e vi \stringvalue{filetoedit}} +\end{paste} +\end{patch} + +@ +\subsection{HTXAdvPage1xPatch2A patch} +\label{HTXAdvPage1xPatch2A} +\index{patch!HTXAdvPage1xPatch2A!htxadvpage1.ht} +\index{htxadvpage1.ht!patch!HTXAdvPage1xPatch2A} +\index{HTXAdvPage1xPatch2A!htxadvpage1.ht!patch} +<>= +\begin{patch}{HTXAdvPage1xPatch2A} +\begin{paste}{HTXAdvPage1xPaste2B}{HTXAdvPage1xPatch2} +\pastebutton{HTXAdvPage1xPaste2B}{Interpret} +\newline +{\tt File to edit \\tab\{16\}}\newline +{\tt \\inputstring\{filetoedit\}\{30\}\{/etc/passwd\}}\newline +{\tt \\newline}\newline +{\tt \\unixcommand\{Ready!\}\{xterm -e vi \\stringvalue\{filetoedit\}\}} +\end{paste} +\end{patch} + +@ +\section{htxadvpage2.ht} +\subsection{Radio buttons} +\label{HTXAdvPage2} +See HTXAdvPage3 \ref{HTXAdvPage3} on page~\pageref{HTXAdvPage3} +\index{pages!HTXAdvPage2!htxadvpage2.ht} +\index{htxadvpage2.ht!pages!HTXAdvPage2} +\index{HTXAdvPage2!htxadvpage2.ht!pages} +<>= +\begin{page}{HTXAdvPage2}{Radio buttons} +\centerline{\fbox{{\tt \thispage}}}\newline +\begin{scroll} + + +If you just want to make a multiple-choice +type selection, why not use the {\it radio buttons}. + +You need to use bitmaps for the active areas (the buttons) but +\HyperName{} will keep track of the currently activated button. +You can use this boolean information somewhere else on your page. +The commands to use are: +\beginImportant +\newline +{\tt \\radioboxes\{{\it group name}\}\{{\it bitmap file1}\}\{{\it bitmap file0}\}} +\newline +{\tt \\radiobox[{\it initial state}]\{{\it label}\}\{{\it group name}\}} +\newline +{\tt \\boxvalue\{{\it label}\}} +\endImportant + +The {\tt \\radioboxes} command sets up a group of {\tt \\radiobox} +buttons. The {\it group name} is a label for the group. The filenames +for the bitmaps are specified in {\it bitmap file1} and {\it bitmap file0}. +The first one should denote an activated button and the second +a de-activated one. + +To display each button in a group, use {\tt \\radiobox}. +The {\it initial state} should be either {\tt 1} or {\tt 0} +depending on whether the button should first be displayed as activated or not. +The second {\it label} argument defines the name by which the +current state of the button can be referred to. +The third argument specifies which group this button belongs to. + +The {\tt \\boxvalue} command can then be used in various +actions. The value of it will +be either {\tt t} or {\tt nil}. + +In the example below, we use the {\tt \\htbmfile} macro +defined in {\bf util.ht} so that we do not have to write +the full bitmap file pathnames. + +This is how we set up the group. The {\tt \\radioboxes} command does not display +anything. +Note that these commands cannot be included in a {\it patch}. +This is why we display this time the source and the result +at the same time. +\beginImportant +\newline +{\tt \\radioboxes\{group\}\{\\htbmfile\{pick\}\}\{\\htbmfile\{unpick\}\}}\newline +{\tt \\newline \\table\{}\newline +{\tt \{\\radiobox[1]\{b1\}\{group\}\}}\newline +{\tt \{\\radiobox[0]\{b2\}\{group\}\}}\newline +{\tt \{\\radiobox[0]\{b3\}\{group\}\}\}}\newline +{\tt \\newline}\newline +{\tt \\lispcommand\{lisp\}\{(pprint (list}\newline +{\tt \\boxvalue\{b1\} \\boxvalue\{b2\} \\boxvalue\{b3\}))\}}\newline +{\tt \\newline}\newline +{\tt \\unixcommand\{unix\}\{echo '\\boxvalue\{b1\}}\newline +{\tt \\boxvalue\{b2\} \\boxvalue\{b3\}'\}} +\endImportant +\radioboxes{group}{\htbmfile{pick}}{\htbmfile{unpick}} +\table{ +{\radiobox[1]{b1}{group}} +{\radiobox[0]{b2}{group}} +{\radiobox[0]{b3}{group}}} +\newline +\lispcommand{lisp}{(pprint (list +\boxvalue{b1} \boxvalue{b2} \boxvalue{b3}))} +\newline +\unixcommand{unix}{echo '\boxvalue{b1} +\boxvalue{b2} \boxvalue{b3}'} +\endImportant + + + + +You can only set one radio button at a time. If you want +a non--exclusive selection, try {\tt \\inputbox}. +The syntax for this command is +\beginImportant +\newline +{\tt \\inputbox[{\it initial state}]\{{\it label}\}\{{\it bitmap file1}\}\{{\it bitmap file0}\}} +\endImportant + +There is no group command for these. +\beginImportant +\newline +{\tt \\table\{}\newline +{\tt \{\\inputbox[1]\{c1\}\{\\htbmfile\{pick\}\}\{\\htbmfile\{unpick\}\}\}}\newline +{\tt \{\\inputbox\{c2\}\{\\htbmfile\{pick\}\}\{\\htbmfile\{unpick\}\}\}}\newline +{\tt \{\\inputbox[1]\{c3\}\{\\htbmfile\{pick\}\}\{\\htbmfile\{unpick\}\}\}\}}\newline +{\tt \\newline}\newline +{\tt \\lispcommand\{lisp\}\{(pprint (list}\newline +{\tt \\boxvalue\{c1\} \\boxvalue\{c2\} \\boxvalue\{c3\}))\}}\newline +{\tt \\newline}\newline +{\tt \\unixcommand\{unix\}\{echo }\newline +{\tt '\\boxvalue\{c1\} \\boxvalue\{c2\} \\boxvalue\{c3\}'\}}\newline +\endImportant +\table{ +{\inputbox[1]{c1}{\htbmfile{pick}}{\htbmfile{unpick}}} +{\inputbox{c2}{\htbmfile{pick}}{\htbmfile{unpick}}} +{\inputbox[1]{c3}{\htbmfile{pick}}{\htbmfile{unpick}}}} +\newline +\lispcommand{lisp}{(pprint (list +\boxvalue{c1} \boxvalue{c2} \boxvalue{c3}))} +\newline +\unixcommand{unix}{echo +'\boxvalue{c1} \boxvalue{c2} \boxvalue{c3}'} +\endImportant + + +Note that the {\it initial state} is an +optional argument. If omitted +the button will initially +be deactivated. + +\end{scroll} +\beginmenu +\menulink{Next Page --- Macros}{HTXAdvPage3} +\endmenu + +\end{page} + +@ +\section{htxadvpage3.ht} +\subsection{Macros} +\label{HTXAdvPage3} +See HTXAdvPage4 \ref{HTXAdvPage4} on page~\ref{HTXAdvPage4}p +\index{pages!HTXAdvPage3!htxadvpage3.ht} +\index{htxadvpage3.ht!pages!HTXAdvPage3} +\index{HTXAdvPage3!htxadvpage3.ht!pages} +<>= +\begin{page}{HTXAdvPage3}{Macros} +\centerline{\fbox{{\tt \thispage}}}\newline +\begin{scroll} + + +Sometimes you may find yourself having to +write +almost the same piece of \HyperName{} +text many times. Thankfully, there is a command to ease +the work. +It is the {\tt \\newcommand} command and provides +a macro facility for \HyperName{}. +In this way, you can give a short name to a sequence of \HyperName{} +text and use that name to include the sequence in your pages. +The way this works is the following +\beginImportant +\newline +\centerline{{\tt \\newcommand\{\\{\it name}\}[{\it number of arguments}]\{{\it \HyperName{} text}\}}} +\endImportant +and here is an example from {\bf util.ht} +\beginImportant +\newline +{\tt \\newcommand\{\\axiomSig\}[2]\{\\axiomType\{\#1\} \{\\tt ->\} \\axiomType\{\#2\}\}} +\newline +{\tt \\newcommand\{\\axiomType\}[1]\{\\lispdownlink\{\#1\}\{(|spadType| '|\#1|)\}\}} +\endImportant + +You see that a macro's definition can invoke another. +Don't create a circular definition though! +Notice how the arguments of the macro are used +in the definition. The {\tt \#{\it n}} construct +is the place--holder of the {\it n}'th argument. + +To use the macro, just treat it as an ordinary command. +For instance +\beginImportant +\newline +{\tt \\axiomSig\{Integer\}\{List Integer\}} +\endImportant +displays and acts like this +\beginImportant +\newline +\axiomSig{Integer}{List Integer} +\endImportant + +The best way to familiarise yourself to +macros is to study the macros defined in +\centerline{ +{\bf \env{AXIOM}/doc/hypertex/pages/util.ht} +} +It is highly probable that a good many of them +will prove useful to you. +Clever use of macros will allow you to +create \HyperName{} text that can be +formatted by other programs (such as TeX). +The Axiom User Guide was written +in such a way as to make translation in +\HyperName{} form and TeX form a mechanical process. + + + + +\end{scroll} +\beginmenu +\menulink{Next Page --- Patch and Paste}{HTXAdvPage4} +\endmenu + +\end{page} + +@ +\section{htxadvpage4.ht} +\subsection{Patch and Paste} +\label{HTXAdvPage4} +See HTXAdvPage5 \ref{HTXAdvPage5} on page~\pageref{HTXAdvPage5} +\index{pages!HTXAdvPage4!htxadvpage4.ht} +\index{htxadvpage4.ht!pages!HTXAdvPage4} +\index{HTXAdvPage4!htxadvpage4.ht!pages} +<>= +\begin{page}{HTXAdvPage4}{Patch and Paste} +\centerline{\fbox{{\tt \thispage}}}\newline +\begin{scroll} + + +A powerful \HyperName{} feature is +the ability to {\it replace} +part of a displayed page with another part +when an active area is clicked. The group commands +{\it patch} and {\it paste} offer this facility. +A {\it paste} region can appear anywhere +within a page or a {\it patch}. A {\it patch} region must be defined +outside a page definition. + +We need a few objects to define the {\it paste} +region. These are a {\it name} with which to +refer to it, some way of specifying what it +is to be replaced by and a {\it trigger} for the +replacement. A {\it patch} is how we specify the +second of these objects. +The {\it patch} is generally a sequence of \HyperName{} +text. + +If we want to have the option of returning to the original +(or ,indeed, proceeding to a {\it third} alternative) +we clearly must include a {\it paste} in the {\it patch}. + +Let us start with a simple example. We wish to +have the word {\tt initial} somewhere on the page replaced by the +word {\tt final} at a click of a button. +Let us first define the {\it patch}. It will just contain +the word {\tt final}. Here is a definition of a +patch called {\tt patch1} (note that +the actual definition must be outside this page's definition). +\beginImportant +\newline +{\tt \\begin\{patch\}\{patch1\}} \newline +{\tt final}\newline +{\tt \\end\{patch\}} +\endImportant +We now define a {\it paste} region exactly where we +want the word {\tt initial} to appear. +\beginImportant +\newline +{\tt \\begin\{paste\}\{paste1\}\{patch1\}}\newline +{\tt initial}\newline +{\tt \\end\{paste\}} +\centerline{{\it results in}} +\begin{paste}{paste1}{patch1} +initial +\end{paste} +\endImportant +We have specified first the name of the {\it paste} region +which is {\tt paste1} and then the name of the +replacement {\it patch} which is {\tt patch1}. +Something is missing -- the trigger. +To include a trigger we write +\beginImportant +\newline +{\tt \\pastebutton\{paste1\}\{trigger\} +\centerline{{\it results in}} +\pastebutton{paste1}{trigger} +\endImportant +This new command {\tt \\pastebutton} displays the second argument +as an active area. The first argument specifies the {\it paste} +region it refers to. Clicking on {\tt trigger} above will +replace the word {\tt initial} with the word {\tt final}. + +We can, if we like, include the {\tt \\pastebutton} in the {\it paste} +region. +Let us improve on the situation by providing a way +of going back to the original word {\tt initial} on the page. +The {\it patch} must itself include a {\it paste}. +What will the replacement {\it patch} for this new {\it paste} +be ? Well, we have to define a second {\it patch} +that contains all the stuff in the original {\it paste} +region. Here is the updated {\it patch} for the first replacement. +The {\tt \\MenuDotBitmap} macro is defined in {\bf util.ht}. +It displays a button bitmap. +This time we put the {\tt \\pastebutton} +inside the {\it paste}. +\beginImportant +\newline +{\tt \\begin\{patch\}\{Patch1\}}\newline +{\tt \\begin\{paste\}\{Paste2\}\{Patch2\}}\newline +{\tt \\pastebutton\{Paste2\}\{\\MenuDotBitmap\}}\newline +{\tt final}\newline +{\tt \\end\{paste\}}\newline +{\tt \\end\{patch\}}\newline +\endImportant +and the new {\tt Patch2} {\it patch} +\beginImportant +\newline +{\tt \\begin\{patch\}\{Patch2\}}\newline +{\tt \\begin\{paste\}\{Paste3\}\{Patch1\}}\newline +{\tt \\pastebutton\{Paste3\}\{\\MenuDotBitmap\}}\newline +{\tt initial}\newline +{\tt \\end\{paste\}}\newline +{\tt \\end\{patch\}}\newline +\endImportant + +Remember that these {\it patch} definitons must +occur outside a {\tt \\begin\{page\} - \\end\{page\}} group. +What is left now is to define the starting {\it paste} +region. +\beginImportant +\newline +{\tt \\begin\{paste\}\{Paste1\}\{Patch1\}}\newline +{\tt \\pastebutton\{Paste1\}\{\\MenuDotBitmap\}}\newline +{\tt initial}\newline +{\tt \\end\{paste\}} +\centerline{{\it results in}} +\begin{paste}{Paste1}{Patch1} +\pastebutton{Paste1}{\MenuDotBitmap} +initial +\end{paste} +\endImportant + +Clicking on the button above next to {\tt initial} +will replace the {\tt Paste1} region with +{\tt Patch1}. That {\it patch} +also contains a {\it paste} region ({\tt Paste2}). +Clicking on {\it its} button will put up +{\tt Patch2} which has a {\it paste} region ({\tt Paste3}). +Clicking on {\it its} button will put up {\tt Patch1} +again. In that way, we close the chain of replacements. + + + + +\end{scroll} +\beginmenu +\menulink{Next Page --- Axiom paste-ins}{HTXAdvPage5} +\endmenu + +\end{page} + +@ +\subsection{patch1 patch} +\label{patch1} +\index{patch!patch1!htxadvpage4.ht} +\index{htxadvpage4.ht!patch!patch1} +\index{patch1!htxadvpage4.ht!patch} +<>= +\begin{patch}{patch1} +final +\end{patch} + +@ +\subsection{Patch1 patch} +\label{Patch1} +\index{patch!Patch1!htxadvpage4.ht} +\index{htxadvpage4.ht!patch!Patch1} +\index{Patch1!htxadvpage4.ht!patch} +<>= +\begin{patch}{Patch1} +\begin{paste}{Paste2}{Patch2} +\pastebutton{Paste2}{\MenuDotBitmap} +final +\end{paste} +\end{patch} + +@ +\subsection{Patch2 patch} +\label{Patch2} +\index{patch!Patch2!htxadvpage4.ht} +\index{htxadvpage4.ht!patch!Patch2} +\index{Patch2!htxadvpage4.ht!patch} +<>= +\begin{patch}{Patch2} +\begin{paste}{Paste3}{Patch1} +\pastebutton{Paste3}{\MenuDotBitmap} +initial +\end{paste} +\end{patch} + +@ +\section{htxadvpage5.ht} +\subsection{Axiom paste-ins} +\label{HTXAdvPage5} +See HTXAdvPage6 \ref{HTXAdvPage6} on page~\pageref{HTXAdvPage6} +\index{pages!HTXAdvPage5!htxadvpage5.ht} +\index{htxadvpage5.ht!pages!HTXAdvPage5} +\index{HTXAdvPage5!htxadvpage5.ht!pages} +<>= +\begin{page}{HTXAdvPage5}{Axiom paste-ins} +\centerline{\fbox{{\tt \thispage}}}\newline +\begin{scroll} + +The {\it paste} and {\it patch} facility (see \downlink{previous page}{HTXAdvPage4}) +is used to display (or hide) the Axiom output +of an Axiom command ({\tt\\axiomcommand}) +included in a +\HyperName{} page. + +A mechanism has been set up to {\it automatically} generate +these paste-ins. It amounts to replacing an +{\tt \\axiomcommand} by a {\tt \\pastecommand} in the +\HyperName{} page. + +In the case of a \axiomOp{draw} Axiom command +, where the result is to create an interactive viewport, +the appropriate command to use is {\tt \\pastegraph}. +The effect of this is to include (as the output) +the Axiom generated {\it image} of the graph +as an active area. Clicking on it will put up an +interactive viewport. +The {\tt \\pastegraph} command should be used only when +the result of the associated Axiom operation is +to {\it create} an interactive viewport. It is {\it not} necessarily +appropriate for all commands whose result +is a \axiomType{Two{}Dimensional{}Viewport} or \axiomType{Three{}Dimensional{}Viewport}. +The {\tt \\pastecommand} and {\tt \\pastegraph} +are macros defined in {\bf util.ht}. + + + +There is no automatic paste-in generation facility +for Axiom piles (the {\tt \\begin\{axiomsrc\}} command). + + +The automatic paste-in generation mechanism works +by invoking Axiom with a particular option. +\HyperName{} is also started automatically. It reads +the {\tt \\pastecommand} and {\tt \\pastegraph} +commands in all pages in a specified +{\bf somefile.ht} and passes them to +Axiom for evaluation. \HyperName{} +captures the output and writes it out (to a +file called {\bf somefile.pht}) as the body of +some {\it patch} definitions. The commands +encountered are written to a file +{\bf somefile.input} which you can {\tt )read} from an Axiom session. +It also creates directories for the graphics +images encountered. Those files and directories +will be written under the {\it current} directory. +The idea is that you then include the {\it patch} definitions +in {\bf somefile.pht} in your local database using {\bf htadd}. + + +You can try this feature now. Edit a file called, say, {\bf trypaste.ht} +in a directory, say {\bf /tmp}. Put the following \HyperName{} text +in it. +\beginImportant +\newline +{\tt \\begin\{page\}\{TryPaste\}\{Trying out paste-in generation\}}\newline +{\tt \\begin\{scroll\}}\newline +{\tt \\pastecommand\{f z == z**2 \\bound\{f\}\}}\newline +{\tt \\pastegraph\{draw(f,-1..1) \\free\{f\}\}}\newline +{\tt \\pastecommand\{x:= f 3 \\free\{f\}\}}\newline +{\tt \\end\{scroll\}}\newline +{\tt \\end\{page\}}\newline +\endImportant + +From the directory that contains the {\bf trypaste.ht}, +issue +\centerline{ +{\tt htadd -l ./trypaste.ht} +} +You will get the +{\bf ht.db} database file. +Set the environment variable {\tt HTPATH} so that +it points first to your directory and then the system directory. +In the {\bf /bin/csh}, you might use +\centerline{ +{\tt setenv HTPATH /tmp:\$AXIOM/doc/hypertex/pages} +} +Make sure that no {\bf trypaste.input} or {\bf trypaste.pht} +files exist in the directory. +Then issue +\centerline{ +{\tt axiom -paste trypaste.ht} +} + and wait for +Axiom to finish. + +There is a modification you will wish to make to +the {\bf trypaste.pht} file. +This is because the generated \HyperName{} text will assume that +the {\it viewport} data will be located in the +{\it system} directory. +\centerline{{\bf \env{AXIOM}/doc/viewports}} +You may want to place your images in a different directory, +say, {\bf /u/sugar/viewports}. +If so, then change all occurences of +\beginImportant +\newline +{\tt \\env\{AXIOM\}/doc/viewports/} +\centerline{{\it by}} +{\tt /u/sugar/viewports/} +\endImportant +in the file {\bf trypaste.pht}. The last step +is to include the {\it patch} definitions +in {\bf trypaste.pht} to your local database. +Issue +\centerline{ +{\tt htadd -l ./trypaste.pht} +} +to update the database. If you have provided +a link to your pages from the {\tt RootPage} +via the {\tt \\localinfo} macro, you should now +be able to start Axiom or \HyperName{} +and see the computed Axiom output whenever you +click on the buttons to the left of each command. + + + +\end{scroll} +\beginmenu +\menulink{Next Page --- Miscellaneous}{HTXAdvPage6} +\endmenu + +\end{page} + +@ +\section{htxadvpage6.ht} +\subsection{Miscellaneous} +\label{HTXAdvPage6} +See HTXAdvTopPage \ref{HTXAdvTopPage} on page~\pageref{HTXAdvTopPage} +\index{pages!HTXAdvPage6!htxadvpage6.ht} +\index{htxadvpage6.ht!pages!HTXAdvPage6} +\index{HTXAdvPage6!htxadvpage6.ht!pages} +<>= +\begin{page}{HTXAdvPage6}{Miscellaneous} +\centerline{\fbox{{\tt \thispage}}}\newline +\begin{scroll} + + +We present here a few commands that may be of some use +to you. +You may want to know certain parameters so that you can pass +them to one of the action {\tt \\command}s. + +The {\tt \\thispage} command shows the name of the +current page. + +\beginImportant +\begin{paste}{HTXAdvPage6xPaste1}{HTXAdvPage6xPatch1} +\pastebutton{HTXAdvPage6xPaste1}{Interpret} +\newline +{\tt \\thispage}\newline +{\tt \\newline}\newline +{\tt \\lispcommand\{Lisp\}\{(pprint "\\thispage")\}}\newline +\end{paste} +\endImportant + + +The {\tt \\windowid} command shows the X Windows +{\it WindowID} of the current window. + +\beginImportant +\begin{paste}{HTXAdvPage6xPaste2}{HTXAdvPage6xPatch2} +\pastebutton{HTXAdvPage6xPaste2}{Interpret} +\newline +{\tt \\windowid}\newline +{\tt \\newline}\newline +{\tt \\lispcommand\{Lisp\}\{(pprint \\windowid)\}}\newline +\end{paste} +\endImportant + +% \examplenumber not documented + +The {\tt \\env} command gets the value of an environment +variable. It is an error to use {\tt \\env} with an undefined +environment variable. + +\beginImportant +\begin{paste}{HTXAdvPage6xPaste3}{HTXAdvPage6xPatch3} +\pastebutton{HTXAdvPage6xPaste3}{Interpret} +\newline +{\tt \\env\{AXIOM\}} +\end{paste} +\endImportant + + +The {\tt \\nolines} command, if included somewhere +in the page, eliminates the horizontal lines that +delimit the header and footer regions. + + +% \beep not documented + +%\returnbutton{homebutton}{ReturnPage} + +%\upbutton{upbutton}{UpPage} + + +\end{scroll} +\beginmenu +\menulink{Back to Menu}{HTXAdvTopPage} +\endmenu + +\end{page} + +@ +\subsection{HTXAdvPage6xPatch1 patch} +\label{HTXAdvPage6xPatch1} +\index{patch!HTXAdvPage6xPatch1!htxadvpage6.ht} +\index{htxadvpage6.ht!patch!HTXAdvPage6xPatch1} +\index{HTXAdvPage6xPatch1!htxadvpage6.ht!patch} +<>= +\begin{patch}{HTXAdvPage6xPatch1} +\begin{paste}{HTXAdvPage6xPaste1A}{HTXAdvPage6xPatch1A} +\pastebutton{HTXAdvPage6xPaste1A}{Source} +\newline +\thispage +\newline +\lispcommand{Lisp}{(pprint "\thispage")} +\end{paste} +\end{patch} + +@ +\subsection{HTXAdvPage6xPatch1A patch} +\label{HTXAdvPage6xPatch1A} +\index{patch!HTXAdvPage6xPatch1A!htxadvpage6.ht} +\index{htxadvpage6.ht!patch!HTXAdvPage6xPatch1A} +\index{HTXAdvPage6xPatch1A!htxadvpage6.ht!patch} +<>= +\begin{patch}{HTXAdvPage6xPatch1A} +\begin{paste}{HTXAdvPage6xPaste1B}{HTXAdvPage6xPatch1} +\pastebutton{HTXAdvPage6xPaste1B}{Interpret} +\newline +{\tt \\thispage}\newline +{\tt \\newline}\newline +{\tt \\lispcommand\{Lisp\}\{(pprint "\\thispage")\}}\newline +\end{paste} +\end{patch} + +@ +\subsection{HTXAdvPage6xPatch2 patch} +\label{HTXAdvPage6xPatch2} +\index{patch!HTXAdvPage6xPatch2!htxadvpage6.ht} +\index{htxadvpage6.ht!patch!HTXAdvPage6xPatch2} +\index{HTXAdvPage6xPatch2!htxadvpage6.ht!patch} +<>= +\begin{patch}{HTXAdvPage6xPatch2} +\begin{paste}{HTXAdvPage6xPaste2A}{HTXAdvPage6xPatch2A} +\pastebutton{HTXAdvPage6xPaste2A}{Source} +\newline +\windowid +\newline +\lispcommand{Lisp}{(pprint \windowid)} +\end{paste} +\end{patch} + +@ +\subsection{HTXAdvPage6xPatch2A patch} +\label{HTXAdvPage6xPatch2A} +\index{patch!HTXAdvPage6xPatch2A!htxadvpage6.ht} +\index{htxadvpage6.ht!patch!HTXAdvPage6xPatch2A} +\index{HTXAdvPage6xPatch2A!htxadvpage6.ht!patch} +<>= +\begin{patch}{HTXAdvPage6xPatch2A} +\begin{paste}{HTXAdvPage6xPaste2B}{HTXAdvPage6xPatch2} +\pastebutton{HTXAdvPage6xPaste2B}{Interpret} +\newline +{\tt \\windowid}\newline +{\tt \\newline}\newline +{\tt \\lispcommand\{Lisp\}\{(pprint \\windowid)\}}\newline +\end{paste} +\end{patch} + +@ +\subsection{HTXAdvPage6xPatch3 patch} +\label{HTXAdvPage6xPatch3} +\index{patch!HTXAdvPage6xPatch3!htxadvpage6.ht} +\index{htxadvpage6.ht!patch!HTXAdvPage6xPatch3} +\index{HTXAdvPage6xPatch3!htxadvpage6.ht!patch} +<>= +\begin{patch}{HTXAdvPage6xPatch3} +\begin{paste}{HTXAdvPage6xPaste3A}{HTXAdvPage6xPatch3A} +\pastebutton{HTXAdvPage6xPaste3A}{Source} +\newline +\env{AXIOM} +\end{paste} +\end{patch} + +@ +\subsection{HTXAdvPage6xPatch3A patch} +\label{HTXAdvPage6xPatch3A} +\index{patch!HTXAdvPage6xPatch3A!htxadvpage6.ht} +\index{htxadvpage6.ht!patch!HTXAdvPage6xPatch3A} +\index{HTXAdvPage6xPatch3A!htxadvpage6.ht!patch} +<>= +\begin{patch}{HTXAdvPage6xPatch3A} +\begin{paste}{HTXAdvPage6xPaste3B}{HTXAdvPage6xPatch3} +\pastebutton{HTXAdvPage6xPaste3B}{Interpret} +\newline +{\tt \\env\{AXIOM\}}\newline +\end{paste} +\end{patch} + +@ +\section{htxadvtoppage.ht} +\subsection{Advanced features in Hyperdoc} +\label{HTXAdvTopPage} +\begin{itemize} +\item HTXAdvPage1 \ref{HTXAdvPage1} on page~\pageref{HTXAdvPage1} +\item HTXAdvPage2 \ref{HTXAdvPage2} on page~\pageref{HTXAdvPage2} +\item HTXAdvPage3 \ref{HTXAdvPage3} on page~\pageref{HTXAdvPage3} +\item HTXAdvPage4 \ref{HTXAdvPage4} on page~\pageref{HTXAdvPage4} +\item HTXAdvPage5 \ref{HTXAdvPage5} on page~\pageref{HTXAdvPage5} +\item HTXAdvPage6 \ref{HTXAdvPage6} on page~\pageref{HTXAdvPage6} +\end{itemize} +\index{pages!HTXAdvTopPage!htxadvtoppage.ht} +\index{htxadvtoppage.ht!pages!HTXAdvTopPage} +\index{HTXAdvTopPage!htxadvtoppage.ht!pages} +<>= +\begin{page}{HTXAdvTopPage}{Advanced features in Hyperdoc} +\centerline{\fbox{{\tt \thispage}}}\newline +\beginscroll +\beginmenu +\menudownlink{Creating input areas}{HTXAdvPage1} +\menudownlink{Creating radio boxes}{HTXAdvPage2} +\menudownlink{Define new macros }{HTXAdvPage3} +\menudownlink{Using patch and paste}{HTXAdvPage4} +\menudownlink{Generate paste-ins for Axiom commands}{HTXAdvPage5} +\menudownlink{Miscellaneous}{HTXAdvPage6} +\endmenu +\endscroll +\end{page} + +@ +\section{htxformatpage1.ht} +\subsection{Using the special characters} +\label{HTXFormatPage1} +See HTXFormatPage2 \ref{HTXFormatPage2} on page~\pageref{HTXFormatPage2} +\index{pages!HTXFormatPage1!htxformatpage1.ht} +\index{htxformatpage1.ht!pages!HTXFormatPage1} +\index{HTXFormatPage1!htxformatpage1.ht!pages} +<>= +\begin{page}{HTXFormatPage1}{Using the special characters} +\centerline{\fbox{{\tt \thispage}}}\newline +\begin{scroll} + +You can display the special characters ({\tt \$ \\ \{ \{ \[ \] \% \#}) +by simply inserting the backslash {\tt \\} character just before any +one of them. So, +\beginImportant +\begin{paste}{HTXFormatPage1xPaste1}{HTXFormatPage1xPatch1} +\pastebutton{HTXFormatPage1xPaste1}{Interpret} +\newline +{\tt the characters \\\$, \\\\, \\\{, \\\}, \\\[, \\\], \\\%, \\\# } +\end{paste} +\endImportant + + +The {\tt \%} character is used in \HyperName{} as a comment marker. +If it is encountered on any line (without being preceded by the {\tt \\} + character, +of course), \HyperName{} will ignore all remaining characters on that line. +\beginImportant +\begin{paste}{HTXFormatPage1xPaste2}{HTXFormatPage1xPatch2} +\pastebutton{HTXFormatPage1xPaste2}{Interpret} +\newline +{\tt the latest figures indicate \% GET THE LATEST FIGURES}\newline +{\tt a steady rise}\indent{0} +\end{paste} +\endImportant + +Earlier versions of \HyperName{} merged the words "indicate" and "a" +into one word in the example above. This no longer occurs. + +%The two lines below are from Release 1.x +%Note that you must leave a space at the beginning of the line after the comment, +%otherwise \HyperName{} would treat "indicate" and "a" as one word. + + +\end{scroll} +\beginmenu +\menulink{Next -- Formatting without commands}{HTXFormatPage2} +\endmenu + +\end{page} + +@ +\subsection{HTXFormatPage1xPatch1 patch} +\label{HTXFormatPage1xPatch1} +\index{patch!HTXFormatPage1xPatch1!htxformatpage1.ht} +\index{htxformatpage1.ht!patch!HTXFormatPage1xPatch1} +\index{HTXFormatPage1xPatch1!htxformatpage1.ht!patch} +<>= +\begin{patch}{HTXFormatPage1xPatch1} +\begin{paste}{HTXFormatPage1xPaste1A}{HTXFormatPage1xPatch1A} +\pastebutton{HTXFormatPage1xPaste1A}{Source} +\newline +the characters \$, \\, \{, \}, \[, \], \%, \# +\end{paste} +\end{patch} +\begin{patch}{HTXFormatPage1xPatch1A} +\begin{paste}{HTXFormatPage1xPaste1B}{HTXFormatPage1xPatch1} +\pastebutton{HTXFormatPage1xPaste1B}{Interpret} +\newline +{\tt the characters \\\$, \\\\, \\\{, \\\}, \\\[, \\\], \\\%, \\\# } +\end{paste} +\end{patch} + +@ +\subsection{HTXFormatPage1xPatch2 patch} +\label{HTXFormatPage1xPatch2} +\index{patch!HTXFormatPage1xPatch2!htxformatpage1.ht} +\index{htxformatpage1.ht!patch!HTXFormatPage1xPatch2} +\index{HTXFormatPage1xPatch2!htxformatpage1.ht!patch} +<>= +\begin{patch}{HTXFormatPage1xPatch2} +\begin{paste}{HTXFormatPage1xPaste2A}{HTXFormatPage1xPatch2A} +\pastebutton{HTXFormatPage1xPaste2A}{Source} +\newline +the latest figures indicate % GET THE LATEST FIGURES +a steady rise +\end{paste} +\end{patch} +\begin{patch}{HTXFormatPage1xPatch2A} +\begin{paste}{HTXFormatPage1xPaste2B}{HTXFormatPage1xPatch2} +\pastebutton{HTXFormatPage1xPaste2B}{Interpret} +\newline +{\tt the latest figures indicate \% GET THE LATEST FIGURES}\newline +{\tt a steady rise}\indent{0} +\end{paste} +\end{patch} + +@ +\section{htxformatpage2.ht} +\subsection{Formatting without commands} +\label{HTXFormatPage2} +\index{pages!HTXFormatPage2!htxformatpage2.ht} +\index{htxformatpage2.ht!pages!HTXFormatPage2} +\index{HTXFormatPage2!htxformatpage2.ht!pages} +<>= +\begin{page}{HTXFormatPage2}{Formatting without commands} +\centerline{\fbox{{\tt \thispage}}}\newline +\begin{scroll} + +\HyperName{} will interpret normal text in a {\em source file} +according to the following rules. \newline +\menuitemstyle{\indentrel{4} +Spaces mark the ends of words. The number of spaces between +words is {\em not} significant, that is, you cannot control word spacing +by inserting or removing extra space characters. +\indentrel{-4}} +\beginImportant +\begin{paste}{HTXFormatPage2xxPaste1}{HTXFormatPage2xPatch1} +\pastebutton{HTXFormatPage2xxPaste1}{Interpret} +\begin{verbatim} +word spacing is not important +\end{verbatim} +\end{paste} +\endImportant +\menuitemstyle{\indentrel{4} +End-of-line characters are not significant. +You can break up lines in the source file as you like. The end-of-line +will be interpreted as a space. +Take advantage of this feature to improve the readability of the +source file. +\indentrel{-4}} +\beginImportant +\begin{paste}{HTXFormatPage2xxPaste2}{HTXFormatPage2xPatch2} +\pastebutton{HTXFormatPage2xxPaste2}{Interpret} +\begin{verbatim} +This is +one +sentence. +\end{verbatim} +\end{paste} +\endImportant +\menuitemstyle{\indentrel{4} +A blank line marks the end of a paragraph. +Leaving more blank lines that necessary has no effect. +\indentrel{-4}} +\beginImportant +\begin{paste}{HTXFormatPage2xxPaste3}{HTXFormatPage2xPatch3} +\pastebutton{HTXFormatPage2xxPaste3}{Interpret} +\begin{verbatim} +some end.% A COMMENT + + +Start a paragraph + +Start another paragraph. +\end{verbatim} +\end{paste} +\endImportant +\menuitemstyle{\indentrel{4} +The two-character combination {\tt \{\}} can be used to indicate +possible breaking of long words. It does not affect the formatting +in any other way. +\indentrel{-4}} +\beginImportant +\begin{paste}{HTXFormatPage2xxPaste4}{HTXFormatPage2xPatch4} +\pastebutton{HTXFormatPage2xxPaste4}{Interpret} +\begin{verbatim} +Generalized{}Multivariate{}Factorize +One{}Dimensional{}Array{}Aggregate +Elementary{}Function{}Definite{}Integration +Elementary{}Functions{}Univariate{}Puiseux{}Series +Finite{}Field{}Cyclic{}Group{}Extension{}ByPolynomial +\end{verbatim} +\end{paste} +\endImportant + + + + +\end{scroll} +\beginmenu +\menulink{Next -- Using different fonts}{HTXFormatPage3} +\endmenu + +\end{page} + +@ +\subsection{HTXFormatPage2xPatch1 patch} +\label{HTXFormatPage2xPatch1} +\index{patch!HTXFormatPage2xPatch1!htxformatpage2.ht} +\index{htxformatpage2.ht!patch!HTXFormatPage2xPatch1} +\index{HTXFormatPage2xPatch1!htxformatpage2.ht!patch} +<>= +\begin{patch}{HTXFormatPage2xPatch1} +\begin{paste}{HTXFormatPage2xxPaste1A}{HTXFormatPage2xPatch1A} +\pastebutton{HTXFormatPage2xxPaste1A}{Source} +\newline +word spacing is not important +\end{paste} +\end{patch} +\begin{patch}{HTXFormatPage2xPatch1A} +\begin{paste}{HTXFormatPage2xxPaste1B}{HTXFormatPage2xPatch1} +\pastebutton{HTXFormatPage2xxPaste1B}{Interpret} +\begin{verbatim} +word spacing is not important +\end{verbatim} +\end{paste} +\end{patch} + +@ +\subsection{HTXFormatPage2xPatch2 patch} +\label{HTXFormatPage2xPatch2} +\index{patch!HTXFormatPage2xPatch2!htxformatpage2.ht} +\index{htxformatpage2.ht!patch!HTXFormatPage2xPatch2} +\index{HTXFormatPage2xPatch2!htxformatpage2.ht!patch} +<>= +\begin{patch}{HTXFormatPage2xPatch2} +\begin{paste}{HTXFormatPage2xxPaste2A}{HTXFormatPage2xPatch2A} +\pastebutton{HTXFormatPage2xxPaste2A}{Source} +\newline +This is +one +sentence. +\end{paste} +\end{patch} + +@ +\subsection{HTXFormatPage2xPatch2A patch} +\label{HTXFormatPage2xPatch2A} +\index{patch!HTXFormatPage2xPatch2A!htxformatpage2.ht} +\index{htxformatpage2.ht!patch!HTXFormatPage2xPatch2A} +\index{HTXFormatPage2xPatch2A!htxformatpage2.ht!patch} +<>= +\begin{patch}{HTXFormatPage2xPatch2A} +\begin{paste}{HTXFormatPage2xxPaste2B}{HTXFormatPage2xPatch2} +\pastebutton{HTXFormatPage2xxPaste2B}{Interpret} +\begin{verbatim} +This is +one +sentence. +\end{verbatim} +\end{paste} +\end{patch} + +@ +\subsection{HTXFormatPage2xPatch3 patch} +\label{HTXFormatPage2xPatch3} +\index{patch!HTXFormatPage2xPatch3!htxformatpage2.ht} +\index{htxformatpage2.ht!patch!HTXFormatPage2xPatch3} +\index{HTXFormatPage2xPatch3!htxformatpage2.ht!patch} +<>= +\begin{patch}{HTXFormatPage2xPatch3} +\begin{paste}{HTXFormatPage2xxPaste3A}{HTXFormatPage2xPatch3A} +\pastebutton{HTXFormatPage2xxPaste3A}{Source} +\newline +some end.% A COMMENT + + +Start a paragraph. + +Start another paragraph. +\end{paste} +\end{patch} + +@ +\subsection{HTXFormatPage2xPatch3A patch} +\label{HTXFormatPage2xPatch3A} +\index{patch!HTXFormatPage2xPatch3A!htxformatpage2.ht} +\index{htxformatpage2.ht!patch!HTXFormatPage2xPatch3A} +\index{HTXFormatPage2xPatch3A!htxformatpage2.ht!patch} +<>= +\begin{patch}{HTXFormatPage2xPatch3A} +\begin{paste}{HTXFormatPage2xxPaste3B}{HTXFormatPage2xPatch3} +\pastebutton{HTXFormatPage2xxPaste3B}{Interpret} +\begin{verbatim} +some end.% A COMMENT + + +Start a paragraph + +Start another paragraph. +\end{verbatim} +\end{paste} +\end{patch} + +@ +\subsection{HTXFormatPage2xPatch4 patch} +\label{HTXFormatPage2xPatch4} +\index{patch!HTXFormatPage2xPatch4!htxformatpage2.ht} +\index{htxformatpage2.ht!patch!HTXFormatPage2xPatch4} +\index{HTXFormatPage2xPatch4!htxformatpage2.ht!patch} +<>= +\begin{patch}{HTXFormatPage2xPatch4} +\begin{paste}{HTXFormatPage2xxPaste4A}{HTXFormatPage2xPatch4A} +\pastebutton{HTXFormatPage2xxPaste4A}{Source} +\newline +Generalized{}Multivariate{}Factorize +One{}Dimensional{}Array{}Aggregate +Elementary{}Function{}Definite{}Integration +Elementary{}Functions{}Univariate{}Puiseux{}Series +Finite{}Field{}Cyclic{}Group{}Extension{}ByPolynomial +\end{paste} +\end{patch} + +@ +\subsection{HTXFormatPage2xPatch4A patch} +\label{HTXFormatPage2xPatch4A} +\index{patch!HTXFormatPage2xPatch4A!htxformatpage2.ht} +\index{htxformatpage2.ht!patch!HTXFormatPage2xPatch4A} +\index{HTXFormatPage2xPatch4A!htxformatpage2.ht!patch} +<>= +\begin{patch}{HTXFormatPage2xPatch4A} +\begin{paste}{HTXFormatPage2xxPaste4B}{HTXFormatPage2xPatch4} +\pastebutton{HTXFormatPage2xxPaste4B}{Interpret} +\begin{verbatim} +Generalized{}Multivariate{}Factorize +One{}Dimensional{}Array{}Aggregate +Elementary{}Function{}Definite{}Integration +Elementary{}Functions{}Univariate{}Puiseux{}Series +Finite{}Field{}Cyclic{}Group{}Extension{}ByPolynomial +\end{verbatim} +\end{paste} +\end{patch} + +@ +\section{htxformatpage3.ht} +\subsection{Using different fonts} +\label{HTXFormatPage3} +\index{pages!HTXFormatPage3!htxformatpage3.ht} +\index{htxformatpage3.ht!pages!HTXFormatPage3} +\index{HTXFormatPage3!htxformatpage3.ht!pages} +<>= +\begin{page}{HTXFormatPage3}{Using different fonts} +\centerline{\fbox{{\tt \thispage}}}\newline +\begin{scroll} + +You can use various fonts for the text. \HyperName{} makes +four {\em logical} fonts available to you: a {\em roman} font, an +{\em emphasised} font, a {\em bold} font and a +{\em typewriter} font. The actual font that corresponds to +each logical font is determined by the end user via a +defaults file. The colour for each of these fonts can also +be specified. + +Normal text is displayed with the roman font. +If you want to emphasize some text, use the {\tt \\em} +command in the following way. +\beginImportant +\begin{paste}{HTXFormatPage3xPaste1}{HTXFormatPage3xPatch1} +\pastebutton{HTXFormatPage3xPaste1}{Interpret} +\newline +{\tt this is \{\\em emphasised\} text} +\end{paste} +\endImportant + +Note the use of the braces to enclose command and "arguments". +All font commands are specified in the same way. The {\tt \\em} command +will in fact {\em switch} between roman and emphasised +font every time it is used. +\beginImportant +\begin{paste}{HTXFormatPage3xPaste2}{HTXFormatPage3xPatch2} +\pastebutton{HTXFormatPage3xPaste2}{Interpret} +\newline +{\tt \{\\em this is \{\\em emphasised\} text\}} +\end{paste} +\endImportant + +If you want to be sure that the emphasized font will be used, specify +the {\tt \\it} command. Similarly, you can explicitly select the roman font +with the {\tt \\rm} command. +\beginImportant +\begin{paste}{HTXFormatPage3xPaste3}{HTXFormatPage3xPatch3} +\pastebutton{HTXFormatPage3xPaste3}{Interpret} +\newline +{\tt \{\\em this is \{\\it emphasised\} text and this is \{\\rm roman\}\}} +\end{paste} +\endImportant + + +The bold font is selected with the {\tt \\bf} command and the typewriter +font with the {\tt \\tt} command. All these commands can be applied to +individual characters, words, sentences etc. +\beginImportant +\begin{paste}{HTXFormatPage3xPaste4}{HTXFormatPage3xPatch4} +\pastebutton{HTXFormatPage3xPaste4}{Interpret} +\newline +{\tt \{\\bf U\}\{\\tt g\}\{\\it l\}\{\\rm y\}} +\end{paste} +\endImportant + + +Currently, \HyperName{} does not adjust its internal spacing rules +to each font individually. This means that, for consistent results, +users are encouraged to specify (in the defaults file) +"character-cell" fonts that are not +too small or too large for \HyperName{}. Here is the correspondence +between the above font commands and the defaults names:\newline +\menuitemstyle{RmFont \tab{26} {\tt \\rm} or {\tt \\em} }\newline +\menuitemstyle{BoldFont \tab{26} {\tt \\bf} }\newline +\menuitemstyle{EmphasizeFont \tab{26} {\tt \\it} or {\tt \\em} }\newline +\menuitemstyle{Ttfont \tab{26} {\tt \\tt} }\newline + +\HyperName{} uses two more logical fonts that can be specified by +the end user : AxiomFont and ActiveFont. However, you cannot +explicitly use these fonts in your text. The ActiveFont is automatically +used for active area text and the AxiomFont is reserved for +active Axiom commands. + + + + +\end{scroll} +\beginmenu +\menulink{Next -- Indentation}{HTXFormatPage4} +\endmenu + +\end{page} + +@ +\subsection{HTXFormatPage3xPatch1 patch} +\label{HTXFormatPage3xPatch1} +\index{patch!HTXFormatPage3xPatch1!htxformatpage3.ht} +\index{htxformatpage3.ht!patch!HTXFormatPage3xPatch1} +\index{HTXFormatPage3xPatch1!htxformatpage3.ht!patch} +<>= +\begin{patch}{HTXFormatPage3xPatch1} +\begin{paste}{HTXFormatPage3xPaste1A}{HTXFormatPage3xPatch1A} +\pastebutton{HTXFormatPage3xPaste1A}{Source} +\newline +this is {\em emphasised} text +\end{paste} +\end{patch} +\begin{patch}{HTXFormatPage3xPatch1A} +\begin{paste}{HTXFormatPage3xPaste1B}{HTXFormatPage3xPatch1} +\pastebutton{HTXFormatPage3xPaste1B}{Interpret} +\newline +{\tt this is \{\\em emphasised\} text} +\end{paste} +\end{patch} + +@ +\subsection{HTXFormatPage3xPatch2 patch} +\label{HTXFormatPage3xPatch2} +\index{patch!HTXFormatPage3xPatch2!htxformatpage3.ht} +\index{htxformatpage3.ht!patch!HTXFormatPage3xPatch2} +\index{HTXFormatPage3xPatch2!htxformatpage3.ht!patch} +<>= +\begin{patch}{HTXFormatPage3xPatch2} +\begin{paste}{HTXFormatPage3xPaste2A}{HTXFormatPage3xPatch2A} +\pastebutton{HTXFormatPage3xPaste2A}{Source} +\newline +{\em this is {\em emphasised} text} +\end{paste} +\end{patch} +\begin{patch}{HTXFormatPage3xPatch2A} +\begin{paste}{HTXFormatPage3xPaste2B}{HTXFormatPage3xPatch2} +\pastebutton{HTXFormatPage3xPaste2B}{Interpret} +\newline +{\tt \{\\em this is \{\\em emphasised\} text\}} +\end{paste} +\end{patch} + +@ +\subsection{HTXFormatPage3xPatch3 patch} +\label{HTXFormatPage3xPatch3} +\index{patch!HTXFormatPage3xPatch3!htxformatpage3.ht} +\index{htxformatpage3.ht!patch!HTXFormatPage3xPatch3} +\index{HTXFormatPage3xPatch3!htxformatpage3.ht!patch} +<>= +\begin{patch}{HTXFormatPage3xPatch3} +\begin{paste}{HTXFormatPage3xPaste3A}{HTXFormatPage3xPatch3A} +\pastebutton{HTXFormatPage3xPaste3A}{Source} +\newline +{\em this is {\it emphasised} text and this is {\rm roman}} +\end{paste} +\end{patch} +\begin{patch}{HTXFormatPage3xPatch3A} +\begin{paste}{HTXFormatPage3xPaste3B}{HTXFormatPage3xPatch3} +\pastebutton{HTXFormatPage3xPaste3B}{Interpret} +\newline +{\tt \{\\em this is \{\\it emphasised\} text and this is \{\\rm roman\}\}} +\end{paste} +\end{patch} + +@ +\subsection{HTXFormatPage3xPatch4 patch} +\label{HTXFormatPage3xPatch4} +\index{patch!HTXFormatPage3xPatch4!htxformatpage3.ht} +\index{htxformatpage3.ht!patch!HTXFormatPage3xPatch4} +\index{HTXFormatPage3xPatch4!htxformatpage3.ht!patch} +<>= +\begin{patch}{HTXFormatPage3xPatch4} +\begin{paste}{HTXFormatPage3xPaste4A}{HTXFormatPage3xPatch4A} +\pastebutton{HTXFormatPage3xPaste4A}{Source} +\newline +{\bf U}{\tt g}{\it l}{\rm y} +\end{paste} +\end{patch} +\begin{patch}{HTXFormatPage3xPatch4A} +\begin{paste}{HTXFormatPage3xPaste4B}{HTXFormatPage3xPatch4} +\pastebutton{HTXFormatPage3xPaste4B}{Interpret} +\newline +{\tt \{\\bf U\}\{\\tt g\}\{\\it l\}\{\\rm y\}} +\end{paste} +\end{patch} + +@ +\section{htxformatpage4.ht} +\subsection{Indentation} +\label{HTXFormatPage4} +\index{pages!HTXFormatPage4!htxformatpage4.ht} +\index{htxformatpage4.ht!pages!HTXFormatPage4} +\index{HTXFormatPage4!htxformatpage4.ht!pages} +<>= +\begin{page}{HTXFormatPage4}{Indentation} +\centerline{\fbox{{\tt \thispage}}}\newline +\begin{scroll} + +You can control the indentation of lines of text in \HyperName{} with +some useful commands. +Use the command {\tt \\par} to force a new paragraph if you don't want +to use the blank-line rule. The first line of a new paragraph +will normally be indented by a standard small amount. If you +just want to start on a new line, use the {\tt \\newline} +command. +\beginImportant +\begin{paste}{HTXFormatPage4xPaste1}{HTXFormatPage4xPatch1} +\pastebutton{HTXFormatPage4xPaste1}{Interpret} +\newline +{\tt let us start a new line \\newline here } +\end{paste} +\endImportant + +The command {\tt \\indent\{{\it value}\}} will +set the left-page-margin {\it value} characters +to the right of the standard left-page-margin. +The initial (standard) state of a page can be reset by the +{\tt \\indent\{0\}} command. +The first lines of paragraphs will be indented +by the {\it extra} standard amount. The {\tt \\indent\{{\it value}\}} +command does {\em not} force a new line or paragraph. + +You can also use the {\tt \\indentrel\{{\it value}\}} command. +Here, the {\it value} argument is a {\em relative} indentation +which can be positive or negative. +Otherwise, it behaves in the same way as the {\tt \\indent} command. +\beginImportant +\begin{paste}{HTXFormatPage4xPaste2}{HTXFormatPage4xPatch2} +\pastebutton{HTXFormatPage4xPaste2}{Interpret} +\newline +{\tt let us start a new line \\newline \\indent\{10\} here }\newline +{\tt \\newline \\indentrel\{-5\} there}\newline +{\tt \\newline \\indentrel\{-5\} back} +\end{paste} +\endImportant + +The {\tt \\centerline\{{\it some text}\}} command will center its +argument between the current left and right margins. The argument of +the command should not be more than a paragraph of text and should not contain +any commands that change the left margin. The centered text will +start on a new line. +\beginImportant +\begin{paste}{HTXFormatPage4xPaste3}{HTXFormatPage4xPatch3} +\pastebutton{HTXFormatPage4xPaste3}{Interpret} +\newline +{\tt previous text. \\centerline\{This could}\newline +{\tt be some heading.\} Carry on} +\end{paste} +\endImportant + +Placing text in vertically aligned columns is easily done with the +{\tt \\tab\{{\it value}\}} command. The {\tt \\tab} command has the +immediate effect of placing the next word {\it value} characters to +the right of the current left margin. +\beginImportant +\begin{paste}{HTXFormatPage4xPaste4}{HTXFormatPage4xPatch4} +\pastebutton{HTXFormatPage4xPaste4}{Interpret} +\newline +{\tt \\indent\{5\}\\newline}\newline +{\tt Team A \\tab\{17\}Score\\tab\{25\}Team B\\tab\{42\}Score\\newline}\newline +{\tt 012345678901234567890123456789012345678901234567890\\newline}\newline +{\tt Green-Red\\tab\{17\}4\\tab\{25\}Blue-Black\\tab\{42\}6\\newline}\newline +{\tt \\indent\{0\}} +\end{paste} +\endImportant + +If you wish to preserve the indentation of a piece of text +you can use the {\tt verbatim} group command. Simply place +a {\tt \\begin\{verbatim\}} and {\tt \\end\{verbatim\}} around +the text. Note that \HyperName{} commands will +not be interpreted within the {\tt verbatim} group. +\beginImportant +\begin{paste}{HTXFormatPage4xPaste5}{HTXFormatPage4xPatch5} +\pastebutton{HTXFormatPage4xPaste5}{Interpret} +\newline +{\tt \\begin\{verbatim\}} +\begin{verbatim} +This spacing will be preserved + {\bf is} preserved +\end{verbatim} +{\tt \\end\{verbatim\}}\newline +\end{paste} + + + + + + + +\end{scroll} +\beginmenu +\menulink{Next -- Creating Lists and Tables}{HTXFormatPage5} +\endmenu + +\end{page} + +@ +\subsection{HTXFormatPage4xPatch1 patch} +\label{HTXFormatPage4xPatch1} +\index{patch!HTXFormatPage4xPatch1!htxformatpage4.ht} +\index{htxformatpage4.ht!patch!HTXFormatPage4xPatch1} +\index{HTXFormatPage4xPatch1!htxformatpage4.ht!patch} +<>= +\begin{patch}{HTXFormatPage4xPatch1} +\begin{paste}{HTXFormatPage4xPaste1A}{HTXFormatPage4xPatch1A} +\pastebutton{HTXFormatPage4xPaste1A}{Source} +\newline +let us start a new line \newline here +\end{paste} +\end{patch} + +@ +\subsection{HTXFormatPage4xPatch1A patch} +\label{HTXFormatPage4xPatch1A} +\index{patch!HTXFormatPage4xPatch1A!htxformatpage4.ht} +\index{htxformatpage4.ht!patch!HTXFormatPage4xPatch1A} +\index{HTXFormatPage4xPatch1A!htxformatpage4.ht!patch} +<>= +\begin{patch}{HTXFormatPage4xPatch1A} +\begin{paste}{HTXFormatPage4xPaste1B}{HTXFormatPage4xPatch1} +\pastebutton{HTXFormatPage4xPaste1B}{Interpret} +\newline +{\tt let us start a new line \\newline here } +\end{paste} +\end{patch} + +@ +\subsection{HTXFormatPage4xPatch2 patch} +\label{HTXFormatPage4xPatch2} +\index{patch!HTXFormatPage4xPatch2!htxformatpage4.ht} +\index{htxformatpage4.ht!patch!HTXFormatPage4xPatch2} +\index{HTXFormatPage4xPatch2!htxformatpage4.ht!patch} +<>= +\begin{patch}{HTXFormatPage4xPatch2} +\begin{paste}{HTXFormatPage4xPaste2A}{HTXFormatPage4xPatch2A} +\pastebutton{HTXFormatPage4xPaste2A}{Source} +\newline +let us start a new line\newline\indent{10} here +\newline\indentrel{-5} there +\newline\indentrel{-5} back +\end{paste} +\end{patch} + +@ +\subsection{HTXFormatPage4xPatch2A patch} +\label{HTXFormatPage4xPatch2A} +\index{patch!HTXFormatPage4xPatch2A!htxformatpage4.ht} +\index{htxformatpage4.ht!patch!HTXFormatPage4xPatch2A} +\index{HTXFormatPage4xPatch2A!htxformatpage4.ht!patch} +<>= +\begin{patch}{HTXFormatPage4xPatch2A} +\begin{paste}{HTXFormatPage4xPaste2B}{HTXFormatPage4xPatch2} +\pastebutton{HTXFormatPage4xPaste2B}{Interpret} +\newline +{\tt let us start a new line \\newline \\indent\{10\} here }\newline +{\tt \\newline \\indentrel\{-5\} there}\newline +{\tt \\newline \\indentrel\{-5\} back} +\end{paste} +\end{patch} + +@ +\subsection{HTXFormatPage4xPatch3 patch} +\label{HTXFormatPage4xPatch3} +\index{patch!HTXFormatPage4xPatch3!htxformatpage4.ht} +\index{htxformatpage4.ht!patch!HTXFormatPage4xPatch3} +\index{HTXFormatPage4xPatch3!htxformatpage4.ht!patch} +<>= +\begin{patch}{HTXFormatPage4xPatch3} +\begin{paste}{HTXFormatPage4xPaste3A}{HTXFormatPage4xPatch3A} +\pastebutton{HTXFormatPage4xPaste3A}{Source} +\newline +previous text. \centerline{This could +be some heading.} Carry on +\end{paste} +\end{patch} + +@ +\subsection{HTXFormatPage4xPatch3A patch} +\label{HTXFormatPage4xPatch3A} +\index{patch!HTXFormatPage4xPatch3A!htxformatpage4.ht} +\index{htxformatpage4.ht!patch!HTXFormatPage4xPatch3A} +\index{HTXFormatPage4xPatch3A!htxformatpage4.ht!patch} +<>= +\begin{patch}{HTXFormatPage4xPatch3A} +\begin{paste}{HTXFormatPage4xPaste3B}{HTXFormatPage4xPatch3} +\pastebutton{HTXFormatPage4xPaste3B}{Interpret} +\newline +{\tt previous text. \\centerline\{This could}\newline +{\tt be some heading.\} Carry on} +\end{paste} +\end{patch} + +@ +\subsection{HTXFormatPage4xPatch4 patch} +\label{HTXFormatPage4xPatch4} +\index{patch!HTXFormatPage4xPatch4!htxformatpage4.ht} +\index{htxformatpage4.ht!patch!HTXFormatPage4xPatch4} +\index{HTXFormatPage4xPatch4!htxformatpage4.ht!patch} +<>= +\begin{patch}{HTXFormatPage4xPatch4} +\begin{paste}{HTXFormatPage4xPaste4A}{HTXFormatPage4xPatch4A} +\pastebutton{HTXFormatPage4xPaste4A}{Source} +\newline +\indent{5}\newline +Team A \tab{17}Score\tab{25}Team B\tab{42}Score\newline +012345678901234567890123456789012345678901234567890\newline +Green-Red\tab{17}4\tab{25}Blue-Black\tab{42}6\newline +\indent{0} +\end{paste} +\end{patch} +\begin{patch}{HTXFormatPage4xPatch4A} +\begin{paste}{HTXFormatPage4xPaste4B}{HTXFormatPage4xPatch4} +\pastebutton{HTXFormatPage4xPaste4B}{Interpret} +\newline +{\tt \\indent\{5\}\\newline}\newline +{\tt Team A \\tab\{17\}Score\\tab\{25\}Team B\\tab\{42\}Score\\newline}\newline +{\tt 012345678901234567890123456789012345678901234567890\\newline}\newline +{\tt Green-Red\\tab\{17\}4\\tab\{25\}Blue-Black\\tab\{42\}6\\newline}\newline +{\tt \\indent\{0\}} +\end{paste} +\end{patch} + +@ +\subsection{HTXFormatPage4xPatch5 patch} +\label{HTXFormatPage4xPatch5} +\index{patch!HTXFormatPage4xPatch5!htxformatpage4.ht} +\index{htxformatpage4.ht!patch!HTXFormatPage4xPatch5} +\index{HTXFormatPage4xPatch5!htxformatpage4.ht!patch} +<>= +\begin{patch}{HTXFormatPage4xPatch5} +\begin{paste}{HTXFormatPage4xPaste5A}{HTXFormatPage4xPatch5A} +\pastebutton{HTXFormatPage4xPaste5A}{Source} +\begin{verbatim} +This spacing will be preserved + {\bf is} preserved +\end{verbatim} +\end{paste} +\end{patch} + +@ +\subsection{HTXFormatPage4xPatch5A patch} +\label{HTXFormatPage4xPatch5A} +\index{patch!HTXFormatPage4xPatch5A!htxformatpage4.ht} +\index{htxformatpage4.ht!patch!HTXFormatPage4xPatch5A} +\index{HTXFormatPage4xPatch5A!htxformatpage4.ht!patch} +<>= +\begin{patch}{HTXFormatPage4xPatch5A} +\begin{paste}{HTXFormatPage4xPaste5B}{HTXFormatPage4xPatch5} +\pastebutton{HTXFormatPage4xPaste5B}{Interpret} +\newline +{\tt \\begin\{verbatim\}} +\begin{verbatim} +This spacing will be preserved + {\bf is} preserved +\end{verbatim} +{\tt \\end\{verbatim\}}\newline +\end{paste} +\end{patch} + +@ +\section{htxformatpage5.ht} +\subsection{Creating Lists and Tables} +\label{HTXFormatPage5} +See HTXFormatPage6 \ref{HTXFormatPage6} on page~\pageref{HTXFormatPage6} +\index{pages!HTXFormatPage5!htxformatpage5.ht} +\index{htxformatpage5.ht!pages!HTXFormatPage5} +\index{HTXFormatPage5!htxformatpage5.ht!pages} +<>= +\begin{page}{HTXFormatPage5}{Creating Lists and Tables} +\centerline{\fbox{{\tt \thispage}}}\newline +\begin{scroll} + +The {\tt \\begin\{items\} {\rm -} \\end\{items\}} +group command constructs itemized lists. +The {\tt \\item} command separates the items in the list. +The indentation rules for the list group are different from those +of a paragraph. The first line of an item will +normally extend further to the left than the rest of the lines. +Both commands accept {\em optional} arguments. +Optional arguments are enclosed in square brackets ({\tt \[ \]}) rather +than braces. + +The indentation of subsequent lines in an item is determined by the +optional argument {\it some text} in the +{\tt \\begin\{items\}\[{\it some text}\]} +command. The optional argument is {\em not} displayed. Its width is calculated +and used to indent all subsequent lines in the group except from the +first line of each new item. This indentation rule applies to all text +{\em before} the first {\tt \\item} command as well. + +The {\tt \\item\[{\it some text}\]} command specifies the start of a new item. +The {\it some text} optional argument will be displayed in {\em bold} +font at the current left-page-margin. Then, the text following the command +will be displayed in normal fashion with the above indentation rule. +\beginImportant +\begin{paste}{HTXFormatPage5xPaste1}{HTXFormatPage5xPatch1} +\pastebutton{HTXFormatPage5xPaste1}{Interpret} +\newline +{\tt \\indent\{5\}}\newline +{\tt \\begin\{items\}\[how wide am I\]}\newline +{\tt Here we carry on but a \\newline} \newline +{\tt new line will be indented } \newline +{\tt \\item\[how wide am I\] fits nicely. +Here is a \\newline new line in an item.}\newline +{\tt \\item\[again\] to show another item}\newline +{\tt \\item\[\\\\tab\]\\tab\{0\} can be used \\tab\{15\} effectively}\newline +{\tt \\end\{items\}}\newline +{\tt \\indent\{0\}}\newline +\end{paste} +\endImportant + + +Note that the {\tt \\begin\{items\}} command immediately sets the left- +page-margin to a new value. Subsequent +{\tt \\tab} or {\tt \\centerline} commands +refer to this new margin. +Any explicit margin setting commands included +in the group {\em will} have the normal effect. +The {\tt \\par} command does not produce +the standard paragraph indentation within a list group --- it behaves +instead like {\tt \\newline}. + + +You can nest list groups like the following example suggests. +\beginImportant +\begin{paste}{HTXFormatPage5xPaste2}{HTXFormatPage5xPatch2} +\pastebutton{HTXFormatPage5xPaste2}{Interpret} +\newline +{\tt \\begin\{items\}\[quitealot\]}\newline +{\tt A nested list:}\newline +{\tt \\item\[The first\] item of an itemized list is on this line.}\newline +{\tt \\item\[The second\] item of the list starts here. It contains another}\newline +{\tt list nested inside it.}\newline +{\tt \\begin\{items\}\[somuchmore\]}\newline +{\tt \\item \[First\]\\tab\{0\}This is the first item of an enumerated}\newline +{\tt list that is nested within the itemized list.}\newline +{\tt \\item \[Second\]\\tab\{0\}This is the second item of the inner list.}\newline +{\tt \\end\{items\}}\newline +{\tt This is the rest of the second item of the outer list. It}\newline +{\tt is no more interesting than any other part of the item.}\newline +{\tt \\item\[The third\] item of the list.}\newline +{\tt \\end\{items\}}\newline +\end{paste} +\endImportant + +Another facility for presenting lists is the {\tt \\table} command. +The correct syntax for it is : {\tt \\table\{\{{\it item a}\} \{{\it item b}\} {\it ..}\}}. +The items in the braces will be placed in as many aligned columns +as is possible for the current window dimensions or page width. +If one item is particularly long there will probably be only one column +in the table. Here is a table of color names. +\beginImportant +\begin{paste}{HTXFormatPage5xPaste3}{HTXFormatPage5xPatch3} +\pastebutton{HTXFormatPage5xPaste3}{Interpret} +\newline +{\tt +\\table\{ +\{Dark Orchid\} \{Dark Salmon\} \{Dark Sea Green\} \{Dark Slate Blue\} +\{Dark Slate Gray\} \{Dark Turquoise\} \{Dark Violet\} \{Deep Pink\} +\{Deep Sky Blue\} \{Dodger Blue\} \{Floral White\} \{Forest Green\} +\{Ghost White\} \{Hot Pink\} \{Indian Red\} \{Lavender Blush\} +\} +} +\end{paste} +\endImportant + + + + +\end{scroll} +\beginmenu +\menulink{Next -- Boxes and Lines}{HTXFormatPage6} +\endmenu + +\end{page} + +@ +\subsection{HTXFormatPage5xPatch1 patch} +\label{HTXFormatPage5xPatch1} +\index{patch!HTXFormatPage5xPatch1!htxformatpage5.ht} +\index{htxformatpage5.ht!patch!HTXFormatPage5xPatch1} +\index{HTXFormatPage5xPatch1!htxformatpage5.ht!patch} +<>= +\begin{patch}{HTXFormatPage5xPatch1} +\begin{paste}{HTXFormatPage5xPaste1A}{HTXFormatPage5xPatch1A} +\pastebutton{HTXFormatPage5xPaste1A}{Source} +\newline +\indent{5} +\begin{items}[how wide am I] +Here we carry on but a \newline +new line will be indented +\item[how wide am I] fits nicely. Here is a \newline new line in an item. +\item[again] to show another item +\item[\\tab]\tab{0} can be used \tab{15} effectively +\end{items} +\indent{0} +\end{paste} +\end{patch} + +@ +\subsection{HTXFormatPage5xPatch1A patch} +\label{HTXFormatPage5xPatch1A} +\index{patch!HTXFormatPage5xPatch1A!htxformatpage5.ht} +\index{htxformatpage5.ht!patch!HTXFormatPage5xPatch1A} +\index{HTXFormatPage5xPatch1A!htxformatpage5.ht!patch} +<>= +\begin{patch}{HTXFormatPage5xPatch1A} +\begin{paste}{HTXFormatPage5xPaste1B}{HTXFormatPage5xPatch1} +\pastebutton{HTXFormatPage5xPaste1B}{Interpret} +\newline +{\tt \\indent\{5\}}\newline +{\tt \\begin\{items\}\[how wide am I\]}\newline +{\tt Here we carry on but a \\newline} \newline +{\tt new line will be indented } \newline +{\tt \\item\[how wide am I\] fits nicely. Here is a \\newline new line in an item.}\newline +{\tt \\item\[again\] to show another item}\newline +{\tt \\item\[\\\\tab\]\\tab\{0\} can be used \\tab\{15\} effectively}\newline +{\tt \\end\{items\}}\newline +{\tt \\indent\{0\}}\newline +\end{paste} +\end{patch} + +@ +\subsection{HTXFormatPage5xPatch2 patch} +\label{HTXFormatPage5xPatch2} +\index{patch!HTXFormatPage5xPatch2!htxformatpage5.ht} +\index{htxformatpage5.ht!patch!HTXFormatPage5xPatch2} +\index{HTXFormatPage5xPatch2!htxformatpage5.ht!patch} +<>= +\begin{patch}{HTXFormatPage5xPatch2} +\begin{paste}{HTXFormatPage5xPaste2A}{HTXFormatPage5xPatch2A} +\pastebutton{HTXFormatPage5xPaste2A}{Source} +\newline +\begin{items}[quitealot] +A nested list: +\item[The first] item of an itemized list is on this line. +\item[The second] item of the list starts here. It contains another +list nested inside it. +\begin{items}[somuchmore] +\item [First]\tab{0}This is the first item of the +list that is nested within the itemized list. +\item [Second]\tab{0}This is the second item of the inner list. +\end{items} +This is the rest of the second item of the outer list. It +is no more interesting than any other part of the item. +\item [The third] item of the list. +\end{items} +\end{paste} +\end{patch} + +@ +\subsection{HTXFormatPage5xPatch2A patch} +\label{HTXFormatPage5xPatch2A} +\index{patch!HTXFormatPage5xPatch2A!htxformatpage5.ht} +\index{htxformatpage5.ht!patch!HTXFormatPage5xPatch2A} +\index{HTXFormatPage5xPatch2A!htxformatpage5.ht!patch} +<>= +\begin{patch}{HTXFormatPage5xPatch2A} +\begin{paste}{HTXFormatPage5xPaste2B}{HTXFormatPage5xPatch2} +\pastebutton{HTXFormatPage5xPaste2B}{Interpret} +\newline +{\tt \\begin\{items\}\[quitealot\]}\newline +{\tt A nested list:}\newline +{\tt \\item\[The first\] item of an itemized list is on this line.}\newline +{\tt \\item\[The second\] item of the list starts here. It contains another}\newline +{\tt list nested inside it.}\newline +{\tt \\begin\{items\}\[somuchmore\]}\newline +{\tt \\item \[First\]\\tab\{0\}This is the first item of an enumerated}\newline +{\tt list that is nested within the itemized list.}\newline +{\tt \\item \[Second\]\\tab\{0\}This is the second item of the inner list.}\newline +{\tt \\end\{items\}}\newline +{\tt This is the rest of the second item of the outer list. It}\newline +{\tt is no more interesting than any other part of the item.}\newline +{\tt \\item\[The third\] item of the list.}\newline +{\tt \\end\{items\}}\newline +\end{paste} +\end{patch} + +@ +\subsection{HTXFormatPage5xPatch3 patch} +\label{HTXFormatPage5xPatch3} +\index{patch!HTXFormatPage5xPatch3!htxformatpage5.ht} +\index{htxformatpage5.ht!patch!HTXFormatPage5xPatch3} +\index{HTXFormatPage5xPatch3!htxformatpage5.ht!patch} +<>= +\begin{patch}{HTXFormatPage5xPatch3} +\begin{paste}{HTXFormatPage5xPaste3A}{HTXFormatPage5xPatch3A} +\pastebutton{HTXFormatPage5xPaste3A}{Source} +\newline +\table{ +{Dark Orchid} {Dark Salmon} {Dark Sea Green} {Dark Slate Blue} {Dark Slate Gray} +{Dark Turquoise} {Dark Violet} {Deep Pink} {Deep Sky Blue} {Dodger Blue} +{Floral White} {Forest Green} {Ghost White} {Hot Pink} {Indian Red} +{Lavender Blush} +} +\end{paste} +\end{patch} + +@ +\subsection{HTXFormatPage5xPatch3A patch} +\label{HTXFormatPage5xPatch3A} +\index{patch!HTXFormatPage5xPatch3A!htxformatpage5.ht} +\index{htxformatpage5.ht!patch!HTXFormatPage5xPatch3A} +\index{HTXFormatPage5xPatch3A!htxformatpage5.ht!patch} +<>= +\begin{patch}{HTXFormatPage5xPatch3A} +\begin{paste}{HTXFormatPage5xPaste3B}{HTXFormatPage5xPatch3} +\pastebutton{HTXFormatPage5xPaste3B}{Interpret} +\newline +{\tt +\\table\{ +\{Dark Orchid\} \{Dark Salmon\} \{Dark Sea Green\} \{Dark Slate Blue\} +\{Dark Slate Gray\} \{Dark Turquoise\} \{Dark Violet\} \{Deep Pink\} +\{Deep Sky Blue\} \{Dodger Blue\} \{Floral White\} \{Forest Green\} +\{Ghost White\} \{Hot Pink\} \{Indian Red\} \{Lavender Blush\} +\} +} +\end{paste} +\end{patch} + +@ +\section{htxformatpage6} +\subsection{Boxes and Lines} +\label{HTXFormatPage6} +\index{pages!HTXFormatPage6!htxformatpage6.ht} +\index{htxformatpage6.ht!pages!HTXFormatPage6} +\index{HTXFormatPage6!htxformatpage6.ht!pages} +<>= +\begin{page}{HTXFormatPage6}{Boxes and Lines} +\centerline{\fbox{{\tt \thispage}}}\newline +\begin{scroll} + +The {\tt \\fbox} command can be used to place a box around one or more +words. The argument of the {\tt \\fbox} command is the text that will be +placed in the box. This command should only be used for text that can fit +in one line. + +\beginImportant +\begin{paste}{HTXFormatPage6xPaste1}{HTXFormatPage6xPatch1} +\pastebutton{HTXFormatPage6xPaste1}{Interpret} +\newline +{\tt \\fbox\{Boxed!\}}\newline +\end{paste} +\endImportant + + +Use the {\tt \\horizontalline} command to draw a horizontal line +across the window. This might be useful for added emphasis. + +\beginImportant +\begin{paste}{HTXFormatPage6xPaste2}{HTXFormatPage6xPatch2} +\pastebutton{HTXFormatPage6xPaste2}{Interpret} +\newline +{\tt \\horizontalline}\newline +\end{paste} +\endImportant + +\end{scroll} +\beginmenu +\menulink{Next -- Micro-Spacing}{HTXFormatPage7} +\endmenu + +\end{page} + +@ +\subsection{HTXFormatPage6xPatch1 patch} +\label{HTXFormatPage6xPatch1} +\index{patch!HTXFormatPage6xPatch1!htxformatpage6.ht} +\index{htxformatpage6.ht!patch!HTXFormatPage6xPatch1} +\index{HTXFormatPage6xPatch1!htxformatpage6.ht!patch} +<>= +\begin{patch}{HTXFormatPage6xPatch1} +\begin{paste}{HTXFormatPage6xPaste1A}{HTXFormatPage6xPatch1A} +\pastebutton{HTXFormatPage6xPaste1A}{Source} +\newline +\fbox{Boxed!} +\end{paste} +\end{patch} +\begin{patch}{HTXFormatPage6xPatch1A} +\begin{paste}{HTXFormatPage6xPaste1B}{HTXFormatPage6xPatch1} +\pastebutton{HTXFormatPage6xPaste1B}{Interpret} +\newline +{\tt \\fbox\{Boxed!\}}\newline +\end{paste} +\end{patch} + +@ +\subsection{HTXFormatPage6xPatch2 patch} +\label{HTXFormatPage6xPatch2} +\index{patch!HTXFormatPage6xPatch2!htxformatpage6.ht} +\index{htxformatpage6.ht!patch!HTXFormatPage6xPatch2} +\index{HTXFormatPage6xPatch2!htxformatpage6.ht!patch} +<>= +\begin{patch}{HTXFormatPage6xPatch2} +\begin{paste}{HTXFormatPage6xPaste2A}{HTXFormatPage6xPatch2A} +\pastebutton{HTXFormatPage6xPaste2A}{Source} +\newline +\horizontalline +\end{paste} +\end{patch} +\begin{patch}{HTXFormatPage6xPatch2A} +\begin{paste}{HTXFormatPage6xPaste2B}{HTXFormatPage6xPatch2} +\pastebutton{HTXFormatPage6xPaste2B}{Interpret} +\newline +{\tt \\horizontalline}\newline +\end{paste} +\end{patch} + +@ +\section{htxformatpage7} +\subsection{Micro-Spacing} +\label{HTXFormatPage7} +\index{pages!HTXFormatPage7!htxformatpage7.ht} +\index{htxformatpage7.ht!pages!HTXFormatPage7} +\index{HTXFormatPage7!htxformatpage7.ht!pages} +<>= +\begin{page}{HTXFormatPage7}{Micro-Spacing} +\centerline{\fbox{{\tt \thispage}}}\newline +\begin{scroll} + +There are three commands that one can use to exercise finer control +over the appearance of text on a page: {\tt \\space}, {\tt \\hspace} +and {\tt \\vspace}. + +The {\tt \\space\{{\it value}\}} command accepts an integer argument and simply +changes the position of the next character to the right or to the left. +A negative argument will move the next character to the left and a +positive one to the right. The unit of movement is {\it the width +of a character}. In this way one can overstrike characters to produce +various effects. + +\beginImportant +\begin{paste}{HTXFormatPage7xPaste1}{HTXFormatPage7xPatch1} +\pastebutton{HTXFormatPage7xPaste1}{Interpret} +\newline +{\tt 0\\space\{-1\}\\}\newline +{\tt underlined\\space\{-10\}__________}\newline +\end{paste} +\endImportant + + +The {\tt \\hspace\{{\it value}\}} command +is similar to the {\tt \\space\{{\it value}\}} command. +It also accepts an integer argument and +changes the position of the next character to the right or to the left. +A negative argument will move the next character to the left and a +positive one to the right. The unit of movement is {\it a pixel}. +The {\it value} argument specifies an offset from the default placement +of the character. + +\beginImportant +\begin{paste}{HTXFormatPage7xPaste2}{HTXFormatPage7xPatch2} +\pastebutton{HTXFormatPage7xPaste2}{Interpret} +\newline +{\tt x\\hspace\{-4\}x\\hspace\{-3\}x\\hspace\{-2\}x\\hspace\{-1\}x\%}\newline +{\tt x\\hspace\{1\}x\\hspace\{2\}x\\hspace\{3\}x\\hspace\{4\}x} +\end{paste} +\endImportant + +The {\tt \\vspace\{{\it value}\}} command is similar to the +{\tt \\hspace\{{\it value}\}} command but (as the name suggests) +works in the vertical direction. The unit of movement is {\it a +pixel}. The {\it value} argument specifies an offset from {\it +the next line}. A negative argument moves the next character up +and a positive down. This command can be used for subscripts and +superscripts. One drawback in the use of {\tt \\vspace} is that +it can only work with a particular font at a time. This is +because the inter-line spacing depends on the font being used +and the value of it is needed to get "back" on the line. +In general, the command {\tt \\vspace\{{\it - ils}\}} will +have a null effect when {\it ils} = ( font ascent + font descent + 5 ). +The example below assumes that {\it ils} = 25 e.g. the Rom14 font +on the RISC System/6000. + +\beginImportant +\begin{paste}{HTXFormatPage7xPaste3}{HTXFormatPage7xPatch3} +\pastebutton{HTXFormatPage7xPaste3}{Interpret} +\newline +{\tt CO\\vspace\{-18\}2\\vspace\{-32\} + CaO ->}\newline +{\tt CaCO\\vspace\{-18\}3\\vspace\{-32\}\\newline}\newline +{\tt R\\space\{-1\}~\\vspace\{-18\}æv\\vspace\{-32\}}\newline +{\tt \\hspace\{4\}-\\hspace\{8\}---\\hspace\{-12\}}\newline +{\tt \\vspace\{-32\}1\\space\{-1\}\\vspace\{-7\}2}\newline +{\tt \\vspace\{-36\}\\hspace\{8\}g\\space\{-1\}~}\newline +{\tt \\vspace\{-18\}æv\\vspace\{-32\}\\hspace\{2\}R}\newline +{\tt \\space\{-1\}~ = T\\vspace\{-18\}æv\\vspace\{-32\}}\newline +{\tt \\vspace\{-25\}} +\end{paste} +\endImportant + + +\end{scroll} +\beginmenu +\menulink{Next -- Bitmaps and Images}{HTXFormatPage8} +\endmenu + +\end{page} + +@ +\subsection{HTXFormatPage7xPatch1 patch} +\label{HTXFormatPage7xPatch1} +\index{patch!HTXFormatPage7xPatch1!htxformatpage7.ht} +\index{htxformatpage7.ht!patch!HTXFormatPage7xPatch1} +\index{HTXFormatPage7xPatch1!htxformatpage7.ht!patch} +<>= +\begin{patch}{HTXFormatPage7xPatch1} +\begin{paste}{HTXFormatPage7xPaste1A}{HTXFormatPage7xPatch1A} +\pastebutton{HTXFormatPage7xPaste1A}{Source} +\newline +0\space{-1}\\ +underlined\space{-10}__________ +\end{paste} +\end{patch} +\begin{patch}{HTXFormatPage7xPatch1A} +\begin{paste}{HTXFormatPage7xPaste1B}{HTXFormatPage7xPatch1} +\pastebutton{HTXFormatPage7xPaste1B}{Interpret} +\newline +{\tt 0\\space\{-1\}\\}\newline +{\tt underlined\\space\{-10\}__________}\newline +\end{paste} +\end{patch} + +@ +\subsection{HTXFormatPage7xPatch2 patch} +\label{HTXFormatPage7xPatch2} +\index{patch!HTXFormatPage7xPatch2!htxformatpage7.ht} +\index{htxformatpage7.ht!patch!HTXFormatPage7xPatch2} +\index{HTXFormatPage7xPatch2!htxformatpage7.ht!patch} +<>= +\begin{patch}{HTXFormatPage7xPatch2} +\begin{paste}{HTXFormatPage7xPaste2A}{HTXFormatPage7xPatch2A} +\pastebutton{HTXFormatPage7xPaste2A}{Source} +\newline +x\hspace{-4}x\hspace{-3}x\hspace{-2}x\hspace{-1}x% +x\hspace{1}x\hspace{2}x\hspace{3}x\hspace{4}x +\end{paste} +\end{patch} + +@ +\subsection{HTXFormatPage7xPatch2A patch} +\label{HTXFormatPage7xPatch2A} +\index{patch!HTXFormatPage7xPatch2A!htxformatpage7.ht} +\index{htxformatpage7.ht!patch!HTXFormatPage7xPatch2A} +\index{HTXFormatPage7xPatch2A!htxformatpage7.ht!patch} +<>= +\begin{patch}{HTXFormatPage7xPatch2A} +\begin{paste}{HTXFormatPage7xPaste2B}{HTXFormatPage7xPatch2} +\pastebutton{HTXFormatPage7xPaste2B}{Interpret} +\newline +{\tt x\\hspace\{-4\}x\\hspace\{-3\}x\\hspace\{-2\}x\\hspace\{-1\}x\%}\newline +{\tt x\\hspace\{1\}x\\hspace\{2\}x\\hspace\{3\}x\\hspace\{4\}x} +\end{paste} +\end{patch} + +@ +\subsection{HTXFormatPage7xPatch3 patch} +\label{HTXFormatPage7xPatch3} +\index{patch!HTXFormatPage7xPatch3!htxformatpage7.ht} +\index{htxformatpage7.ht!patch!HTXFormatPage7xPatch3} +\index{HTXFormatPage7xPatch3!htxformatpage7.ht!patch} +<>= +\begin{patch}{HTXFormatPage7xPatch3} +\begin{paste}{HTXFormatPage7xPaste3A}{HTXFormatPage7xPatch3A} +\pastebutton{HTXFormatPage7xPaste3A}{Source} +\newline +CO\vspace{-18}2\vspace{-32} + CaO -> +CaCO\vspace{-18}3\vspace{-32}\newline +R\space{-1}~\vspace{-18}æv\vspace{-32} +\hspace{4}-\hspace{8}---\hspace{-12} +\vspace{-32}1\space{-1}\vspace{-7}2 +\vspace{-36}\hspace{8}g\space{-1}~ +\vspace{-18}æv\vspace{-32}\hspace{2}R +\space{-1}~ = T\vspace{-18}æv\vspace{-32} +\vspace{-25} +\end{paste} +\end{patch} + +@ +\subsection{HTXFormatPage7xPatch3A patch} +\label{HTXFormatPage7xPatch3A} +\index{patch!HTXFormatPage7xPatch3A!htxformatpage7.ht} +\index{htxformatpage7.ht!patch!HTXFormatPage7xPatch3A} +\index{HTXFormatPage7xPatch3A!htxformatpage7.ht!patch} +<>= +\begin{patch}{HTXFormatPage7xPatch3A} +\begin{paste}{HTXFormatPage7xPaste3B}{HTXFormatPage7xPatch3} +\pastebutton{HTXFormatPage7xPaste3B}{Interpret} +\newline +{\tt CO\\vspace\{-18\}2\\vspace\{-32\} + CaO ->}\newline +{\tt CaCO\\vspace\{-18\}3\\vspace\{-32\}\\newline}\newline +{\tt R\\space\{-1\}~\\vspace\{-18\}æv\\vspace\{-32\}}\newline +{\tt \\hspace\{4\}-\\hspace\{8\}---\\hspace\{-12\}}\newline +{\tt \\vspace\{-32\}1\\space\{-1\}\\vspace\{-7\}2}\newline +{\tt \\vspace\{-36\}\\hspace\{8\}g\\space\{-1\}~}\newline +{\tt \\vspace\{-18\}æv\\vspace\{-32\}\\hspace\{2\}R}\newline +{\tt \\space\{-1\}~ = T\\vspace\{-18\}æv\\vspace\{-32\}}\newline +{\tt \\vspace\{-25\}} +\end{paste} +\end{patch} + +@ +\section{htxformatpage8} +\subsection{Bitmaps and Images} +\label{HTXFormatPage8} +\index{pages!HTXFormatPage8!htxformatpage8.ht} +\index{htxformatpage8.ht!pages!HTXFormatPage8} +\index{HTXFormatPage8!htxformatpage8.ht!pages} +<>= +\begin{page}{HTXFormatPage8}{Bitmaps and Images} +\centerline{\fbox{{\tt \thispage}}}\newline +\begin{scroll} + +The commands {\tt \\inputbitmap\{{\it filename}\}} +and {\tt \\inputimage\{{\it filename}\}} +allow you to include an X11 bitmap or an +Axiom-generated viewport in a \HyperName{} +page. + +In the case of the {\tt \\inputbitmap} command +the {\it filename} parameter must be the full pathname +of an X11 bitmap file. + +\beginImportant +\begin{paste}{HTXFormatPage8xPaste1}{HTXFormatPage8xPatch1} +\pastebutton{HTXFormatPage8xPaste1}{Interpret} +\newline +{\tt \\inputbitmap\{\env{AXIOM}/doc/hypertex/bitmaps/sup.bitmap\} } +\end{paste} +\endImportant + +The {\it filename} parameter of the {\tt \\inputimage} +command must be the full pathname of a {\it compressed XPM image} file without the name extensions. +\HyperName{} always adds ".xpm.Z" to whatever filename you give and looks for the augmented filename. +Such files can be generated by Axiom command +\axiomOp{write} with the {\tt "image"} or {\tt "pixmap"} +options. + +\beginImportant +\begin{paste}{HTXFormatPage8xPaste2}{HTXFormatPage8xPatch2} +\pastebutton{HTXFormatPage8xPaste2}{Interpret} +\newline +{\tt \\inputimage\{\env{AXIOM}/doc/viewports/ugProblemNumericPage30.view/image\}} +\end{paste} +\endImportant + +Be careful not to break the pathname across lines. + +The {\tt \\inputimage} command will automatically select +the {\it image.xpm} or the {\it image.bm} file for you +based on the capabilities of your X server. + +For your convenience, there are two macros defined +in \centerline{ {\bf \env{AXIOM}{}/doc/hypertex/pages/util.ht}.} +The {\tt \\viewport} macro eliminates the need to specify +the {\tt .view/image} part and the +{\tt \\axiomViewport} macro automatically selects viewport +files in the system directories. The above {\tt \\inputimage} +could have been written +\beginImportant +{\tt \\viewport\{\env{AXIOM}/doc/viewports/ugProblemNumericPage30\}} +\endImportant +or +\beginImportant +{\tt \\axiomViewport\{ugProblemNumericPage30\}} +\endImportant + + + +\end{scroll} +\beginmenu +\menulink{Back to Formatting menu}{HTXFormatTopPage} +\endmenu + +\end{page} + +@ +\subsection{HTXFormatPage8xPatch1 patch} +\label{HTXFormatPage8xPatch1} +\index{patch!HTXFormatPage8xPatch1!htxformatpage8.ht} +\index{htxformatpage8.ht!patch!HTXFormatPage8xPatch1} +\index{HTXFormatPage8xPatch1!htxformatpage8.ht!patch} +<>= +\begin{patch}{HTXFormatPage8xPatch1} +\begin{paste}{HTXFormatPage8xPaste1A}{HTXFormatPage8xPatch1A} +\pastebutton{HTXFormatPage8xPaste1A}{Source} +\newline +\inputbitmap{\env{AXIOM}/doc/hypertex/bitmaps/sup.bitmap} +\end{paste} +\end{patch} +\begin{patch}{HTXFormatPage8xPatch1A} +\begin{paste}{HTXFormatPage8xPaste1B}{HTXFormatPage8xPatch1} +\pastebutton{HTXFormatPage8xPaste1B}{Interpret} +\newline +{\tt \\inputbitmap\{\env{AXIOM}/doc/hypertex/bitmaps/sup.bitmap\} } +\end{paste} +\end{patch} + +@ +\subsection{HTXFormatPage8xPatch2 patch} +\label{HTXFormatPage8xPatch2} +\index{patch!HTXFormatPage8xPatch2!htxformatpage8.ht} +\index{htxformatpage8.ht!patch!HTXFormatPage8xPatch2} +\index{HTXFormatPage8xPatch2!htxformatpage8.ht!patch} +<>= +\begin{patch}{HTXFormatPage8xPatch2} +\begin{paste}{HTXFormatPage8xPaste2A}{HTXFormatPage8xPatch2A} +\pastebutton{HTXFormatPage8xPaste2A}{Source} +\newline +\inputimage{\env{AXIOM}/doc/viewports/ugProblemNumericPage30.view/image} +\end{paste} +\end{patch} +@ +\subsection{HTXFormatPage8xPatch2A patch} +\label{HTXFormatPage8xPatch2A} +\index{patch!HTXFormatPage8xPatch2A!htxformatpage8.ht} +\index{htxformatpage8.ht!patch!HTXFormatPage8xPatch2A} +\index{HTXFormatPage8xPatch2A!htxformatpage8.ht!patch} +<>= +\begin{patch}{HTXFormatPage8xPatch2A} +\begin{paste}{HTXFormatPage8xPaste2B}{HTXFormatPage8xPatch2} +\pastebutton{HTXFormatPage8xPaste2B}{Interpret} +\newline +{\tt \\inputimage\{\env{AXIOM}/doc/viewports/ugProblemNumericPage30.view/image\}} +\end{paste} +\end{patch} + +@ +\section{htxformattoppage.ht} +\subsection{Formatting in Hyperdoc} +\begin{itemize} +\item HTXFormatPage1 \ref{HTXFormatPage1} on page~\pageref{HTXFormatPage1} +\item HTXFormatPage2 \ref{HTXFormatPage2} on page~\pageref{HTXFormatPage2} +\item HTXFormatPage3 \ref{HTXFormatPage3} on page~\pageref{HTXFormatPage3} +\item HTXFormatPage4 \ref{HTXFormatPage4} on page~\pageref{HTXFormatPage4} +\item HTXFormatPage5 \ref{HTXFormatPage5} on page~\pageref{HTXFormatPage5} +\item HTXFormatPage6 \ref{HTXFormatPage6} on page~\pageref{HTXFormatPage6} +\item HTXFormatPage7 \ref{HTXFormatPage7} on page~\pageref{HTXFormatPage7} +\item HTXFormatPage8 \ref{HTXFormatPage8} on page~\pageref{HTXFormatPage8} +\end{itemize} +\label{HTXFormatTopPage} +\index{pages!HTXFormatTopPage!htxformattoppage.ht} +\index{htxformattoppage.ht!pages!HTXFormatTopPage} +\index{HTXFormatTopPage!htxformattoppage.ht!pages} +<>= +\begin{page}{HTXFormatTopPage}{Formatting in Hyperdoc} +\centerline{\fbox{{\tt \thispage}}}\newline + +\HyperName{} offers various facilities for formatting text and images. +You can learn about these facilities by clicking on the topics below. +\begin{scroll} +\beginmenu +\menudownlink{Special Characters}{HTXFormatPage1} +\menudownlink{Formatting without commands}{HTXFormatPage2} +\menudownlink{Using different fonts}{HTXFormatPage3} +\menudownlink{Indentation}{HTXFormatPage4} +\menudownlink{Creating Lists and Tables}{HTXFormatPage5} +\menudownlink{Boxes and Lines}{HTXFormatPage6} +\menudownlink{Micro-Spacing}{HTXFormatPage7} +\menudownlink{Bitmaps and Images}{HTXFormatPage8} +\endmenu +\end{scroll} +\end{page} + + +@ +\section{htxintropage1.ht} +\subsection{What Hyperdoc does} +\label{HTXIntroPage1} +\begin{itemize} +\item HTXIntroPage2 \ref{HTXIntroPage2} on page~\pageref{HTXIntroPage2} +\item ugHyperPage \ref{ugHyperPage} on page~\pageref{ugHyperPage} +\end{itemize} +\index{pages!HTXIntroPage1!htxintropage1.ht} +\index{htxintropage1.ht!pages!HTXIntroPage1} +\index{HTXIntroPage1!htxintropage1.ht!pages} +<>= +\begin{page}{HTXIntroPage1}{What Hyperdoc does} +\centerline{\fbox{{\tt \thispage}}}\newline +\beginscroll + +Take a close look at the objects in the \HyperName{} window you are now reading. +Most of them are text. Resize the window using the window manager +facilities. The text is reformatted to fit the window +border. This action is performed by \HyperName{}. At the simplest +level, it provides a method for {\em formatting} text in a window. +In fact, it can place other things on the window as well, such as +bitmaps or color images. The {\em buttons} you see at either +side at the top of the window are bitmaps. + +Move the cursor so that it rests on one of those buttons. You notice that +the cursor has changed appearance. This indicates that there is an action associated +with the button. This action will be performed when you click the mouse button +over the {\em active area}. If you are familiar with \HyperName{}, you know +that the active area can be words, bitmaps, images or {\em input areas}. In +fact, anything that can be displayed in a \HyperName{} window can be an +active area. + +So, what can the action associated with an active area be? \HyperName{} +allows quite a bit of freedom in defining that action. We will have a close +look at this issue \downlink{later on}{HTXLinkTopPage}. For now, recall +the various actions that you have encountered so far --- executing Axiom +commands, popping up new windows, providing parameters for other active areas, +and replacing or changing the contents of the window. The most common action +is to bring up some \HyperName{} text in the same or a different window. +This lets us create {\em links} between pieces of text and images. A +system with such capability is usually called a {\em hypertext} system. +\HyperName{} is in fact much more. + +\endscroll +\beginmenu +\menulink{Next -- How \HyperName{} does it}{HTXIntroPage2} +\menuwindowlink{Review some features of \HyperName{}}{ugHyperPage} +\endmenu + +\helppage{ugHyperPage} +\end{page} + +@ +\section{htxintropage2.ht} +\subsection{How Hyperdoc does it} +\label{HTXIntroPage2} +See HTXIntroPage3 \ref{HTXIntroPage3} on page~\pageref{HTXIntroPage3} +\index{pages!HTXIntroPage2!htxintropage2.ht} +\index{htxintropage2.ht!pages!HTXIntroPage2} +\index{HTXIntroPage2!htxintropage2.ht!pages} +<>= +\begin{page}{HTXIntroPage2}{How Hyperdoc does it} +{\centerline{\fbox{{\tt \thispage}}}\newline} +\beginscroll + +\HyperName{} can read the {\em hypertext} information from standard text +files. This means that you can create or change this information with any +text editor. Once this information has been entered into the files, a +special program, called {\bf htadd}, scans these files and produces +a database (another file called {\bf ht.db}) of {\em objects} +encountered in the files. \HyperName{} +consults this database when it first starts and so knows where it might +find the definitions of these objects. You can maintain several such +databases on different directories. You indicate which database you +want \HyperName{} to consult by setting an {\em environment variable} +called {\bf HTPATH}. + +In general, hypertext must obviously use some kind of special (that is, +non-textual) marks for all the extra functionality it provides. In +\HyperName{}, these marks are some special characters --- special +in the sense that they are not interpreted as ordinary displayable text. +These characters, however, are part of the standard ASCII set. +There is also a way to display these special characters as text . +The \HyperName{} special characters are : + +\beginImportant +\noindent{\em Special Characters}: {\tt \table{{\$}{\\}{\{}{\}}{\[}{\]}{\%}{\#}}} +\endImportant + + + +\HyperName{} uses the special characters to distinguish between +{\em text} and {\em commands} (by {\em text}, we mean here anything +displayable). The commands are instructions to +\HyperName{} to treat some text in a particular way. Some commands +define special \HyperName{} objects. The most important objects +are {\em pages}, {\em patches}, and {\em macros}. +A {\em page} is a description of the contents of a +\HyperName{} window. A {\em patch} is a portion of a page. +A {\em macro} is a user-defined new \HyperName{} command. +Some commands allow special text {\em formatting} and others +associate some text with an action. + +In order to display anything at all in \HyperName, you must define a +{\em page}. The next section explains how to define a {\em page} and put +some simple text into it. +\endscroll +\beginmenu +\menudownlink{Next -- Define a simple text page}{HTXIntroPage3} +\endmenu + +\end{page} + + +@ +\section{htxintropage3.ht} +\subsection{A simple text page} +\begin{itemize} +\item HTXLinkPage6 \ref{HTXLinkPage6} on page~\pageref{HTXLinkPage6} +\item HTXTryPage \ref{HTXTryPage} on page~\pageref{HTXTryPage} +\item HTXFormatTopPage \ref{HTXFormatTopPage} on +page~\pageref{HTXFormatTopPage} +\end{itemize} +\label{HTXIntroPage3} +\index{pages!HTXIntroPage3!htxintropage3.ht} +\index{htxintropage3.ht!pages!HTXIntroPage3} +\index{HTXIntroPage3!htxintropage3.ht!pages} +<>= +\begin{page}{HTXIntroPage3}{A simple text page} +{\centerline{\fbox{{\tt \thispage}}}\newline} +\begin{scroll} + + +A page is defined by a {\em group} command. Group commands are used to +delimit a group, that is, to declare where a group starts and where it +ends. The proper syntax for a page definition is as follows: +\beginImportant +{\tt \\begin\{page\}\{{\it name}\}\{{\it a title}\}} +\newline +. +\newline +. +\newline +. +\newline +{\tt \\end\{page\}} +\beginImportant + +Note the use of the special characters {\tt \\}, {\tt \{} and {\tt +\}}. The {\tt \\} (backslash) character introduces a command, in this +case, {\tt begin}. The {\tt \{ \}} (braces) delimit the {\em +parameters} to the command. The first parameter (the word {\tt page}) +specifies this as a page definition command. + +The second parameter can be any single unbroken word consisting of +alphanumeric characters only, and specifies the name of the page by which +it can be referred to by other commands. You should choose +this internal name with care so as to avoid potential conflict with +page names that are defined by the Axiom system. This caveat only +applies in the case where you have started \HyperName{} with the Axiom +database --- see \downlink{later on}{HTXLinkPage6}. It is suggested that +the page names you define start with the letters {\tt UX} (standing for +{\tt U}ser e{\tt X}tensions). You can have a look at the Axiom +system database file {\centerline{\bf \env{AXIOM}/doc/hypertex/pages/ht.db} } +which contains the names of all pages, macros and patches used by Axiom. + +The third parameter specifies a title for the page. +The title of a page is the area at the very top +of the window, between the buttons. Virtually anything +that can be put in the main page can also be put in the +title. As an example, {\em this} page's +declaration is like this:\newline +{\tt \\begin\{page\}\{\thispage\}\{A simple text page\}} + +Everything you type between the {\tt \\begin\{page\}} command and the next +{\tt \\end\{page\}} command will become the body of the page. It is +an error to insert another {\tt \\begin\{page\}} between the two, that is, +this group command cannot be nested. + +There is another useful group command that should be mentioned here +--- the {\em scroll} command. It controls the portion of the page that +will be scrollable. \HyperName{} will split a page in three sections: +a {\em header}, a {\em scroll region} and a {\em footer}. \HyperName{} +will always try to keep the header and footer regions visible on the +page; the header at the top and the footer at the bottom. The middle +scroll region will be truncated and a scroll bar will be automatically +provided if the window becomes too small for the whole contents of the +page. Only one scroll region can be defined in a page and the correct +syntax is as follows: +\beginImportant +{\tt \\begin\{scroll\}} +\newline +. +\newline +. +\newline +. +\newline +{\tt \\end\{scroll\}} +\beginImportant + +This group should be placed inside the relevant page group. The text +between the {\tt \\begin\{page\}} and {\tt \\begin\{scroll\}} commands +defines the header region, the text inside the scroll group defines +the scroll region and the text between the {\tt \\end\{scroll\}} and +{\tt \\end\{page\}} commands defines the footer region. It is +important to keep the header and footer areas small. Use them to +display information that might be needed at any time by the user. If +you don't define a scroll region in your page, you may find that a +portion of the page is truncated. + +You are now ready to experiment with a page of your own. If you just +want to display some text on a page, you don't need any other +\HyperName{} commands. Just make sure that the text you type for the +title, header, scroll and footer regions does not contain (for the +moment) any of the \HyperName{} special characters. + +\end{scroll} +\beginmenu +\menuwindowlink{Try out what you learned}{HTXTryPage} +\menudownlink{Next -- Learn how to format text}{HTXFormatTopPage} +\endmenu +\end{page} + + +@ +\section{htxintrotoppage.ht} +\subsection{First Steps} +\begin{itemize} +\item HTXIntroPage1 \ref{HTXIntroPage1} on page~\pageref{HTXIntroPage1} +\item HTXIntroPage2 \ref{HTXIntroPage2} on page~\pageref{HTXIntroPage2} +\item HTXIntroPage3 \ref{HTXIntroPage3} on page~\pageref{HTXIntroPage3} +\end{itemize} +\label{HTXIntroTopPage} +\index{pages!HTXIntroTopPage!htxintrotoppage.ht} +\index{htxintrotoppage.ht!pages!HTXIntroTopPage} +\index{HTXIntroTopPage!htxintrotoppage.ht!pages} +<>= +\begin{page}{HTXIntroTopPage}{First Steps} +\centerline{\fbox{{\tt \thispage}}}\newline +\beginscroll + +\HyperName{} is both a way of presenting information and +a customisable front-end. Axiom uses +it for its own purpose as a front-end and documentation system. +\HyperName{} has special facilities that allow it to interact +very closely with Axiom. The \Browse{} facility, the Basic +Commands section and the ability to execute Axiom commands +by clicking on \HyperName{} text are witness to this. + +These pages will show you the features of \HyperName{} that might +make it appropriate for your own use in, for example, providing +documentation for Axiom code that you write or some other purpose. + +It is recommended that you get familiar with the {\em use} of +\HyperName{} before proceeding. + +\endscroll +\beginmenu +\menudownlink{What \HyperName{} does}{HTXIntroPage1} +\menudownlink{How \HyperName{} does it}{HTXIntroPage2} +\menudownlink{Define a simple text page}{HTXIntroPage3} +\endmenu + +\end{page} + +@ +\section{htxlinkpage1.ht} +\subsection{Linking to a named page} +\label{HTXLinkPage1} +\begin{itemize} +\item HTXLinkTopPage \ref{HTXLinkTopPage} on page~\pageref{HTXLinkTopPage} +\item TestHelpPage \ref{TestHelpPage} on page~\pageref{TestHelpPage} +\item HTXLinkPage2 \ref{HTXLinkPage2} on page~pageref{HTXLinkPage2} +\end{itemize} +\index{pages!HTXLinkPage1!htxlinkpage1.ht} +\index{htxlinkpage1.ht!pages!HTXLinkPage1} +\index{HTXLinkPage1!htxlinkpage1.ht!pages} +<>= +\begin{page}{HTXLinkPage1}{Linking to a named page} +\centerline{\fbox{{\tt \thispage}}}\newline +\begin{scroll} + +In \HyperName{}, hypertext links are specified by different +flavors of the {\tt \\link} command. These commands take two +arguments. One argument specifies the active area, that +is, the {\it trigger} of the link. The second argument specifies the +{\it target} of the link, that is, a page. The trigger can be quite arbitrary +\HyperName{} text and can include images or whole paragraphs. The trigger +text will be formatted in the normal fashion but its default font will be +the font specified by the ActiveFont resource. + +The simplest kind of \HyperName{} link is a link to a named page. +Clicking on the trigger will cause the named page to appear in a +\HyperName{} window. +There are three flavors for such a link. +\begin{items}[123456] +\item\menuitemstyle{{\tt \\windowlink\{{\it trigger}\}\{{\it page name}\}}} +\newline +This link command, when activated, will create a new window for the named page. +\newline +There will be no \centerline{\UpBitmap{} or \ReturnBitmap{}} +buttons on the new page. +The new page will have a \centerline{\ExitBitmap{}} button, however. +The original page containing the {\tt \\windowlink} command will be unaffected. +\item\menuitemstyle{{\tt \\downlink\{{\it trigger}\}\{{\it page name}\}} } +\newline This link command, when activated, will cause the +current page to be replaced by the target page +in the same \HyperName{} window. +A \centerline{\UpBitmap{}} button will automatically be placed +on the new page allowing you to get back to the page +containing the {\tt \\downlink} command. +If the current page has a \centerline{\ReturnBitmap{}} button then +the target page will also carry it. The associated +target page of that button will be the same as it is +in the current page. +\item\menuitemstyle{{\tt \\memolink\{{\it trigger}\}\{{\it page name}\}}} +\newline This link command is similar to the {\tt \\downlink} command. +In addition, it will cause a \centerline{\ReturnBitmap{}} +button to be included in +the target page and all pages {\tt \\downlink}ed from it. +This button will act as a +direct link to the page containing the {\tt \\memolink} command allowing +a short-cut to be taken. +\end{items} + +\beginImportant +\begin{paste}{HTXLinkPage1xPaste1}{HTXLinkPage1xPatch1} +\pastebutton{HTXLinkPage1xPaste1}{Interpret} +\newline +{\tt \\windowlink\{windowlink to Actions menu\}\{HTXLinkTopPage\}\\newline}\newline +{\tt \\downlink\{downlink to Actions menu\}\{HTXLinkTopPage\}\\newline}\newline +{\tt \\memolink\{memolink to Actions menu\}\{HTXLinkTopPage\}} +\end{paste} +\endImportant + + +There is a fourth button that can appear at the top of the page +next to the \centerline{\ExitBitmap{}} button. +Its purpose is to provide access to a particular {\it help page} +associated with the current page. +That is the \centerline{\HelpBitmap{}} button. The command to use +is +\centerline{{\tt \\helppage\{{\it help page name}\}}} +The {\tt \\helppage} command {\it must } be placed +just before the {\tt \\end\{page\}} command. +For instance, to get a help button on this page +the following command is used. +\centerline{{\tt {\\helppage\{TestHelpPage\}}}} +Clicking on the help button at the top +will display the {\tt TestHelpPage} page in a new window. + +\end{scroll} +\beginmenu +\menulink{Next -- Standard Pages}{HTXLinkPage2} +\endmenu + +\helppage{TestHelpPage} +\end{page} + +@ +\subsection{HTXLinkPage1xPatch1 patch} +\label{HTXLinkPage1xPatch1} +\index{patch!HTXLinkPage1xPatch1!htxlinkpage1.ht} +\index{htxlinkpage1.ht!patch!HTXLinkPage1xPatch1} +\index{HTXLinkPage1xPatch1!htxlinkpage1.ht!patch} +<>= +\begin{patch}{HTXLinkPage1xPatch1} +\begin{paste}{HTXLinkPage1xPaste1A}{HTXLinkPage1xPatch1A} +\pastebutton{HTXLinkPage1xPaste1A}{Source} +\newline +\windowlink{windowlink to Actions +menu}{HTXLinkTopPage}\newline +\downlink{downlink to Actions menu}{HTXLinkTopPage}\newline +\memolink{memolink to Actions menu}{HTXLinkTopPage} +\end{paste} +\end{patch} + +@ +\subsection{HTXLinkPage1xPatch1A patch} +\label{HTXLinkPage1xPatch1A} +\index{patch!HTXLinkPage1xPatch1A!htxlinkpage1.ht} +\index{htxlinkpage1.ht!patch!HTXLinkPage1xPatch1A} +\index{HTXLinkPage1xPatch1A!htxlinkpage1.ht!patch} +<>= +\begin{patch}{HTXLinkPage1xPatch1A} +\begin{paste}{HTXLinkPage1xPaste1B}{HTXLinkPage1xPatch1} +\pastebutton{HTXLinkPage1xPaste1B}{Interpret} +\newline +{\tt \\windowlink\{windowlink to Actions menu\}\{HTXLinkTopPage\}\\newline}\newline +{\tt \\downlink\{downlink to Actions menu\}\{HTXLinkTopPage\}\\newline}\newline +{\tt \\memolink\{memolink to Actions menu\}\{HTXLinkTopPage\}} +\end{paste} +\end{patch} + +@ +\subsection{Test Help Page} +\label{TestHelpPage} +\index{pages!TestHelpPage!htxlinkpage1.ht} +\index{htxlinkpage1.ht!pages!TestHelpPage} +\index{TestHelpPage!htxlinkpage1.ht!pages} +<>= +\begin{page}{TestHelpPage}{Test Help Page} +\begin{scroll} + +\vspace{100} +\centerline{Is this any help?} +\end{scroll} +\end{page} + +@ +\section{htxlinkpage2.ht} +\subsection{Standard Pages} +\label{HTXLinkPage2} +\begin{itemize} +\item HTXLinkPage6 \ref{HTXLinkPage6} on page~\pageref{HTXLinkPage6} +\item SpadNotConnectedPage \ref{SpadNotConnectedPage} on +page~\pageref{SpadNotConnectedPage} +\item UnknownPage \ref{UnknownPage} on page~\pageref{UnknownPage} +\item ErrorPage \ref{ErrorPage} on page~\pageref{ErrorPage} +\item ProtectedQuitPage \ref{ProtectedQuitPage} on +page~\pageref{ProtectedQuitPage} +\item HTXLinkPage3 \ref{HTXLinkPage3} on page~\pageref{HTXLinkPage3} +\end{itemize} +\index{pages!HTXLinkPage2!htxlinkpage2.ht} +\index{htxlinkpage2.ht!pages!HTXLinkPage2} +\index{HTXLinkPage2!htxlinkpage2.ht!pages} +<>= +\begin{page}{HTXLinkPage2}{Standard Pages} +\centerline{\fbox{{\tt \thispage}}}\newline +\begin{scroll} + +You have reached this page after performing +a series of mouse clicks on \HyperName{} +active areas. Each time, a {\tt \\link} +command was activated. Well, how does it all +start? +The answer is that \HyperName{} always puts up +a particular page called {\tt RootPage} when +it starts up. If this page is not found in the database, +\HyperName{} will immediately exit. +It is, of course, desirable that the {\tt RootPage} +contains links to other pages! +It is possible to override +Axiom's choice of {\tt RootPage} and provide your own +to \HyperName{}. This is done in the same way as +you would override any Axiom-defined page and is +discussed in \downlink{How to use your pages with \HyperName{}}{HTXLinkPage6}. + + + +You may have noticed that \HyperName{} +uses some pages when certain events occur. +There is a page that is put up, for instance, +whenever \HyperName{} cannot connect to Axiom. +Another page is put up whenever there is a formatting +error and yet another when a request for an unknown page +is made. Finally, there is a page that prompts +for confirmation when you press the +exit button on the initial page. + + +These pages have standard names and must be provided +in the \HyperName{} page database. +They are already defined in the Axiom system +\HyperName{} page database so that you do not have to +define them yourself. + +Here are the pages required by \HyperName{}. You can click on any of these +to see their contents. Click on their exit buttons when you are finished. + +\beginImportant +\begin{paste}{HTXLinkPage2xPaste1}{HTXLinkPage2xPatch1} +\pastebutton{HTXLinkPage2xPaste1}{Interpret} +\newline +{\tt \\table\{}\newline +{\tt \{\\windowlink\{SpadNotConnectedPage\}\{SpadNotConnectedPage\}\}}\newline +{\tt \{\\windowlink\{UnknownPage\}\{UnknownPage\}\}}\newline +{\tt \{\\windowlink\{ErrorPage\}\{ErrorPage\}\}}\newline +{\tt \{\\windowlink\{ProtectedQuitPage\}\{ProtectedQuitPage\}\}}\newline +{\tt \}}\newline +\end{paste} +\endImportant + + +In addition, \HyperName{} uses certain bitmaps for its buttons. +They are also provided in the Axiom system +bitmap directory and \HyperName{} knows where to find them. + +The bitmap files required by \HyperName{} are the following. +\newline +\tab{7}{\it exit.bitmap}\tab{22} = \tab{25}{\ExitBitmap{}} \newline +\tab{7}{\it help2.bitmap}\tab{22} = \tab{25}{\HelpBitmap{}} \newline +\tab{7}{\it up3.bitmap}\tab{22} = \tab{25}{\UpBitmap{}}\newline +\tab{7}{\it return3.bitmap}\tab{22} = \tab{25}{\ReturnBitmap{}}\newline +\tab{7}{\it noop.bitmap}\tab{22} = \tab{25}{\NoopBitmap{}} + +These files must exist in your current directory if +the {\tt AXIOM} environment variable is not set. +If it is, then \HyperName{} will assume that it points +to the Axiom system directory and will look for +these files in +{\bf \$AXIOM/doc/hypertex/bitmaps}. + + + + +\end{scroll} +\beginmenu +\menulink{Next -- Active Axiom commands}{HTXLinkPage3} +\endmenu + +\end{page} + +@ +\subsection{HTXLinkPage2xPatch1 patch} +\label{HTXLinkPage2xPatch1} +\index{patch!HTXLinkPage2xPatch1!htxlinkpage2.ht} +\index{htxlinkpage2.ht!patch!HTXLinkPage2xPatch1} +\index{HTXLinkPage2xPatch1!htxlinkpage2.ht!patch} +<>= +\begin{patch}{HTXLinkPage2xPatch1} +\begin{paste}{HTXLinkPage2xPaste1A}{HTXLinkPage2xPatch1A} +\pastebutton{HTXLinkPage2xPaste1A}{Source} +\newline +\table{ +{\windowlink{SpadNotConnectedPage}{SpadNotConnectedPage}} +{\windowlink{UnknownPage}{UnknownPage}} +{\windowlink{ErrorPage}{ErrorPage}} +{\windowlink{ProtectedQuitPage}{ProtectedQuitPage}} +} +\end{paste} +\end{patch} + +@ +\subsection{HTXLinkPage2xPatch1A patch} +\label{HTXLinkPage2xPatch1A} +\index{patch!HTXLinkPage2xPatch1A!htxlinkpage2.ht} +\index{htxlinkpage2.ht!patch!HTXLinkPage2xPatch1A} +\index{HTXLinkPage2xPatch1A!htxlinkpage2.ht!patch} +<>= +\begin{patch}{HTXLinkPage2xPatch1A} +\begin{paste}{HTXLinkPage2xPaste1B}{HTXLinkPage2xPatch1} +\pastebutton{HTXLinkPage2xPaste1B}{Interpret} +\newline +{\tt \\table\{}\newline +{\tt \{\\windowlink\{SpadNotConnectedPage\}\{SpadNotConnectedPage\}\}}\newline +{\tt \{\\windowlink\{UnknownPage\}\{UnknownPage\}\}}\newline +{\tt \{\\windowlink\{ErrorPage\}\{ErrorPage\}\}}\newline +{\tt \{\\windowlink\{ProtectedQuitPage\}\{ProtectedQuitPage\}\}}\newline +{\tt \}}\newline +\end{paste} +\end{patch} + +@ +\section{htxlinkpage3.ht} +\subsection{Active Axiom commands} +\label{HTXLinkPage3} +See HTXLinkPage4 \ref{HTXLinkPage4} on page~\pageref{HTXLinkPage4} +\index{pages!HTXLinkPage3!htxlinkpage3.ht.ht} +\index{htxlinkpage3.ht.ht!pages!HTXLinkPage3} +\index{HTXLinkPage3!htxlinkpage3.ht.ht!pages} +<>= +\begin{page}{HTXLinkPage3}{Active Axiom commands} +\centerline{\fbox{{\tt \thispage}}}\newline +\begin{scroll} + +This section explains how to include Axiom +commands in your page. The commands we will +introduce are actually {\it macros} that are defined +in +\centerline{{\bf \env{AXIOM}/doc/hypertex/pages/util.ht}} +This means that you can use them only if you include +this file in your \HyperName{} database. + +The first command to learn is +\horizontalline +{\tt \\axiomcommand\{ {\it command }{\tt \ \\free\{}{\it var1 var2 ...}{\tt \}\ \\bound\{}{\it var}{\tt \}\ \}} } +\horizontalline + + +The {\tt \\free\{\}} and {\tt \\bound\{\}} directives are optional. +We will come to them in a minute. The {\it command} above is the +text of the Axiom command. Only single line commands are allowed +here. +This text will be displayed in the reserved AxiomFont logical +font. The area of the text will be active and clicking on it +will attempt to send the command to Axiom for evaluation. +A new Axiom interpreter window (and Axiom frame) +will be created if this was the first Axiom command +activated in the current page. If not, the command will be sent +to the already opened Axiom interpreter window for the current page. +Note that it {\it is} necessary to escape special +\HyperName{} characters with the {\tt '\\'} backslash character. +The exceptions are the characters {\tt \[\]}; they do not +need to be escaped in this context. + +\beginImportant +\begin{paste}{HTXLinkPage3xPaste1}{HTXLinkPage3xPatch1} +\pastebutton{HTXLinkPage3xPaste1}{Interpret} +\newline +{\tt \\axiomcommand\{ l:=brace[1,2,3] ; length:=\\\# l ; m:=[1,2]\}} +\end{paste} +\endImportant + + +The optional {\tt \\free\{\}} and {\tt \\bound\{\}} directives +provide dependency control. The reader of a \HyperName{} +page is not forced to click on the commands in the +order in which in they appear on the page. If the +correct {\tt \\free\{\}} and {\tt \\bound\{\}} +specifications are made, clicking on a command +will result in execution of all those other +commands that should be executed before it. +This will {\it only} happen the first time the command is +clicked. + +So, how are the dependencies specified? +The arguments of the {\tt \\free\{\}} directive must +be space-separated words (labels). The argument of {\tt \\bound\{\}} +must be a single (unique for the page) label. Each label in the {\tt \\free\{\}} list +must exist as an argument to one (and only one) {\tt \\bound\{\}} directive +somewhere in the current page. +When the command is activated, \HyperName{} will look +in the {\tt \\free\{\}} list and check each label. +For each label, it will find the command that specifies that label +in its {\tt \\bound\{\}} directive and +execute it if it has not been already executed. +The order of labels in the {\tt \\free\{\}} directive list +is respected. \HyperName{} will follow all +dependency links recursively. + +Here is an example. +Clicking on the third command will automatically +execute all of them in the correct sequence. +Note that in this case the order of labels in the last +line is immaterial since {\tt v2} explicitly depends on {\tt v1}. + +\beginImportant +\begin{paste}{HTXLinkPage3xPaste2}{HTXLinkPage3xPatch2} +\pastebutton{HTXLinkPage3xPaste2}{Interpret} +\newline +{\tt \\axiomcommand\{a:=1;d:=4 \\bound\{v1\}\}}\newline +{\tt \\newline}\newline +{\tt \\axiomcommand\{b:=a+3 \\free\{v1\} \\bound\{v2\}\}}\newline +{\tt \\newline}\newline +{\tt \\axiomcommand\{c:=b+d \\free\{v1 v2\} \\bound\{v3\}\}}\newline +\end{paste} +\endImportant + +The second command deals with multi-line Axiom +code. This is the command to use for execution of +an Axiom {\it pile}. It is a {\it group} +command. The proper syntax for it is as follows: +\horizontalline +{\tt \\begin\{spadsrc\}\ [\\free\{{\it var1 var2} ...\}\ \\bound\{{\it var}\}]} +\newline +. +\newline +. +\newline +{\tt \\end\{spadsrc\}} +\horizontalline + +Again, the {\tt \\free} and {\tt \\bound} directives are +optional. If they are specified (in exactly the same way +as {\tt \\axiomcommand}), they must be enclosed in +square brackets {\tt []}. +The lines between the {\tt \\begin} and {\tt \\end} +contain the Axiom statements. Indentation +will be respected. \HyperName{} will +actually save this part in a temporary file +and instruct Axiom to read the file +with the {\tt )read} system command. + +Here is an example. The execution of the following +fragment is dependent on the {\tt v3} label. +Make sure that previous commands are active (and +hence the label {\tt v3} is "visible") before +trying to execute it. If the label {\tt v3} +is not seen in the page, \HyperName{} will +print an error message on standard output +and ignore the dependency. + + +\beginImportant +\begin{paste}{HTXLinkPage3xPaste3}{HTXLinkPage3xPatch3} +\pastebutton{HTXLinkPage3xPaste3}{Interpret} +\newline +{\tt \\begin\{spadsrc\}\ [\\free\{v3\}\ \\bound\{v4\}]}\newline +{\tt f\ x\ ==}\newline +{\tt \ \ \ x+c}\newline +{\tt f\ 3}\newline +{\tt \\end\{spadsrc\}} +\end{paste} +\endImportant + +There is, in fact, more that one can do +with Axiom commands. In pages elsewhere +in the system, Axiom commands appear next +to button like this \ \MenuDotBitmap{}. +Clicking on this button, one can see the output +for that command. The output has been +pre-computed and is also stored in +\HyperName{} files. This is done using +{\it patch} and {\it paste}. +It is the same mechanism that +is used to alternatively display +\HyperName{} source and interpreted +result in this and other pages. +It is explained \downlink{later on}{HTXAdvPage5}. + + +\end{scroll} +\beginmenu +\menulink{Next -- Linking to Lisp}{HTXLinkPage4} +\endmenu + +\end{page} + +@ +\subsection{HTXLinkPage3xPatch1 patch} +\label{HTXLinkPage3xPatch1} +\index{patch!HTXLinkPage3xPatch1!htxlinkpage3.ht} +\index{htxlinkpage3.ht!patch!HTXLinkPage3xPatch1} +\index{HTXLinkPage3xPatch1!htxlinkpage3.ht!patch} +<>= +\begin{patch}{HTXLinkPage3xPatch1} +\begin{paste}{HTXLinkPage3xPaste1A}{HTXLinkPage3xPatch1A} +\pastebutton{HTXLinkPage3xPaste1A}{Source} +\newline +\axiomcommand{ l:=brace[1,2,3] ; length:=\# l ; m:=[1,2]} +\end{paste} +\end{patch} + +@ +\subsection{HTXLinkPage3xPatch1A patch} +\label{HTXLinkPage3xPatch1A} +\index{patch!HTXLinkPage3xPatch1A!htxlinkpage3.ht} +\index{htxlinkpage3.ht!patch!HTXLinkPage3xPatch1A} +\index{HTXLinkPage3xPatch1A!htxlinkpage3.ht!patch} +<>= +\begin{patch}{HTXLinkPage3xPatch1A} +\begin{paste}{HTXLinkPage3xPaste1B}{HTXLinkPage3xPatch1} +\pastebutton{HTXLinkPage3xPaste1B}{Interpret} +\newline +{\tt \\axiomcommand\{ l:=brace[1,2,3] ; length:=\\\# l ; m:=[1,2]\}} +\end{paste} +\end{patch} + +@ +\subsection{HTXLinkPage3xPatch2 patch} +\label{HTXLinkPage3xPatch2} +\index{patch!HTXLinkPage3xPatch2!htxlinkpage3.ht} +\index{htxlinkpage3.ht!patch!HTXLinkPage3xPatch2} +\index{HTXLinkPage3xPatch2!htxlinkpage3.ht!patch} +<>= +\begin{patch}{HTXLinkPage3xPatch2} +\begin{paste}{HTXLinkPage3xPaste2A}{HTXLinkPage3xPatch2A} +\pastebutton{HTXLinkPage3xPaste2A}{Source} +\newline +\axiomcommand{a:=1;d:=4 \bound{v1}} +\newline +\axiomcommand{b:=a+3 \free{v1} \bound{v2}} +\newline +\axiomcommand{c:=b+d \free{v1 v2} \bound{v3}} +\end{paste} +\end{patch} + +@ +\subsection{HTXLinkPage3xPatch2A patch} +\label{HTXLinkPage3xPatch2A} +\index{patch!HTXLinkPage3xPatch2A!htxlinkpage3.ht} +\index{htxlinkpage3.ht!patch!HTXLinkPage3xPatch2A} +\index{HTXLinkPage3xPatch2A!htxlinkpage3.ht!patch} +<>= +\begin{patch}{HTXLinkPage3xPatch2A} +\begin{paste}{HTXLinkPage3xPaste2B}{HTXLinkPage3xPatch2} +\pastebutton{HTXLinkPage3xPaste2B}{Interpret} +\newline +{\tt \\axiomcommand\{a:=1;d:=4 \\bound\{v1\}\}}\newline +{\tt \\newline}\newline +{\tt \\axiomcommand\{b:=a+3 \\free\{v1\} \\bound\{v2\}\}}\newline +{\tt \\newline}\newline +{\tt \\axiomcommand\{c:=b+d \\free\{v1 v2\} \\bound\{v3\}\}}\newline +\end{paste} +\end{patch} + +@ +\subsection{HTXLinkPage3xPatch3 patch} +\label{HTXLinkPage3xPatch3} +\index{patch!HTXLinkPage3xPatch3!htxlinkpage3.ht} +\index{htxlinkpage3.ht!patch!HTXLinkPage3xPatch3} +\index{HTXLinkPage3xPatch3!htxlinkpage3.ht!patch} +<>= +\begin{patch}{HTXLinkPage3xPatch3} +\begin{paste}{HTXLinkPage3xPaste3A}{HTXLinkPage3xPatch3A} +\pastebutton{HTXLinkPage3xPaste3A}{Source} +\newline +\begin{spadsrc} [\free{v3} \bound{v4}] +f x == + x+c +f 3 +\end{spadsrc} +\end{paste} +\end{patch} + +@ +\subsection{HTXLinkPage3xPatch3A patch} +\label{HTXLinkPage3xPatch3A} +\index{patch!HTXLinkPage3xPatch3A!htxlinkpage3.ht} +\index{htxlinkpage3.ht!patch!HTXLinkPage3xPatch3A} +\index{HTXLinkPage3xPatch3A!htxlinkpage3.ht!patch} +<>= +\begin{patch}{HTXLinkPage3xPatch3A} +\begin{paste}{HTXLinkPage3xPaste3B}{HTXLinkPage3xPatch3} +\pastebutton{HTXLinkPage3xPaste3B}{Interpret} +\newline +{\tt \\begin\{spadsrc\}\ [\\free\{v3\}\ \\bound\{v4\}]}\newline +{\tt f\ x\ ==}\newline +{\tt \ \ \ x+c}\newline +{\tt f\ 3}\newline +{\tt \\end\{spadsrc\}} +\end{paste} +\end{patch} + +@ +\section{htxlinkpage4.ht} +\subsection{Linking to Lisp} +\label{HTXLinkPage4} +See HTXLinkPage5 \ref{HTXLinkPage5} on page~\pageref{HTXLinkPage5} +\index{pages!HTXLinkPage4!htxlinkpage4.ht} +\index{htxlinkpage4.ht!pages!HTXLinkPage4} +\index{HTXLinkPage4!htxlinkpage4.ht!pages} +<>= +\begin{page}{HTXLinkPage4}{Linking to Lisp} +\centerline{\fbox{{\tt \thispage}}}\newline +\begin{scroll} + +Another feature of the Axiom\hspace{2}--\HyperName{} +link is the ability to execute {\it Lisp} +code at a click of a button. +There are two things one can do. + +The first is to cause the evaluation +of a {\it Lisp} form and ignore (as far as \HyperName{} +is concerned) its value. The evaluation of the function +might have an effect however on your Axiom session. + +The command for this is +\horizontalline +\centerline{ {\tt \\lispcommand\{{\it text}\}\{{\it Lisp form}\}}} +\horizontalline + +Here is an example. We will first define a {\it Lisp} function +and then execute it. Notice that the \HyperName{} +special characters must be escaped (this is on top +of {\it Lisp} escaping conventions). + + +\beginImportant +\begin{paste}{HTXLinkPage4xPaste1}{HTXLinkPage4xPatch1} +\pastebutton{HTXLinkPage4xPaste1}{Interpret} +\newline +{\tt \\lispcommand\{Definition\}\{(defun HTXTESTFUNCTION ()}\newline +{\tt (print "Hello from HyperDoc \\\\\\\\ \\\% \\\{ \\\}"))\}} \newline +{\tt \\newline}\newline +{\tt \\lispcommand\{Execution\}\{(HTXTESTFUNCTION)\}} \newline +\end{paste} +\endImportant + +Your command will be executed as soon as +Axiom completes any computation it might be +carrying out. + + +%\axiomcommand{)lisp (defun f () (pprint "hello"))} +%\lispcommand{f}{(|f|)} + + +The second thing you can do is quite powerful. It allows you +to delegate to a {\it Lisp} function +the {\it dynamic} creation of a page. This is used +in \Browse{} to present +the Axiom Library in a hypertext form. + +The command to use is a lot like the {\tt link} commands +you encountered \downlink{earlier}{HTXLinkPage1} and comes in three flavours. +\centerline{{\tt \\lispwindowlink\{{\it trigger}\}\{{\it Lisp form}\}}} +\centerline{{\tt \\lispdownlink\{{\it trigger}\}\{{\it Lisp form}\}}} +\centerline{{\tt \\lispmemolink\{{\it trigger}\}\{{\it Lisp form}\}}} + +The difference between the three versions is the same as before. +When such a link is activated, \HyperName{} issues the +{\it Lisp form} to Axiom and waits for a full +page definition. An important point to note is that +\HyperName{} does {\it not} use +the value of the {\it Lisp form} but, instead, it +depends on its {\it side-effects}. +What {\it must} happen during evaluation +of the form is enough evaluations of a special {\it Lisp} +function called {\bf issueHT} to define a page. +The argument of {\bf issueHT} is a string +containing \HyperName{} text. Perhaps an example will clarify +matters. + +First we will define a {\it Lisp} function that accepts +a string argument and calls {\bf issueHT} a few times. +The strings that are passed to {\bf issueHT} construct +a \HyperName{} page that would just contain our +original argument centered roughly on the page. +Then we write the {\tt \\lisplink} with a call to +the function. Finally, we execute a {\it Lisp} +command that just pretty--prints the function's definition. + + + +\beginImportant +\begin{paste}{HTXLinkPage4xPaste2}{HTXLinkPage4xPatch2} +\pastebutton{HTXLinkPage4xPaste2}{Interpret} +\newline +{\tt \\lispcommand\{Definition\}\{(defun HTXTESTPAGE (x) (|issueHT|}\newline +{\tt "\\\\\\\\begin\\\{page\\\}\\\{LispTestPage\\\}\\\{Lisp Test Page\\\}}\newline +{\tt \\\\\\\\vspace\\\{150\\\} \\\\\\\\centerline\\\{") (|issueHT| x) (|issueHT|}\newline +{\tt "\\\} \\\\\\\\end\\\{page\\\}" ) ) \}}\newline +{\tt \\newline}\newline +{\tt \\lispwindowlink\{Link to it\}\{(HTXTESTPAGE "Hi there")\}}\newline +{\tt \\newline}\newline +{\tt \\lispcommand\{Show Lisp definition\}\{(pprint (symbol-function 'HTXTESTPAGE))\}}\newline +\end{paste} +\endImportant + +The {\tt '\\\{'} and {\tt '\\\}'} is required to escape +\HyperName{}'s special characters {\tt '\{'} and {\tt '\}'}. +The {\tt '\\\\\\\\'} has the following rationale. +We need to send to \HyperName{} (from {\it Lisp}) the sequence +{\tt \\begin}. But {\tt '\\'} is a special {\it Lisp} +character. Therefore the {\it Lisp} string must be +{\tt '\\\\begin'}. But to specify this +in \HyperName{} we need to escape the two {\tt '\\'}. +Therefore, we write {\tt '\\\\\\\\begin'}. + + +The definition of {\tt HTXTESTPAGE} would have been written in {\it Lisp} +as follows. +\begin{verbatim} +(defun HTXTESTPAGE (X) + (|issueHT| + "\\begin{page}{LispTestPage}{Lisp Test Page} \\vspace{200} \\centerline{") + (|issueHT| X) + (|issueHT| "} \\end{page}")) +\end{verbatim} + + + +You should not execute {\tt HTXTESTPAGE} in the +{\it Lisp} environment manually. It is meant to +be executed {\it only} in response to a +\HyperName{} request. + +Can you pop-up a named page from {\it Lisp} regardless of +user action? Yes --- use {\it Lisp} function {\bf linkToHTPage} +with the page name as a string argument. Click on the +{\tt \\axiomcommand} below. Then, in your Axiom +session, you can repeat it if you like. + +\beginImportant +\begin{paste}{HTXLinkPage4xPaste3}{HTXLinkPage4xPatch3} +\pastebutton{HTXLinkPage4xPaste3}{Interpret} +\newline +{\tt \\axiomcommand\{)lisp (|linkToHTPage| "RootPage")\}} +\end{paste} +\endImportant + +You can also pop-up a {\it dynamic} page regardless of user action. +To do this, make sure you evaluate the {\it Lisp form} +{\bf (|startHTPage| 50)} before using {\bf issueHT}. +The example below requires the {\tt HTXTESTPAGE} function +to be defined in {\it Lisp} so you should make sure +you have executed the command above that defines it. + +\beginImportant +\begin{paste}{HTXLinkPage4xPaste4}{HTXLinkPage4xPatch4} +\pastebutton{HTXLinkPage4xPaste4}{Interpret} +\newline +{\tt \\axiomcommand\{)lisp (progn (|startHTPage| 50)(HTXTESTPAGE "Immediately"))\}} +\end{paste} +\endImportant + +Now, the most important use of this facility +so far has been in the \Browse{} and Basic Commands components of +\HyperName{}. Instead of giving you details of the various +\Browse{} {\it Lisp} functions, a few macros are defined in +\centerline{{\bf \$AXIOM/doc/hypertex/pages/util.ht}} + +The most important defined macros are +\beginImportant +\table{ +{ {\tt \\axiomType\{{\it constructor}\}} } +{ {\tt \\axiomOp\{{\it operation}\}} } +{ {\tt \\axiomOpFrom\{{\it operation }\}\{{\it constructor}\}}} +} +\endImportant + +Here are some examples of their use. +\beginImportant +\begin{paste}{HTXLinkPage4xPaste5}{HTXLinkPage4xPatch5} +\pastebutton{HTXLinkPage4xPaste5}{Interpret} +\newline +{\tt \\axiomType\{Expression Integer\}}\newline +{\tt \newline}\newline +{\tt \\axiomType\{Expression\}}\newline +{\tt \newline}\newline +{\tt \\axiomType\{EXPR\}}\newline +{\tt \newline}\newline +{\tt \\axiomOp\{reduce\}}\newline +{\tt \newline}\newline +{\tt \\axiomOp\{as*\}}\newline +{\tt \newline}\newline +{\tt \\axiomOpFrom\{reduce\}\{Expression\}}\newline +\end{paste} +\endImportant + +The macro {\tt \\axiomType} brings up the \Browse{} +constructor page for the constructor specified. +You can specify a full name, or an abbreviation +or just the top level name. +The macro {\tt \\axiomOp} brings up a list of operations +matching the argument. +The macro {\tt \\axiomOpFrom} shows documentation +about the specified operation whose origin is +constructor. No wildcard in the operation name +or type abbreviation is +allowed here. You should also specify just the top level type. + + + + + + + +\end{scroll} +\beginmenu +\menulink{Next -- Linking to Unix}{HTXLinkPage5} +\endmenu + +\end{page} + +@ +\subsection{HTXLinkPage4xPatch1 patch} +\label{HTXLinkPage4xPatch1} +\index{patch!HTXLinkPage4xPatch1!htxlinkpage4.ht} +\index{htxlinkpage4.ht!patch!HTXLinkPage4xPatch1} +\index{HTXLinkPage4xPatch1!htxlinkpage4.ht!patch} +<>= +\begin{patch}{HTXLinkPage4xPatch1} +\begin{paste}{HTXLinkPage4xPaste1A}{HTXLinkPage4xPatch1A} +\pastebutton{HTXLinkPage4xPaste1A}{Source} +\newline +\lispcommand{Definition}{(defun HTXTESTFUNCTION () +(print "Hello from HyperDoc \\\\ \% \{ \}"))} +\newline +\lispcommand{Execution}{(HTXTESTFUNCTION)} +\end{paste} +\end{patch} + +@ +\subsection{HTXLinkPage4xPatch1A patch} +\label{HTXLinkPage4xPatch1A} +\index{patch!HTXLinkPage4xPatch1A!htxlinkpage4.ht} +\index{htxlinkpage4.ht!patch!HTXLinkPage4xPatch1A} +\index{HTXLinkPage4xPatch1A!htxlinkpage4.ht!patch} +<>= +\begin{patch}{HTXLinkPage4xPatch1A} +\begin{paste}{HTXLinkPage4xPaste1B}{HTXLinkPage4xPatch1} +\pastebutton{HTXLinkPage4xPaste1B}{Interpret} +\newline +{\tt \\lispcommand\{Definition\}\{(defun HTXTESTFUNCTION () (print "Hello from HyperDoc \\\\\\\\ \\\% \\\{ \\\}"))\}} \newline +{\tt \\newline}\newline +{\tt \\lispcommand\{Execution\}\{(HTXTESTFUNCTION)\}} \newline +\end{paste} +\end{patch} + +@ +\subsection{HTXLinkPage4xPatch2 patch} +\label{HTXLinkPage4xPatch2} +\index{patch!HTXLinkPage4xPatch2!htxlinkpage4.ht} +\index{htxlinkpage4.ht!patch!HTXLinkPage4xPatch2} +\index{HTXLinkPage4xPatch2!htxlinkpage4.ht!patch} +<>= +\begin{patch}{HTXLinkPage4xPatch2} +\begin{paste}{HTXLinkPage4xPaste2A}{HTXLinkPage4xPatch2A} +\pastebutton{HTXLinkPage4xPaste2A}{Source} +\newline +\lispcommand{Definition}{(defun HTXTESTPAGE (x) (|issueHT| +"\\\\begin\{page\}\{LispTestPage\}\{Lisp Test Page\} +\\\\vspace\{150\} \\\\centerline\{") (|issueHT| x) (|issueHT| +"\} \\\\end\{page\}" ) ) } +\newline +\lispwindowlink{Link to it}{(HTXTESTPAGE "Hi there")} +\newline +\lispcommand{Show Lisp definition}{(pprint (symbol-function 'HTXTESTPAGE))} +\end{paste} +\end{patch} + +@ +\subsection{HTXLinkPage4xPatch2A patch} +\label{HTXLinkPage4xPatch2A} +\index{patch!HTXLinkPage4xPatch2A!htxlinkpage4.ht} +\index{htxlinkpage4.ht!patch!HTXLinkPage4xPatch2A} +\index{HTXLinkPage4xPatch2A!htxlinkpage4.ht!patch} +<>= +\begin{patch}{HTXLinkPage4xPatch2A} +\begin{paste}{HTXLinkPage4xPaste2B}{HTXLinkPage4xPatch2} +\pastebutton{HTXLinkPage4xPaste2B}{Interpret} +\newline +{\tt \\lispcommand\{Definition\}\{(defun HTXTESTPAGE (x) (|issueHT|}\newline +{\tt "\\\\\\\\begin\\\{page\\\}\\\{LispTestPage\\\}\\\{Lisp Test Page\\\}}\newline +{\tt \\\\\\\\vspace\\\{150\\\} \\\\\\\\centerline\\\{") (|issueHT| x) (|issueHT|}\newline +{\tt "\\\} \\\\\\\\end\\\{page\\\}" ) ) \}}\newline +{\tt \\newline}\newline +{\tt \\lispwindowlink\{Link to it\}\{(HTXTESTPAGE "Hi there")\}}\newline +{\tt \\newline}\newline +{\tt \\lispcommand\{Show Lisp definition\}\{(pprint (symbol-function 'HTXTESTPAGE))\}}\newline +\end{paste} +\end{patch} + +@ +\subsection{HTXLinkPage4xPatch3 patch} +\label{HTXLinkPage4xPatch3} +\index{patch!HTXLinkPage4xPatch3!htxlinkpage4.ht} +\index{htxlinkpage4.ht!patch!HTXLinkPage4xPatch3} +\index{HTXLinkPage4xPatch3!htxlinkpage4.ht!patch} +<>= +\begin{patch}{HTXLinkPage4xPatch3} +\begin{paste}{HTXLinkPage4xPaste3A}{HTXLinkPage4xPatch3A} +\pastebutton{HTXLinkPage4xPaste3A}{Source} +\newline +\axiomcommand{)lisp (|linkToHTPage| "RootPage")} +\end{paste} +\end{patch} + +@ +\subsection{HTXLinkPage4xPatch3A patch} +\label{HTXLinkPage4xPatch3A} +\index{patch!HTXLinkPage4xPatch3A!htxlinkpage4.ht} +\index{htxlinkpage4.ht!patch!HTXLinkPage4xPatch3A} +\index{HTXLinkPage4xPatch3A!htxlinkpage4.ht!patch} +<>= +\begin{patch}{HTXLinkPage4xPatch3A} +\begin{paste}{HTXLinkPage4xPaste3B}{HTXLinkPage4xPatch3} +\pastebutton{HTXLinkPage4xPaste3B}{Interpret} +\newline +{\tt \\axiomcommand\{)lisp (|linkToHTPage| "RootPage")\}} +\end{paste} +\end{patch} + +@ +\subsection{HTXLinkPage4xPatch4 patch} +\label{HTXLinkPage4xPatch4} +\index{patch!HTXLinkPage4xPatch4!htxlinkpage4.ht} +\index{htxlinkpage4.ht!patch!HTXLinkPage4xPatch4} +\index{HTXLinkPage4xPatch4!htxlinkpage4.ht!patch} +<>= +\begin{patch}{HTXLinkPage4xPatch4} +\begin{paste}{HTXLinkPage4xPaste4A}{HTXLinkPage4xPatch4A} +\pastebutton{HTXLinkPage4xPaste4A}{Source} +\newline +\axiomcommand{)lisp (progn (|startHTPage| 50)(HTXTESTPAGE "Immediately"))} +\end{paste} +\end{patch} + +@ +\subsection{HTXLinkPage4xPatch4A patch} +\label{HTXLinkPage4xPatch4A} +\index{patch!HTXLinkPage4xPatch4A!htxlinkpage4.ht} +\index{htxlinkpage4.ht!patch!HTXLinkPage4xPatch4A} +\index{HTXLinkPage4xPatch4A!htxlinkpage4.ht!patch} +<>= +\begin{patch}{HTXLinkPage4xPatch4A} +\begin{paste}{HTXLinkPage4xPaste4B}{HTXLinkPage4xPatch4} +\pastebutton{HTXLinkPage4xPaste4B}{Interpret} +\newline +{\tt \\axiomcommand\{)lisp (progn (|startHTPage| 50)(HTXTESTPAGE "Immediately"))\}} +\end{paste} +\end{patch} + +@ +\subsection{HTXLinkPage4xPatch5 patch} +\label{HTXLinkPage4xPatch5} +\index{patch!HTXLinkPage4xPatch5!htxlinkpage4.ht} +\index{htxlinkpage4.ht!patch!HTXLinkPage4xPatch5} +\index{HTXLinkPage4xPatch5!htxlinkpage4.ht!patch} +<>= +\begin{patch}{HTXLinkPage4xPatch5} +\begin{paste}{HTXLinkPage4xPaste5A}{HTXLinkPage4xPatch5A} +\pastebutton{HTXLinkPage4xPaste5A}{Source} +\newline +\axiomType{Expression Integer} +\newline +\axiomType{Expression} +\newline +\axiomType{EXPR} +\newline +\axiomOp{reduce} +\newline +\axiomOp{as*} +\newline +\axiomOpFrom{reduce}{Expression} +\end{paste} +\end{patch} + +@ +\subsection{HTXLinkPage4xPatch5A patch} +\label{HTXLinkPage4xPatch5A} +\index{patch!HTXLinkPage4xPatch5A!htxlinkpage4.ht} +\index{htxlinkpage4.ht!patch!HTXLinkPage4xPatch5A} +\index{HTXLinkPage4xPatch5A!htxlinkpage4.ht!patch} +<>= +\begin{patch}{HTXLinkPage4xPatch5A} +\begin{paste}{HTXLinkPage4xPaste5B}{HTXLinkPage4xPatch5} +\pastebutton{HTXLinkPage4xPaste5B}{Interpret} +\newline +{\tt \\axiomType\{Expression Integer\}}\newline +{\tt \newline}\newline +{\tt \\axiomType\{Expression\}}\newline +{\tt \newline}\newline +{\tt \\axiomType\{EXPR\}}\newline +{\tt \newline}\newline +{\tt \\axiomOp\{reduce\}}\newline +{\tt \newline}\newline +{\tt \\axiomOp\{as*\}}\newline +{\tt \newline}\newline +{\tt \\axiomOpFrom\{reduce\}\{Expression\}}\newline +\end{paste} +\end{patch} + +@ +\section{htxlinkpage5.ht} +\subsection{Linking to Unix} +\label{HTXLinkPage5} +\index{pages!HTXLinkPage5!htxlinkpage5.ht} +\index{htxlinkpage5.ht!pages!HTXLinkPage5} +\index{HTXLinkPage5!htxlinkpage5.ht!pages} +<>= +\begin{page}{HTXLinkPage5}{Linking to Unix} +\centerline{\fbox{{\tt \thispage}}}\newline +\begin{scroll} + +Let us conclude the tour of \HyperName{} +actions that can be triggered with a click of a button +with two more facilities. These are +\beginImportant +\table{ +{ {\tt \\unixcommand\{{\it trigger text}\}\{{\it unix command}\}}} +{ {\tt \\unixlink\{{\it trigger text}\}\{{\it unix command}\}}} +} +\endImportant + + +The first one, {\tt \\unixcommand}, is very much like +{\tt \\axiomcommand} and {\tt \\lispcommand}. +The trigger text becomes an active area. Clicking on it +will force \HyperName{} to pass the second argument +to the system as a shell command to be executed. +The shell used is {\bf /bin/sh}. +\HyperName{} ignores the output of the command. + + +\beginImportant +\begin{paste}{HTXLinkPage5xPaste1}{HTXLinkPage5xPatch1} +\pastebutton{HTXLinkPage5xPaste1}{Interpret} +\newline +{\tt \\unixcommand\{List \\\$HOME directory\}\{ls \\\$HOME\}}\newline +\end{paste} +\endImportant + +The {\tt \\unixlink} command delegates to a another +program the creation of a dynamic page. When the trigger +text is activated, \HyperName{} will invoke the command +specified in the second argument. It will then start reading +the {\it standard output} of the command until +a complete page has been received. It is important that +a single page and nothing more is written by the command. +This command is essentially a {\tt \\downlink}, i.e. +the new page replaces the current page in the window. +There aren't any other flavours of {\tt \\unixlink}. +A trivial example is to use {\bf cat} on a \HyperName{} +file known to contain just one page. + +\beginImportant +\begin{paste}{HTXLinkPage5xPaste2}{HTXLinkPage5xPatch2} +\pastebutton{HTXLinkPage5xPaste2}{Interpret} +\newline +{\tt \\unixlink\{Some file\}} \newline +{\tt \{cat\\ \\env\{AXIOM\}/doc/hypertex/pages/HTXplay.ht\}} +\end{paste} +\endImportant + + +Two things to notice in the second argument of +{\tt \\unixlink}: You must use a {\it hard space} +{\tt '\\\ '} to preserve the spacing in the command. +Also, the {\tt \\env} command allows you to use +an environment variable in \HyperName{} text. + +With a little ingenuity (and maybe some shell and {\bf awk} scripts !) +, one can use these +facilities to create, say, a point-and-click +directory viewer which allows you to edit +a file by clicking on its name. + +\end{scroll} +\beginmenu +\menulink{Next -- How to use your pages with \HyperName{}}{HTXLinkPage6} +\endmenu + +\end{page} + +@ +\subsection{HTXLinkPage5xPatch1 patch} +\label{HTXLinkPage5xPatch1} +\index{patch!HTXLinkPage5xPatch1!htxlinkpage5.ht} +\index{htxlinkpage5.ht!patch!HTXLinkPage5xPatch1} +\index{HTXLinkPage5xPatch1!htxlinkpage5.ht!patch} +<>= +\begin{patch}{HTXLinkPage5xPatch1} +\begin{paste}{HTXLinkPage5xPaste1A}{HTXLinkPage5xPatch1A} +\pastebutton{HTXLinkPage5xPaste1A}{Source} +\newline +\unixcommand{List \$HOME directory}{ls \$HOME} +\end{paste} +\end{patch} + +@ +\subsection{HTXLinkPage5xPatch1A patch} +\label{HTXLinkPage5xPatch1A} +\index{patch!HTXLinkPage5xPatch1A!htxlinkpage5.ht} +\index{htxlinkpage5.ht!patch!HTXLinkPage5xPatch1A} +\index{HTXLinkPage5xPatch1A!htxlinkpage5.ht!patch} +<>= +\begin{patch}{HTXLinkPage5xPatch1A} +\begin{paste}{HTXLinkPage5xPaste1B}{HTXLinkPage5xPatch1} +\pastebutton{HTXLinkPage5xPaste1B}{Interpret} +\newline +{\tt \\unixcommand\{List \\\$HOME directory\}\{ls \\\$HOME\}}\newline +\end{paste} +\end{patch} + +@ +\subsection{HTXLinkPage5xPatch2 patch} +\label{HTXLinkPage5xPatch2} +\index{patch!HTXLinkPage5xPatch2!htxlinkpage5.ht} +\index{htxlinkpage5.ht!patch!HTXLinkPage5xPatch2} +\index{HTXLinkPage5xPatch2!htxlinkpage5.ht!patch} +<>= +\begin{patch}{HTXLinkPage5xPatch2} +\begin{paste}{HTXLinkPage5xPaste2A}{HTXLinkPage5xPatch2A} +\pastebutton{HTXLinkPage5xPaste2A}{Source} +\newline +\unixlink{Some file} +{cat\ \env{AXIOM}/doc/hypertex/pages/HTXplay.ht} +\end{paste} +\end{patch} + +@ +\subsection{HTXLinkPage5xPatch2A patch} +\label{HTXLinkPage5xPatch2A} +\index{patch!HTXLinkPage5xPatch2A!htxlinkpage5.ht} +\index{htxlinkpage5.ht!patch!HTXLinkPage5xPatch2A} +\index{HTXLinkPage5xPatch2A!htxlinkpage5.ht!patch} +<>= +\begin{patch}{HTXLinkPage5xPatch2A} +\begin{paste}{HTXLinkPage5xPaste2B}{HTXLinkPage5xPatch2} +\pastebutton{HTXLinkPage5xPaste2B}{Interpret} +\newline +{\tt \\unixlink\{Some file\}} \newline +{\tt \{cat\\ \\env\{AXIOM\}/doc/hypertex/pages/HTXplay.ht\}} +\end{paste} +\end{patch} + +@ +\section{htxlinkpage6.ht} +\subsection{How to use your pages with Hyperdoc} +\label{HTXLinkPage6} +\index{pages!HTXLinkPage6!htxlinkpage6.ht} +\index{htxlinkpage6.ht!pages!HTXLinkPage6} +\index{HTXLinkPage6!htxlinkpage6.ht!pages} +<>= +\begin{page}{HTXLinkPage6}{How to use your pages with Hyperdoc} +\centerline{\fbox{{\tt \thispage}}}\newline +\begin{scroll} + +Let us say that you have written a few \HyperName{} +pages and you would like to incorporate them in the system. +Here is what you should do. + +Put all your files in some directory and make sure that +they all have the {\bf .ht} extension. + +You will need a way of "hooking" into a system--defined +\HyperName{} page. The proper way to do this is to use +the {\tt \\localinfo} macro. The Axiom system +\HyperName{} page database includes, as it should, +a {\tt RootPage}. This is the page that first comes up +when you start \HyperName{}. This page contains +a line like this. +\beginImportant +\newline +{\tt \\localinfo} +\endImportant + +This macro is defined in +\centerline{ {\bf \env{AXIOM}/doc/hypertex/pages/util.ht}} +to be (see \downlink{Macros}{HTXAdvPage3} to learn how to define macros): +\beginImportant +\newline +{\tt \\newcommand\{\\localinfo\}\{\}} +\endImportant +which is an empty definition (the second argument of {\tt \\newcommand}). +The idea then is that you {\it override} this definition of the macro +with your own. +To do that, include a definition like the following in one (possibly the +one that contains your top--level page) of your files. You can +put this definition in its own file if you like. +\beginImportant +\newline +{\tt \\newcommand\{\\localinfo\}\{\\menuwindowlink\{{\it active text}\}} \newline +{\tt \{{\it page name}\} \\tab\{16\}{\it short description}\}} +\endImportant + +If you have a look at the initial \HyperName{} page, you will +probably be able to decipher what this does. The macro +{\tt \\menuwindowlink} is defined (again in {\bf util.ht}) +and is responsible for putting the little square to the left of the +active area. +Specify a word or two for {\it active text}. That will become the +trigger of the {\tt \\link}. Specify the page name of your top--level page +in {\it page name}. Finally, you can give a comment about the topic +under {\it short description}. That will appear to the right of the +{\it active text}. + +The next thing you need to do is to create a {\it local database} +for your files. You will use the {\bf \env{AXIOM}/bin/htadd} program. +This program will create an {\bf ht.db} file that summarises your +definitions and acts as an index. Let us present an example +of its use. Suppose you have two files {\bf user1.ht} and {\bf user2.ht} +in directory {\bf /u/sugar/\HyperName{}}. You should create the {\bf ht.db} +in that same directory. To create the {\bf ht.db} file you issue to +the unix shell: +\beginImportant +\newline +{\tt htadd -f /u/sugar/\HyperName{} /u/sugar/\HyperName{}/user1.ht /u/sugar/\HyperName{}/user2.ht} +\centerline{or ,if you are already in /u/sugar/\HyperName{}} +{\tt htadd -l ./user1.ht ./user2.ht} +\endImportant + + +The options and conventions for {\bf htadd} will be explained below. +To start \HyperName{} with your own pages, you now need to tell +it where to search for {\bf ht.db} files and \HyperName{} {\bf .ht} +files. To do this, define the shell environment variable +{\bf HTPATH}. The value should be a colon {\tt ':'} separated +list of directory full pathnames. +The order of the directories is respected with earlier entries overriding +later ones. Since we want all the Axiom pages but need to override the +{\tt \\localinfo} macro, we should use the value +\centerline{{\bf /u/sugar/\HyperName{}:\env{AXIOM}/doc/hypertex/pages}} +The way that you define environment variables depends on the shell +you are using. In the {\bf /bin/csh}, it would be +\newline +{\bf setenv HTPATH /u/sugar/\HyperName{}:\env{AXIOM}{}/doc{}/hypertex{}/pages} + + + +\beginImportant +\begin{paste}{HTXLinkPage6xPaste1}{HTXLinkPage6xPatch1} +\pastebutton{HTXLinkPage6xPaste1}{Options for {\bf htadd}} +\newline +\end{paste} +\endImportant + + +\beginImportant +\begin{paste}{HTXLinkPage6xPaste2}{HTXLinkPage6xPatch2} +\pastebutton{HTXLinkPage6xPaste2}{Where does \HyperName{} look for files} +\newline +\end{paste} +\endImportant + + + +\end{scroll} +\beginmenu +\menulink{Back to Actions menu}{HTXLinkTopPage} +\endmenu + +\end{page} + +@ +\subsection{HTXLinkPage6xPatch1 patch} +\label{HTXLinkPage6xPatch1} +\index{patch!HTXLinkPage6xPatch1!htxlinkpage6.ht} +\index{htxlinkpage6.ht!patch!HTXLinkPage6xPatch1} +\index{HTXLinkPage6xPatch1!htxlinkpage6.ht!patch} +<>= +\begin{patch}{HTXLinkPage6xPatch1} +\begin{paste}{HTXLinkPage6xPaste1A}{HTXLinkPage6xPatch1A} +\pastebutton{HTXLinkPage6xPaste1A}{Hide} +\newline +Name: + +{\tt htadd - create or modify a \HyperName{} database} +\vspace{} +\newline +Syntax: + +{\tt htadd [ -l | -s | -f\ }{\it path}{\tt ] [ -d | -n ]\ }{\it filename ...} +\vspace{} +\newline +Options:\indentrel{4}\newline +{\tt -l}\tab{8}\indentrel{8} +build {\bf ht.db} database in current working directory. +This is the default behaviour if no {\tt -l}, {\tt -s} or {\tt -f} +is specified. + +\indentrel{-8}\newline +{\tt -s}\tab{8}\indentrel{8} +build {\bf ht.db} database in {\it system} directory. The +system directory is built as follows. If the {\tt AXIOM} +variable is defined, the {\bf \$AXIOM/doc/hypertex/pages} directory +is used. If {\tt AXIOM} is not defined, the +{\bf /usr/local/axiom/doc/hypertex/pages} directory is used. + + +\indentrel{-8}\newline +{\tt -f\ }{\it path}\newline\tab{8}\indentrel{8} +build {\bf ht.db} database in specified {\it path}. + +\indentrel{-8}\newline +{\tt -d}\tab{8}\indentrel{8} +delete the entries in the specified files from {\bf ht.db}. + +\indentrel{-8}\newline +{\tt -n}\tab{8}\indentrel{8} +delete {\bf ht.db} and create a new one using only the files +specified. + +If none of {\tt -n} and {\tt -d} is specified, the {\bf ht.db} +is updated with the entries in the file specified. + + +\indentrel{-8} +\indentrel{-4} +\vspace{}\newline +Filename interpretation : +\indentrel{12}\newline +A full pathname (i.e. anything that has a {\tt '/'} in it) +will be taken do be a completely specified file. +Otherwise, the following interpretation will occur: +If the {\tt HTPATH} variable is defined, the directories +specified in it will be tried in order. If {\tt HTPATH} +is not defined, then, if {\tt AXIOM} is defined, the +{\bf \$AXIOM/doc/hypertex/pages} will be tried, else +the file will be deemed missing and {\bf htadd} will fail. +\indentrel{-12} +\end{paste} +\end{patch} + +@ +\subsection{HTXLinkPage6xPatch1A patch} +\label{HTXLinkPage6xPatch1A} +\index{patch!HTXLinkPage6xPatch1A!htxlinkpage6.ht} +\index{htxlinkpage6.ht!patch!HTXLinkPage6xPatch1A} +\index{HTXLinkPage6xPatch1A!htxlinkpage6.ht!patch} +<>= +\begin{patch}{HTXLinkPage6xPatch1A} +\begin{paste}{HTXLinkPage6xPaste1B}{HTXLinkPage6xPatch1} +\pastebutton{HTXLinkPage6xPaste1B}{Options for {\bf htadd}} +\newline +\end{paste} +\end{patch} + +@ +\subsection{HTXLinkPage6xPatch2 patch} +\label{HTXLinkPage6xPatch2} +\index{patch!HTXLinkPage6xPatch2!htxlinkpage6.ht} +\index{htxlinkpage6.ht!patch!HTXLinkPage6xPatch2} +\index{HTXLinkPage6xPatch2!htxlinkpage6.ht!patch} +<>= +\begin{patch}{HTXLinkPage6xPatch2} +\begin{paste}{HTXLinkPage6xPaste2A}{HTXLinkPage6xPatch2A} +\pastebutton{HTXLinkPage6xPaste2A}{Hide} +\indentrel{12}\newline +The \HyperName{} program is +\centerline{{\bf \env{AXIOM}/lib/hypertex}} +If {\tt AXIOM} is defined and {\tt HTPATH} is not +(this is the case when Axiom starts \HyperName{}) +\HyperName{} will look in +\centerline{{\bf \env{AXIOM}/doc/hypertex/pages}} +for the {\bf ht.db} file and all \HyperName{} pages. +If {\tt HTPATH} is defined, it is assumed that +it alone points to the directories to be searched +(the above default will NOT be searched unless +explicitly specified in {\tt HTPATH}). +For each directory in {\tt HTPATH}, the {\bf ht.db} +file, if there, will be read. +Each file listed in {\bf ht.db} will +then be searched for in the complete sequence +of directories in {\tt HTPATH}. Note that +the {\bf ht.db} does not keep full pathnames + of files. +If a {\it page}, {\it macro} or {\it patch} +(specified in some {\bf ht.db}) happens +to be (in a file) in more than one of the directories +specified in {\tt HTPATH}, \HyperName{} +will print a warning and explain which version +in which file is ignored. Generally, earlier +directories in {\tt HTPATH} are preferred over later +ones. +\indentrel{-12}\newline +\end{paste} +\end{patch} + +@ +\subsection{HTXLinkPage6xPatch2A patch} +\label{HTXLinkPage6xPatch2A} +\index{patch!HTXLinkPage6xPatch2A!htxlinkpage6.ht} +\index{htxlinkpage6.ht!patch!HTXLinkPage6xPatch2A} +\index{HTXLinkPage6xPatch2A!htxlinkpage6.ht!patch} +<>= +\begin{patch}{HTXLinkPage6xPatch2A} +\begin{paste}{HTXLinkPage6xPaste2B}{HTXLinkPage6xPatch2} +\pastebutton{HTXLinkPage6xPaste2B}{Where does \HyperName{} look for files}} +\newline +\end{paste} +\end{patch} + +@ +\section{htxlinktoppage.ht} +\subsection{Actions in Hyperdoc} +\label{HTXLinkTopPage} +\begin{itemize} +\item HTXLinkPage1 \ref{HTXLinkPage1} on page~\pageref{HTXLinkPage1} +\item HTXLinkPage2 \ref{HTXLinkPage2} on page~\pageref{HTXLinkPage2} +\item HTXLinkPage3 \ref{HTXLinkPage3} on page~\pageref{HTXLinkPage3} +\item HTXLinkPage4 \ref{HTXLinkPage4} on page~\pageref{HTXLinkPage4} +\item HTXLinkPage5 \ref{HTXLinkPage5} on page~\pageref{HTXLinkPage5} +\item HTXLinkPage6 \ref{HTXLinkPage6} on page~\pageref{HTXLinkPage6} +\end{itemize} +\index{pages!HTXLinkTopPage!htxlinktoppage.ht} +\index{htxlinktoppage.ht!pages!HTXLinkTopPage} +\index{HTXLinkTopPage!htxlinktoppage.ht!pages} +<>= +\begin{page}{HTXLinkTopPage}{Actions in Hyperdoc} +\centerline{\fbox{{\tt \thispage}}}\newline +\HyperName{} offers various types of hypertext links. +You can learn about these facilities by clicking on the topics below. +\begin{scroll} +\beginmenu +\menudownlink{Linking to a named page}{HTXLinkPage1} +\menudownlink{Standard pages}{HTXLinkPage2} +\menudownlink{Active Axiom commands}{HTXLinkPage3} +\menudownlink{Linking to Lisp}{HTXLinkPage4} +\menudownlink{Linking to Unix}{HTXLinkPage5} +\menudownlink{How to use your pages with \HyperName{}}{HTXLinkPage6} +\endmenu +\end{scroll} +\end{page} + +@ +\section{htxtoppage.ht} +\subsection{Extending Hyperdoc} +\label{HTXTopPage} +\begin{itemize} +\item HTXIntroTopPage \ref{HTXIntroTopPage} on page~\pageref{HTXIntroTopPage} +\item HTXFormatTopPage \ref{HTXFormatTopPage} on +page~\pageref{HTXFormatTopPage} +\item HTXLinkTopPage \ref{HTXLinkTopPage} on page~\pageref{HTXLinkTopPage} +\item HTXAdvTopPage \ref{HTXAdvTopPage} on page~\pageref{HTXAdvTopPage} +\item HTXTryPage \ref{HTXTryTopPage} on page~\pageref{HTXTryPage} +\end{itemize} +\index{pages!HTXTopPage!htxtoppage.ht} +\index{htxtoppage.ht!pages!HTXTopPage} +\index{HTXTopPage!htxtoppage.ht!pages} +<>= +\begin{page}{HTXTopPage}{Extending Hyperdoc} +\centerline{\fbox{{\tt \thispage}}}\newline +This is a guide to extending \HyperName{}. You can learn +how to write your own \HyperName{} pages and link them to the +\HyperName{} page database that Axiom uses. +\begin{scroll} +\beginmenu +\menumemolink{Introduction}{HTXIntroTopPage} \tab{20} An easy start. +\menumemolink{Formatting}{HTXFormatTopPage} \tab{20} Learn how to format text. +\menumemolink{Actions}{HTXLinkTopPage} \tab{20} Learn how to define actions. +\menumemolink{Advanced features}{HTXAdvTopPage} \tab{20} More effects. +\menuwindowlink{Try it!}{HTXTryPage} \tab{20} Try out what you learn. +\endmenu +\end{scroll} +\end{page} + +@ +\section{htxtrypage.ht} +\subsection{Try out Hyperdoc} +\label{HTXTryPage} +\index{pages!HTXTryPage!htxtrypage.ht} +\index{htxtrypage.ht!pages!HTXTryPage} +\index{HTXTryPage!htxtrypage.ht!pages} +<>= +\begin{page}{HTXTryPage}{Try out Hyperdoc} +\centerline{\fbox{{\tt \thispage}}}\newline + +This page allows you to quickly experiment with \HyperName{}. +It is a good idea to keep it handy as you learn about various commands. + +\beginscroll + +We are going to use here the \HyperName{} facilities that allow +us to communicate with external programs and files. For more +information see \downlink{later on}{HTXLinkPage5}. +\beginmenu +\item\menuitemstyle{ +In order to use the buttons at the bottom of this page, you must +first specify a name for the file you are going to use to hold +\HyperName{} commands. Edit the input area below to change the +name of the file.} +\item\menuitemstyle{ +If the file you specified does not yet exist, click on the +{\bf Initialize} button below. This action will fill the file +with the minimum of \HyperName{} commands necessary to define a page.} +\item\menuitemstyle{ +If you want to edit the file, just click on the {\bf Edit} button. +This action will pop up a window, and invoke the {\it vi} +editor on the file. Alternatively, use an editor of your choice.} +\item\menuitemstyle{ +Once you have finished making the changes to the file, update it and +click on the {\bf Link} button. \HyperName{} will then read +the file, interpret it as a new page, and display the page on +this window. If you change the file and want to display it again, +just get back to this page and click on {\bf Link} again. } +\endmenu +\endscroll +\beginmenu +{\it Filename: }{\inputstring{filename}{40}{\env{HOME}/HTXplay.ht}} +\menuunixcommand{Initialize}{cp\space{1}\env{AXIOM}/doc/hypertex/pages/HTXplay.ht \stringvalue{filename}} \tab{20} Get a fresh copy from the system. +\menuunixcommand{Edit}{xterm -T "\stringvalue{filename}" -e vi \stringvalue{filename}} \tab{20} Edit the file. +\menuunixwindow{Link}{cat \space{1}\stringvalue{filename}} \tab{20} Link to the page defined in the file. +\endmenu +{\it Important : The file must contain +one and only one page definition and must not contain any macro or patch +definitions.} +\end{page} + +@ +\section{hyperdoc.ht} +\subsection{Creating Hyperdoc Pages} +\label{Hyperdoc} +\begin{itemize} +\item ViewportPage \ref{ViewportPage} on +page~\pageref{ViewportPage} +\item BitMaps \ref{BitMaps} on +page~\pageref{BitMaps} +\item CPHelp \ref{CPHelp} on +page~\pageref{CPHelp} +\end{itemize} +\index{pages!Hyperdoc!hyperdoc.ht} +\index{hyperdoc.ht!pages!Hyperdoc} +\index{Hyperdoc!hyperdoc.ht!pages} +<>= +\begin{page}{Hyperdoc}{Creating Hyperdoc Pages} + +\beginscroll +This document tells how to create \HyperName pages. +To start with, it is rather meager but it will grow with time. +\beginmenu +\menulink{Viewports}{ViewportPage} Including live graphics in documents. +\menulink{Gadjets}{BitMaps} Bitmaps for use in macros. +\menulink{Control Panel Bits}{CPHelp} Development page for help +facility for viewports. yuck. +%\menulink{Test Pages}{TestPage} Some test pages left by J.M. +%\menulink{Paste Pages}{PastePage} Examples of how to use paste in areas. +\endmenu + +\endscroll +\autobuttons +\end{page} + +@ +\section{int.ht} +<>= +\newcommand{\IntegerXmpTitle}{Integer} +\newcommand{\IntegerXmpNumber}{9.34} + +@ +\subsection{Integer} +\label{IntegerXmpPage} +\begin{itemize} +\item ugIntroNumbersPage \ref{ugIntroNumbersPage} on +page~pageref{ugIntroNumbersPage} +\item IntegerNumberTheoryFunctionsXmpPage +\ref{IntegerNumberTheoryFunctionsXmpPage} on +page~pageref{IntegerNumberTheoryFunctionsXmpPage} +\item DecimalExpansionXmpPage \ref{DecimalExpansionXmpPage} on +page~pageref{DecimalExpansionXmpPage} +\item BinaryExpansionXmpPage \ref{BinaryExpansionXmpPage} on +page~pageref{BinaryExpansionXmpPage} +\item HexadecimalExpansionXmpPage \ref{HexadecimalExpansionXmpPage} on +page~pageref{HexadecimalExpansionXmpPage} +\item RadixExpansionXmpPage \ref{RadixExpansionXmpPage} on +page~pageref{RadixExpansionXmpPage} +\item ugxIntegerBasicPage \ref{ugxIntegerBasicPage} on +page~pageref{ugxIntegerBasicPage} +\item ugxIntegerPrimesPage \ref{ugxIntegerPrimesPage} on +page~pageref{ugxIntegerPrimesPage} +\item ugxIntegerNtPage \ref{ugxIntegerNtPage} on +page~pageref{ugxIntegerNtPage} +\end{itemize} +\index{pages!IntegerXmpPage!int.ht} +\index{int.ht!pages!IntegerXmpPage} +\index{IntegerXmpPage!int.ht!pages} +<>= +\begin{page}{IntegerXmpPage}{Integer} +\beginscroll +Axiom provides many operations for manipulating arbitrary +precision integers. +In this section we will show some of those that come from \spadtype{Integer} +itself plus some that are implemented in other packages. +More examples of using integers are in the following sections: +\downlink{``\ugIntroNumbersTitle''}{ugIntroNumbersPage} in section \ugIntroNumbersNumber +\downlink{`IntegerNumberTheoryFunctions'}{IntegerNumberTheoryFunctionsXmpPage}\ignore{IntegerNumberTheoryFunctions}, +\downlink{`DecimalExpansion'}{DecimalExpansionXmpPage}\ignore{DecimalExpansion}, +\downlink{`BinaryExpansion'}{BinaryExpansionXmpPage}\ignore{BinaryExpansion}, +\downlink{`HexadecimalExpansion'}{HexadecimalExpansionXmpPage}\ignore{HexadecimalExpansion}, and +\downlink{`RadixExpansion'}{RadixExpansionXmpPage}\ignore{RadixExpansion}. + +\beginmenu + \menudownlink{{9.34.1. Basic Functions}}{ugxIntegerBasicPage} + \menudownlink{{9.34.2. Primes and Factorization}}{ugxIntegerPrimesPage} + \menudownlink{{9.34.3. Some Number Theoretic Functions}}{ugxIntegerNTPage} +\endmenu +\endscroll +\autobuttons +\end{page} + +@ +<>= +\newcommand{\ugxIntegerBasicTitle}{Basic Functions} +\newcommand{\ugxIntegerBasicNumber}{9.34.1.} + +@ +\subsection{Basic Functions} +\label{ugxIntegerBasicPage} +\begin{itemize} +\item FractionXmpPage \ref{FractionXmpPage} on +page~pageref{FractionXmpPage} +\item ugTypesUnionsPage \ref{ugTypesUnionsPage} on +page~pageref{ugTypesUnionsPage} +\item ugTypesRecordsPage \ref{ugTypesRecordsPage} on +page~pageref{ugTypesRecordsPage} +\end{itemize} +\index{pages!ugxIntegerBasicPage!int.ht} +\index{int.ht!pages!ugxIntegerBasicPage} +\index{ugxIntegerBasicPage!int.ht!pages} +<>= +\begin{page}{ugxIntegerBasicPage}{Basic Functions} +\beginscroll + +\labelSpace{3pc} +\xtc{ +The size of an integer in Axiom is only limited by the amount of +computer storage you have available. +The usual arithmetic operations are available. +}{ +\spadpaste{2**(5678 - 4856 + 2 * 17)} +} + +\xtc{ +There are a number of ways of working with the sign of an integer. +Let's use this \spad{x} as an example. +}{ +\spadpaste{x := -101 \bound{x}} +} +\xtc{ +First of all, there is the absolute value function. +}{ +\spadpaste{abs(x) \free{x}} +} +\xtc{ +The \spadfunFrom{sign}{Integer} operation returns \spad{-1} if its argument +is negative, \spad{0} if zero and \spad{1} if positive. +}{ +\spadpaste{sign(x) \free{x}} +} +% +\xtc{ +You can determine if an integer is negative in several other ways. +}{ +\spadpaste{x < 0 \free{x}} +} +\xtc{ +}{ +\spadpaste{x <= -1 \free{x}} +} +\xtc{ +}{ +\spadpaste{negative?(x) \free{x}} +} +% +\xtc{ +Similarly, you can find out if it is positive. +}{ +\spadpaste{x > 0 \free{x}} +} +\xtc{ +}{ +\spadpaste{x >= 1 \free{x}} +} +\xtc{ +}{ +\spadpaste{positive?(x) \free{x}} +} +\xtc{ +This is the recommended way of determining whether an integer is zero. +}{ +\spadpaste{zero?(x) \free{x}} +} + +\beginImportant +Use the \spadfunFrom{zero?}{Integer} operation whenever you are testing any +mathematical object for equality with zero. +This is usually more efficient that using \spadop{=} (think of matrices: +it is easier to tell if a matrix is zero by just checking term by term +than constructing another ``zero'' matrix and comparing the two +matrices term by term) and also avoids the problem that \spadop{=} is +usually used for creating equations. +\endImportant + +\xtc{ +This is the recommended way of determining +whether an integer is equal to one. +}{ +\spadpaste{one?(x) \free{x}} +} + +\xtc{ +This syntax is used to test equality using \spadopFrom{=}{Integer}. +It says that you want a \spadtype{Boolean} (\spad{true} or +\spad{false}) answer rather than an equation. +}{ +\spadpaste{(x = -101)@Boolean \free{x}} +} + +\xtc{ +The operations \spadfunFrom{odd?}{Integer} and +\spadfunFrom{even?}{Integer} determine whether an integer is odd or even, +respectively. +They each return a \spadtype{Boolean} object. +}{ +\spadpaste{odd?(x) \free{x}} +} +\xtc{ +}{ +\spadpaste{even?(x) \free{x}} +} +\xtc{ +The operation \spadfunFrom{gcd}{Integer} computes the greatest common divisor of +two integers. +}{ +\spadpaste{gcd(56788,43688)} +} +\xtc{ +The operation +\spadfunFrom{lcm}{Integer} computes their least common multiple. +}{ +\spadpaste{lcm(56788,43688)} +} +\xtc{ +To determine the maximum of two integers, use \spadfunFrom{max}{Integer}. +}{ +\spadpaste{max(678,567)} +} +\xtc{ +To determine the minimum, use \spadfunFrom{min}{Integer}. +}{ +\spadpaste{min(678,567)} +} + +\xtc{ +The \spadfun{reduce} operation is used to extend +binary operations to more than two arguments. +For example, you can use \spadfun{reduce} to find the maximum integer in +a list or compute the least common multiple of all integers in the list. +}{ +\spadpaste{reduce(max,[2,45,-89,78,100,-45])} +} +\xtc{ +}{ +\spadpaste{reduce(min,[2,45,-89,78,100,-45])} +} +\xtc{ +}{ +\spadpaste{reduce(gcd,[2,45,-89,78,100,-45])} +} +\xtc{ +}{ +\spadpaste{reduce(lcm,[2,45,-89,78,100,-45])} +} + +\xtc{ +The infix operator ``/'' is {\it not} used to compute the quotient +of integers. +Rather, it is used to create rational numbers as described in +\downlink{`Fraction'}{FractionXmpPage}\ignore{Fraction}. +}{ +\spadpaste{13 / 4} +} +\xtc{ +The infix operation \spadfunFrom{quo}{Integer} computes the integer +quotient. +}{ +\spadpaste{13 quo 4} +} +\xtc{ +The infix operation \spadfunFrom{rem}{Integer} computes the integer +remainder. +}{ +\spadpaste{13 rem 4} +} +\xtc{ +One integer is evenly divisible by another if the remainder is zero. +The operation \spadfunFrom{exquo}{Integer} can also be used. +See \downlink{``\ugTypesUnionsTitle''}{ugTypesUnionsPage} in Section +\ugTypesUnionsNumber\ignore{ugTypesUnions} for an example. +}{ +\spadpaste{zero?(167604736446952 rem 2003644)} +} + +\xtc{ +The operation \spadfunFrom{divide}{Integer} returns a record of the +quotient and remainder and thus is more efficient when both are needed. +}{ +\spadpaste{d := divide(13,4) \bound{d}} +} +\xtc{ +}{ +\spadpaste{d.quotient \free{d}} +} +\xtc{ +Records are discussed in detail in +\downlink{``\ugTypesRecordsTitle''}{ugTypesRecordsPage} in Section +\ugTypesRecordsNumber\ignore{ugTypesRecords}. +}{ +\spadpaste{d.remainder \free{d}} +} + +\endscroll +\autobuttons +\end{page} + +@ +<>= +\newcommand{\ugxIntegerPrimesTitle}{Primes and Factorization} +\newcommand{\ugxIntegerPrimesNumber}{9.34.2.} + +@ +\subsection{Primes and Factorization} +\label{ugxIntegerPrimesPage} +\begin{itemize} +\item FactoredXmpPage \ref{FactoredXmpPage} on +page~pageref{FactoredXmpPage} +\item ComplexXmpPage \ref{ComplexXmpPage} on +page~pageref{ComplexXmpPage} +\end{itemize} +\index{pages!ugxIntegerPrimesPage!int.ht} +\index{int.ht!pages!ugxIntegerPrimesPage} +\index{ugxIntegerPrimesPage!int.ht!pages} +<>= +\begin{page}{ugxIntegerPrimesPage}{Primes and Factorization} +\beginscroll + +\labelSpace{3pc} +\xtc{ +Use the operation \spadfunFrom{factor}{Integer} to factor integers. +It returns an object of type \spadtype{Factored Integer}. +See \downlink{`Factored'}{FactoredXmpPage}\ignore{Factored} +for a discussion of the +manipulation of factored objects. +}{ +\spadpaste{factor 102400} +} + +\xtc{ +The operation \spadfunFrom{prime?}{Integer} returns \spad{true} or \spad{false} depending +on whether its argument is a prime. +}{ +\spadpaste{prime? 7} +} +\xtc{ +}{ +\spadpaste{prime? 8} +} +\xtc{ +The operation \spadfunFrom{nextPrime}{IntegerPrimesPackage} returns the +least prime number greater than its argument. +}{ +\spadpaste{nextPrime 100} +} +\xtc{ +The operation +\spadfunFrom{prevPrime}{IntegerPrimesPackage} returns the greatest prime +number less than its argument. +}{ +\spadpaste{prevPrime 100} +} +\xtc{ +To compute all primes between two integers (inclusively), use the +operation \spadfunFrom{primes}{IntegerPrimesPackage}. +}{ +\spadpaste{primes(100,175)} +} +\xtc{ +You might sometimes want to see the factorization of an integer +when it is considered a {\it Gaussian integer}. +See \downlink{`Complex'}{ComplexXmpPage}\ignore{Complex} for more details. +}{ +\spadpaste{factor(2 :: Complex Integer)} +} + +\endscroll +\autobuttons +\end{page} + +@ +<>= +\newcommand{\ugxIntegerNTTitle}{Some Number Theoretic Functions} +\newcommand{\ugxIntegerNTNumber}{9.34.3.} + +@ +\subsection{Some Number Theoretic Functions} +\label{ugxIntegerNTPage} +\index{pages!ugxIntegerNTPage!int.ht} +\index{int.ht!pages!ugxIntegerNTPage} +\index{ugxIntegerNTPage!int.ht!pages} +<>= +\begin{page}{ugxIntegerNTPage}{Some Number Theoretic Functions} +\beginscroll + +Axiom provides several number theoretic operations for integers. +More examples are in \downlink{`IntegerNumberTheoryFunctions'} +{IntegerNumberTheoryFunctionsXmpPage}\ignore{IntegerNumberTheoryFunctions}. + +\labelSpace{1pc} +\xtc{ +The operation \spadfunFrom{fibonacci}{IntegerNumberTheoryFunctions} +computes the Fibonacci numbers. +The algorithm has running time +\texht{$O\,(\log^3(n))$}{O(\spad{log(n)**3})} for argument \spad{n}. +}{ +\spadpaste{[fibonacci(k) for k in 0..]} +} +\xtc{ +The operation \spadfunFrom{legendre}{IntegerNumberTheoryFunctions} +computes the Legendre symbol for its two integer arguments where the +second one is prime. +If you know the second argument to be prime, use +\spadfunFrom{jacobi}{IntegerNumberTheoryFunctions} instead where no check +is made. +}{ +\spadpaste{[legendre(i,11) for i in 0..10]} +} +\xtc{ +The operation \spadfunFrom{jacobi}{IntegerNumberTheoryFunctions} computes +the Jacobi symbol for its two integer arguments. +By convention, \spad{0} is returned if the greatest common divisor of the +numerator and denominator is not \spad{1}. +}{ +\spadpaste{[jacobi(i,15) for i in 0..9]} +} +\xtc{ +The operation \spadfunFrom{eulerPhi}{IntegerNumberTheoryFunctions} computes +the values of Euler's \texht{$\phi$}{phi}-function where +\texht{$\phi(n)$}{\spad{phi(n)}} +equals the number of positive integers +less than or equal to \spad{n} that are relatively prime to +the positive integer \spad{n}. +}{ +\spadpaste{[eulerPhi i for i in 1..]} +} +\xtc{ +The operation \spadfunFrom{moebiusMu}{IntegerNumberTheoryFunctions} +computes the \texht{M\"{o}bius $\mu$}{Moebius mu} function. +}{ +\spadpaste{[moebiusMu i for i in 1..]} +} + +\xtc{ +Although they have somewhat limited utility, Axiom provides +Roman numerals. +}{ +\spadpaste{a := roman(78) \bound{a}} +} +\xtc{ +}{ +\spadpaste{b := roman(87) \bound{b}} +} +\xtc{ +}{ +\spadpaste{a + b \free{a}\free{b}} +} +\xtc{ +}{ +\spadpaste{a * b \free{a}\free{b}} +} +\xtc{ +}{ +\spadpaste{b rem a \free{a}\free{b}} +} +\endscroll +\autobuttons +\end{page} +% + +@ +\section{intheory.ht} +<>= +\newcommand{\IntegerNumberTheoryFunctionsXmpTitle} +{IntegerNumberTheoryFunctions} +\newcommand{\IntegerNumberTheoryFunctionsXmpNumber}{9.36} + +@ +\subsection{IntegerNumberTheoryFunctions} +\label{IntegerNumberTheoryFunctionsXmpPage} +\index{pages!IntegerNumberTheoryFunctionsXmpPage!intheory.ht} +\index{intheory.ht!pages!IntegerNumberTheoryFunctionsXmpPage} +\index{IntegerNumberTheoryFunctionsXmpPage!intheory.ht!pages} +<>= +\begin{page}{IntegerNumberTheoryFunctionsXmpPage}{IntegerNumberTheoryFunctions} +\beginscroll + +The \spadtype{IntegerNumberTheoryFunctions} package contains a variety of +operations of interest to number theorists. +Many of these operations deal with divisibility properties of integers. +(Recall that an integer \spad{a} divides an integer \spad{b} if there is +an integer \spad{c} such that \spad{b = a * c}.) + +\xtc{ +The operation \spadfunFrom{divisors}{IntegerNumberTheoryFunctions} +returns a list of the divisors of an integer. +}{ +\spadpaste{div144 := divisors(144) \bound{div144}} +} +\xtc{ +You can now compute the number of divisors of \spad{144} and the sum of +the divisors of \spad{144} by counting and summing the elements of the +list we just created. +}{ +\spadpaste{\#(div144) \free{div144}} +} +\xtc{ +}{ +\spadpaste{reduce(+,div144) \free{div144}} +} + +Of course, you can compute the number of divisors of an integer \spad{n}, +usually denoted \spad{d(n)}, and the sum of the divisors of an integer +\spad{n}, usually denoted \spad{\texht{$\sigma$}{sigma}(n)}, +without ever listing the divisors of \spad{n}. + +\xtc{ +In Axiom, you can simply call the operations +\spadfunFrom{numberOfDivisors}{IntegerNumberTheoryFunctions} and +\spadfunFrom{sumOfDivisors}{IntegerNumberTheoryFunctions}. +}{ +\spadpaste{numberOfDivisors(144)} +} +\xtc{ +}{ +\spadpaste{sumOfDivisors(144)} +} + +The key is that \spad{d(n)} and +\spad{\texht{$\sigma$}{sigma}(n)} +are ``multiplicative functions.'' +This means that when \spad{n} and \spad{m} are relatively prime, that is, when +\spad{n} and \spad{m} have no prime factor in common, then +\spad{d(nm) = d(n) d(m)} and +\spad{\texht{$\sigma$}{sigma}(nm) = \texht{$\sigma$}{sigma}(n) +\texht{$\sigma$}{sigma}(m)}. +Note that these functions are trivial to compute when \spad{n} is a prime +power and are computed for general \spad{n} from the prime factorization +of \spad{n}. +Other examples of multiplicative functions are +\spad{\texht{$\sigma_k$}{sigma_k}(n)}, the sum of the \eth{\spad{k}} powers of +the divisors of \spad{n} and \texht{$\varphi(n)$}{\spad{phi(n)}}, the +number of integers between 1 and \spad{n} which are prime to \spad{n}. +The corresponding Axiom operations are called +\spadfunFrom{sumOfKthPowerDivisors}{IntegerNumberTheoryFunctions} and +\spadfunFrom{eulerPhi}{IntegerNumberTheoryFunctions}. + +An interesting function is \spad{\texht{$\mu$}{mu}(n)}, +the \texht{M\"{o}bius $\mu$}{Moebius mu} function, defined +as follows: +\spad{\texht{$\mu$}{mu}(1) = 1}, \spad{\texht{$\mu$}{mu}(n) = 0}, +when \spad{n} is divisible by a +square, and +\spad{\texht{$\mu = {(-1)}^k$}{mu(n) = (-1) ** k}}, when \spad{n} +is the product of \spad{k} distinct primes. +The corresponding Axiom operation is +\spadfunFrom{moebiusMu}{IntegerNumberTheoryFunctions}. +This function occurs in the following theorem: + +\noindent +{\bf Theorem} (\texht{M\"{o}bius}{Moebius} Inversion Formula): \newline +%\texht{\begin{quotation}\noindent}{\newline\indent{5}} +Let \spad{f(n)} be a function on the positive integers and let \spad{F(n)} +be defined by +\texht{\narrowDisplay{F(n) = \sum_{d \mid n} f(n)}}{\spad{F(n) =} +sum of \spad{f(n)} over \spad{d | n}} +where the sum is taken over the positive divisors of \spad{n}. +Then the values of \spad{f(n)} can be recovered from the values of +\spad{F(n)}: +\texht{\narrowDisplay{f(n) = \sum_{d \mid n} \mu(n) F({{n}\over {d}})}} +{\spad{f(n) =} +sum of \spad{mu(n) F(n/d)} over \spad{d | n},} +where again the sum is taken over the positive divisors of \spad{n}. + +\xtc{ +When \spad{f(n) = 1}, then \spad{F(n) = d(n)}. +Thus, if you sum \spad{\texht{$\mu$}{mu}(d) \texht{$\cdot$}{*} d(n/d)} +over the positive divisors +\spad{d} of \spad{n}, you should always get \spad{1}. +}{ +\spadpaste{f1(n) == reduce(+,[moebiusMu(d) * numberOfDivisors(quo(n,d)) for d in divisors(n)]) \bound{f1}} +} +\xtc{ +}{ +\spadpaste{f1(200) \free{f1}} +} +\xtc{ +}{ +\spadpaste{f1(846) \free{f1}} +} +\xtc{ +Similarly, when \spad{f(n) = n}, then \spad{F(n) = \texht{$\sigma$}{sigma}(n)}. +Thus, if you sum \spad{\texht{$\mu$}{mu}(d) \texht{$\cdot$}{*} +\texht{$\sigma$}{sigma}(n/d)} over the positive divisors +\spad{d} of \spad{n}, you should always get \spad{n}. +}{ +\spadpaste{f2(n) == reduce(+,[moebiusMu(d) * sumOfDivisors(quo(n,d)) for d in divisors(n)]) \bound{f2}} +} +\xtc{ +}{ +\spadpaste{f2(200) \free{f2}} +} +\xtc{ +}{ +\spadpaste{f2(846) \free{f2}} +} + + +The Fibonacci numbers are defined by \spad{F(1) = F(2) = 1} and +\spad{F(n) = F(n-1) + F(n-2)} for \spad{n = 3,4, ...}. +\xtc{ +The operation \spadfunFrom{fibonacci}{IntegerNumberTheoryFunctions} +computes the \eth{\spad{n}} Fibonacci number. +}{ +\spadpaste{fibonacci(25)} +} +\xtc{ +}{ +\spadpaste{[fibonacci(n) for n in 1..15]} +} +\xtc{ +Fibonacci numbers can also be expressed as sums of binomial coefficients. +}{ +\spadpaste{fib(n) == reduce(+,[binomial(n-1-k,k) for k in 0..quo(n-1,2)]) \bound{fib}} +} +\xtc{ +}{ +\spadpaste{fib(25) \free{fib}} +} +\xtc{ +}{ +\spadpaste{[fib(n) for n in 1..15] \free{fib}} +} + +Quadratic symbols can be computed with the operations +\spadfunFrom{legendre}{IntegerNumberTheoryFunctions} and +\spadfunFrom{jacobi}{IntegerNumberTheoryFunctions}. +The Legendre symbol +\texht{$\left({a \over p}\right)$}{\spad{(a/p)}} +is defined for integers \spad{a} and +\spad{p} with \spad{p} an odd prime number. +By definition, \texht{$\left({a \over p}\right)$}{\spad{(a/p) = +1}}, +when \spad{a} is a square \spad{(mod p)}, +\texht{$\left({a \over p}\right)$}{\spad{(a/p) = -1}}, +when \spad{a} is not a square \spad{(mod p)}, and +\texht{$\left({a \over p}\right)$}{\spad{(a/p) = 0}}, +when \spad{a} is divisible by \spad{p}. +\xtc{ +You compute \texht{$\left({a \over p}\right)$}{\spad{(a/p)}} +via the command \spad{legendre(a,p)}. +}{ +\spadpaste{legendre(3,5)} +} +\xtc{ +}{ +\spadpaste{legendre(23,691)} +} +The Jacobi symbol \texht{$\left({a \over n}\right)$}{\spad{(a/n)}} +is the usual extension of the Legendre +symbol, where \spad{n} is an arbitrary integer. +The most important property of the Jacobi symbol is the following: +if \spad{K} is a quadratic field with discriminant \spad{d} and quadratic +character \texht{$\chi$}{\spad{chi}}, +then \texht{$\chi$}{\spad{chi}}\spad{(n) = (d/n)}. +Thus, you can use the Jacobi symbol +to compute, say, the class numbers of +imaginary quadratic fields from a standard class number formula. +\xtc{ +This function computes the class number of the imaginary +quadratic field with discriminant \spad{d}. +}{ +\spadpaste{h(d) == quo(reduce(+, [jacobi(d,k) for k in 1..quo(-d, 2)]), 2 - jacobi(d,2)) \bound{h}} +} +\xtc{ +}{ +\spadpaste{h(-163) \free{h}} +} +\xtc{ +}{ +\spadpaste{h(-499) \free{h}} +} +\xtc{ +}{ +\spadpaste{h(-1832) \free{h}} +} + +\endscroll +\autobuttons +\end{page} + +@ +\section{kafile.ht} +<>= +\newcommand{\KeyedAccessFileXmpTitle}{KeyedAccessFile} +\newcommand{\KeyedAccessFileXmpNumber}{9.38} + +@ +\subsection{KeyedAccessFile} +\label{KeyedAccessFileXmpPage} +\begin{itemize} +\item FileXmpPage \ref{FileXmpPage} on +page~pageref{FileXmpPage} +\item TextFileXmpPage \ref{TextFileXmpPage} on +page~pageref{TextFileXmpPage} +\item LibraryXmpPage \ref{LibraryXmpPage} on +page~pageref{LibraryXmpPage} +\end{itemize} +\index{pages!KeyedAccessFileXmpPage!kafile.ht} +\index{kafile.ht!pages!KeyedAccessFileXmpPage} +\index{KeyedAccessFileXmpPage!kafile.ht!pages} +<>= +\begin{page}{KeyedAccessFileXmpPage}{KeyedAccessFile} +\beginscroll + +The domain \spadtype{KeyedAccessFile(S)} provides files which can be used as +associative tables. +Data values are stored in these files and can be retrieved according to +their keys. +The keys must be strings so this type behaves very much like the +\spadtype{StringTable(S)} domain. +The difference is that keyed access files reside in secondary storage while +string tables are kept in memory. +For more information on table-oriented operations, see the description of +\spadtype{Table}. + +\xtc{ +Before a keyed access file can be used, it must first be opened. +A new file can be created by opening it for output. +}{ +\spadpaste{ey: KeyedAccessFile(Integer) := open("/tmp/editor.year", "output") \bound{ey}} +} +\xtc{ +Just as for vectors, tables or lists, values are saved in a keyed access file +by setting elements. +}{ +\spadpaste{ey."Char" := 1986 \free{ey}\bound{eya}} +} +\xtc{ +}{ +\spadpaste{ey."Caviness" := 1985 \free{ey}\bound{eyb}} +} +\xtc{ +}{ +\spadpaste{ey."Fitch" := 1984 \free{ey}\bound{eyc}} +} +\xtc{ +Values are retrieved using application, in any of its syntactic forms. +}{ +\spadpaste{ey."Char" \free{eya}} +} +\xtc{ +}{ +\spadpaste{ey("Char") \free{eya}} +} +\xtc{ +}{ +\spadpaste{ey "Char" \free{eya}} +} +\xtc{ +Attempting to retrieve a non-existent element in this way causes an error. +If it is not known whether a key exists, you should use the +\spadfunFrom{search}{KeyedAccessFile} operation. +}{ +\spadpaste{search("Char", ey) \free{eya,eyb,eyc}\bound{eyaa}} +} +\xtc{ +}{ +\spadpaste{search("Smith", ey) \free{eyaa}} +} +\xtc{ +When an entry is no longer needed, it can be removed from the file. +}{ +\spadpaste{remove!("Char", ey) \free{eyaa}\bound{eybb}} +} +\xtc{ +The \spadfunFrom{keys}{KeyedAccessFile} operation returns a list of all the +keys for a given file. +}{ +\spadpaste{keys ey \free{eybb}} +} +\xtc{ +The \spadfunFrom{\#}{KeyedAccessFile} operation gives the +number of entries. +}{ +\spadpaste{\#ey \free{eybb}} +} + +The table view of keyed access files provides safe operations. +That is, if the Axiom program is terminated between file operations, +the file is left in a consistent, current state. +This means, however, that the operations are somewhat costly. +For example, after each update the file is closed. +\xtc{ +Here we add several more items to the file, then check its contents. +}{ +\spadpaste{KE := Record(key: String, entry: Integer) \bound{KE}} +} +\xtc{ +}{ +\spadpaste{reopen!(ey, "output") \free{eybb,KE}\bound{eycc}} +} +\xtc{ +If many items are to be added to a file at the same time, then +it is more efficient to use the +\spadfunFromX{write}{KeyedAccessFile} operation. +}{ +\spadpaste{write!(ey, ["van Hulzen", 1983]\$KE) \bound{eyccA}\free{eycc}} +} +\xtc{ +}{ +\spadpaste{write!(ey, ["Calmet", 1982]\$KE) \bound{eyccB}\free{eycc}} +} +\xtc{ +}{ +\spadpaste{write!(ey, ["Wang", 1981]\$KE) \bound{eyccC}\free{eycc}} +} +\xtc{ +}{ +\spadpaste{close! ey \free{eyccA,eyccB,eyccC}\bound{eydd}} +} +\xtc{ +The \spadfunFromX{read}{KeyedAccessFile} operation is also available from +the file view, but it returns elements in a random order. +It is generally clearer and more efficient to use the +\spadfunFrom{keys}{KeyedAccessFile} operation and to extract elements by +key. +}{ +\spadpaste{keys ey \free{eydd}} +} +\xtc{ +}{ +\spadpaste{members ey \free{eydd}} +} +\noOutputXtc{ +}{ +\spadpaste{)system rm -r /tmp/editor.year \free{ey}} +} + +For more information on related topics, see +\downlink{`File'}{FileXmpPage}\ignore{File}, +\downlink{`TextFile'}{TextFileXmpPage}\ignore{TextFile}, and +\downlink{`Library'}{LibraryXmpPage}\ignore{Library}. +% +\showBlurb{KeyedAccessFile} +\endscroll +\autobuttons +\end{page} + +@ +\section{kernel.ht} +<>= +\newcommand{\KernelXmpTitle}{Kernel} +\newcommand{\KernelXmpNumber}{9.37} + +@ +\subsection{Kernel} +\label{KernelXmpPage} +\begin{itemize} +\item BasicOperatorXmpPage \ref{BasicOperatorXmpPage} on +page~pageref{BasicOperatorXmpPage} +\item ExpressionXmpPage \ref{ExpressionXmpPage} on +page~pageref{ExpressionXmpPage} +\end{itemize} +\index{pages!KernelXmpPage!kernel.ht} +\index{kernel.ht!pages!KernelXmpPage} +\index{KernelXmpPage!kernel.ht!pages} +<>= +\begin{page}{KernelXmpPage}{Kernel} +\beginscroll + +A {\it kernel} is a symbolic function application (such as +\spad{sin(x + y)}) or a symbol (such as \spad{x}). +More precisely, a non-symbol kernel over a set {\it S} is an operator applied +to a given list of arguments from {\it S}. +The operator has type \axiomType{BasicOperator} +(see +\downlink{`BasicOperator'}{BasicOperatorXmpPage}\ignore{BasicOperator}) +and the kernel object is usually part of +an expression object (see +\downlink{`Expression'}{ExpressionXmpPage}\ignore{Expression}). + +Kernels are created implicitly for you when you +create expressions. +\xtc{ +}{ +\spadpaste{x :: Expression Integer} +} +\xtc{ +You can directly create a ``symbol'' kernel by using the +\axiomFunFrom{kernel}{Kernel} operation. +}{ +\spadpaste{kernel x} +} +\xtc{ +This expression has two different kernels. +}{ +\spadpaste{sin(x) + cos(x) \bound{sincos}} +} +\xtc{ +The operator \axiomFunFrom{kernels}{Expression} returns a list +of the kernels in an object of type \axiomType{Expression}. +}{ +\spadpaste{kernels \% \free{sincos}} +} +\xtc{ +This expression also has two different kernels. +}{ +\spadpaste{sin(x)**2 + sin(x) + cos(x) \bound{sincos2}} +} +\xtc{ +The \spad{sin(x)} kernel is used twice. +}{ +\spadpaste{kernels \% \free{sincos2}} +} +\xtc{ +An expression need not contain any kernels. +}{ +\spadpaste{kernels(1 :: Expression Integer)} +} +\xtc{ +If one or more kernels are present, one of them is +designated the {\it main} kernel. +}{ +\spadpaste{mainKernel(cos(x) + tan(x))} +} +\xtc{ +Kernels can be nested. Use \axiomFunFrom{height}{Kernel} to determine +the nesting depth. +}{ +\spadpaste{height kernel x} +} +\xtc{ +This has height 2 because the \spad{x} has height 1 and then we apply +an operator to that. +}{ +\spadpaste{height mainKernel(sin x)} +} +\xtc{ +}{ +\spadpaste{height mainKernel(sin cos x)} +} +\xtc{ +}{ +\spadpaste{height mainKernel(sin cos (tan x + sin x))} +} +\xtc{ +Use the \axiomFunFrom{operator}{Kernel} operation to extract the +operator component of the kernel. +The operator has type \axiomType{BasicOperator}. +}{ +\spadpaste{operator mainKernel(sin cos (tan x + sin x))} +} +\xtc{ +Use the \axiomFunFrom{name}{Kernel} operation to extract the name of the +operator component of the kernel. +The name has type \axiomType{Symbol}. +This is really just a shortcut for a two-step process of extracting the +operator and then calling \axiomFunFrom{name}{BasicOperator} on the +operator. +}{ +\spadpaste{name mainKernel(sin cos (tan x + sin x))} +} +Axiom knows about functions such as \axiomFun{sin}, \axiomFun{cos} +and so on and can make kernels and then expressions using them. +To create a kernel and expression using an arbitrary operator, use +\axiomFunFrom{operator}{BasicOperator}. +\xtc{ +Now \spad{f} can be used to create symbolic function applications. +}{ +\spadpaste{f := operator 'f \bound{f}} +} +\xtc{ +}{ +\spadpaste{e := f(x, y, 10) \free{f}\bound{e}} +} +\xtc{ +Use the \axiomFunFrom{is?}{Kernel} operation to learn if the +operator component of a kernel is equal to a given operator. +}{ +\spadpaste{is?(e, f) \free{f e}} +} +\xtc{ +You can also use a symbol or a string as the second +argument to \axiomFunFrom{is?}{Kernel}. +}{ +\spadpaste{is?(e, 'f) \free{e}} +} +\xtc{ +Use the \axiomFunFrom{argument}{Kernel} operation to get a list containing +the argument component of a kernel. +}{ +\spadpaste{argument mainKernel e \free{f}\free{e}} +} + +Conceptually, an object of type \axiomType{Expression} can be thought +of a quotient of multivariate polynomials, where the ``variables'' +are kernels. +The arguments of the kernels are again expressions and so the +structure recurses. +See +\downlink{`Expression'}{ExpressionXmpPage}\ignore{Expression} +for examples of using kernels to +take apart expression objects. +\endscroll +\autobuttons +\end{page} + +@ +\section{lazm3pk.ht} +<>= +\newcommand{\LazardSetSolvingPackageXmpTitle}{LazardSetSolvingPackage} +\newcommand{\LazardSetSolvingPackageXmpNumber}{9.40} + +@ +\subsection{LazardSetSolvingPackage} +\label{LazardSetSolvingPackageXmpPage} +\index{pages!LazardSetSolvingPackageXmpPage!lazm3pk.ht} +\index{lazm3pk.ht!pages!LazardSetSolvingPackageXmpPage} +\index{LazardSetSolvingPackageXmpPage!lazm3pk.ht!pages} +<>= +\begin{page}{LazardSetSolvingPackageXmpPage}{LazardSetSolvingPackage} +\beginscroll +The \spadtype{LazardSetSolvingPackage} package constructor solves +polynomial systems by means of Lazard triangular sets. +However one condition is relaxed: +Regular triangular sets whose saturated ideals have positive dimension +are not necessarily normalized. + +The decompositions are computed in two steps. +First the algorithm of Moreno Maza (implemented in +the \spadtype{RegularTriangularSet} domain constructor) +is called. +Then the resulting decompositions are converted into lists +of square-free regular triangular sets +and the redundant components are removed. +Moreover, zero-dimensional regular triangular sets are normalized. + +Note that the way of understanding triangular decompositions +is detailed in the example of the \spadtype{RegularTriangularSet} +constructor. + +The \spadtype{LazardSetSolvingPackage} constructor takes six arguments. +The first one, {\bf R}, is the coefficient ring of the polynomials; +it must belong to the category \spadtype{GcdDomain}. +The second one, {\bf E}, is the exponent monoid of the polynomials; +it must belong to the category \spadtype{OrderedAbelianMonoidSup}. +the third one, {\bf V}, is the ordered set of variables; +it must belong to the category \spadtype{OrderedSet}. +The fourth one is the polynomial ring; +it must belong to the category \spadtype{RecursivePolynomialCategory(R,E,V)}. +The fifth one is a domain of the category \spadtype{RegularTriangularSetCategory(R,E,V,P)} +and the last one is a domain of the category \spadtype{SquareFreeRegularTriangularSetCategory(R,E,V,P)}. +The abbreviation for \spadtype{LazardSetSolvingPackage} is \spad{LAZM3PK}. + +{\bf N.B.} For the purpose of solving zero-dimensional algebraic systems, +see also \spadtype{LexTriangularPackage} and \spadtype{ZeroDimensionalSolvePackage}. +These packages are easier to call than \spad{LAZM3PK}. +Moreover, the \spadtype{ZeroDimensionalSolvePackage} +package provides operations +to compute either the complex roots or the real roots. + +We illustrate now the use of the \spadtype{LazardSetSolvingPackage} package +constructor with two examples (Butcher and Vermeer). + + +\xtc{ +Define the coefficient ring. +}{ +\spadpaste{R := Integer \bound{R}} +} + + +\xtc{ +Define the list of variables, +}{ +\spadpaste{ls : List Symbol := [b1,x,y,z,t,v,u,w] \bound{ls}} +} + +\xtc{ +and make it an ordered set; +}{ +\spadpaste{V := OVAR(ls) \free{ls} \bound{V}} +} + +\xtc{ +then define the exponent monoid. +}{ +\spadpaste{E := IndexedExponents V \free{V} \bound{E}} +} + +\xtc{ +Define the polynomial ring. +}{ +\spadpaste{P := NSMP(R, V) \free{R} \free{V} \bound{P}} +} + +\xtc{ +Let the variables be polynomial. +}{ +\spadpaste{b1: P := 'b1 \free{P} \bound{b1}} +} +\xtc{ +}{ +\spadpaste{x: P := 'x \free{P} \bound{x}} +} +\xtc{ +}{ +\spadpaste{y: P := 'y \free{P} \bound{y}} +} +\xtc{ +}{ +\spadpaste{z: P := 'z \free{P} \bound{z}} +} +\xtc{ +}{ +\spadpaste{t: P := 't \free{P} \bound{t}} +} +\xtc{ +}{ +\spadpaste{u: P := 'u \free{P} \bound{u}} +} +\xtc{ +}{ +\spadpaste{v: P := 'v \free{P} \bound{v}} +} +\xtc{ +}{ +\spadpaste{w: P := 'w \free{P} \bound{w}} +} + +\xtc{ +Now call the \spadtype{RegularTriangularSet} domain constructor. +}{ +\spadpaste{T := REGSET(R,E,V,P) \free{R} \free{E} \free{V} \free{P} \bound{T} } +} + +\xtc{ +Define a polynomial system (the Butcher example). +}{ +\spadpaste{p0 := b1 + y + z - t - w \free{b1} \free{x} \free{y} \free{z} \free{t} \free{u} \free{v} \free{w} \bound{p0}} +} +\xtc{ +}{ +\spadpaste{p1 := 2*z*u + 2*y*v + 2*t*w - 2*w**2 - w - 1 \free{b1} \free{x} \free{y} \free{z} \free{t} \free{u} \free{v} \free{w} \bound{p1}} +} +\xtc{ +}{ +\spadpaste{p2 := 3*z*u**2 + 3*y*v**2 - 3*t*w**2 + 3*w**3 + 3*w**2 - t + 4*w \free{b1} \free{x} \free{y} \free{z} \free{t} \free{u} \free{v} \free{w} \bound{p2}} +} +\xtc{ +}{ +\spadpaste{p3 := 6*x*z*v - 6*t*w**2 + 6*w**3 - 3*t*w + 6*w**2 - t + 4*w \free{b1} \free{x} \free{y} \free{z} \free{t} \free{u} \free{v} \free{w} \bound{p3}} +} +\xtc{ +}{ +\spadpaste{p4 := 4*z*u**3+ 4*y*v**3+ 4*t*w**3- 4*w**4 - 6*w**3+ 4*t*w- 10*w**2- w- 1 \free{b1} \free{x} \free{y} \free{z} \free{t} \free{u} \free{v} \free{w} \bound{p4}} +} +\xtc{ +}{ +\spadpaste{p5 := 8*x*z*u*v +8*t*w**3 -8*w**4 +4*t*w**2 -12*w**3 +4*t*w -14*w**2 -3*w -1 \free{b1} \free{x} \free{y} \free{z} \free{t} \free{u} \free{v} \free{w} \bound{p5}} +} +\xtc{ +}{ +\spadpaste{p6 := 12*x*z*v**2+12*t*w**3 -12*w**4 +12*t*w**2 -18*w**3 +8*t*w -14*w**2 -w -1 \free{b1} \free{x} \free{y} \free{z} \free{t} \free{u} \free{v} \free{w} \bound{p6}} +} +\xtc{ +}{ +\spadpaste{p7 := -24*t*w**3 + 24*w**4 - 24*t*w**2 + 36*w**3 - 8*t*w + 26*w**2 + 7*w + 1 \free{b1} \free{x} \free{y} \free{z} \free{t} \free{u} \free{v} \free{w} \bound{p7}} +} +\xtc{ +}{ +\spadpaste{lp := [p0, p1, p2, p3, p4, p5, p6, p7] \free{p0} \free{p1} \free{p2} \free{p3} \free{p4} \free{p5} \free{p6} \free{p7} \bound{lp}} +} + +\xtc{ +First of all, let us solve this system in the sense of Lazard by means of the \spadtype{REGSET} +constructor: +}{ +\spadpaste{lts := zeroSetSplit(lp,false)$T \free{lp} \free{T} \bound{lts}} +} + +\xtc{ +We can get the dimensions of each component +of a decomposition as follows. +}{ +\spadpaste{[coHeight(ts) for ts in lts] \free{lts}} +} + +The first five sets have a simple shape. +However, the last one, which has dimension zero, can be simplified +by using Lazard triangular sets. + +\xtc{ +Thus we call the \spadtype{SquareFreeRegularTriangularSet} domain constructor, +}{ +\spadpaste{ST := SREGSET(R,E,V,P) \free{R} \free{E} \free{V} \free{P} \bound{ST} } +} + +\xtc{ +and set the \spadtype{LAZM3PK} package constructor to our situation. +}{ +\spadpaste{pack := LAZM3PK(R,E,V,P,T,ST) \free{R} \free{E} \free{V} \free{P} \free{T} \free{ST} \bound{pack} } +} + +\xtc{ +We are ready to solve the system by means of Lazard triangular sets: +}{ +\spadpaste{zeroSetSplit(lp,false)$pack \free{lp} \free{pack}} +} + +We see the sixth triangular set is {\em nicer} now: +each one of its polynomials has a constant initial. + +We follow with the Vermeer example. The ordering is the usual one +for this system. + +\xtc{ +Define the polynomial system. +}{ +\spadpaste{f0 := (w - v) ** 2 + (u - t) ** 2 - 1 \free{b1} \free{x} \free{y} \free{z} \free{t} \free{u} \free{v} \free{w} \bound{f0}} +} +\xtc{ +}{ +\spadpaste{f1 := t ** 2 - v ** 3 \free{b1} \free{x} \free{y} \free{z} \free{t} \free{u} \free{v} \free{w} \bound{f1}} +} +\xtc{ +}{ +\spadpaste{f2 := 2 * t * (w - v) + 3 * v ** 2 * (u - t) \free{b1} \free{x} \free{y} \free{z} \free{t} \free{u} \free{v} \free{w} \bound{f2}} +} +\xtc{ +}{ +\spadpaste{f3 := (3 * z * v ** 2 - 1) * (2 * z * t - 1) \free{b1} \free{x} \free{y} \free{z} \free{t} \free{u} \free{v} \free{w} \bound{f3}} +} +\xtc{ +}{ +\spadpaste{lf := [f0, f1, f2, f3] \free{f0} \free{f1} \free{f2} \free{f3} \bound{lf}} +} + + +\xtc{ +First of all, let us solve this system in the sense of Kalkbrener by +means of the \spadtype{REGSET} +constructor: +}{ +\spadpaste{zeroSetSplit(lf,true)$T \free{lf} \free{T}} +} + +We have obtained one regular chain (i.e. regular triangular set) with +dimension 1. +This set is in fact a characterist set of the (radical of) of the ideal +generated by the input system {\bf lf}. +Thus we have only the {\em generic points} of the variety associated +with {\bf lf} +(for the elimination ordering given by {\bf ls}). + +So let us get now a full description of this variety. +\xtc{ +Hence, we solve this system in the sense of Lazard by means of the +\spadtype{REGSET} +constructor: +}{ +\spadpaste{zeroSetSplit(lf,false)$T \free{lf} \free{T}} +} + +We retrieve our regular chain of dimension 1 and we get three regular chains +of dimension 0 corresponding to the {\em degenerated cases}. +We want now to simplify these zero-dimensional regular chains +by using Lazard triangular sets. +Moreover, this will allow us to prove that the above decomposition has +no redundant component. +{\bf N.B.} Generally, decompositions computed by the \spadtype{REGSET} +constructor do not have redundant components. +However, to be sure that no redundant component occurs one needs to use +the \spadtype{SREGSET} or \spadtype{LAZM3PK} constructors. + +\xtc{ +So let us solve the input system in the sense of Lazard by means of +the \spadtype{LAZM3PK} constructor: +}{ +\spadpaste{zeroSetSplit(lf,false)$pack \free{lf} \free{pack}} +} +Due to square-free factorization, we obtained now four zero-dimensional +regular chains. +Moreover, each of them is normalized (the initials are constant). +Note that these zero-dimensional components may be investigated further +with the \spadtype{ZeroDimensionalSolvePackage} package constructor. + +\endscroll +\autobuttons +\end{page} + +@ +\section{lexp.ht} +<>= +\newcommand{\LieExponentialsXmpTitle}{LieExponentials} +\newcommand{\LieExponentialsXmpNumber}{9.42} + +@ +\subsection{LieExponentials} +\label{LieExponentialsXmpPage} +\index{pages!LieExponentialsXmpPage!lexp.ht} +\index{lexp.ht!pages!LieExponentialsXmpPage} +\index{LieExponentialsXmpPage!lexp.ht!pages} +<>= +\begin{page}{LieExponentialsXmpPage}{LieExponentials} +% ===================================================================== +\beginscroll +\xtc{ +}{ +\spadpaste{ a: Symbol := 'a \bound{a}} +} +\xtc{ +}{ +\spadpaste{ b: Symbol := 'b \bound{b}} +} + +Declarations of domains +\xtc{ +}{ +\spadpaste{ coef := Fraction(Integer) \bound{coef}} +} +\xtc{ +}{ +\spadpaste{ group := LieExponentials(Symbol, coef, 3) \free{coef} \bound{group}} +} +\xtc{ +}{ +\spadpaste{ lpoly := LiePolynomial(Symbol, coef) \free{coef} \bound{lpoly}} +} +\xtc{ +}{ +\spadpaste{ poly := XPBWPolynomial(Symbol, coef) \free{coef} \bound{poly}} +} + +Calculations +\xtc{ +}{ +\spadpaste{ ea := exp(a::lpoly)$group \free{a} \free{lpoly} \free{group} \bound{ea}} +} +\xtc{ +}{ +\spadpaste{ eb := exp(b::lpoly)$group \free{b} \free{lpoly} \free{group} \bound{eb}} +} +\xtc{ +}{ +\spadpaste{ g: group := ea*eb \free{ea} \free{eb} \bound{g}} +} +\xtc{ +}{ +\spadpaste{ g :: poly \free{g} \free{poly}} +} +\xtc{ +}{ +\spadpaste{ log(g)$group \free{g} \free{group}} +} +\xtc{ +}{ +\spadpaste{ g1: group := inv(g) \free{g} \free{group} \bound{g1}} +} +\xtc{ +}{ +\spadpaste{ g*g1 \free{g} \free{g1}} +} + +\endscroll +\autobuttons +\end{page} +% + +@ +\section{lextripk.ht} +<>= +\newcommand{\LexTriangularPackageXmpTitle}{LexTriangularPackage} +\newcommand{\LexTriangularPackageXmpNumber}{9.39} + +@ +\subsection{LexTriangularPackage} +\label{LexTriangularPackageXmpPage} +\index{pages!LexTriangularPackageXmpPage!lextripk.ht} +\index{lextripk.ht!pages!LexTriangularPackageXmpPage} +\index{LexTriangularPackageXmpPage!lextripk.ht!pages} +<>= +\begin{page}{LexTriangularPackageXmpPage}{LexTriangularPackage} +\beginscroll +The \spadtype{LexTriangularPackage} package constructor provides +an implementation of the {\em lexTriangular} algorithm +(D. Lazard "Solving Zero-dimensional Algebraic Systems", +J. of Symbol. Comput., 1992). +This algorithm decomposes a zero-dimensional variety into zero-sets of regular +triangular sets. +Thus the input system must have a finite number of complex solutions. +Moreover, this system needs to be a lexicographical Groebner basis. + +This package takes two arguments: +the coefficient-ring {\bf R} of the polynomials, +which must be a \spadtype{GcdDomain} +and their set of variables given by {\bf ls} a \spadtype{List Symbol}. +The type of the input polynomials must be +\spadtype{NewSparseMultivariatePolynomial(R,V)} +where {\bf V} is \spadtype{OrderedVariableList(ls)}. +The abbreviation for \spadtype{LexTriangularPackage} is \spadtype{LEXTRIPK}. +The main operations are \axiomOpFrom{lexTriangular}{LexTriangularPackage} +and \axiomOpFrom{squareFreeLexTriangular}{LexTriangularPackage}. +The later provide decompositions by means of +square-free regular triangular sets, +built with the \spadtype{SREGSET} constructor, +whereas the former uses the \spadtype{REGSET} constructor. +Note that these constructors also implement another algorithm +for solving algebraic systems by means of regular triangular sets; +in that case no computations of Groebner bases are needed and the input +system may have any dimension (i.e. it may have an infinite number +of solutions). + +The implementation of the {\em lexTriangular} algorithm +provided in the \spadtype{LexTriangularPackage} constructor +differs from that reported in "Computations of gcd over +algebraic towers of simple extensions" by M. Moreno Maza and R. Rioboo +(in proceedings of AAECC11, Paris, 1995). +Indeed, the \axiomOpFrom{squareFreeLexTriangular}{LexTriangularPackage} +operation +removes all multiplicities of the solutions (i.e. the computed solutions +are pairwise different) and the +\axiomOpFrom{lexTriangular}{LexTriangularPackage} +operation may keep some multiplicities; +this latter operation runs generally faster +than the former. + +The interest of the {\em lexTriangular} algorithm is due +to the following experimental remark. +For some examples, a triangular decomposition +of a zero-dimensional variety can be computed faster +via a lexicographical Groebner basis computation than +by using a direct method (like that of \spadtype{SREGSET} +and \spadtype{REGSET}). +This happens typically when the total degree of the system +relies essentially on its smallest variable (like in the +{\em Katsura} systems). +When this is not the case, the direct method may give better timings +(like in the {\em Rose} system). + +Of course, the direct method can also be applied to a +lexicographical Groebner basis. +However, the {\em lexTriangular} algorithm takes advantage +of the structure of this basis and avoids many unnecessary +computations which are performed by the direct method. + +For this purpose of solving algebraic systems with a +finite number of solutions, +see also the \spadtype{ZeroDimensionalSolvePackage}. +It allows to use both strategies (the lexTriangular algorithm and the direct +method) for computing either the complex or real roots of a system. + +Note that the way of understanding triangular decompositions +is detailed in the example of the \spadtype{RegularTriangularSet} +constructor. + +Since the \spadtype{LEXTRIPK} package constructor is limited +to zero-dimensional systems, it provides a +\axiomOpFrom{zeroDimensional?}{LexTriangularPackage} +operation to check whether this requirement holds. +There is also a \axiomOpFrom{groebner}{LexTriangularPackage} +operation to compute the lexicographical Groebner basis +of a set of polynomials with type +\spadtype{NewSparseMultivariatePolynomial(R,V)}. +The elimination ordering is that given by {\bf ls} +(the greatest variable being the first element of {\bf ls}). +This basis is computed by the {\em FLGM} algorithm +(Faugere et al. "Efficient Computation of Zero-Dimensional Groebner Bases by + Change of Ordering" , J. of Symbol. Comput., 1993) +implemented in the \spadtype{LinGroebnerPackage} package constructor. +Once a lexicographical Groebner basis is computed, +then one can call the operations +\axiomOpFrom{lexTriangular}{LexTriangularPackage} +and \axiomOpFrom{squareFreeLexTriangular}{LexTriangularPackage}. +Note that these operations admit an optional argument +to produce normalized triangular sets. +There is also a \axiomOpFrom{zeroSetSplit}{LexTriangularPackage} operation +which does all the job from the input system; +an error is produced if this system is not zero-dimensional. + +Let us illustrate the facilities of the \spadtype{LEXTRIPK} constructor +by a famous example, the {\em cyclic-6 root} system. +\xtc{ +Define the coefficient ring. +}{ +\spadpaste{R := Integer \bound{R}} +} +\xtc{ +Define the list of variables, +}{ +\spadpaste{ls : List Symbol := [a,b,c,d,e,f] \bound{ls}} +} +\xtc{ +and make it an ordered set. +}{ +\spadpaste{V := OVAR(ls) \free{ls} \bound{V}} +} +\xtc{ +Define the polynomial ring. +}{ +\spadpaste{P := NSMP(R, V) \free{R} \free{V} \bound{P}} +} +\xtc{ +Define the polynomials. +}{ +\spadpaste{p1: P := a*b*c*d*e*f - 1 \free{P} \bound{p1}} +} +\xtc{ +}{ +\spadpaste{p2: P := a*b*c*d*e +a*b*c*d*f +a*b*c*e*f +a*b*d*e*f +a*c*d*e*f +b*c*d*e*f \free{P} \bound{p2}} +} +\xtc{ +}{ +\spadpaste{p3: P := a*b*c*d + a*b*c*f + a*b*e*f + a*d*e*f + b*c*d*e + c*d*e*f \free{P} \bound{p3}} +} +\xtc{ +}{ +\spadpaste{p4: P := a*b*c + a*b*f + a*e*f + b*c*d + c*d*e + d*e*f \free{P} \bound{p4}} +} +\xtc{ +}{ +\spadpaste{p5: P := a*b + a*f + b*c + c*d + d*e + e*f \free{P} \bound{p5}} +} +\xtc{ +}{ +\spadpaste{p6: P := a + b + c + d + e + f \free{P} \bound{p6}} +} +\xtc{ +}{ +\spadpaste{lp := [p1, p2, p3, p4, p5, p6] \free{p1} \free{p2} \free{p3} \free{p4} \free{p5} \free{p6} \bound{lp}} +} +\xtc{ +Now call \spadtype{LEXTRIPK} . +}{ +\spadpaste{lextripack := LEXTRIPK(R,ls) \free{R} \free{ls} \bound{lextripack}} +} +\xtc{ +Compute the lexicographical Groebner basis of the system. +This may take between 5 minutes and one hour, depending on your machine. +}{ +\spadpaste{lg := groebner(lp)$lextripack \free{lp} \free{lextripack} \bound{lg}} +} +\xtc{ +Apply lexTriangular to compute a decomposition into regular triangular sets. +This should not take more than 5 seconds. +}{ +\spadpaste{lexTriangular(lg,false)$lextripack \free{lg} \free{lextripack}} +} +Note that the first set of the decomposition is normalized +(all initials are integer numbers) but not the second one +(normalized triangular sets are defined in the +description of the \spadtype{NormalizedTriangularSetCategory} constructor). +\xtc{ +So apply now lexTriangular to produce normalized triangular sets. +}{ +\spadpaste{lts := lexTriangular(lg,true)$lextripack \free{lg} \free{lextripack} \bound{lts}} +} +\xtc{ +We check that all initials are constant. +}{ +\spadpaste{[[init(p) for p in (ts :: List(P))] for ts in lts] \free{lts}} +} +Note that each triangular set in {\bf lts} is a lexicographical Groebner basis. +Recall that a point belongs to the variety +associated with {\bf lp} if and only if +it belongs to that associated with one triangular set {\bf ts} in {\bf lts}. + +By running the +\axiomOpFrom{squareFreeLexTriangular}{LexTriangularPackage} operation, +we retrieve the above decomposition. +\xtc{ +}{ +\spadpaste{squareFreeLexTriangular(lg,true)$lextripack \free{lg} \free{lextripack}} +} +Thus the solutions given by {\bf lts} are pairwise different. +\xtc{ +We count them as follows. +}{ +\spadpaste{reduce(+,[degree(ts) for ts in lts]) \free{lts}} +} + +We can investigate the triangular decomposition {\bf lts} by +using the \spadtype{ZeroDimensionalSolvePackage}. +\xtc{ +This requires to add an extra variable (smaller than the others) as follows. +}{ +\spadpaste{ls2 : List Symbol := concat(ls,new()$Symbol) \free{ls} \bound{ls2}} +} + +\xtc{ +Then we call the package. +}{ +\spadpaste{zdpack := ZDSOLVE(R,ls,ls2) \free{R} \free{ls} \free{ls2} \bound{zdpack}} +} + + +\xtc{ +We compute a univariate representation of the variety associated with the input +system as follows. +}{ +\spadpaste{concat [univariateSolve(ts)$zdpack for ts in lts] \free{lts} \free{zdpack}} +} +Since the +\axiomOpFrom{univariateSolve}{ZeroDimensionalSolvePackage} operation may +split a regular set, it returns a list. This explains the use +of \axiomOpFrom{concat}{List}. + +Look at the last item of the result. It consists of two parts. +For any complex root {\bf ?} of the univariate polynomial in the first part, +we get a tuple of +univariate polynomials (in {\bf a}, ..., {\bf f} respectively) +by replacing {\bf \%A} by {\bf ?} in the second part. +Each of these tuples {\bf t} +describes a point of the variety associated with {\bf lp} +by equaling to zero the polynomials in {\bf t}. + +Note that the way of reading these univariate representations is explained also +in the example illustrating the +\spadtype{ZeroDimensionalSolvePackage} constructor. + +\xtc{ +Now, we compute the points of the variety with real coordinates. +}{ +\spadpaste{concat [realSolve(ts)$zdpack for ts in lts] \free{lts} \free{zdpack}} +} +We obtain 24 points given by lists of elements in the \spadtype{RealClosure} +of \spadtype{Fraction} of {\bf R}. +In each list, the first value corresponds to the indeterminate {\bf f}, +the second to {\bf e} and so on. +See \spadtype{ZeroDimensionalSolvePackage} to learn more about +the \axiomOpFrom{realSolve}{ZeroDimensionalSolvePackage} operation. + +\endscroll +\autobuttons +\end{page} + +@ +\section{lib.ht} +<>= +\newcommand{\LibraryXmpTitle}{Library} +\newcommand{\LibraryXmpNumber}{9.41} + +@ +\subsection{Library} +\label{LibraryXmpPage} +\begin{itemize} +\item FileXmpPage \ref{FileXmpPage} on +page~pageref{FileXmpPage} +\item TextFileXmpPage \ref{TextFileXmpPage} on +page~pageref{TextFileXmpPage} +\item KeyedAccessFileXmpPage \ref{KeyedAccessFileXmpPage} on +page~pageref{KeyedAccessFileXmpPage} +\end{itemize} +\index{pages!LibraryXmpPage!lib.ht} +\index{lib.ht!pages!LibraryXmpPage} +\index{LibraryXmpPage!lib.ht!pages} +<>= +\begin{page}{LibraryXmpPage}{Library} +\beginscroll + +The \spadtype{Library} domain provides a simple way to store Axiom +values in a file. +This domain is similar to \spadtype{KeyedAccessFile} but fewer declarations +are needed and items of different types can be saved together in the same +file. + +\xtc{ +To create a library, you supply a file name. +}{ +\spadpaste{stuff := library "/tmp/Neat.stuff" \bound{stuff}} +} +\xtc{ +Now values can be saved by key in the file. +The keys should be mnemonic, just as the field names are for records. +They can be given either as strings or symbols. +}{ +\spadpaste{stuff.int := 32**2 \free{stuff}\bound{stuffa}} +} +\xtc{ +}{ +\spadpaste{stuff."poly" := x**2 + 1 \free{stuffa}\bound{stuffb}} +} +\xtc{ +}{ +\spadpaste{stuff.str := "Hello" \free{stuffb}\bound{stuffc}} +} +\xtc{ +You obtain +the set of available keys using the \spadfunFrom{keys}{Library} operation. +}{ +\spadpaste{keys stuff \free{stuffa,stuffb,stuffc}\bound{stuffabc}} +} +\xtc{ +You extract values by giving the desired key in this way. +}{ +\spadpaste{stuff.poly \free{stuffb}} +} +\xtc{ +}{ +\spadpaste{stuff("poly") \free{stuffb}} +} +\noOutputXtc{ +When the file is no longer needed, you should remove it from the +file system. +}{ +\spadpaste{)system rm -rf /tmp/Neat.stuff \free{stuff}\bound{rmstuff}} +} + +For more information on related topics, see +\downlink{`File'}{FileXmpPage}\ignore{File}, +\downlink{`TextFile'}{TextFileXmpPage}\ignore{TextFile}, and +\downlink{`KeyedAccessFile'}{KeyedAccessFileXmpPage}\ignore{KeyedAccessFile}. +\showBlurb{Library} +\endscroll +\autobuttons +\end{page} +% + +@ +\section{link.ht} +\subsection{The AXIOM Link to NAG Software} +\begin{itemize} +\item nagLinkIntroPage \ref{nagLinkIntroPage} on +page~\pageref{nagLinkIntroPage} +\item htxl1 \ref{htxl1} on +page~\pageref{htxl1} +\item FoundationLibraryDocPage \ref{FoundationLibraryDocPage} on +page~\pageref{FoundationLibraryDocPage} +\end{itemize} +\label{htxl} +\index{pages!htxl!link.ht} +\index{link.ht!pages!htxl} +\index{htxl!link.ht!pages} +<>= +\begin{page}{htxl}{The AXIOM Link to NAG Software} +\beginscroll +\beginmenu +\item \menumemolink{Introduction to the NAG Library Link}{nagLinkIntroPage} +\menumemolink{Access the Link from HyperDoc}{htxl1} +\menulispmemolink{Browser pages for individual routines}{(|kSearch| "Nag*")} +\menumemolink{NAG Library Documentation}{FoundationLibraryDocPage} +\endmenu +\endscroll +\end{page} + +@ +\subsection{Use of the Link from HyperDoc} +\label{htx11} +\begin{itemize} +\item c02 \ref{c02} on +page~\pageref{c02} +\item c05 \ref{c05} on +page~\pageref{c05} +\item c06 \ref{c06} on +page~\pageref{c06} +\item d01 \ref{d01} on +page~\pageref{d01} +\item d02 \ref{d02} on +page~\pageref{d02} +\item d03 \ref{d03} on +page~\pageref{d03} +\item e01 \ref{e01} on +page~\pageref{e01} +\item e02 \ref{e02} on +page~\pageref{e02} +\item e04 \ref{e04} on +page~\pageref{e04} +\item f01 \ref{f01} on +page~\pageref{f01} +\item f02 \ref{f02} on +page~\pageref{f02} +\item f04 \ref{f04} on +page~\pageref{f04} +\item f07 \ref{f07} on +page~\pageref{f07} +\item s \ref{s} on +page~\pageref{s} +\end{itemize} +\index{pages!htx11!link.ht} +\index{link.ht!pages!htx11} +\index{htx11!link.ht!pages} +<>= +\begin{page}{htxl1}{Use of the Link from HyperDoc} +Click on the chapter of routines that you would like to use. +\beginscroll +\beginmenu +\menumemolink{C02}{c02}\tab{8} Zeros of Polynomials +\menumemolink{C05}{c05}\tab{8} Roots of One or More Transcendental Equations +\menumemolink{C06}{c06}\tab{8} Summation of Series +\menumemolink{D01}{d01}\tab{8} Quadrature +\menumemolink{D02}{d02}\tab{8} Ordinary Differential Equations +\menumemolink{D03}{d03}\tab{8} Partial Differential Equations +\menumemolink{E01}{e01}\tab{8} Interpolation +\menumemolink{E02}{e02}\tab{8} Curve and Surface Fitting +\menumemolink{E04}{e04}\tab{8} Minimizing or Maximizing a Function +\menumemolink{F01}{f01}\tab{8} Matrix Operations, Including Inversion +\menumemolink{F02}{f02}\tab{8} Eigenvalues and Eigenvectors +\menumemolink{F04}{f04}\tab{8} Simultaneous Linear Equations +\menumemolink{F07}{f07}\tab{8} Linear Equations (LAPACK) +\menumemolink{S}{s}\tab{8} Approximations of Special Functions +\endmenu +\endscroll +\end{page} + +@ +\subsection{C02 Zeros of Polynomials} +\label{c02} +\begin{itemize} +\item manpageXXc02 \ref{manpageXXc02} on +page~\pageref{manpageXXc02} +\end{itemize} +\index{pages!c02!link.ht} +\index{link.ht!pages!c02} +\index{c02!link.ht!pages} +<>= +\begin{page}{c02}{C02 Zeros of Polynomials} +\beginscroll +\centerline{What would you like to do?} +\newline +\beginmenu +\item Read +\menuwindowlink{Foundation Library Chapter c02 Manual Page}{manpageXXc02} +\item or +\menulispwindowlink{Browse}{(|kSearch| "NagPolynomialRootsPackage")}\tab{10} through this chapter +\item or use the routines: +\menulispdownlink{C02AFF}{(|c02aff|)}\space{} +\tab{10} All zeros of a complex polynomial +\menulispdownlink{C02AGF}{(|c02agf|)}\space{} +\tab{10} All zeros of a real polynomial +\endmenu +\endscroll +\autobuttons +\end{page} + +@ +\subsection{C05 Roots of One or More Transcendental Equations} +\label{c05} +\begin{itemize} +\item manpageXXc05 \ref{manpageXXc05} on +page~\pageref{manpageXXc05} +\end{itemize} +\index{pages!c05!link.ht} +\index{link.ht!pages!c05} +\index{c05!link.ht!pages} +<>= +\begin{page}{c05}{C05 Roots of One or More Transcendental Equations} +\beginscroll +\centerline{What would you like to do?} +\newline +\beginmenu +\item Read +\menuwindowlink{Foundation Library Chapter c05 Manual Page}{manpageXXc05} +\item or +\menulispwindowlink{Browse}{(|kSearch| "NagRootFindingPackage")}\tab{10} through this chapter +\item or use the routines: +\menulispdownlink{C05ADF}{(|c05adf|)}\space{} +\tab{10} Zero of continuous function in given interval, Bus and Dekker algorithm +\menulispdownlink{C05NBF}{(|c05nbf|)}\space{} +\tab{10} Solution of system of nonlinear equations using function values only +\menulispdownlink{C05PBF}{(|c05pbf|)}\space{} +\tab{10} Solution of system of nonlinear equations using 1st derivatives +\endmenu +\endscroll +\autobuttons +\end{page} + +@ +\subsection{C06 Summation of Series} +\label{c06} +\begin{itemize} +\item manpageXXc06 \ref{manpageXXc06} on +page~\pageref{manpageXXc06} +\end{itemize} +\index{pages!c06!link.ht} +\index{link.ht!pages!c06} +\index{c06!link.ht!pages} +<>= +\begin{page}{c06}{C06 Summation of Series} +\beginscroll +\centerline{What would you like to do?} +\newline +\beginmenu +\item Read +\menuwindowlink{Foundation Library Chapter c06 Manual Page}{manpageXXc06} +\item or +\menulispwindowlink{Browse}{(|kSearch| "NagSeriesSummationPackage")}\tab{10} through this chapter +\item or use the routines: +\menulispdownlink{C06EAF}{(|c06eaf|)}\space{} +\tab{10} Single 1-D real discrete Fourier transform, no extra workspace +\menulispdownlink{C06EBF}{(|c06ebf|)}\space{} +\tab{10} Single 1-D Hermitian discrete Fourier transform, no extra workspace +\menulispdownlink{C06ECF}{(|c06ecf|)}\space{} +\tab{10} Single 1-D complex discrete Fourier transform, no extra workspace +\menulispdownlink{C06EKF}{(|c06ekf|)}\space{} +\tab{10} Circular convolution or correlation of two real vectors, no extra +workspace +\menulispdownlink{C06FPF}{(|c06fpf|)}\space{} +\tab{10} Multiple 1-D real discrete Fourier transforms +\menulispdownlink{C06FQF}{(|c06fqf|)}\space{} +\tab{10} Multiple 1-D Hermitian discrete Fourier transforms +\menulispdownlink{C06FRF}{(|c06frf|)}\space{} +\tab{10} Multiple 1-D complex discrete Fourier transforms +\menulispdownlink{C06FUF}{(|c06fuf|)}\space{} +\tab{10} 2-D complex discrete Fourier transforms +\menulispdownlink{C06GBF}{(|c06gbf|)}\space{} +\tab{10} Complex conjugate of Hermitian sequence +\menulispdownlink{C06GCF}{(|c06gcf|)}\space{} +\tab{10} Complex conjugate of complex sequence +\menulispdownlink{C06GQF}{(|c06gqf|)}\space{} +\tab{10} Complex conjugate of multiple Hermitian sequences +\menulispdownlink{C06GSF}{(|c06gsf|)}\space{} +\tab{10} Convert Hermitian sequences to general complex sequences +\endmenu +\endscroll +\autobuttons +\end{page} + +@ +\subsection{D01 Quadrature} +\label{d01} +\begin{itemize} +\item manpageXXd01 \ref{manpageXXd01} on +page~\pageref{manpageXXd01} +\end{itemize} +\index{pages!d01!link.ht} +\index{link.ht!pages!d01} +\index{d01!link.ht!pages} +<>= +\begin{page}{d01}{D01 Quadrature} +\beginscroll +\centerline{What would you like to do?} +\newline +\beginmenu +\item Read +\menuwindowlink{Foundation Library Chapter d01 Manual Page}{manpageXXd01} +\item or +\menulispwindowlink{Browse}{(|kSearch| "NagIntegrationPackage")}\tab{10} through this chapter +\item or use the routines: +\menulispdownlink{D01AJF}{(|d01ajf|)}\space{} +\tab{10} 1-D quadrature, adaptive, finite interval, strategy due to Plessens +and de Doncker, allowing for badly-behaved integrands +\menulispdownlink{D01AKF}{(|d01akf|)}\space{} +\tab{10} 1-D quadrature, adaptive, finite interval, method suitable for +oscillating functions +\menulispdownlink{D01ALF}{(|d01alf|)}\space{} +\tab{10} 1-D quadrature, adaptive, finite interval, allowing for +singularities at user specified points +\menulispdownlink{D01AMF}{(|d01amf|)}\space{} +\tab{10} 1-D quadrature, adaptive, infinite or semi-finite interval +\menulispdownlink{D01ANF}{(|d01anf|)}\space{} +\tab{10} 1-D quadrature, adaptive, finite interval, weight function + cos(\omega x) or sin(\omega x) +\menulispdownlink{D01APF}{(|d01apf|)}\space{} +\tab{10} 1-D quadrature, adaptive, finite interval, weight function +with end point singularities of algebraico-logarithmic type +\menulispdownlink{D01AQF}{(|d01aqf|)}\space{} +\tab{10} 1-D quadrature, adaptive, finite interval, weight function +1/(x-c), Cauchy principle value (Hilbert transform) +\menulispdownlink{D01ASF}{(|d01asf|)}\space{} +\tab{10} 1-D quadrature, adaptive, semi-infinite interval, weight function +cos(\omega x) or sin(\omega x) +\menulispdownlink{D01BBF}{(|d01bbf|)}\space{} +\tab{10} Pre-computed weights and abscissae for Gaussian quadrature rules, +restricted choice of rule +\menulispdownlink{D01FCF}{(|d01fcf|)}\space{} +\tab{10} Multi-dimensional adaptive quadrature over hyper-rectangle +\menulispdownlink{D01GAF}{(|d01gaf|)}\space{} +\tab{10} 1-D quadrature, integration of function defined by data values, +Gill-Miller method +\menulispdownlink{D01GBF}{(|d01gbf|)}\space{} +\tab{10} Multi-dimensional quadrature over hyper-rectangle, Monte Carlo method +\endmenu +\endscroll +\autobuttons +\end{page} + +@ +\subsection{D02 Ordinary Differential Equations} +\label{d02} +\begin{itemize} +\item manpageXXd02 \ref{manpageXXd02} on +page~\pageref{manpageXXd02} +\end{itemize} +\index{pages!d02!link.ht} +\index{link.ht!pages!d02} +\index{d02!link.ht!pages} +<>= +\begin{page}{d02}{D02 Ordinary Differential Equations} +\beginscroll +\centerline{What would you like to do?} +\newline +\beginmenu +\item Read +\menuwindowlink{Foundation Library Chapter d02 Manual Page}{manpageXXd02} +\item or +\menulispwindowlink{Browse}{(|kSearch| "NagOrdinaryDifferentialEquationsPackage")}\tab{10} through this chapter +\item or use the routines: +\menulispdownlink{D02BBF}{(|d02bbf|)}\space{} +\tab{10} ODEs, IVP, Runge-Kutta-Merson method, over a range, +intermediate output +\menulispdownlink{D02BHF}{(|d02bhf|)}\space{} +\tab{10} ODEs, IVP, Runge-Kutta-Merson method, until function of +solution is zero +\menulispdownlink{D02CJF}{(|d02cjf|)}\space{} +\tab{10} ODEs, IVP, Adams method, until function of solution is zero, +intermediate output +\menulispdownlink{D02EJF}{(|d02ejf|)}\space{} +\tab{10} ODEs, stiff IVP, BDF method, until function of solution is zero, +intermediate output +\menulispdownlink{D02GAF}{(|d02gaf|)}\space{} +\tab{10} ODEs, boundary value problem, finite difference technique with +deferred correction, simple nonlinear problem +\menulispdownlink{D02GBF}{(|d02gbf|)}\space{} +\tab{10} ODEs, boundary value problem, finite difference technique with +deferred correction, general nonlinear problem +\menulispdownlink{D02KEF}{(|d02kef|)}\space{} +\tab{10} 2nd order Sturm-Liouville problem, regular/singular system, +finite/infinite range, eigenvalue and eigenfunction, user-specified +break-points +\menulispdownlink{D02RAF}{(|d02raf|)}\space{} +\tab{10} ODEs, general nonlinear boundary value problem, finite difference +technique with deferred correction, continuation facility +\endmenu +\endscroll +\autobuttons +\end{page} + +@ +\subsection{D03 Partial Differential Equations} +\label{d03} +\begin{itemize} +\item manpageXXd03 \ref{manpageXXd03} on +page~\pageref{manpageXXd03} +\end{itemize} +\index{pages!d03!link.ht} +\index{link.ht!pages!d03} +\index{d03!link.ht!pages} +<>= +\begin{page}{d03}{D03 Partial Differential Equations} +\beginscroll +\centerline{What would you like to do?} +\newline +\beginmenu +\item Read +\menuwindowlink{Foundation Library Chapter d03 Manual Page}{manpageXXd03} +\item or +\menulispwindowlink{Browse}{(|kSearch| "NagPartialDifferentialEquationsPackage")}\tab{10} through this chapter +\item or use the routines: +\menulispdownlink{D03EDF}{(|d03edf|)}\space{} +\tab{10} Elliptic PDE, solution of finite difference equations by a multigrid +technique +\menulispdownlink{D03EEF}{(|d03eef|)}\space{} +\tab{10} Discretize a 2nd order elliptic PDE on a rectangle +\menulispdownlink{D03FAF}{(|d03faf|)}\space{} +\tab{10} Elliptic PDE, Helmholtz equation, 3-D Cartesian co-ordinates +\endmenu +\endscroll +\autobuttons +\end{page} + +@ +\subsection{E01 Interpolation} +\label{e01} +\begin{itemize} +\item manpageXXe01 \ref{manpageXXe01} on +page~\pageref{manpageXXe01} +\end{itemize} +\index{pages!e01!link.ht} +\index{link.ht!pages!e01} +\index{e01!link.ht!pages} +<>= +\begin{page}{e01}{E01 Interpolation} +\beginscroll +\centerline{What would you like to do?} +\beginmenu +\item Read +\menuwindowlink{Foundation Library Chapter e01 Manual Page}{manpageXXe01} +\item or +\menulispwindowlink{Browse}{(|kSearch| "NagInterpolationPackage")}\tab{10} through this chapter +\item or use the routines: +\menulispdownlink{E01BAF}{(|e01baf|)}\space{} +\tab{10} Interpolating functions, cubic spline interpolant, one variable +\menulispdownlink{E01BEF}{(|e01bef|)}\space{} +\tab{10} Interpolating functions, monotonicity-preserving, piecewise +cubic Hermite, one variable +\menulispdownlink{E01BFF}{(|e01bff|)}\space{} +\tab{10} Interpolated values, interpolant computed by E01BEF, function +only, one variable +\menulispdownlink{E01BGF}{(|e01bgf|)}\space{} +\tab{10} Interpolated values, interpolant computed by E01BEF, function +and 1st derivative, one variable +\menulispdownlink{E01BHF}{(|e01bhf|)}\space{} +\tab{10} Interpolated values, interpolant computed by E01BEF, definite +integral, one variable +\menulispdownlink{E01DAF}{(|e01daf|)}\space{} +\tab{10} Interpolating functions, fitting bicubic spline, data on a +rectangular grid +\menulispdownlink{E01SAF}{(|e01saf|)}\space{} +\tab{10} Interpolating functions, method of Renka and Cline, two variables +\menulispdownlink{E01SEF}{(|e01sef|)}\space{} +\tab{10} Interpolating functions, modified Shepherd's method, two variables +\endmenu +\endscroll +\autobuttons +\end{page} + +@ +\subsection{E02 Curve and Surface Fitting} +\label{e02} +\begin{itemize} +\item manpageXXe02 \ref{manpageXXe02} on +page~\pageref{manpageXXe02} +\end{itemize} +\index{pages!e02!link.ht} +\index{link.ht!pages!e02} +\index{e02!link.ht!pages} +<>= +\begin{page}{e02}{E02 Curve and Surface Fitting} +\beginscroll +\centerline{What would you like to do?} +\beginmenu +\item Read +\menuwindowlink{Foundation Library Chapter e02 Manual Page}{manpageXXe02} +\item or +\menulispwindowlink{Browse}{(|kSearch| "NagFittingPackage")}\tab{10} through this chapter +\item or use the routines: +\menulispdownlink{E02ADF}{(|e02adf|)}\space{} +\tab{10} Least-squares curve fit, by polynomials, arbitrary data points +\menulispdownlink{E02AEF}{(|e02aef|)}\space{} +\tab{10} Evaluation of fitted polynomial in one variable from Chebyshev series +form (simplified parameter list) +\menulispdownlink{E02AGF}{(|e02agf|)}\space{} +\tab{10} Least-squares polynomial fit, values and derivatives may be +constrained, arbitrary data points +\menulispdownlink{E02AHF}{(|e02ahf|)}\space{} +\tab{10} Derivative of fitted polynomial in Chebyshev series form +\menulispdownlink{E02AJF}{(|e02ajf|)}\space{} +\tab{10} Integral of fitted polynomial in Chebyshev series form +\menulispdownlink{E02AKF}{(|e02akf|)}\space{} +\tab{10} Evaluation of fitted polynomial in one variable, from Chebyshev +series form +\menulispdownlink{E02BAF}{(|e02baf|)}\space{} +\tab{10} Least-squares curve cubic spline fit (including interpolation) +\menulispdownlink{E02BBF}{(|e02bbf|)}\space{} +\tab{10} Evaluation of fitted cubic spline, function only +\menulispdownlink{E02BCF}{(|e02bcf|)}\space{} +\tab{10} Evaluation of fitted cubic spline, function and derivatives +\menulispdownlink{E02BDF}{(|e02bdf|)}\space{} +\tab{10} Evaluation of fitted cubic spline, definite integral +\menulispdownlink{E02BEF}{(|e02bef|)}\space{} +\tab{10} Least-squares curve cubic spline fit, automatic knot placement +\menulispdownlink{E02DAF}{(|e02daf|)}\space{} +\tab{10} Least-squares surface fit, bicubic splines +\menulispdownlink{E02DCF}{(|e02dcf|)}\space{} +\tab{10} Least-squares surface fit by bicubic splines with automatic knot +placement, data on a rectangular grid +\menulispdownlink{E02DDF}{(|e02ddf|)}\space{} +\tab{10} Least-squares surface fit by bicubic splines with automatic knot +placement, scattered data +\menulispdownlink{E02DEF}{(|e02def|)}\space{} +\tab{10} Evaluation of a fitted bicubic spline at a vector of points +\menulispdownlink{E02DFF}{(|e02dff|)}\space{} +\tab{10} Evaluation of a fitted bicubic spline at a mesh of points +\menulispdownlink{E02GAF}{(|e02gaf|)}\space{} +\tab{10} \htbitmap{l1}-approximation by general linear function +\menulispdownlink{E02ZAF}{(|e02zaf|)}\space{} +\tab{10} Sort 2-D sata into panels for fitting bicubic splines +\endmenu +\endscroll +\autobuttons +\end{page} + +@ +\subsection{E04 Minimizing or Maximizing a Function} +\label{e04} +\begin{itemize} +\item manpageXXe04 \ref{manpageXXe04} on +page~\pageref{manpageXXe04} +\end{itemize} +\index{pages!e04!link.ht} +\index{link.ht!pages!e04} +\index{e04!link.ht!pages} +<>= +\begin{page}{e04}{E04 Minimizing or Maximizing a Function} +\beginscroll +\centerline{What would you like to do?} +\newline +\beginmenu +\item Read +\menuwindowlink{Foundation Library Chapter e04 Manual Page}{manpageXXe04} +\item or +\menulispwindowlink{Browse}{(|kSearch| "NagOptimisationPackage")}\tab{10} through this chapter +\item or use the routines: +\menulispdownlink{E04DGF}{(|e04dgf|)}\space{} +\tab{10} Unconstrained minimum, pre-conditioned conjugate gradient algorithm, +function of several variables using 1st derivatives +\menulispdownlink{E04FDF}{(|e04fdf|)}\space{} +\tab{10} Unconstrained minimum of a sum of squares, combined Gauss-Newton +and modified Newton algorithm using function values only +\menulispdownlink{E04GCF}{(|e04gcf|)}\space{} +\tab{10} Unconstrained minimum, of a sum of squares, combined Gauss-Newton +and modified Newton algorithm using 1st derivatives +\menulispdownlink{E04JAF}{(|e04jaf|)}\space{} +\tab{10} Minimum, function of several variables, quasi-Newton algorithm, +simple bounds, using function values only +\menulispdownlink{E04MBF}{(|e04mbf|)}\space{} +\tab{10} Linear programming problem +\menulispdownlink{E04NAF}{(|e04naf|)}\space{} +\tab{10} Quadratic programming problem +\menulispdownlink{E04UCF}{(|e04ucf|)}\space{} +\tab{10} Minimum, function of several variables, sequential QP method, +nonlinear constraints, using function values and optionally 1st derivatives +\menulispdownlink{E04YCF}{(|e04ycf|)}\space{} +\tab{10} Covariance matrix for non-linear least-squares problem +\endmenu +\endscroll +\autobuttons +\end{page} + +@ +\subsection{F01 Matrix Operations - Including Inversion} +\label{f01} +\begin{itemize} +\item manpageXXf \ref{manpageXXf} on +page~\pageref{manpageXXf} +\item manpageXXf01 \ref{manpageXXf01} on +page~\pageref{manpageXXf01} +\end{itemize} +\index{pages!f01!link.ht} +\index{link.ht!pages!f01} +\index{f01!link.ht!pages} +<>= +\begin{page}{f01}{F01 Matrix Operations - Including Inversion} +\beginscroll +\centerline{What would you like to do?} +\beginmenu +\item Read +\menuwindowlink{Foundation Library Chapter f Manual Page}{manpageXXf} +\menuwindowlink{Foundation Library Chapter f01 Manual Page}{manpageXXf01} +\item or +\menulispwindowlink{Browse}{(|kSearch| "NagMatrixOperationsPackage")}\tab{10} through this chapter +\item or use the routines: +\menulispdownlink{F01BRF}{(|f01brf|)}\space{} +\tab{10} {\it LU} factorization of real sparse matrix +\menulispdownlink{F01BSF}{(|f01bsf|)}\space{} +\tab{10} {\it LU} factorization of real sparse matrix with known sparsity +pattern +\menulispdownlink{F01MAF}{(|f01maf|)}\space{} +\tab{10} \htbitmap{llt} factorization of real sparse +symmetric positive-definite matrix +\menulispdownlink{F01MCF}{(|f01mcf|)}\space{} +\tab{10} \htbitmap{ldlt} factorization of real +symmetric positive-definite variable-bandwith matrix +\menulispdownlink{F01QCF}{(|f01qcf|)}\space{} +\tab{10} {\it QR} factorization of real {\it m} by {\it n} matrix +(m \htbitmap{great=} n) +\menulispdownlink{F01QDF}{(|f01qdf|)}\space{} +\tab{10} Operations with orthogonal matrices, compute {\it QB} or +\htbitmap{f01qdf} after factorization by F01QCF or F01QFF +\menulispdownlink{F01QEF}{(|f01qef|)}\space{} +\tab{10} Operations with orthogonal matrices, form columns of {\it Q} +after factorization by F01QCF or F01QFF +\menulispdownlink{F01RCF}{(|f01rcf|)}\space{} +\tab{10} {\it QR} factorization of complex {\it m} by {\it n} matrix +(m \htbitmap{great=} n) +\menulispdownlink{F01RDF}{(|f01rdf|)}\space{} +\tab{10} Operations with unitary matrices, compute {\it QB} or +\htbitmap{f01rdf} after factorization by F01RCF +\menulispdownlink{F01REF}{(|f01ref|)}\space{} +\tab{10} Operations with unitary matrices, form columns of {\it Q} +after factorization by F01RCF +\endmenu +\endscroll +\autobuttons +\end{page} + +@ +\subsection{F02 Eigenvalues and Eigenvectors} +\label{f02} +\begin{itemize} +\item manpageXXf \ref{manpageXXf} on +page~\pageref{manpageXXf} +\item manpageXXf02 \ref{manpageXXf02} on +page~\pageref{manpageXXf02} +\end{itemize} +\index{pages!f02!link.ht} +\index{link.ht!pages!f02} +\index{f02!link.ht!pages} +<>= +\begin{page}{f02}{F02 Eigenvalues and Eigenvectors} +\beginscroll +\centerline{What would you like to do?} +\beginmenu +\item Read +\menuwindowlink{Foundation Library Chapter f Manual Page}{manpageXXf} +\menuwindowlink{Foundation Library Chapter f02 Manual Page}{manpageXXf02} +\item or +\menulispwindowlink{Browse}{(|kSearch| "NagEigenPackage")}\tab{10} through this chapter +\item or use the routines: +\menulispdownlink{F02AAF}{(|f02aaf|)}\space{} +\tab{10} All eigenvalues of real symmetric matrix (Black box) +\menulispdownlink{F02ABF}{(|f02abf|)}\space{} +\tab{10} All eigenvalues and eigenvectors of real symmetric matrix (Black box) +\menulispdownlink{F02ADF}{(|f02adf|)}\space{} +\tab{10} All eigenvalues of generalized real eigenproblem of the form +Ax = \lambda Bx where A and B are symmetric and B is positive definite +\menulispdownlink{F02AEF}{(|f02aef|)}\space{} +\tab{10} All eigenvalues and eigenvectors of generalized real eigenproblem +of the form Ax = \lambda Bx where A and B are symmetric and B is positive +definite +\menulispdownlink{F02AFF}{(|f02aff|)}\space{} +\tab{10} All eigenvalues of real matrix (Black box) +\menulispdownlink{F02AGF}{(|f02agf|)}\space{} +\tab{10} All eigenvalues and eigenvectors of real matrix (Black box) +\menulispdownlink{F02AJF}{(|f02ajf|)}\space{} +\tab{10} All eigenvalues of complex matrix (Black box) +\menulispdownlink{F02AKF}{(|f02akf|)}\space{} +\tab{10} All eigenvalues and eigenvectors of complex matrix (Black box) +\menulispdownlink{F02AWF}{(|f02awf|)}\space{} +\tab{10} All eigenvalues of complex Hermitian matrix (Black box) +\menulispdownlink{F02AXF}{(|f02axf|)}\space{} +\tab{10} All eigenvalues and eigenvectors of complex Hermitian +matrix (Black box) +\menulispdownlink{F02BBF}{(|f02bbf|)}\space{} +\tab{10} Selected eigenvalues and eigenvectors of real symmetric +matrix (Black box) +\menulispdownlink{F02BJF}{(|f02bjf|)}\space{} +\tab{10} All eigenvalues and optionally eigenvectors of generalized +eigenproblem by {\it QZ} algorithm, real matrices (Black box) +\menulispdownlink{F02FJF}{(|f02fjf|)}\space{} +\tab{10} Selected eigenvalues and eigenvectors of sparse symmetric +eigenproblem +\menulispdownlink{F02WEF}{(|f02wef|)}\space{} +\tab{10} SVD of real matrix +\menulispdownlink{F02XEF}{(|f02xef|)}\space{} +\tab{10} SVD of complex matrix +\endmenu +\endscroll +\autobuttons +\end{page} + +@ +\subsection{F04 Simultaneous Linear Equations} +\label{f04} +\begin{itemize} +\item manpageXXf \ref{manpageXXf} on +page~\pageref{manpageXXf} +\item manpageXXf04 \ref{manpageXXf04} on +page~\pageref{manpageXXf04} +\end{itemize} +\index{pages!f04!link.ht} +\index{link.ht!pages!f04} +\index{f04!link.ht!pages} +<>= +\begin{page}{f04}{F04 Simultaneous Linear Equations} +\beginscroll +\centerline{What would you like to do?} +\beginmenu +\item Read +\menuwindowlink{Foundation Library Chapter f Manual Page}{manpageXXf} +\menuwindowlink{Foundation Library Chapter f04 Manual Page}{manpageXXf04} +\item or +\menulispwindowlink{Browse}{(|kSearch| "NagLinearEquationSolvingPackage")}\tab{10} through this chapter +\item or use the routines: +\menulispdownlink{F04ADF}{(|f04adf|)}\space{} +\tab{10} Solution of complex simultaneous linear equations, with multiple +right-hand sides (Black box) +\menulispdownlink{F04ARF}{(|f04arf|)}\space{} +\tab{10} Solution of real simultaneous linear equations, one right-hand side +(Black box) +\menulispdownlink{F04ASF}{(|f04asf|)}\space{} +\tab{10} Solution of real symmetric positive-definite simultaneous linear +equations, one right-hand side using iterative refinement (Black box) +\menulispdownlink{F04ATF}{(|f04atf|)}\space{} +\tab{10} Solution of real simultaneous linear equations, one right-hand side +using iterative refinement (Black box) +\menulispdownlink{F04AXF}{(|f04axf|)}\space{} +\tab{10} Approximate solution of real sparse simultaneous linear equations +(coefficient matrix already factorized by F01BRF or F01BSF) +\menulispdownlink{F04FAF}{(|f04faf|)}\space{} +\tab{10} Solution of real symmetric positive-definite tridiagonal +simultaneous linear equations, one right-hand side (Black box) +\menulispdownlink{F04JGF}{(|f04jgf|)}\space{} +\tab{10} Least-squares (if rank = n) or minimal least-squares (if rank < n) +solution of m real equations in n unknowns, rank \htbitmap{less=} n, + m \htbitmap{great=} n +\menulispdownlink{F04MAF}{(|f04maf|)}\space{} +\tab{10} Real sparse symmetric positive-definite simultaneous linear +equations(coefficient matrix already factorized) +\menulispdownlink{F04MBF}{(|f04mbf|)}\space{} +\tab{10} Real sparse symmetric simultaneous linear equations +\menulispdownlink{F04MCF}{(|f04mcf|)}\space{} +\tab{10} Approximate solution of real symmetric positive-definite +variable-bandwidth simultaneous linear equations (coefficient matrix +already factorized) +\menulispdownlink{F04QAF}{(|f04qaf|)}\space{} +\tab{10} Sparse linear least-squares problem, {\it m} real equations +in {\it n} unknowns +\endmenu +\endscroll +\autobuttons +\end{page} + +@ +\subsection{F07 Linear Equations (LAPACK)} +\label{f07} +\begin{itemize} +\item manpageXXf \ref{manpageXXf} on +page~\pageref{manpageXXf} +\item manpageXXf07 \ref{manpageXXf07} on +page~\pageref{manpageXXf07} +\end{itemize} +\index{pages!f07!link.ht} +\index{link.ht!pages!f07} +\index{f07!link.ht!pages} +<>= +\begin{page}{f07}{F07 Linear Equations (LAPACK)} +\beginscroll +\centerline{What would you like to do?} +\beginmenu +\item Read +\menuwindowlink{Foundation Library Chapter f Manual Page}{manpageXXf} +\menuwindowlink{Foundation Library Chapter f07 Manual Page}{manpageXXf07} +\item or +\menulispwindowlink{Browse}{(|kSearch| "NagLapack")}\tab{10} through this chapter +\item or use the routines: +\menulispdownlink{F07ADF}{(|f07adf|)}\space{} +\tab{10} (DGETRF) {\it LU} factorization of real {\it m} by {\it n} matrix +\menulispdownlink{F07AEF}{(|f07aef|)}\space{} +\tab{10} (DGETRS) Solution of real system of linear equations, multiple +right hand sides, matrix factorized by F07ADF +\menulispdownlink{F07FDF}{(|f07fdf|)}\space{} +\tab{10} (DPOTRF) Cholesky factorization of real symmetric positive-definite +matrix +\menulispdownlink{F07FEF}{(|f07fef|)}\space{} +\tab{10} (DPOTRS) Solution of real symmetric positive-definite system of +linear equations, multiple right-hand sides, matrix already factorized by +F07FDF +\endmenu +\endscroll +\autobuttons +\end{page} + + +@ +\subsection{S \space{2} Approximations of Special Functions} +\label{s} +\begin{itemize} +\item manpageXXs \ref{manpageXXs} on +page~\pageref{manpageXXs} +\end{itemize} +\index{pages!s!link.ht} +\index{link.ht!pages!s} +\index{s!link.ht!pages} +<>= +\begin{page}{s}{S \space{2} Approximations of Special Functions} +\beginscroll +\centerline{What would you like to do?} +\beginmenu +\item Read +\menuwindowlink{Foundation Library Chapter s Manual Page}{manpageXXs} +\item or +\menulispwindowlink{Browse}{(|kSearch| "NagSpecialFunctionsPackage")}\tab{10} through this chapter +\item or use the routines: +\menulispdownlink{S01EAF}{(|s01eaf|)}\space{} +\tab{10} Complex exponential {\em exp(z)} +\menulispdownlink{S13AAF}{(|s13aaf|)}\space{} +\tab{10} Exponential integral \htbitmap{s13aaf2} +\menulispdownlink{S13ACF}{(|s13acf|)}\space{} +\tab{10} Cosine integral {\em Ci(x)} +\menulispdownlink{S13ADF}{(|s13adf|)}\space{} +\tab{10} Sine integral {\em Si(x)} +\menulispdownlink{S14AAF}{(|s14aaf|)}\space{} +\tab{10} Gamma function \Gamma +\menulispdownlink{S14ABF}{(|s14abf|)}\space{} +\tab{10} Log Gamma function {\em ln \Gamma} +\menulispdownlink{S14BAF}{(|s14baf|)}\space{} +\tab{10} Incomplete gamma functions P(a,x) and Q(a,x) +\menulispdownlink{S15ADF}{(|s15adf|)}\space{} +\tab{10} Complement of error function {\em erfc x } +\menulispdownlink{S15AEF}{(|s15aef|)}\space{} +\tab{10} Error function {\em erf x} +\menulispdownlink{S17ACF}{(|s17acf|)}\space{} +\tab{10} Bessel function \space{1} \htbitmap{s17acf} +\menulispdownlink{S17ADF}{(|s17adf|)}\space{} +\tab{10} Bessel function \space{1} \htbitmap{s17adf} +\menulispdownlink{S17AEF}{(|s17aef|)}\space{} +\tab{10} Bessel function \space{1} \htbitmap{s17aef1} +\menulispdownlink{S17AFF}{(|s17aff|)}\space{} +\tab{10} Bessel function \space{1} \htbitmap{s17aff1} +\menulispdownlink{S17AGF}{(|s17agf|)} +\tab{10} Airy function {\em Ai(x)} +\menulispdownlink{S17AHF}{(|s17ahf|)} +\tab{10} Airy function {\em Bi(x)} +\menulispdownlink{S17AJF}{(|s17ajf|)} +\tab{10} Airy function {\em Ai'(x)} +\menulispdownlink{S17AKF}{(|s17akf|)} +\tab{10} Airy function {\em Bi'(x)} +\menulispdownlink{S17DCF}{(|s17dcf|)} +\tab{10} Bessel function \htbitmap{s17dcf}, real a \space{1} +\htbitmap{great=} 0, complex z, v = 0,1,2,... +\menulispdownlink{S17DEF}{(|s17def|)} +\tab{10} Bessel function \htbitmap{s17def}, real a \space{1} +\htbitmap{great=} 0, complex z, v = 0,1,2,... +\menulispdownlink{S17DGF}{(|s17dgf|)} +\tab{10} Airy function {\em Ai(z)} and {\em Ai'(z)}, complex z +\menulispdownlink{S17DHF}{(|s17dhf|)} +\tab{10} Airy function {\em Bi(z)} and {\em Bi'(z)}, complex z +\menulispdownlink{S17DLF}{(|s17dlf|)} +\tab{10} Hankel function \vspace{-32} \htbitmap{s17dlf} +\vspace{-37}, j = 1,2, real a \space{1} \htbitmap{great=} 0, +complex z, v = 0,1,2,... \newline +\menulispdownlink{S18ACF}{(|s18acf|)} +\tab{10} Modified Bessel function \space{1} \htbitmap{s18acf1} +\menulispdownlink{S18ADF}{(|s18adf|)} +\tab{10} Modified Bessel function \space{1} \htbitmap{s18adf1} +\menulispdownlink{S18AEF}{(|s18aef|)} +\tab{10} Modified Bessel function \space{1} \htbitmap{s18aef1} +\menulispdownlink{S18AFF}{(|s18aff|)} +\tab{10} Modified Bessel function \space{1} \htbitmap{s18aff1} +\menulispdownlink{S18DCF}{(|s18dcf|)} +\tab{10} Modified bessel function \htbitmap{s18dcf}, real a \space{1} +\htbitmap{great=} 0, complex z, v = 0,1,2,... +\menulispdownlink{S18DEF}{(|s18def|)} +\tab{10} Modified bessel function \htbitmap{s18def}, real a \space{1} +\htbitmap{great=} 0, complex z, v = 0,1,2,... +\menulispdownlink{S19AAF}{(|s19aaf|)} +\tab{10} Kelvin function {\em ber x} +\menulispdownlink{S19ABF}{(|s19abf|)} +\tab{10} Kelvin function {\em bei x} +\menulispdownlink{S19ACF}{(|s19acf|)} +\tab{10} Kelvin function {\em ker x} +\menulispdownlink{S19ADF}{(|s19adf|)} +\tab{10} Kelvin function {\em kei x} +\menulispdownlink{S20ACF}{(|s20acf|)} +\tab{10} Fresnel integral {\em S(x)} +\menulispdownlink{S20ADF}{(|s20adf|)} +\tab{10} Fresnel integral {\em C(x)} +\menulispdownlink{S21BAF}{(|s21baf|)} +\tab{10} Degenerate symmetrised elliptic integral of 1st kind +\space{1} \htbitmap{s21baf1} +\menulispdownlink{S21BBF}{(|s21bbf|)} +\tab{10} Symmetrised elliptic integral of 1st kind \space{1} +\vspace{-28} \htbitmap{s21bbf1} \vspace{-40} +\menulispdownlink{S21BCF}{(|s21bcf|)} +\tab{10} Symmetrised elliptic integral of 2nd kind \space{1} +\vspace{-28} \htbitmap{s21bcf1} \vspace{-40} +\menulispdownlink{S21BDF}{(|s21bdf|)} +\tab{10} Symmetrised elliptic integral of 3rd kind \space{1} +\vspace{-26} \htbitmap{s21bdf1} \vspace{-40} +\endmenu +\endscroll +\autobuttons +\end{page} + +@ +\section{list.ht} +<>= +\newcommand{\ListXmpTitle}{List} +\newcommand{\ListXmpNumber}{9.47} + +@ +\subsection{List} +\label{ListXmpPage} +\begin{itemize} +\item ugxListCreatePage \ref{ugxListCreatePage} on +page~pageref{ugxListCreatePage} +\item ugxListAccessPage \ref{ugxListAccessPage} on +page~pageref{ugxListAccessPage} +\item ugxListChangePage \ref{ugxListChangePage} on +page~pageref{ugxListChangePage} +\item ugxListOtherPage \ref{ugxListOtherPage} on +page~pageref{ugxListOtherPage} +\item ugxListDotPage \ref{ugxListDotPage} on +page~pageref{ugxListDotPage} +\end{itemize} +\index{pages!ListXmpPage!list.ht} +\index{list.ht!pages!ListXmpPage} +\index{ListXmpPage!list.ht!pages} +<>= +\begin{page}{ListXmpPage}{List} +\beginscroll +A \spadgloss{list} is a finite collection of elements in a specified +order that can contain duplicates. +A list is a convenient structure to work with because it is easy +to add or remove elements and the length need not be constant. +There are many different kinds of lists in Axiom, but the +default types (and those used most often) are created by the +\spadtype{List} constructor. +For example, there are objects of type \spadtype{List Integer}, +\spadtype{List Float} and \spadtype{List Polynomial Fraction Integer}. +Indeed, you can even have \spadtype{List List List Boolean} +(that is, lists of lists of lists of Boolean values). +You can have lists of any type of Axiom object. + +\beginmenu + \menudownlink{{9.47.1. Creating Lists}}{ugxListCreatePage} + \menudownlink{{9.47.2. Accessing List Elements}}{ugxListAccessPage} + \menudownlink{{9.47.3. Changing List Elements}}{ugxListChangePage} + \menudownlink{{9.47.4. Other Functions}}{ugxListOtherPage} + \menudownlink{{9.47.5. Dot, Dot}}{ugxListDotPage} +\endmenu +\endscroll +\autobuttons +\end{page} + +@ +<>= +\newcommand{\ugxListCreateTitle}{Creating Lists} +\newcommand{\ugxListCreateNumber}{9.47.1.} + +@ +\subsection{Creating Lists} +\label{ugxListCreatePage} +\index{pages!ugxListCreatePage!list.ht} +\index{list.ht!pages!ugxListCreatePage} +\index{ugxListCreatePage!list.ht!pages} +<>= +\begin{page}{ugxListCreatePage}{Creating Lists} +\beginscroll + +The easiest way to create a list with, for example, the elements +\spad{2, 4, 5, 6} is to enclose the elements with square +brackets and separate the elements with commas. +\xtc{ +The spaces after the commas are optional, but they do improve the +readability. +}{ +\spadpaste{[2, 4, 5, 6]} +} +\xtc{ +To create a list with the single element \spad{1}, you can use +either \spad{[1]} or the operation \spadfunFrom{list}{List}. +}{ +\spadpaste{[1]} +} +\xtc{ +}{ +\spadpaste{list(1)} +} +\xtc{ +Once created, two lists \spad{k} and \spad{m} can be +concatenated by issuing \spad{append(k,m)}. +\spadfunFrom{append}{List} does {\it not} physically join the lists, +but rather produces a new list with the elements coming from the two +arguments. +}{ +\spadpaste{append([1,2,3],[5,6,7])} +} +\xtc{ +Use \spadfunFrom{cons}{List} to append an element onto the front of a +list. +}{ +\spadpaste{cons(10,[9,8,7])} +} + +\endscroll +\autobuttons +\end{page} + +@ +<>= +\newcommand{\ugxListAccessTitle}{Accessing List Elements} +\newcommand{\ugxListAccessNumber}{9.47.2.} + +@ +\subsection{Accessing List Elements} +\label{ugxListAccessPage} +\index{pages!ugxListAccessPage!list.ht} +\index{list.ht!pages!ugxListAccessPage} +\index{ugxListAccessPage!list.ht!pages} +<>= +\begin{page}{ugxListAccessPage}{Accessing List Elements} +\beginscroll + +\labelSpace{4pc} +\xtc{ +To determine whether a list has any elements, use the operation +\spadfunFrom{empty?}{List}. +}{ +\spadpaste{empty? [x+1]} +} +\xtc{ +Alternatively, equality with the list constant \spadfunFrom{nil}{List} can +be tested. +}{ +\spadpaste{([] = nil)@Boolean} +} + +\xtc{ +We'll use this in some of the following examples. +}{ +\spadpaste{k := [4,3,7,3,8,5,9,2] \bound{k}} +} +\xtc{ +Each of the next four expressions extracts the \spadfunFrom{first}{List} +element of \spad{k}. +}{ +\spadpaste{first k \free{k}} +} +\xtc{ +}{ +\spadpaste{k.first \free{k}} +} +\xtc{ +}{ +\spadpaste{k.1 \free{k}} +} +\xtc{ +}{ +\spadpaste{k(1) \free{k}} +} +The last two forms generalize to \spad{k.i} and \spad{k(i)}, +respectively, where +\texht{$ 1 \leq i \leq n$}{\spad{1 <= i <= n}} +and +\spad{n} equals the length of \spad{k}. +\xtc{ +This length is calculated by \spadopFrom{\#}{List}. +}{ +\spadpaste{n := \#k \free{k}} +} +Performing an operation such as \spad{k.i} is sometimes +referred to as {\it indexing into k} or +{\it elting into k}. +The latter phrase comes about because the name of the operation +that extracts elements is called \spadfunFrom{elt}{List}. +That is, \spad{k.3} is just alternative syntax for \spad{elt(k,3)}. +It is important to remember that list indices +begin with 1. +If we issue \spad{k := [1,3,2,9,5]} then \spad{k.4} +returns \spad{9}. +It is an error to use an index that is not in the range from +\spad{1} to the length of the list. + +\xtc{ +The last element of a list is extracted by any of the +following three expressions. +}{ +\spadpaste{last k \free{k}} +} +\xtc{ +}{ +\spadpaste{k.last \free{k}} +} +\xtc{ +This form computes the index of the last element and then +extracts the element from the list. +}{ +\spadpaste{k.(\#k) \free{k}} +} + +\endscroll +\autobuttons +\end{page} + +@ +<>= +\newcommand{\ugxListChangeTitle}{Changing List Elements} +\newcommand{\ugxListChangeNumber}{9.47.3.} + +@ +\subsection{Changing List Elements} +\label{ugxListChangePage} +\index{pages!ugxListChangePage!list.ht} +\index{list.ht!pages!ugxListChangePage} +\index{ugxListChangePage!list.ht!pages} +<>= +\begin{page}{ugxListChangePage}{Changing List Elements} +\beginscroll + +\labelSpace{4pc} +\xtc{ +We'll use this in some of the following examples. +}{ +\spadpaste{k := [4,3,7,3,8,5,9,2] \bound{k}} +} +\xtc{ +List elements are reset by using the \spad{k.i} form on +the left-hand side of an assignment. +This expression resets the first element of \spad{k} to \spad{999}. +}{ +\spadpaste{k.1 := 999 \free{k}\bound{k1}} +} +\xtc{ +As with indexing into a list, it is an error to use an index +that is not within the proper bounds. +Here you see that \spad{k} was modified. +}{ +\spadpaste{k \free{k1}} +} + +The operation that performs the assignment of an element to a particular +position in a list is called \spadfunFrom{setelt}{List}. +This operation is {\it destructive} in that it changes the list. +In the above example, the assignment returned the value \spad{999} and +\spad{k} was modified. +For this reason, lists are called \spadglos{mutable} objects: it is +possible to change part of a list (mutate it) rather than always returning +a new list reflecting the intended modifications. +\xtc{ +Moreover, since lists can share structure, changes to one list can +sometimes affect others. +}{ +\spadpaste{k := [1,2] \bound{k2}} +} +\xtc{ +}{ +\spadpaste{m := cons(0,k) \free{k2}\bound{m}} +} +\xtc{ +Change the second element of \spad{m}. +}{ +\spadpaste{m.2 := 99 \free{m}\bound{m2}} +} +\xtc{ +See, \spad{m} was altered. +}{ +\spadpaste{m \free{m2}} +} +\xtc{ +But what about \spad{k}? +It changed too! +}{ +\spadpaste{k \free{m2 k2}} +} + +\endscroll +\autobuttons +\end{page} + +@ +<>= +\newcommand{\ugxListOtherTitle}{Other Functions} +\newcommand{\ugxListOtherNumber}{9.47.4.} + +@ +\subsection{Other Functions} +\label{ugxListOtherPage} +\index{pages!ugxListOtherPage!list.ht} +\index{list.ht!pages!ugxListOtherPage} +\index{ugxListOtherPage!list.ht!pages} +<>= +\begin{page}{ugxListOtherPage}{Other Functions} +\beginscroll + +\labelSpace{3pc} +\xtc{ +An operation that is used frequently in list processing is that +which returns all elements in a list after the first element. +}{ +\spadpaste{k := [1,2,3] \bound{k}} +} +\xtc{ +Use the \spadfunFrom{rest}{List} operation to do this. +}{ +\spadpaste{rest k \free{k}} +} + +\xtc{ +To remove duplicate elements in a list \spad{k}, use +\spadfunFrom{removeDuplicates}{List}. +}{ +\spadpaste{removeDuplicates [4,3,4,3,5,3,4]} +} +\xtc{ +To get a list with elements in the order opposite to those in +a list \spad{k}, use \spadfunFrom{reverse}{List}. +}{ +\spadpaste{reverse [1,2,3,4,5,6]} +} +\xtc{ +To test whether an element is in a list, use \spadfunFrom{member?}{List}: +\spad{member?(a,k)} returns \spad{true} or \spad{false} +depending on whether \spad{a} is in \spad{k} or not. +}{ +\spadpaste{member?(1/2,[3/4,5/6,1/2])} +} +\xtc{ +}{ +\spadpaste{member?(1/12,[3/4,5/6,1/2])} +} + +As an exercise, the reader should determine how to get a +list containing all but the last of the elements in a given non-empty +list \spad{k}.\footnote{\spad{reverse(rest(reverse(k)))} works.} + +\endscroll +\autobuttons +\end{page} + +@ +<>= +\newcommand{\ugxListDotTitle}{Dot, Dot} +\newcommand{\ugxListDotNumber}{9.47.5.} + +@ +\subsection{Dot, Dot} +\label{ugxListDotPage} +\index{pages!ugxListDotPage!list.ht} +\index{list.ht!pages!ugxListDotPage} +\index{ugxListDotPage!list.ht!pages} +<>= +\begin{page}{ugxListDotPage}{Dot, Dot} +\beginscroll + +Certain lists are used so often that Axiom provides an easy way +of constructing them. +If \spad{n} and \spad{m} are integers, then \spad{expand [n..m]} +creates a list containing \spad{n, n+1, ... m}. +If \spad{n > m} then the list is empty. +It is actually permissible to leave off the \spad{m} in the +dot-dot construction (see below). + +\xtc{ +The dot-dot notation can be used more than once in a list +construction and with specific elements being given. +Items separated by dots are called {\it segments.} +}{ +\spadpaste{[1..3,10,20..23]} +} +\xtc{ +Segments can be expanded into the range of items between the +endpoints by using \spadfunFrom{expand}{Segment}. +}{ +\spadpaste{expand [1..3,10,20..23]} +} +\xtc{ +What happens if we leave off a number on the right-hand side of +\spadopFrom{..}{UniversalSegment}? +}{ +\spadpaste{expand [1..]} +} +What is created in this case is a \spadtype{Stream} which is a +generalization of a list. +See \downlink{`Stream'}{StreamXmpPage}\ignore{Stream} for more information. +\endscroll +\autobuttons +\end{page} + +@ +\section{lodo.ht} +<>= +\newcommand{\LinearOrdinaryDifferentialOperatorXmpTitle}{LinearOrdinaryDifferentialOperator} +\newcommand{\LinearOrdinaryDifferentialOperatorXmpNumber}{9.44} + +@ +\subsection{LinearOrdinaryDifferentialOperator} +\label{LinearOrdinaryDifferentialOperatorXmpPage} +See ugxLinearOrdinaryDifferentialOperatorSeriesPage +\ref{ugxLinearOrdinaryDifferentialOperatorSeriesPage} on +page~\pageref{ugxLinearOrdinaryDifferentialOperatorSeriesPage} +\index{pages!LinearOrdinaryDifferentialOperatorXmpPage!lodo.ht} +\index{lodo.ht!pages!LinearOrdinaryDifferentialOperatorXmpPage} +\index{LinearOrdinaryDifferentialOperatorXmpPage!lodo.ht!pages} +<>= +\begin{page}{LinearOrdinaryDifferentialOperatorXmpPage} +{LinearOrdinaryDifferentialOperator} +\beginscroll + +\spadtype{LinearOrdinaryDifferentialOperator(A, diff)} is the domain of linear +ordinary differential operators with coefficients in a ring +\spad{A} with a given derivation. +%This includes the cases of operators which are polynomials in \spad{D} +%acting upon scalar or vector expressions of a single variable. +%The coefficients of the operator polynomials can be integers, rational +%functions, matrices or elements of other domains. +\showBlurb{LinearOrdinaryDifferentialOperator} + +\beginmenu + \menudownlink{{9.44.1. Differential Operators with Series Coefficients}} +{ugxLinearOrdinaryDifferentialOperatorSeriesPage} +\endmenu +\endscroll +\autobuttons +\end{page} + +@ +<>= +\newcommand{\ugxLinearOrdinaryDifferentialOperatorSeriesTitle}{Differential Operators with Series Coefficients} +\newcommand{\ugxLinearOrdinaryDifferentialOperatorSeriesNumber}{9.44.1.} + +@ +\subsection{Differential Operators with Series Coefficients} +\label{ugxLinearOrdinaryDifferentialOperatorSeriesPage} +\index{pages!ugxLinearOrdinaryDifferentialOperatorSeriesPage!lodo.ht} +\index{lodo.ht!pages!ugxLinearOrdinaryDifferentialOperatorSeriesPage} +\index{ugxLinearOrdinaryDifferentialOperatorSeriesPage!lodo.ht!pages} +<>= +\begin{page}{ugxLinearOrdinaryDifferentialOperatorSeriesPage} +{Differential Operators with Series Coefficients} +\beginscroll + +\noindent +{\bf Problem:} +Find the first few coefficients of \spad{exp(x)/x**i} of \spad{Dop phi} where +\begin{verbatim} +Dop := D**3 + G/x**2 * D + H/x**3 - 1 +phi := sum(s[i]*exp(x)/x**i, i = 0..) +\end{verbatim} + +\noindent +{\bf Solution:} +\xtc{ +Define the differential. +}{ +\spadpaste{Dx: LODO(EXPR INT, f +-> D(f, x)) \bound{Dxd}} +} +\xtc{ +}{ +\spadpaste{Dx := D() \free{Dxd}\bound{Dx}} +} +\xtc{ +Now define the differential operator \spad{Dop}. +}{ +\spadpaste{Dop:= Dx**3 + G/x**2*Dx + H/x**3 - 1 \free{Dx}\bound{Dop}} +} +\xtc{ +}{ +\spadpaste{n == 3 \bound{n3}} +} +\xtc{ +}{ +\spadpaste{phi == reduce(+,[subscript(s,[i])*exp(x)/x**i for i in 0..n]) \bound{phi}} +} +\xtc{ +}{ +\spadpaste{phi1 == Dop(phi) / exp x \bound{phi1}\free{Dop phi}} +} +\xtc{ +}{ +\spadpaste{phi2 == phi1 *x**(n+3) \bound{phi2}\free{phi1}} +} +\xtc{ +}{ +\spadpaste{phi3 == retract(phi2)@(POLY INT) \bound{phi3}\free{phi2}} +} +\xtc{ +}{ +\spadpaste{pans == phi3 ::UP(x,POLY INT) \free{phi3}\bound{pans}} +} +\xtc{ +}{ +\spadpaste{pans1 == [coefficient(pans, (n+3-i) :: NNI) for i in 2..n+1] \bound{pans1}\free{pans}} +} +\xtc{ +}{ +\spadpaste{leq == solve(pans1,[subscript(s,[i]) for i in 1..n]) \bound{leq}\free{pans1}} +} +\xtc{ +Evaluate this for several values of \spad{n}. +}{ +\spadpaste{leq \free{n3 leq}} +} +\xtc{ +}{ +\spadpaste{n==4 \bound{n4}} +} +\xtc{ +}{ +\spadpaste{leq \free{n4 leq}} +} +\xtc{ +}{ +\spadpaste{n==7 \bound{n7}} +} +\xtc{ +}{ +\spadpaste{leq \free{n7 leq}} +} + +\endscroll +\autobuttons +\end{page} + +@ +\section{lodo1.ht} +<>= +\newcommand{\LinearOrdinaryDifferentialOperatorOneXmpTitle}{LinearOrdinaryDifferentialOperator1} +\newcommand{\LinearOrdinaryDifferentialOperatorOneXmpNumber}{9.45} + +@ +\subsection{LinearOrdinaryDifferentialOperator1} +\label{LinearOrdinaryDifferentialOperatorOneXmpPage} +See ugxLinearOrdinaryDifferentialOperatorOneRatPage +\ref{ugxLinearOrdinaryDifferentialOperatorOneRatPage} on +page~\pageref{ugxLinearOrdinaryDifferentialOperatorOneRatPage} +\index{pages!LinearOrdinaryDifferentialOperatorOneXmpPage!lodo1.ht} +\index{lodo1.ht!pages!LinearOrdinaryDifferentialOperatorOneXmpPage} +\index{LinearOrdinaryDifferentialOperatorOneXmpPage!lodo1.ht!pages} +<>= +\begin{page}{LinearOrdinaryDifferentialOperatorOneXmpPage} +{LinearOrdinaryDifferentialOperator1} +\beginscroll + +\spadtype{LinearOrdinaryDifferentialOperator1(A)} is the domain of linear +ordinary differential operators with coefficients in the differential ring +\spad{A}. +%This includes the cases of operators which are polynomials in \spad{D} +%acting upon scalar or vector expressions of a single variable. +%The coefficients of the operator polynomials can be integers, rational +%functions, matrices or elements of other domains. +\showBlurb{LinearOrdinaryDifferentialOperator1} + +\beginmenu + \menudownlink{ +{9.45.1. Differential Operators with Rational Function Coefficients}} +{ugxLinearOrdinaryDifferentialOperatorOneRatPage} +\endmenu +\endscroll +\autobuttons +\end{page} + +<>= +\newcommand{\ugxLinearOrdinaryDifferentialOperatorOneRatTitle}{Differential Operators with Rational Function Coefficients} +\newcommand{\ugxLinearOrdinaryDifferentialOperatorOneRatNumber}{9.45.1.} + +@ +\subsection{Differential Operators with Rational Function Coefficients} +\label{ugxLinearOrdinaryDifferentialOperatorOneRatPage} +\index{pages!ugxLinearOrdinaryDifferentialOperatorOneRatPage!lodo1.ht} +\index{lodo1.ht!pages!ugxLinearOrdinaryDifferentialOperatorOneRatPage} +\index{ugxLinearOrdinaryDifferentialOperatorOneRatPage!lodo1.ht!pages} +<>= +\begin{page}{ugxLinearOrdinaryDifferentialOperatorOneRatPage} +{Differential Operators with Rational Function Coefficients} +\beginscroll + +This example shows differential operators with rational function +coefficients. In this case operator multiplication is non-commutative and, +since the coefficients form a field, an operator division algorithm exists. + +\labelSpace{2pc} +\xtc{ +We begin by defining \spad{RFZ} to be the rational functions in +\spad{x} with integer coefficients and \spad{Dx} to be the differential +operator for \spad{d/dx}. +}{ +\spadpaste{RFZ := Fraction UnivariatePolynomial('x, Integer) \bound{RFZ0}} +} +\xtc{ +}{ +\spadpaste{x : RFZ := 'x \free{RFZ0}\bound{RFZ}} +} +\xtc{ +}{ +\spadpaste{Dx : LODO1 RFZ := D()\bound{Dx}\free{RFZ}} +} +\xtc{ +Operators are created using the usual arithmetic operations. +}{ +\spadpaste{b : LODO1 RFZ := 3*x**2*Dx**2 + 2*Dx + 1/x \free{Dx}\bound{b}} +} +\xtc{ +}{ +\spadpaste{a : LODO1 RFZ := b*(5*x*Dx + 7) \free{b Dx}\bound{a}} +} +\xtc{ +Operator multiplication corresponds to functional composition. +}{ +\spadpaste{p := x**2 + 1/x**2 \bound{p}\free{RFZ}} +} +\xtc{ +Since operator coefficients depend on \spad{x}, the multiplication is +not commutative. +}{ +\spadpaste{(a*b - b*a) p \free{a b p}} +} + +When the coefficients of operator polynomials come from a field, as in this +case, it is possible to define operator division. +Division on the left and division on the right yield different results when +the multiplication is non-commutative. + +The results of +\spadfunFrom{leftDivide}{LinearOrdinaryDifferentialOperator1} and +\spadfunFrom{rightDivide}{LinearOrdinaryDifferentialOperator1} are +quotient-remainder pairs satisfying: \newline +% +\spad{leftDivide(a,b) = [q, r]} such that \spad{a = b*q + r} \newline +\spad{rightDivide(a,b) = [q, r]} such that \spad{a = q*b + r} \newline +% +\xtc{ +In both cases, the \spadfunFrom{degree}{LinearOrdinaryDifferentialOperator1} +of the remainder, \spad{r}, is less than +the degree of \spad{b}. +}{ +\spadpaste{ld := leftDivide(a,b) \bound{ld}\free{a b}} +} +\xtc{ +}{ +\spadpaste{a = b * ld.quotient + ld.remainder \free{a b ld}} +} +\xtc{ +The operations of left and right division +are so-called because the quotient is obtained by dividing +\spad{a} on that side by \spad{b}. +}{ +\spadpaste{rd := rightDivide(a,b) \bound{rd}\free{a b}} +} +\xtc{ +}{ +\spadpaste{a = rd.quotient * b + rd.remainder \free{a b rd}} +} + +\xtc{ +Operations \spadfunFrom{rightQuotient}{LinearOrdinaryDifferentialOperator1} +and \spadfunFrom{rightRemainder}{LinearOrdinaryDifferentialOperator1} +are available if only one of the quotient or remainder are of +interest to you. +This is the quotient from right division. +}{ +\spadpaste{rightQuotient(a,b) \free{a b}} +} +\xtc{ +This is the remainder from right division. +The corresponding ``left'' functions +\spadfunFrom{leftQuotient}{LinearOrdinaryDifferentialOperator1} and +\spadfunFrom{leftRemainder}{LinearOrdinaryDifferentialOperator1} +are also available. +}{ +\spadpaste{rightRemainder(a,b) \free{a b}} +} + +\xtc{ +For exact division, the operations +\spadfunFrom{leftExactQuotient}{LinearOrdinaryDifferentialOperator1} and +\spadfunFrom{rightExactQuotient}{LinearOrdinaryDifferentialOperator1} are supplied. +These return the quotient but only if the remainder is zero. +The call \spad{rightExactQuotient(a,b)} would yield an error. +}{ +\spadpaste{leftExactQuotient(a,b) \free{a b}} +} + +\xtc{ +The division operations allow the computation of left and right greatest +common divisors (\spadfunFrom{leftGcd}{LinearOrdinaryDifferentialOperator1} and +\spadfunFrom{rightGcd}{LinearOrdinaryDifferentialOperator1}) via remainder +sequences, and consequently the computation of left and right least common +multiples (\spadfunFrom{rightLcm}{LinearOrdinaryDifferentialOperator1} and +\spadfunFrom{leftLcm}{LinearOrdinaryDifferentialOperator1}). +}{ +\spadpaste{e := leftGcd(a,b) \bound{e}\free{a b}} +} +\xtc{ +Note that a greatest common divisor doesn't necessarily divide \spad{a} and +{9.45.1.}{Differential Operators with Rational Function Coefficients} +\spad{b} on both sides. +Here the left greatest common divisor does not divide \spad{a} on the right. +}{ +\spadpaste{leftRemainder(a, e) \free{a e}} +} +\xtc{ +}{ +\spadpaste{rightRemainder(a, e) \free{a e}} +} + +\xtc{ +Similarly, a least common multiple +is not necessarily divisible from both sides. +}{ +\spadpaste{f := rightLcm(a,b) \bound{f}\free{a b}} +} +\xtc{ +}{ +\spadpaste{rightRemainder(f, b) \free{f b}} +} +\xtc{ +}{ +\spadpaste{leftRemainder(f, b) \free{f b}} +} + +\endscroll +\autobuttons +\end{page} + +@ +\section{lodo2.ht} +<>= +\newcommand{\LinearOrdinaryDifferentialOperatorTwoXmpTitle}{LinearOrdinaryDifferentialOperator2} +\newcommand{\LinearOrdinaryDifferentialOperatorTwoXmpNumber}{9.46} + +@ +\subsection{LinearOrdinaryDifferentialOperator2} +\label{LinearOrdinaryDifferentialOperatorTwoXmpPage} +\begin{itemize} +\item ugxLinearOrdinaryDifferentialOperatorTwoConstPage +\ref{ugxLinearOrdinaryDifferentialOperatorTwoConstPage} on +page~pageref{ugxLinearOrdinaryDifferentialOperatorTwoConstPage} +\item ugxLinearOrdinaryDifferentialOperatorTwoMatrixPage +\ref{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPage} on +page~pageref{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPage} +\end{itemize} +\index{pages!LinearOrdinaryDifferentialOperatorTwoXmpPage!lodo2.ht} +\index{lodo2.ht!pages!LinearOrdinaryDifferentialOperatorTwoXmpPage} +\index{LinearOrdinaryDifferentialOperatorTwoXmpPage!lodo2.ht!pages} +<>= +\begin{page}{LinearOrdinaryDifferentialOperatorTwoXmpPage} +{LinearOrdinaryDifferentialOperator2} +\beginscroll + +\spadtype{LinearOrdinaryDifferentialOperator2(A, M)} is the domain of linear +ordinary differential operators with coefficients in the differential ring +\spad{A} and operating on \spad{M}, an \spad{A}-module. +This includes the cases of operators which are polynomials in \spad{D} +acting upon scalar or vector expressions of a single variable. +The coefficients of the operator polynomials can be integers, rational +functions, matrices or elements of other domains. +\showBlurb{LinearOrdinaryDifferentialOperator2} + +\beginmenu + \menudownlink{ +{9.46.1. Differential Operators with Constant Coefficients}} +{ugxLinearOrdinaryDifferentialOperatorTwoConstPage} + \menudownlink{ +{9.46.2. Differential Operators with Matrix Coefficients Operating on Vectors}} +{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPage} +\endmenu +\endscroll +\autobuttons +\end{page} + +@ +<>= +\newcommand{\ugxLinearOrdinaryDifferentialOperatorTwoConstTitle}{Differential Operators with Constant Coefficients} +\newcommand{\ugxLinearOrdinaryDifferentialOperatorTwoConstNumber}{9.46.1.} + +@ +\subsection{Differential Operators with Constant Coefficients} +\label{ugxLinearOrdinaryDifferentialOperatorTwoConstPage} +\index{pages!ugxLinearOrdinaryDifferentialOperatorTwoConstPage!lodo2.ht} +\index{lodo2.ht!pages!ugxLinearOrdinaryDifferentialOperatorTwoConstPage} +\index{ugxLinearOrdinaryDifferentialOperatorTwoConstPage!lodo2.ht!pages} +<>= +\begin{page}{ugxLinearOrdinaryDifferentialOperatorTwoConstPage} +{Differential Operators with Constant Coefficients} +\beginscroll + +This example shows differential operators with rational +number coefficients operating on univariate polynomials. + +\labelSpace{3pc} +\xtc{ +We begin by making type assignments so we can conveniently refer +to univariate polynomials in \spad{x} over the rationals. +}{ +\spadpaste{Q := Fraction Integer \bound{Q}} +} +\xtc{ +}{ +\spadpaste{PQ := UnivariatePolynomial('x, Q) \free{Q}\bound{PQ0}} +} +\xtc{ +}{ +\spadpaste{x: PQ := 'x \free{PQ0}\bound{x}} +} +\xtc{ +Now we assign \spad{Dx} to be the differential operator +\spadfunFrom{D}{LinearOrdinaryDifferentialOperator2} +corresponding to \spad{d/dx}. +}{ +\spadpaste{Dx: LODO2(Q, PQ) := D() \free{Q PQ0}\bound{Dx}} +} +\xtc{ +New operators are created as polynomials in \spad{D()}. +}{ +\spadpaste{a := Dx + 1 \bound{a}\free{Dx}} +} +\xtc{ +}{ +\spadpaste{b := a + 1/2*Dx**2 - 1/2 \bound{b}\free{Dx}} +} +\xtc{ +To apply the operator \spad{a} to the value \spad{p} the usual function +call syntax is used. +}{ +\spadpaste{p := 4*x**2 + 2/3 \free{x}\bound{p}} +} +\xtc{ +}{ +\spadpaste{a p \free{a p}} +} +\xtc{ +Operator multiplication is defined by the identity \spad{(a*b) p = a(b(p))} +}{ +\spadpaste{(a * b) p = a b p \free{a b p}} +} +\xtc{ +Exponentiation follows from multiplication. +}{ +\spadpaste{c := (1/9)*b*(a + b)**2 \free{a b}\bound{c}} +} +\xtc{ +Finally, note that operator expressions may be applied directly. +}{ +\spadpaste{(a**2 - 3/4*b + c) (p + 1) \free{a b c p}} +} + +\endscroll +\autobuttons +\end{page} + +@ +<>= +\newcommand{\ugxLinearOrdinaryDifferentialOperatorTwoMatrixTitle}{Differential Operators with Matrix Coefficients Operating on Vectors} +\newcommand{\ugxLinearOrdinaryDifferentialOperatorTwoMatrixNumber}{9.46.2.} + +@ +\subsection{Differential Operators with Matrix Coefficients Operating on Vectors} +\label{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPage} +\index{pages!ugxLinearOrdinaryDifferentialOperatorTwoMatrixPage!lodo2.ht} +\index{lodo2.ht!pages!ugxLinearOrdinaryDifferentialOperatorTwoMatrixPage} +\index{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPage!lodo2.ht!pages} +<>= +\begin{page}{ugxLinearOrdinaryDifferentialOperatorTwoMatrixPage} +{Differential Operators with Matrix Coefficients Operating on Vectors} +\beginscroll + +This is another example of linear ordinary differential +operators with non-commutative multiplication. +Unlike the rational function case, the differential ring of +square matrices (of a given dimension) with univariate +polynomial entries does not form a field. +Thus the number of operations available is more limited. + +\labelSpace{1pc} +\xtc{ +In this section, the operators have three by three +matrix coefficients with polynomial entries. +}{ +\spadpaste{PZ := UnivariatePolynomial(x,Integer)\bound{PZ0}} +} +\xtc{ +}{ +\spadpaste{x:PZ := 'x \free{PZ0}\bound{PZ}} +} +\xtc{ +}{ +\spadpaste{Mat := SquareMatrix(3,PZ)\free{PZ}\bound{Mat}} +} +\xtc{ +The operators act on the vectors considered as a \spad{Mat}-module. +}{ +\spadpaste{Vect := DPMM(3, PZ, Mat, PZ);\free{PZ,Mat}\bound{Vect}} +} +\xtc{ +}{ +\spadpaste{Modo := LODO2(Mat, Vect);\free{Mat Vect}\bound{Modo}} +} +\xtc{ +The matrix \spad{m} is used as a coefficient and the vectors \spad{p} +and \spad{q} are operated upon. +}{ +\spadpaste{m:Mat := matrix [[x**2,1,0],[1,x**4,0],[0,0,4*x**2]]\free{Mat}\bound{m}} +} +\xtc{ +}{ +\spadpaste{p:Vect := directProduct [3*x**2+1,2*x,7*x**3+2*x]\free{Vect}\bound{p}} +} +\xtc{ +}{ +\spadpaste{q: Vect := m * p\free{m p Vect}\bound{q}} +} +\xtc{ +Now form a few operators. +}{ +\spadpaste{Dx : Modo := D()\bound{Dx}\free{Modo}} +} +\xtc{ +}{ +\spadpaste{a : Modo := Dx + m\bound{a}\free{m Dx}} +} +\xtc{ +}{ +\spadpaste{b : Modo := m*Dx + 1\bound{b}\free{m Dx}} +} +\xtc{ +}{ +\spadpaste{c := a*b \bound{c}\free{a b}} +} +\xtc{ +These operators can be applied to vector values. +}{ +\spadpaste{a p \free{p a}} +} +\xtc{ +}{ +\spadpaste{b p \free{p b}} +} +\xtc{ +}{ +\spadpaste{(a + b + c) (p + q) \free{a b c p q}} +} +\endscroll +\autobuttons +\end{page} + +@ +\section{lpoly.ht} +<>= +\newcommand{\LiePolynomialXmpTitle}{LiePolynomial} +\newcommand{\LiePolynomialXmpNumber}{9.43} + +@ +\subsection{LiePolynomial} +\label{LiePolynomialXmpPage} +\index{pages!LiePolynomialXmpPage!lpoly.ht} +\index{lpoly.ht!pages!LiePolynomialXmpPage} +\index{LiePolynomialXmpPage!lpoly.ht!pages} +<>= +\begin{page}{LiePolynomialXmpPage}{LiePolynomial} +% ===================================================================== +\beginscroll +Declaration of domains +\xtc{ +}{ +\spadpaste{RN := Fraction Integer \bound{RN}} +} +\xtc{ +}{ +\spadpaste{Lpoly := LiePolynomial(Symbol,RN) \bound{Lpoly} \free{RN}} +} +\xtc{ +}{ +\spadpaste{Dpoly := XDPOLY(Symbol,RN) \bound{Dpoly} \free{RN}} +} +\xtc{ +}{ +\spadpaste{Lword := LyndonWord Symbol \bound{Lword}} +} + +Initialisation +\xtc{ +}{ +\spadpaste{a:Symbol := 'a \bound{a}} +} +\xtc{ +}{ +\spadpaste{b:Symbol := 'b \bound{b}} +} +\xtc{ +}{ +\spadpaste{c:Symbol := 'c \bound{c}} +} +\xtc{ +}{ +\spadpaste{aa: Lpoly := a \bound{aa} \free{Lpoly} \free{a}} +} +\xtc{ +}{ +\spadpaste{bb: Lpoly := b \bound{bb} \free{Lpoly} \free{b}} +} +\xtc{ +}{ +\spadpaste{cc: Lpoly := c \bound{cc} \free{Lpoly} \free{c}} +} +\xtc{ +}{ +\spadpaste{p : Lpoly := [aa,bb] \bound{p} \free{aa} \free{bb} \free{Lpoly}} +} +\xtc{ +}{ +\spadpaste{q : Lpoly := [p,bb] \bound{q} \free{p} \free{bb} \free{Lpoly}} +} +\xtc{ +All the Lyndon words of order 4 +}{ +\spadpaste{liste : List Lword := LyndonWordsList([a,b], 4) \free{a} \free{b} \free{Lword} \bound{liste}} +} +\xtc{ +}{ +\spadpaste{r: Lpoly := p + q + 3*LiePoly(liste.4)$Lpoly \bound{r} \free{Lpoly} \free{p} \free{q} \free{liste}} +} +\xtc{ +}{ +\spadpaste{s:Lpoly := [p,r] \bound{s} \free{Lpoly} \free{p} \free{r}} +} +\xtc{ +}{ +\spadpaste{t:Lpoly := s + 2*LiePoly(liste.3) - 5*LiePoly(liste.5) \bound{t} \free{Lpoly} \free{s} \free{liste} } +} +\xtc{ +}{ +\spadpaste{degree t \free{t}} +} +\xtc{ +}{ +\spadpaste{mirror t \free{t}} +} + +Jacobi Relation +\xtc{ +}{ +\spadpaste{Jacobi(p: Lpoly, q: Lpoly, r: Lpoly): Lpoly == [[p,q]$Lpoly, r] + [[q,r]$Lpoly, p] + [[r,p]$Lpoly, q] \free{Lpoly} \bound{J}} +} + +Tests +\xtc{ +}{ +\spadpaste{test: Lpoly := Jacobi(a,b,b) \free{J Lpoly a b} \bound{test1}} +} +\xtc{ +}{ +\spadpaste{test: Lpoly := Jacobi(p,q,r) \free{J p q r Lpoly} \bound{test2}} +} +\xtc{ +}{ +\spadpaste{test: Lpoly := Jacobi(r,s,t) \free{J r s t Lpoly} \bound{test3}} +} + +Evaluation +\xtc{ +}{ +\spadpaste{eval(p, a, p)$Lpoly} +} +\xtc{ +}{ +\spadpaste{eval(p, [a,b], [2*bb, 3*aa])$Lpoly \free{p a b bb aa Lpoly}} +} +\xtc{ +}{ +\spadpaste{r: Lpoly := [p,c] \free{p c Lpoly} \bound{rr}} +} +\xtc{ +}{ +\spadpaste{r1: Lpoly := eval(r, [a,b,c], [bb, cc, aa])$Lpoly \free{rr a b c aa bb cc Lpoly} \bound{r1}} +} +\xtc{ +}{ +\spadpaste{r2: Lpoly := eval(r, [a,b,c], [cc, aa, bb])$Lpoly \free{rr a b c cc bb aa Lpoly} \bound{r2}} +} +\xtc{ +}{ +\spadpaste{r + r1 + r2 \free{rr r1 r2}} +} + +\endscroll +\autobuttons +\end{page} +% + +@ +\section{lword.ht} +<>= +\newcommand{\LyndonWordXmpTitle}{LyndonWord} +\newcommand{\LyndonWordXmpNumber}{9.48} + +@ +\subsection{LyndonWord} +\label{LyndonWordXmpPage} +\index{pages!LyndonWordXmpPage!lword.ht} +\index{lword.ht!pages!LyndonWordXmpPage} +\index{LyndonWordXmpPage!lword.ht!pages} +<>= +\begin{page}{LyndonWordXmpPage}{LyndonWord} +\beginscroll +Initialisations +\xtc{ +}{ +\spadpaste{a:Symbol :='a \bound{a}} +} +\xtc{ +}{ +\spadpaste{b:Symbol :='b \bound{b}} +} +\xtc{ +}{ +\spadpaste{c:Symbol :='c \bound{c}} +} +\xtc{ +}{ +\spadpaste{lword:= LyndonWord(Symbol) \bound{lword}} +} +\xtc{ +}{ +\spadpaste{magma := Magma(Symbol) \bound{magma}} +} +\xtc{ +}{ +\spadpaste{word := OrderedFreeMonoid(Symbol) \bound{word}} +} +\xtc{ +All Lyndon words of with a, b, c to order 3 +}{ +\spadpaste{LyndonWordsList1([a,b,c],3)$lword \free{lword} \free{a} \free{b} \free{c} } +} +\xtc{ +All Lyndon words of with a, b, c to order 3 in flat list +}{ +\spadpaste{LyndonWordsList([a,b,c],3)$lword \free{a} \free{b} \free{c} \free{lword}} +} +\xtc{ +All Lyndon words of with a, b to order 5 +}{ +\spadpaste{lw := LyndonWordsList([a,b],5)$lword \free{a} \free{b} \free{lword} \bound{lw}} +} +\xtc{ +}{ +\spadpaste{w1 : word := lw.4 :: word \free{word} \free{lw} \bound{w1}} +} +\xtc{ +}{ +\spadpaste{w2 : word := lw.5 :: word \free{word} \free{lw} \bound{w2}} +} + +Let's try factoring +\xtc{ +}{ +\spadpaste{factor(a::word)$lword \free{a word lword}} +} +\xtc{ +}{ +\spadpaste{factor(w1*w2)$lword \free{ w1 w2 lword}} +} +\xtc{ +}{ +\spadpaste{factor(w2*w2)$lword \free{w2 lword}} +} +\xtc{ +}{ +\spadpaste{factor(w2*w1)$lword \free{w1 w2 lword}} +} + +Checks and coercions +\xtc{ +}{ +\spadpaste{lyndon?(w1)$lword \free{w1 lword}} +} +\xtc{ +}{ +\spadpaste{lyndon?(w1*w2)$lword \free{w1 w2 lword}} +} +\xtc{ +}{ +\spadpaste{lyndon?(w2*w1)$lword \free{w1 w2 lword}} +} +\xtc{ +}{ +\spadpaste{lyndonIfCan(w1)$lword \free{w1 lword}} +} +\xtc{ +}{ +\spadpaste{lyndonIfCan(w2*w1)$lword \free{w1 w2 lword}} +} +\xtc{ +}{ +\spadpaste{lyndon(w1)$lword \free{w1 lword}} +} +\xtc{ +}{ +\spadpaste{lyndon(w1*w2)$lword \free{w1 w2 lword}} +} +\endscroll +\autobuttons +\end{page} + +@ +\section{magma.ht} +<>= +\newcommand{\MagmaXmpTitle}{Magma} +\newcommand{\MagmaXmpNumber}{9.49} + +@ +\subsection{Magma} +\label{MagmaXmpPage} +\index{pages!MagmaXmpPage!magma.ht} +\index{magma.ht!pages!MagmaXmpPage} +\index{MagmaXmpPage!magma.ht!pages} +<>= +\begin{page}{MagmaXmpPage}{Magma} +\beginscroll +Initialisations +\xtc{ +}{ +\spadpaste{x:Symbol :='x \bound{x}} +} +\xtc{ +}{ +\spadpaste{y:Symbol :='y \bound{y}} +} +\xtc{ +}{ +\spadpaste{z:Symbol :='z \bound{z}} +} +\xtc{ +}{ +\spadpaste{word := OrderedFreeMonoid(Symbol) \bound{word}} +} +\xtc{ +}{ +\spadpaste{tree := Magma(Symbol) \bound{tree}} +} + +Let's make some trees +\xtc{ +}{ +\spadpaste{a:tree := x*x \free{x tree} \bound{a}} +} +\xtc{ +}{ +\spadpaste{b:tree := y*y \free{y tree} \bound{b}} +} +\xtc{ +}{ +\spadpaste{c:tree := a*b \free{a b tree} \bound{c}} +} + +Query the trees +\xtc{ +}{ +\spadpaste{left c \free{c}} +} +\xtc{ +}{ +\spadpaste{right c \free{c}} +} +\xtc{ +}{ +\spadpaste{length c \free{c}} +} +\xtc{ +Coerce to the monoid +}{ +\spadpaste{c::word \free{c word}} +} + +Check ordering +\xtc{ +}{ +\spadpaste{a < b \free{a b}} +} +\xtc{ +}{ +\spadpaste{a < c \free{a c}} +} +\xtc{ +}{ +\spadpaste{b < c \free{b c}} +} + +Navigate the tree +\xtc{ +}{ +\spadpaste{first c \free{c}} +} +\xtc{ +}{ +\spadpaste{rest c \free{c}} +} +\xtc{ +}{ +\spadpaste{rest rest c \free{c}} +} + +Check ordering +\xtc{ +}{ +\spadpaste{ax:tree := a*x \free{a x tree} \bound{ax}} +} +\xtc{ +}{ +\spadpaste{xa:tree := x*a \free{a x tree} \bound{xa}} +} +\xtc{ +}{ +\spadpaste{xa < ax \free{ax xa}} +} +\xtc{ +}{ +\spadpaste{lexico(xa,ax) \free{ax xa}} +} +\endscroll +\autobuttons +\end{page} + +@ +\section{man0.ht} +\subsection{Reference Search} +\label{RefSearchPage} +See ugSysCmdPage \ref{ugSysCmdPage} on page~\pageref{ugSysCmdPage} +\index{pages!RefSearchPage!man0.ht} +\index{man0.ht!pages!RefSearchPage} +\index{RefSearchPage!man0.ht!pages} +<>= +\begin{page}{RefSearchPage}{Reference Search} +\beginscroll +Enter search string : +\inputstring{pattern}{40}{} +\newline +\beginmenu +\menuunixlink{Search} + {htsearch "\stringvalue{pattern}"} + \tab{15} Reference documentation ({\em *} wild card is not accepted). +\endmenu +\endscroll +\end{page} + +@ +\subsection{Axiom Browser} +\label{Man0Page} +See ugSysCmdPage \ref{ugSysCmdPage} on page~\pageref{ugSysCmdPage} +\index{pages!Man0Page!man0.ht} +\index{man0.ht!pages!Man0Page} +\index{Man0Page!man0.ht!pages} +<>= +\begin{page}{Man0Page}{A x i o m \ B r o w s e r} + +\beginscroll +Enter search string (use {\em *} for wild card unless counter-indicated): +\inputstring{pattern}{40}{} +\newline +\beginmenu +\menulispmemolink{Constructors} + { (|kSearch| '|\stringvalue{pattern}|) } + \tab{15} Search for \lispmemolink{categories}{(|cSearch| '|\stringvalue{pattern}|)}, \lispmemolink{domains}{(|dSearch| '|\stringvalue{pattern}|)}, or \lispmemolink{packages}{(|pSearch| '|\stringvalue{pattern}|)} +\menulispmemolink{Operations} + { (|oSearch| '|\stringvalue{pattern}|) } + \tab{15} Search for operations. +\menulispmemolink{Attributes} + { (|aSearch| '|\stringvalue{pattern}|) } + \tab{15} Search for attributes. +\menulispmemolink{General} + { (|aokSearch| '|\stringvalue{pattern}|) } + \tab{15} Search for all three of the above. +\menulispmemolink{Documentation} + { (|docSearch| '|\stringvalue{pattern}|) } + \tab{15} Search library documentation. +\menulispmemolink{Complete} + { (|genSearch| '|\stringvalue{pattern}|) } + \tab{15} All of the above. +\menulispmemolink{Selectable} + { (|detailedSearch| '|\stringvalue{pattern}|) } + \tab{15} Detailed search with selectable options. +\horizontalline +\menuunixlink{Reference} + {htsearch "\stringvalue{pattern}"} + \tab{15} Search Reference documentation ({\em *} wild card is not accepted). +\menumemolink{Commands}{ugSysCmdPage} +\tab{15} View system command documentation. +\endmenu +\endscroll +\autobutt{BROWSEhelp} +\end{page} + +@ +\subsection{The Hyperdoc Browse Facility} +\label{BROWSEhelp} +\index{pages!BROWSEhelp!man0.ht} +\index{man0.ht!pages!BROWSEhelp} +\index{BROWSEhelp!man0.ht!pages} +<>= +\begin{page}{BROWSEhelp}{The Hyperdoc Browse Facility} + +\beginscroll +\beginmenu + \menudownlink{{The Front Page: Searching the Library}}{ugBrowseStartPage} + \menudownlink{{The Constructor Page}}{ugBrowseDomainPage} + \menudownlink{{Miscellaneous Features of Browse}}{ugBrowseMiscellaneousFeaturesPage} +\endmenu +\endscroll +\newline +\end{page} + +@ +\section{mapping.ht} +\subsection{Domain {\bf Mapping(T,S,...)}} +\label{DomainMapping} +\index{pages!DomainMapping!mapping.ht} +\index{mapping.ht!pages!DomainMapping} +\index{DomainMapping!mapping.ht!pages} +<>= +\begin{page}{DomainMapping}{Domain {\em Mapping(T,S,...)}} +\beginscroll +{\em Mapping} takes any number of arguments of the form: +\indentrel{2} +\newline \spad{T}, a domain of category \spadtype{SetCategory} +\newline \spad{S}, a domain of category \spadtype{SetCategory} +\newline\tab{10}... +\indentrel{-2}\newline +This constructor is a primitive in Axiom. +\newline +\beginmenu +\item\menulispdownlink{Description} +{(|dbSpecialDescription| '|Mapping|)}\tab{19}General description +\item\menulispdownlink{Operations} +{(|dbSpecialOperations| '|Mapping|)} +\tab{19}All exported operations of \spad{Mapping(T,S)} +%\item\menudownlink{Examples} {MappingExamples} +\tab{19}Examples illustrating use +%\item\menudownlink{Exports} {MappingExports} +\tab{19}Explicit categories and operations +\endmenu +\endscroll\end{page} + +@ +\subsection{Domain Constructor {\bf Mapping}} +\label{MappingDescription} +\index{pages!MappingDescription!mapping.ht} +\index{mapping.ht!pages!MappingDescription} +\index{MappingDescription!mapping.ht!pages} +<>= +\begin{page}{MappingDescription}{Domain Constructor {\em Mapping}} +\beginscroll +\newline\menuitemstyle{}\tab{2}Mapping({\em T},{\em S,...}) +\newline\tab{2}{\em Arguments:}\indent{17}\tab{-2} +{\em T}, a domain of category \spadtype{SetCategory} +\newline\tab{-2} +{\em S}, a domain of category \spadtype{SetCategory} +\newline\tab{10}... +\indent{0}\newline\tab{2}{\em Returns:}\indent{15}\tab{0} +the class of mappings from domain ({\em S,...}) +into domain {\em T} as described below. +\indent{0}\newline\tab{2}{\em Description:}\indent{15}\tab{0} +{\em Mapping(T,S,...)} denotes the class of objects which are +mappings from a source domain ({\em S,...}) into a target domain {\em T}. +The {\em Mapping} constructor can take any number of arguments. +All but the first argument is regarded as part of a source tuple +for the mapping. +For example, {\em Mapping(T,A,B)} denotes the +class of mappings from {\em (A,B)} into {\em T}. +{\em Mapping} is a primitive domain of Axiom which cannot be +defined in the Axiom language. +\endscroll +\end{page} + + +@ +\section{mappkg1.ht} +<>= +\newcommand{\MappingPackageOneXmpTitle}{MappingPackage1} +\newcommand{\MappingPackageOneXmpNumber}{9.51} + +@ +\subsection{MappingPackage1} +\label{MappingPackageOneXmpPage} +\index{pages!MappingPackageOneXmpPage!mappkg1.ht} +\index{mappkg1.ht!pages!MappingPackageOneXmpPage} +\index{MappingPackageOneXmpPage!mappkg1.ht!pages} +<>= +\begin{page}{MappingPackageOneXmpPage}{MappingPackage1} +\beginscroll + +Function are objects of type \pspadtype{Mapping}. +In this section we demonstrate some library operations from the +packages \spadtype{MappingPackage1}, \spadtype{MappingPackage2}, and +\spadtype{MappingPackage3} that manipulate and create functions. +Some terminology: a {\it nullary} function takes no arguments, +a {\it unary} function takes one argument, and +a {\it binary} function takes two arguments. + +\xtc{ +We begin by creating an example function that raises a +rational number to an integer exponent. +}{ +\spadpaste{power(q: FRAC INT, n: INT): FRAC INT == q**n \bound{power}} +} +\xtc{ +}{ +\spadpaste{power(2,3) \free{power}} +} +\xtc{ +The \spadfunFrom{twist}{MappingPackage3} operation +transposes the arguments of a binary function. +Here \spad{rewop(a, b)} is \spad{power(b, a)}. +}{ +\spadpaste{rewop := twist power \free{power}\bound{rewop}} +} +\xtc{ +This is \texht{$2^3.$}{\spad{2**3.}} +}{ +\spadpaste{rewop(3, 2) \free{rewop}} +} +\xtc{ +Now we define \userfun{square} in terms of \userfun{power}. +}{ +\spadpaste{square: FRAC INT -> FRAC INT \bound{squaredec}} +} +\xtc{ +The \spadfunFrom{curryRight}{MappingPackage3} operation creates a unary +function from a binary one by providing a constant +argument on the right. +}{ +\spadpaste{square:= curryRight(power, 2) \free{squaredec poswer}\bound{square}} +} +\xtc{ +Likewise, the +\spadfunFrom{curryLeft}{MappingPackage3} operation provides a constant +argument on the left. +}{ +\spadpaste{square 4 \free{square}} +} +\xtc{ +The \spadfunFrom{constantRight}{MappingPackage3} operation creates +(in a trivial way) a binary function from a unary one: +\spad{constantRight(f)} is the function \spad{g} such that +\spad{g(a,b)= f(a).} +}{ +\spadpaste{squirrel:= constantRight(square)\$MAPPKG3(FRAC INT,FRAC INT,FRAC INT) \free{square}\bound{squirrel}} +} +\xtc{ +Likewise, +\spad{constantLeft(f)} is the function \spad{g} such that \spad{g(a,b)= f(b).} +}{ +\spadpaste{squirrel(1/2, 1/3) \free{squirrel}} +} +\xtc{ +The \spadfunFrom{curry}{MappingPackage2} operation makes a unary function nullary. +}{ +\spadpaste{sixteen := curry(square, 4/1) \free{square}\bound{sixteen}} +} +\xtc{ +}{ +\spadpaste{sixteen() \free{sixteen}} +} +\xtc{ +The \spadopFrom{*}{MappingPackage3} operation +constructs composed functions. +}{ +\spadpaste{square2:=square*square \free{square}\bound{square2}} +} +\xtc{ +}{ +\spadpaste{square2 3 \free{square2}} +} +\xtc{ +Use the \spadopFrom{**}{MappingPackage1} operation to create +functions that are \spad{n}-fold iterations of other functions. +}{ +\spadpaste{sc(x: FRAC INT): FRAC INT == x + 1 \bound{sc}} +} +\xtc{ +This is a list of \pspadtype{Mapping} objects. +}{ +\spadpaste{incfns := [sc**i for i in 0..10] \free{sc}\bound{incfns}} +} +\xtc{ +This is a list of applications of those functions. +}{ +\spadpaste{[f 4 for f in incfns] \free{incfns}} +} +\xtc{ +Use the \spadfunFrom{recur}{MappingPackage1} +operation for recursion: +\spad{g := recur f} means \spad{g(n,x) == f(n,f(n-1,...f(1,x))).} +}{ +\spadpaste{times(n:NNI, i:INT):INT == n*i \bound{rdec}} +} +\xtc{ +}{ +\spadpaste{r := recur(times) \free{rdec}\bound{r}} +} +\xtc{ +This is a factorial function. +}{ +\spadpaste{fact := curryRight(r, 1) \free{r}\bound{fact}} +} +\xtc{ +}{ +\spadpaste{fact 4 \free{fact}} +} +\xtc{ +Constructed functions can be used within other functions. +}{ +\begin{spadsrc}[\free{square}\bound{mto2ton}] +mto2ton(m, n) == + raiser := square**n + raiser m +\end{spadsrc} +} +\xtc{ +This is \texht{$3^{2^3}.$}{\spad{3**(2**3).}} +}{ +\spadpaste{mto2ton(3, 3) \free{mto2ton}} +} +\xtc{ +Here \userfun{shiftfib} is a unary function that modifies its argument. +}{ +\begin{spadsrc}[\bound{shiftfib}] +shiftfib(r: List INT) : INT == + t := r.1 + r.1 := r.2 + r.2 := r.2 + t + t +\end{spadsrc} +} +\xtc{ +By currying over the argument we get a function with private state. +}{ +\spadpaste{fibinit: List INT := [0, 1] \bound{fibinitdec}} +} +\xtc{ +}{ +\spadpaste{fibs := curry(shiftfib, fibinit) \free{shiftfib fibinit}\bound{fibs}} +} +\xtc{ +}{ +\spadpaste{[fibs() for i in 0..30] \free{fibs}} +} +\endscroll +\autobuttons +\end{page} + +@ +\section{mset.ht} +<>= +\newcommand{\MultiSetXmpTitle}{MultiSet} +\newcommand{\MultiSetXmpNumber}{9.53} + +@ +\subsection{MultiSet} +\label{MultiSetXmpPage} +\index{pages!MultiSetXmpPage!mset.ht} +\index{mset.ht!pages!MultiSetXmpPage} +\index{MultiSetXmpPage!mset.ht!pages} +<>= +\begin{page}{MultiSetXmpPage}{MultiSet} +\beginscroll +The domain \spadtype{Multiset(R)} is similar to \spadtype{Set(R)} +except that multiplicities +(counts of duplications) are maintained and displayed. +Use the operation \spadfunFrom{multiset}{Multiset} to create +multisets from lists. +All the standard operations from sets are available for +multisets. +An element with multiplicity greater than one has the +multiplicity displayed first, then a colon, and then the element. + +\xtc{ +Create a multiset of integers. +}{ +\spadpaste{s := multiset [1,2,3,4,5,4,3,2,3,4,5,6,7,4,10]\bound{s}} +} +\xtc{ +The operation \spadfunX{insert} adds an element to a multiset. +}{ +\spadpaste{insert!(3,s)\bound{s1}\free{s}} +} +\xtc{ +Use \spadfunX{remove} to remove an element. +If a third argument is present, it specifies how many instances +to remove. Otherwise all instances of the element are removed. +Display the resulting multiset. +}{ +\spadpaste{remove!(3,s,1); s\bound{s2}\free{s1}} +} +\xtc{ +}{ +\spadpaste{remove!(5,s); s\bound{s2}\free{s1}} +} +\xtc{ +The operation \spadfun{count} returns the number of copies +of a given value. +}{ +\spadpaste{count(5,s)\free{s2}} +} +\xtc{ +A second multiset. +}{ +\spadpaste{t := multiset [2,2,2,-9]\bound{t}} +} +\xtc{ +The \spadfun{union} of two multisets is additive. +}{ +\spadpaste{U := union(s,t)\bound{U}} +} +\xtc{ +The \spadfun{intersect} operation gives the elements that are in +common, with additive multiplicity. +}{ +\spadpaste{I := intersect(s,t)\bound{I}} +} +\xtc{ +The \spadfun{difference} of \spad{s} and \spad{t} consists of +the elements that \spad{s} has but \spad{t} does not. +Elements are regarded as indistinguishable, so that if \spad{s} +and \spad{t} have any element in common, the \spadfun{difference} +does not contain that element. +}{ +\spadpaste{difference(s,t)\free{s2 t}} +} +\xtc{ +The \spadfun{symmetricDifference} is the \spadfun{union} +of \spad{difference(s, t)} and \spad{difference(t, s)}. +}{ +\spadpaste{S := symmetricDifference(s,t)\bound{S}\free{s2 t}} +} +\xtc{ +Check that the \spadfun{union} of the \spadfun{symmetricDifference} and +the \spadfun{intersect} equals the \spadfun{union} of the elements. +}{ +\spadpaste{(U = union(S,I))@Boolean\free{S I U}} +} +\xtc{ +Check some inclusion relations. +}{ +\spadpaste{t1 := multiset [1,2,2,3]; [t1 < t, t1 < s, t < s, t1 <= s]\free{t s2}} +} +\endscroll +\autobuttons +\end{page} + +@ +\section{matrix.ht} +<>= +\newcommand{\MatrixXmpTitle}{Matrix} +\newcommand{\MatrixXmpNumber}{9.52} + +@ +\subsection{Matrix} +\label{MatrixXmpPage} +\begin{itemize} +\item ugxMatrixCreatePage \ref{ugxMatrixCreatePage} on +page~pageref{ugxMatrixCreatePage} +\item ugxMatrixOpsPage \ref{ugxMatrixOpsPage} on +page~pageref{ugxMatrixOpsPage} +\end{itemize} +\index{pages!MatrixXmpPage!matrix.ht} +\index{matrix.ht!pages!MatrixXmpPage} +\index{MatrixXmpPage!matrix.ht!pages} +<>= +\begin{page}{MatrixXmpPage}{Matrix} +\beginscroll + +The \spadtype{Matrix} domain provides arithmetic operations on matrices +and standard functions from linear algebra. +This domain is similar to the \spadtype{TwoDimensionalArray} domain, except +that the entries for \spadtype{Matrix} must belong to a \spadtype{Ring}. + +\beginmenu + \menudownlink{{9.52.1. Creating Matrices}}{ugxMatrixCreatePage} + \menudownlink{{9.52.2. Operations on Matrices}}{ugxMatrixOpsPage} +\endmenu +\endscroll +\autobuttons +\end{page} + +@ +<>= +\newcommand{\ugxMatrixCreateTitle}{Creating Matrices} +\newcommand{\ugxMatrixCreateNumber}{9.52.1.} + +@ +\subsection{Creating Matrices} +\label{ugxMatrixCreatePage} +\index{pages!ugxMatrixCreatePage!matrix.ht} +\index{matrix.ht!pages!ugxMatrixCreatePage} +\index{ugxMatrixCreatePage!matrix.ht!pages} +<>= +\begin{page}{ugxMatrixCreatePage}{Creating Matrices} +\beginscroll + +There are many ways to create a matrix from a collection of +values or from existing matrices. + +\xtc{ +If the matrix has almost all items equal to the same value, +use \spadfunFrom{new}{Matrix} to create a matrix filled with that +value and then reset the entries that are different. +}{ +\spadpaste{m : Matrix(Integer) := new(3,3,0) \bound{m}} +} +\xtc{ +To change the entry in the second row, third column to \spad{5}, use +\spadfunFrom{setelt}{Matrix}. +}{ +\spadpaste{setelt(m,2,3,5) \free{m}\bound{m1}} +} +\xtc{ +An alternative syntax is to use assignment. +}{ +\spadpaste{m(1,2) := 10 \free{m1}\bound{m2}} +} +\xtc{ +The matrix was {\it destructively modified}. +}{ +\spadpaste{m \free{m2}} +} + +\xtc{ +If you already have the matrix entries as a list of lists, +use \spadfunFrom{matrix}{Matrix}. +}{ +\spadpaste{matrix [[1,2,3,4],[0,9,8,7]]} +} + +\xtc{ +If the matrix is diagonal, use +\spadfunFrom{diagonalMatrix}{Matrix}. +}{ +\spadpaste{dm := diagonalMatrix [1,x**2,x**3,x**4,x**5] \bound{dm}} +} +\xtc{ +Use \spadfunFromX{setRow}{Matrix} and +\spadfunFromX{setColumn}{Matrix} to change a row or column of a matrix. +}{ +\spadpaste{setRow!(dm,5,vector [1,1,1,1,1]) \free{dm}\bound{dm1}} +} +\xtc{ +}{ +\spadpaste{setColumn!(dm,2,vector [y,y,y,y,y]) \free{dm1}\bound{dm2}} +} + +% +\xtc{ +Use \spadfunFrom{copy}{Matrix} to make a copy of a matrix. +}{ +\spadpaste{cdm := copy(dm) \free{dm2}\bound{cdm}} +} +\xtc{ +This is useful if you intend to modify a matrix destructively but +want a copy of the original. +}{ +\spadpaste{setelt(dm,4,1,1-x**7) \free{dm2}\bound{setdm}} +} +\xtc{ +}{ +\spadpaste{[dm,cdm] \free{setdm cdm}} +} + +% +\xtc{ +Use \spadfunFrom{subMatrix}{Matrix} to extract part of an +existing matrix. +The syntax is \spad{subMatrix({\it m, firstrow, lastrow, firstcol, +lastcol})}. +}{ +\spadpaste{subMatrix(dm,2,3,2,4) \free{setdm}} +} + +% +\xtc{ +To change a submatrix, use \spadfunFromX{setsubMatrix}{Matrix}. +}{ +\spadpaste{d := diagonalMatrix [1.2,-1.3,1.4,-1.5] \bound{d}} +} +\xtc{ +If \spad{e} is too big to fit where you specify, an error message +is displayed. +Use \spadfunFrom{subMatrix}{Matrix} to extract part of \spad{e}, if +necessary. +}{ +\spadpaste{e := matrix [[6.7,9.11],[-31.33,67.19]] \bound{e}} +} +\xtc{ +This changes the submatrix of \spad{d} whose upper left corner is +at the first row and second column and whose size is that of \spad{e}. +}{ +\spadpaste{setsubMatrix!(d,1,2,e) \free{d e}\bound{d1}} +} +\xtc{ +}{ +\spadpaste{d \free{d1}} +} +% + +% +\xtc{ +Matrices can be joined either horizontally or vertically to make +new matrices. +}{ +\spadpaste{a := matrix [[1/2,1/3,1/4],[1/5,1/6,1/7]] \bound{a}} +} +\xtc{ +}{ +\spadpaste{b := matrix [[3/5,3/7,3/11],[3/13,3/17,3/19]] \bound{b}} +} +\xtc{ +Use \spadfunFrom{horizConcat}{Matrix} to append them side to side. +The two matrices must have the same number of rows. +}{ +\spadpaste{horizConcat(a,b) \free{a b}} +} +\xtc{ +Use \spadfunFrom{vertConcat}{Matrix} to stack one upon the other. +The two matrices must have the same number of columns. +}{ +\spadpaste{vab := vertConcat(a,b) \free{a b}\bound{vab}} +} + +% +\xtc{ +The operation +\spadfunFrom{transpose}{Matrix} is used to create a new matrix by reflection +across the main diagonal. +}{ +\spadpaste{transpose vab \free{vab}} +} + +\endscroll +\autobuttons +\end{page} + +@ +<>= +\newcommand{\ugxMatrixOpsTitle}{Operations on Matrices} +\newcommand{\ugxMatrixOpsNumber}{9.52.2.} + +@ +\subsection{Operations on Matrices} +\label{ugxMatrixOpsPage} +\begin{itemize} +\item ugIntroTwoDimPage \ref{ugIntroTwoDimPage} on +page~pageref{ugIntroTwoDimPage} +\item ugProblemEigenPage \ref{ugProblemEigenPage} on +page~pageref{ugProblemEigenPage} +\item ugxFloatHilbertPage \ref{ugxFloatHilbertPage} on +page~pageref{ugxFloatHilbertPage} +\item PermanentXmpPage \ref{PermanentXmpPage} on +page~pageref{PermanentXmpPage} +\item VectorXmpPage \ref{VectorXmpPage} on +page~pageref{VectorXmpPage} +\item OneDimensionalArrayXmpPage \ref{OneDimensionalArrayXmpPage} on +page~pageref{OneDimensionalArrayXmpPage} +\item TwoDimensionalArrayXmpPage \ref{TwoDimensionalArrayXmpPage} on +page~pageref{TwoDimensionalArrayXmpPage} +\end{itemize} +\index{pages!ugxMatrixOpsPage!matrix.ht} +\index{matrix.ht!pages!ugxMatrixOpsPage} +\index{ugxMatrixOpsPage!matrix.ht!pages} +<>= +\begin{page}{ugxMatrixOpsPage}{Operations on Matrices} +\beginscroll + +\labelSpace{3pc} +\xtc{ +Axiom provides both left and right scalar multiplication. +}{ +\spadpaste{m := matrix [[1,2],[3,4]] \bound{m}} +} +\xtc{ +}{ +\spadpaste{4 * m * (-5)\free{m}} +} +\xtc{ +You can add, subtract, and multiply matrices provided, of course, that +the matrices have compatible dimensions. +If not, an error message is displayed. +}{ +\spadpaste{n := matrix([[1,0,-2],[-3,5,1]]) \bound{n}} +} +\xtc{ +This following product is defined but \spad{n * m} is not. +}{ +\spadpaste{m * n \free{m n}} +} + +The operations \spadfunFrom{nrows}{Matrix} and \spadfunFrom{ncols}{Matrix} +return the number of rows and columns of a matrix. +You can extract a row or a column of a matrix using the operations +\spadfunFrom{row}{Matrix} and \spadfunFrom{column}{Matrix}. +The object returned is a \spadtype{Vector}. +\xtc{ +Here is the third column of the matrix \spad{n}. +}{ +\spadpaste{vec := column(n,3) \free{n} \bound{vec}} +} +\xtc{ +You can multiply a matrix on the left by a ``row vector'' and on the right +by a ``column vector.'' +}{ +\spadpaste{vec * m \free{vec m}} +} +\xtc{ +Of course, the dimensions of the vector and the matrix must be compatible +or an error message is returned. +}{ +\spadpaste{m * vec \free{vec m}} +} + +The operation \spadfunFrom{inverse}{Matrix} computes the inverse of a +matrix if +{Operations on Matrices} +the matrix is invertible, and returns \spad{"failed"} if not. +\xtc{ +This Hilbert matrix is invertible. +}{ +\spadpaste{hilb := matrix([[1/(i + j) for i in 1..3] for j in 1..3]) \bound{hilb}} +} +\xtc{ +}{ +\spadpaste{inverse(hilb) \free{hilb}} +} +\xtc{ +This matrix is not invertible. +}{ +\spadpaste{mm := matrix([[1,2,3,4], [5,6,7,8], [9,10,11,12], [13,14,15,16]]) \bound{mm}} +} +\xtc{ +}{ +\spadpaste{inverse(mm) \free{mm}} +} + +The operation +\spadfunFrom{determinant}{Matrix} computes the determinant of a matrix +{Operations on Matrices} +provided that the entries of the matrix belong to a \spadtype{CommutativeRing}. +\xtc{ +The above matrix \spad{mm} is not invertible and, hence, must have +determinant \spad{0}. +}{ +\spadpaste{determinant(mm) \free{mm}} +} +\xtc{ +The operation +\spadfunFrom{trace}{SquareMatrix} computes the trace of a {\em square} matrix. +{Operations on Matrices} +}{ +\spadpaste{trace(mm) \free{mm}} +} + +\xtc{ +The operation \spadfunFrom{rank}{Matrix} computes the {\it rank} of a +matrix: +the maximal number of linearly independent rows or columns. +}{ +\spadpaste{rank(mm) \free{mm}} +} +\xtc{ +The operation \spadfunFrom{nullity}{Matrix} computes the {\it nullity} of +a matrix: the dimension of its null space. +}{ +\spadpaste{nullity(mm) \free{mm}} +} +\xtc{ +The operation \spadfunFrom{nullSpace}{Matrix} returns a list containing +a basis for the null space of a matrix. +Note that the nullity is the number of elements in a basis for the null space. +}{ +\spadpaste{nullSpace(mm) \free{mm}} +} +\xtc{ +The operation +\spadfunFrom{rowEchelon}{Matrix} returns the row echelon form of a +{Operations on Matrices} +matrix. +It is easy to see that the rank of this matrix is two and that its nullity +is also two. +}{ +\spadpaste{rowEchelon(mm) \free{mm}} +} + +For more information on related topics, see +\downlink{``\ugIntroTwoDimTitle''}{ugIntroTwoDimPage} in Section \ugIntroTwoDimNumber\ignore{ugIntroTwoDim}, +\downlink{``\ugProblemEigenTitle''}{ugProblemEigenPage} in Section \ugProblemEigenNumber\ignore{ugProblemEigen}, +\downlink{``\ugxFloatHilbertTitle''}{ugxFloatHilbertPage} in Section \ugxFloatHilbertNumber\ignore{ugxFloatHilbert}, +\downlink{`Permanent'}{PermanentXmpPage}\ignore{Permanent}, +\downlink{`Vector'}{VectorXmpPage}\ignore{Vector}, +\downlink{`OneDimensionalArray'}{OneDimensionalArrayXmpPage}\ignore{OneDimensionalArray}, and +\downlink{`TwoDimensionalArray'}{TwoDimensionalArrayXmpPage}\ignore{TwoDimensionalArray}. +% +\showBlurb{Matrix} +\endscroll +\autobuttons +\end{page} + +@ +\section{mkfunc.ht} +<>= +\newcommand{\MakeFunctionXmpTitle}{MakeFunction} +\newcommand{\MakeFunctionXmpNumber}{9.50} + +@ +\subsection{MakeFunction} +\label{MakeFunctionXmpPage} +See ugUserMakePage \ref{ugUserMakePage} on page~\pageref{ugUserMakePage} +\index{pages!MakeFunctionXmpPage!mkfunc.ht} +\index{mkfunc.ht!pages!MakeFunctionXmpPage} +\index{MakeFunctionXmpPage!mkfunc.ht!pages} +<>= +\begin{page}{MakeFunctionXmpPage}{MakeFunction} +\beginscroll +It is sometimes useful to be able to define a function given by +the result of a calculation. +% +\xtc{ +Suppose that you have obtained the following expression +after several computations +and that you now want to tabulate the numerical values of \spad{f} +for \spad{x} between \spad{-1} and \spad{+1} with increment +\spad{0.1}. +}{ +\spadpaste{expr := (x - exp x + 1)**2 * (sin(x**2) * x + 1)**3 \bound{expr}} +} +% +You could, of course, use the function +\spadfunFrom{eval}{Expression} within a loop and evaluate +\spad{expr} twenty-one times, but this would be quite slow. +A better way is to create a numerical function \spad{f} such that +\spad{f(x)} is defined by the expression \spad{expr} above, +but without retyping \spad{expr}! +The package \spadtype{MakeFunction} provides the operation +\spadfunFrom{function}{MakeFunction} which does exactly this. +% +\xtc{ +Issue this to create the function \spad{f(x)} given by \spad{expr}. +}{ +\spadpaste{function(expr, f, x) \bound{f}\free{expr}} +} +\xtc{ +To tabulate \spad{expr}, we can now quickly evaluate \spad{f} 21 times. +}{ +\spadpaste{tbl := [f(0.1 * i - 1) for i in 0..20]; \free{f}\bound{tbl}} +} +% +% +\xtc{ +Use the list \spad{[x1,...,xn]} as the +third argument to \spadfunFrom{function}{MakeFunction} +to create a multivariate function \spad{f(x1,...,xn)}. +}{ +\spadpaste{e := (x - y + 1)**2 * (x**2 * y + 1)**2 \bound{e}} +} +\xtc{ +}{ +\spadpaste{function(e, g, [x, y]) \free{e}} +} +% +% +\xtc{ +In the case of just two +variables, they can be given as arguments without making them into a list. +}{ +\spadpaste{function(e, h, x, y) \free{e}\bound{h}} +} +% +% +\xtc{ +Note that the functions created by \spadfunFrom{function}{MakeFunction} +are not limited to floating point numbers, but can be applied to any type +for which they are defined. +}{ +\spadpaste{m1 := squareMatrix [[1, 2], [3, 4]] \bound{m1}} +} +\xtc{ +}{ +\spadpaste{m2 := squareMatrix [[1, 0], [-1, 1]] \bound{m2}} +} +\xtc{ +}{ +\spadpaste{h(m1, m2) \free{h m1 m2}} +} +% +For more information, see \downlink{``\ugUserMakeTitle''}{ugUserMakePage} +in Section \ugUserMakeNumber\ignore{ugUserMake}. +\showBlurb{MakeFunction} +\endscroll +\autobuttons +\end{page} + +@ +\section{mpoly.ht} +<>= +\newcommand{\MultivariatePolynomialXmpTitle}{MultivariatePolynomial} +\newcommand{\MultivariatePolynomialXmpNumber}{9.54} + +@ +\subsection{MultivariatePolynomial} +\label{MultivariatePolynomialXmpPage} +\begin{itemize} +\item PolynomialXmpPage \ref{PolynomialXmpPage} on +page~pageref{PolynomialXmpPage} +\item UnivariatePolynomialXmpPage \ref{UnivariatePolynomialXmpPage} on +page~pageref{UnivariatePolynomialXmpPage} +\item DistributedMultivariatePolynomialXmpPage \ref{DistributedMultivariatePolynomialXmpPage} on +page~pageref{DistributedMultivariatePolynomialXmpPage} +\end{itemize} +\index{pages!MultivariatePolynomialXmpPage!mpoly.ht} +\index{mpoly.ht!pages!MultivariatePolynomialXmpPage} +\index{MultivariatePolynomialXmpPage!mpoly.ht!pages} +<>= +\begin{page}{MultivariatePolynomialXmpPage}{MultivariatePolynomial} +\beginscroll + +The domain constructor \spadtype{MultivariatePolynomial} is similar to +\spadtype{Polynomial} except that it specifies the variables to be used. +\spadtype{Polynomial} are available for \spadtype{MultivariatePolynomial}. +The abbreviation for \spadtype{MultivariatePolynomial} is +\spadtype{MPOLY}. +The type expressions +\centerline{{\spadtype{MultivariatePolynomial([x,y],Integer)}}} +and +\centerline{{\spadtype{MPOLY([x,y],INT)}}} +refer to the domain of multivariate polynomials in the variables +\spad{x} and \spad{y} where the coefficients are restricted to be integers. +The first +variable specified is the main variable and the display of the +polynomial reflects this. +\xtc{ +This polynomial appears with +terms in descending powers of the variable \spad{x}. +}{ +\spadpaste{m : MPOLY([x,y],INT) := (x**2 - x*y**3 +3*y)**2 \bound{m}} +} +\xtc{ +It is easy to see a different variable ordering by doing a conversion. +}{ +\spadpaste{m :: MPOLY([y,x],INT) \free{m}} +} +\xtc{ +You can use other, unspecified variables, by using +\spadtype{Polynomial} in the coefficient type of \spadtype{MPOLY}. +}{ +\spadpaste{p : MPOLY([x,y],POLY INT) \bound{pdec}} +} +\xtc{ +}{ +\spadpaste{p := (a**2*x - b*y**2 + 1)**2 \free{pdec}\bound{p}} +} +\xtc{ +Conversions can be used to re-express such polynomials in terms of +the other variables. For example, you can first push all the +variables into a polynomial with integer coefficients. +}{ +\spadpaste{p :: POLY INT \free{p}\bound{prev}} +} +\xtc{ +Now pull out the variables of interest. +}{ +\spadpaste{\% :: MPOLY([a,b],POLY INT) \free{prev}} +} + +\beginImportant +\noindent {\bf Restriction:} +\texht{\begin{quotation}\noindent}{\newline\indent{5}} +Axiom does not allow you to create types where +\spadtype{MultivariatePolynomial} is contained in the coefficient type of +\spadtype{Polynomial}. Therefore, +\spad{MPOLY([x,y],POLY INT)} is legal but +\spad{POLY MPOLY([x,y],INT)} is not. +\texht{\end{quotation}}{\indent{0}} +\endImportant + +\xtc{ +Multivariate polynomials may be combined with univariate polynomials +to create types with special structures. +}{ +\spadpaste{q : UP(x, FRAC MPOLY([y,z],INT)) \bound{qdec}} +} +\xtc{ +This is a polynomial in \spad{x} whose coefficients are +quotients of polynomials in \spad{y} and \spad{z}. +}{ +\spadpaste{q := (x**2 - x*(z+1)/y +2)**2 \free{qdec}\bound{q}} +} +\xtc{ +Use conversions for structural rearrangements. +\spad{z} does not appear in a denominator and so it can be made +the main variable. +}{ +\spadpaste{q :: UP(z, FRAC MPOLY([x,y],INT)) \free{q}} +} +\xtc{ +Or you can make a multivariate polynomial in \spad{x} and \spad{z} +whose coefficients are fractions in polynomials in \spad{y}. +}{ +\spadpaste{q :: MPOLY([x,z], FRAC UP(y,INT)) \free{q}} +} +A conversion like \spad{q :: MPOLY([x,y], FRAC UP(z,INT))} is not +possible in this example +because \spad{y} appears in the denominator of a fraction. +As you can see, Axiom provides extraordinary flexibility in the +manipulation and display of expressions via its conversion facility. + +For more information on related topics, see +\downlink{`Polynomial'}{PolynomialXmpPage}\ignore{Polynomial}, +\downlink{`UnivariatePolynomial'}{UnivariatePolynomialXmpPage}\ignore{UnivariatePolynomial}, and +\downlink{`DistributedMultivariatePolynomial'}{DistributedMultivariatePolynomialXmpPage}\ignore{DistributedMultivariatePolynomial}. +\showBlurb{MultivariatePolynomial} +\endscroll +\autobuttons +\end{page} + +@ +\section{nagaux.ht} +\subsection{NAG On-line Documentation} +\label{manpageXXonline} +\index{pages!manpageXXonline!nagaux.ht} +\index{nagaux.ht!pages!manpageXXonline} +\index{manpageXXonline!nagaux.ht!pages} +<>= +\begin{page}{manpageXXonline}{NAG On-line Documentation} +\beginscroll +\begin{verbatim} + + + + DOC INTRO(3NAG) Foundation Library (12/10/92) DOC INTRO(3NAG) + + + + Introduction to NAG On-Line Documentation + + + The on-line documentation for the NAG Foundation Library has been + generated automatically from the same base material used to + create the printed Reference Manual. To make the documentation + readable on the widest range of machines, only the basic set of + ascii characters has been used. + + Certain mathematical symbols have been constructed using plain + ascii characters: + + integral signs / + | + + / + + summation signs -- + > + -- + + square root signs ---- + / + \/ + + + Large brackets are constructed using vertical stacks of the + equivalent ascii character: + + ( ) [ ] { } | + ( ) [ ] { } | + ( ) [ ] { } | + + + Fractions are represented as: + + a + --- + x+1 + + + Greek letters are represented by their names enclosed in round + brackets: + + (alpha) (beta) (gamma) ..... + + (Alpha) (Beta) (Gamma) ..... + + + Some characters are accented using: + + ^ ~ + X X X + + Other mathematical symbols are represented as follows: + + * times + + <=> left-right arrow + + <- left arrow + + ~ similar to + + ~= similar or equal to + + == equivalent to + + >= greater than or equal to + + <= less than or equal to + + >> much greater than + + << much less than + + >~ greater than or similar to + + /= not equal to + + dd partial derivative + + +- plus or minus + + (nabla) Nabla + +\end{verbatim} +\endscroll +\end{page} + +@ +\subsection{NAG Documentation: summary} +\label{manpageXXsummary} +\index{pages!manpageXXsummary!nagaux.ht} +\index{nagaux.ht!pages!manpageXXsummary} +\index{manpageXXsummary!nagaux.ht!pages} +<>= +\begin{page}{manpageXXsummary}{NAG Documentation: summary} +\beginscroll +\begin{verbatim} + + + + SUMMARY(3NAG) Foundation Library (12/10/92) SUMMARY(3NAG) + + + + Introduction List of Routines + List of Routines + + The NAG Foundation Library contains three categories of routines + which can be called by users. They are listed separately in the + three sections below. + + Fully Documented Routines + 254 routines, for each of which an individual routine + document is provided. These are regarded as the primary + contents of the Foundation Library. + + Fundamental Support Routines + 83 comparatively simple routines which are documented in + compact form in the relevant Chapter Introductions (F06, + X01, X02). + + Routines from the NAG Fortran Library + An additional 167 routines from the NAG Fortran Library, + which are used as auxiliaries in the Foundation Library. + They are not documented in this publication, but can be + called if you are already familiar with their use in the + Fortran Library. Only their names are given here. + + + Note: all the routines in the above categories have names ending + in 'F'. Occasionally this publication may refer to routines whose + names end in some other letter (e.g. 'Z', 'Y', 'X'). These are + auxiliary routines whose names may be passed as parameters to a + Foundation Library routine; you only need to know their names, + not how to call them directly. + + Fully Documented Routines + + The Foundation Library contains 254 user-callable routines, for + each of which an individual routine document is provided, in the + following chapters: + + C02 -- Zeros of Polynomials + + C02AFF All zeros of complex polynomial, modified Laguerre method + + C02AGF All zeros of real polynomial, modified Laguerre method + + C05 -- Roots of One or More Transcendental Equations + + C05ADF Zero of continuous function in given interval, Bus and + Dekker algorithm + + C05NBF Solution of system of nonlinear equations using function + values only + + C05PBF Solution of system of nonlinear equations using 1st + derivatives + + C05ZAF Check user's routine for calculating 1st derivatives + + C06 -- Summation of Series + + C06EAF Single 1-D real discrete Fourier transform, no extra + workspace + + C06EBF Single 1-D Hermitian discrete Fourier transform, no extra + workspace + + C06ECF Single 1-D complex discrete Fourier transform, no extra + workspace + + C06EKF Circular convolution or correlation of two real vectors, + no extra workspace + + C06FPF Multiple 1-D real discrete Fourier transforms + + C06FQF Multiple 1-D Hermitian discrete Fourier transforms + + C06FRF Multiple 1-D complex discrete Fourier transforms + + C06FUF 2-D complex discrete Fourier transform + + C06GBF Complex conjugate of Hermitian sequence + + C06GCF Complex conjugate of complex sequence + + C06GQF Complex conjugate of multiple Hermitian sequences + + C06GSF Convert Hermitian sequences to general complex sequences + + D01 -- Quadrature + + D01AJF 1-D quadrature, adaptive, finite interval, strategy due + to Piessens and de Doncker, allowing for badly-behaved + integrands + + D01AKF 1-D quadrature, adaptive, finite interval, method + suitable for oscillating functions + + D01ALF 1-D quadrature, adaptive, finite interval, allowing for + singularities at user-specified break-points + + D01AMF 1-D quadrature, adaptive, infinite or semi-infinite + interval + + D01ANF 1-D quadrature, adaptive, finite interval, weight + function cos((omega)x) or sin((omega)x) + + D01APF 1-D quadrature, adaptive, finite interval, weight + function with end-point singularities of algebraico- + logarithmic type + + D01AQF 1-D quadrature, adaptive, finite interval, weight + function 1/(x-c), Cauchy principal value (Hilbert + transform) + + D01ASF 1-D quadrature, adaptive, semi-infinite interval, weight + function cos((omega)x) or sin((omega)x) + + D01BBF Weights and abscissae for Gaussian quadrature rules + + D01FCF Multi-dimensional adaptive quadrature over hyper- + rectangle + + D01GAF 1-D quadrature, integration of function defined by data + values, Gill-Miller method + + D01GBF Multi-dimensional quadrature over hyper-rectangle, Monte + Carlo method + + D02 -- Ordinary Differential Equations + + D02BBF ODEs, IVP, Runge-Kutta-Merson method, over a range, + intermediate output + + D02BHF ODEs, IVP, Runge-Kutta-Merson method, until function of + solution is zero + + D02CJF ODEs, IVP, Adams method, until function of solution is + zero, intermediate output + + D02EJF ODEs, stiff IVP, BDF method, until function of solution + is zero, intermediate output + + D02GAF ODEs, boundary value problem, finite difference technique + with deferred correction, simple nonlinear problem + + D02GBF ODEs, boundary value problem, finite difference technique + with deferred correction, general linear problem + + D02KEF 2nd order Sturm-Liouville problem, regular/singular + system, finite/infinite range, eigenvalue and + eigenfunction, user-specified break-points + + D02RAF ODEs, general nonlinear boundary value problem, finite + difference technique with deferred correction, + continuation facility + + D03 -- Partial Differential Equations + + D03EDF Elliptic PDE, solution of finite difference equations by + a multigrid technique + + D03EEF Discretize a 2nd order elliptic PDE on a rectangle + + D03FAF Elliptic PDE, Helmholtz equation, 3-D Cartesian co- + ordinates + + E01 -- Interpolation + + E01BAF Interpolating functions, cubic spline interpolant, one + variable + + E01BEF Interpolating functions, monotonicity-preserving, + piecewise cubic Hermite, one variable + + E01BFF Interpolated values, interpolant computed by E01BEF, + function only, one variable + + E01BGF Interpolated values, interpolant computed by E01BEF, + function and 1st derivative, one variable + + E01BHF Interpolated values, interpolant computed by E01BEF, + definite integral, one variable + + E01DAF Interpolating functions, fitting bicubic spline, data on + rectangular grid + + E01SAF Interpolating functions, method of Renka and Cline, two + variables + + E01SBF Interpolated values, evaluate interpolant computed by + E01SAF, two variables + + E01SEF Interpolating functions, modified Shepard's method, two + variables + + E01SFF Interpolated values, evaluate interpolant computed by + E01SEF, two variables + + E02 -- Curve and Surface Fitting + + E02ADF Least-squares curve fit, by polynomials, arbitrary data + points + + E02AEF Evaluation of fitted polynomial in one variable from + Chebyshev series form (simplified parameter list) + + E02AGF Least-squares polynomial fit, values and derivatives may + be constrained, arbitrary data points, + + E02AHF Derivative of fitted polynomial in Chebyshev series form + + E02AJF Integral of fitted polynomial in Chebyshev series form + + E02AKF Evaluation of fitted polynomial in one variable, from + Chebyshev series form + + E02BAF Least-squares curve cubic spline fit (including + interpolation) + + E02BBF Evaluation of fitted cubic spline, function only + + E02BCF Evaluation of fitted cubic spline, function and + derivatives + + E02BDF Evaluation of fitted cubic spline, definite integral + + E02BEF Least-squares cubic spline curve fit, automatic knot + placement + + E02DAF Least-squares surface fit, bicubic splines + + E02DCF Least-squares surface fit by bicubic splines with + automatic knot placement, data on rectangular grid + + E02DDF Least-squares surface fit by bicubic splines with + automatic knot placement, scattered data + + E02DEF Evaluation of a fitted bicubic spline at a vector of + points + + E02DFF Evaluation of a fitted bicubic spline at a mesh of points + + E02GAF L -approximation by general linear function + 1 + + E02ZAF Sort 2-D data into panels for fitting bicubic splines + + E04 -- Minimizing or Maximizing a Function + + E04DGF Unconstrained minimum, pre-conditioned conjugate gradient + algorithm, function of several variables using 1st + derivatives + + E04DJF Read optional parameter values for E04DGF from external + file + + E04DKF Supply optional parameter values to E04DGF + + E04FDF Unconstrained minimum of a sum of squares, combined + Gauss-Newton and modified Newton algorithm using function + values only + + E04GCF Unconstrained minimum of a sum of squares, combined + Gauss-Newton and quasi-Newton algorithm, using 1st + derivatives + + E04JAF Minimum, function of several variables, quasi-Newton + algorithm, simple bounds, using function values only + + E04MBF Linear programming problem + + E04NAF Quadratic programming problem + + E04UCF Minimum, function of several variables, sequential QP + method, nonlinear constraints, using function values and + optionally 1st derivatives + + E04UDF Read optional parameter values for E04UCF from external + file + + E04UEF Supply optional parameter values to E04UCF + + E04YCF Covariance matrix for nonlinear least-squares problem + + F01 -- Matrix Factorizations + + F01BRF LU factorization of real sparse matrix + + F01BSF LU factorization of real sparse matrix with known + sparsity pattern + + T + F01MAF LL factorization of real sparse symmetric positive- + definite matrix + + T + F01MCF LDL factorization of real symmetric positive-definite + variable-bandwidth matrix + + F01QCF QR factorization of real m by n matrix (m>=n) + + T + F01QDF Operations with orthogonal matrices, compute QB or Q B + after factorization by F01QCF + + F01QEF Operations with orthogonal matrices, form columns of Q + after factorization by F01QCF + + F01RCF QR factorization of complex m by n matrix (m>=n) + + H + F01RDF Operations with unitary matrices, compute QB or Q B after + factorization by F01RCF + + F01REF Operations with unitary matrices, form columns of Q after + factorization by F01RCF + + F02 -- Eigenvalues and Eigenvectors + + F02AAF All eigenvalues of real symmetric matrix + + F02ABF All eigenvalues and eigenvectors of real symmetric matrix + + F02ADF All eigenvalues of generalized real symmetric-definite + eigenproblem + + F02AEF All eigenvalues and eigenvectors of generalized real + symmetric-definite eigenproblem + + F02AFF All eigenvalues of real matrix + + F02AGF All eigenvalues and eigenvectors of real matrix + + F02AJF All eigenvalues of complex matrix + + F02AKF All eigenvalues and eigenvectors of complex matrix + + F02AWF All eigenvalues of complex Hermitian matrix + + F02AXF All eigenvalues and eigenvectors of complex Hermitian + matrix + + F02BBF Selected eigenvalues and eigenvectors of real symmetric + matrix + + F02BJF All eigenvalues and optionally eigenvectors of + generalized eigenproblem by QZ algorithm, real matrices + + F02FJF Selected eigenvalues and eigenvectors of sparse symmetric + eigenproblem + + F02WEF SVD of real matrix + + F02XEF SVD of complex matrix + + F04 -- Simultaneous Linear Equations + + F04ADF Approximate solution of complex simultaneous linear + equations with multiple right-hand sides + + F04ARF Approximate solution of real simultaneous linear + equations, one right-hand side + + F04ASF Accurate solution of real symmetric positive-definite + simultaneous linear equations, one right-hand side + + F04ATF Accurate solution of real simultaneous linear equations, + one right-hand side + + F04AXF Approximate solution of real sparse simultaneous linear + equations (coefficient matrix already factorized by + F01BRF or F01BSF) + + F04FAF Approximate solution of real symmetric positive-definite + tridiagonal simultaneous linear equations, one right-hand + side + + F04JGF Least-squares (if rank = n) or minimal least-squares (if + rank =n + + F04MAF Real sparse symmetric positive-definite simultaneous + linear equations (coefficient matrix already factorized) + + F04MBF Real sparse symmetric simultaneous linear equations + + F04MCF Approximate solution of real symmetric positive-definite + variable-bandwidth simultaneous linear equations + (coefficient matrix already factorized) + + F04QAF Sparse linear least-squares problem, m real equations in + n unknowns + + F07 -- Linear Equations (LAPACK) + + F07ADF (DGETRF) LU factorization of real m by n matrix + + F07AEF (DGETRS) Solution of real system of linear equations, + multiple right-hand sides, matrix already factorized by + F07ADF + + F07FDF (DPOTRF) Cholesky factorization of real symmetric + positive-definite matrix + + F07FEF (DPOTRS) Solution of real symmetric positive-definite + system of linear equations, multiple right-hand sides, + matrix already factorized by F07FDF + + G01 -- Simple Calculations on Statistical Data + + G01AAF Mean, variance, skewness, kurtosis etc, one variable, + from raw data + + G01ADF Mean, variance, skewness, kurtosis etc, one variable, + from frequency table + + G01AEF Frequency table from raw data + + + G01AFF Two-way contingency table analysis, with (chi) /Fisher's + exact test + + G01ALF Computes a five-point summary (median, hinges and + extremes) + + G01ARF Constructs a stem and leaf plot + + G01EAF Computes probabilities for the standard Normal + distribution + + G01EBF Computes probabilities for Student's t-distribution + + 2 + G01ECF Computes probabilities for (chi) distribution + + G01EDF Computes probabilities for F-distribution + + G01EEF Computes upper and lower tail probabilities and + probability density function for the beta distribution + + G01EFF Computes probabilities for the gamma distribution + + G01FAF Computes deviates for the standard Normal distribution + + G01FBF Computes deviates for Student's t-distribution + + 2 + G01FCF Computes deviates for the (chi) distribution + + G01FDF Computes deviates for the F-distribution + + G01FEF Computes deviates for the beta distribution + + G01FFF Computes deviates for the gamma distribution + + G01HAF Computes probabilities for the bivariate Normal + distribution + + G02 -- Correlation and Regression Analysis + + G02BNF Kendall/Spearman non-parametric rank correlation + coefficients, no missing values, overwriting input data + + G02BQF Kendall/Spearman non-parametric rank correlation + coefficients, no missing values, preserving input data + + G02BXF Computes (optionally weighted) correlation and covariance + matrices + + G02CAF Simple linear regression with constant term, no missing + values + + G02DAF Fits a general (multiple) linear regression model + + G02DGF Fits a general linear regression model for new dependent + variable + + G02DNF Computes estimable function of a general linear + regression model and its standard error + + G02FAF Calculates standardized residuals and influence + statistics + + G02GBF Fits a generalized linear model with binomial errors + + G02GCF Fits a generalized linear model with Poisson errors + + G03 -- Multivariate Methods + + G03AAF Performs principal component analysis + + G03ADF Performs canonical correlation analysis + + G03BAF Computes orthogonal rotations for loading matrix, + generalized orthomax criterion + + G05 -- Random Number Generators + + G05CAF Pseudo-random double precision numbers, uniform + distribution over (0,1) + + G05CBF Initialise random number generating routines to give + repeatable sequence + + G05CCF Initialise random number generating routines to give non- + repeatable sequence + + G05CFF Save state of random number generating routines + + G05CGF Restore state of random number generating routines + + G05DDF Pseudo-random double precision numbers, Normal + distribution + + G05DFF Pseudo-random double precision numbers, Cauchy + distribution + + G05DPF Pseudo-random double precision numbers, Weibull + distribution + + G05DYF Pseudo-random integer from uniform distribution + + G05DZF Pseudo-random logical (boolean) value + + G05EAF Set up reference vector for multivariate Normal + distribution + + G05ECF Set up reference vector for generating pseudo-random + integers, Poisson distribution + + G05EDF Set up reference vector for generating pseudo-random + integers, binomial distribution + + G05EHF Pseudo-random permutation of an integer vector + + G05EJF Pseudo-random sample from an integer vector + + G05EXF Set up reference vector from supplied cumulative + distribution function or probability distribution + function + + G05EYF Pseudo-random integer from reference vector + + G05EZF Pseudo-random multivariate Normal vector from reference + vector + + G05FAF Generates a vector of pseudo-random numbers from a + uniform distribution + + G05FBF Generates a vector of pseudo-random numbers from a + (negative) exponential distribution + + G05FDF Generates a vector of pseudo-random numbers from a Normal + distribution + + G05FEF Generates a vector of pseudo-random numbers from a beta + distribution + + G05FFF Generates a vector of pseudo-random numbers from a gamma + distribution + + G05HDF Generates a realisation of a multivariate time series + from a VARMA model + + G08 -- Nonparameteric Statistics + + G08AAF Sign test on two paired samples + + G08ACF Median test on two samples of unequal size + + G08AEF Friedman two-way analysis of variance on k matched + samples + + G08AFF Kruskal-Wallis one-way analysis of variance on k samples + of unequal size + + G08AGF Performs the Wilcoxon one sample (matched pairs) signed + rank test + + G08AHF Performs the Mann-Whitney U test on two independent + samples + + G08AJF Computes the exact probabilities for the Mann-Whitney U + statistic, no ties in pooled sample + + G08AKF Computes the exact probabilities for the Mann-Whitney U + statistic, ties in pooled sample + + 2 + G08CGF Performs the (chi) goodness of fit test, for standard + continuous distributions + + G13 -- Time Series Analysis + + G13AAF Univariate time series, seasonal and non-seasonal + differencing + + G13ABF Univariate time series, sample autocorrelation function + + G13ACF Univariate time series, partial autocorrelations from + autocorrelations + + G13ADF Univariate time series, preliminary estimation, seasonal + ARIMA model + + G13AFF Univariate time series, estimation, seasonal ARIMA model + + G13AGF Univariate time series, update state set for forecasting + + G13AHF Univariate time series, forecasting from state set + + G13AJF Univariate time series, state set and forecasts, from + fully specified seasonal ARIMA model + + G13ASF Univariate time series, diagnostic checking of residuals, + following G13AFF + + G13BAF Multivariate time series, filtering (pre-whitening) by an + ARIMA model + + G13BCF Multivariate time series, cross correlations + + G13BDF Multivariate time series, preliminary estimation of + transfer function model + + G13BEF Multivariate time series, estimation of multi-input model + + G13BJF Multivariate time series, state set and forecasts from + fully specified multi-input model + + G13CBF Univariate time series, smoothed sample spectrum using + spectral smoothing by the trapezium frequency (Daniell) + window + + G13CDF Multivariate time series, smoothed sample cross spectrum + using spectral smoothing by the trapezium frequency + (Daniell) window + + M01 -- Sorting + + M01CAF Sort a vector, double precision numbers + + M01DAF Rank a vector, double precision numbers + + M01DEF Rank rows of a matrix, double precision numbers + + M01DJF Rank columns of a matrix, double precision numbers + + M01EAF Rearrange a vector according to given ranks, double + precision numbers + + M01ZAF Invert a permutation + + S -- Approximations of Special Functions + + z + S01EAF Complex exponential, e + + S13AAF Exponential integral E (x) + 1 + + S13ACF Cosine integral Ci(x) + + S13ADF Sine integral Si(x) + + S14AAF Gamma function + + S14ABF Log Gamma function + + S14BAF Incomplete gamma functions P(a,x) and Q(a,x) + + S15ADF Complement of error function erfc x + + S15AEF Error function erf x + + S17ACF Bessel function Y (x) + 0 + + S17ADF Bessel function Y (x) + 1 + + S17AEF Bessel function J (x) + 0 + + S17AFF Bessel function J (x) + 1 + + S17AGF Airy function Ai(x) + + S17AHF Airy function Bi(x) + + S17AJF Airy function Ai'(x) + + S17AKF Airy function Bi'(x) + + S17DCF Bessel functions Y (z), real a>=0, complex z, + (nu)+a + (nu)=0,1,2,... + + S17DEF Bessel functions J (z), real a>=0, complex z, + (nu)+a + (nu)=0,1,2,... + + S17DGF Airy functions Ai(z) and Ai'(z), complex z + + S17DHF Airy functions Bi(z) and Bi'(z), complex z + + (j) + S17DLF Hankel functions H (z), j=1,2, real a>=0, complex z, + (nu)+a + (nu)=0,1,2,... + + S18ACF Modified Bessel function K (x) + 0 + + S18ADF Modified Bessel function K (x) + 1 + + S18AEF Modified Bessel function I (x) + 0 + + S18AFF Modified Bessel function I (x) + 1 + + S18DCF Modified Bessel functions K (z), real a>=0, complex + (nu)+a + z, (nu)=0,1,2,... + + S18DEF Modified Bessel functions I (z), real a>=0, complex + (nu)+a + z, (nu)=0,1,2,... + + S19AAF Kelvin function ber x + + S19ABF Kelvin function bei x + + S19ACF Kelvin function ker x + + S19ADF Kelvin function kei x + + S20ACF Fresnel integral S(x) + + S20ADF Fresnel integral C(x) + + S21BAF Degenerate symmetrised elliptic integral of 1st kind + R (x,y) + C + + S21BBF Symmetrised elliptic integral of 1st kind R (x,y,z) + F + + S21BCF Symmetrised elliptic integral of 2nd kind R (x,y,z) + D + + S21BDF Symmetrised elliptic integral of 3rd kind R (x,y,z,r) + J + + X04 -- Input/Output Utilities + + X04AAF Return or set unit number for error messages + + X04ABF Return or set unit number for advisory messages + + X04CAF Print a real general matrix + + X04DAF Print a complex general matrix + + X05 -- Date and Time Utilities + + X05AAF Return date and time as an array of integers + + X05ABF Convert array of integers representing date and time to + character string + + X05ACF Compare two character strings representing date and time + + X05BAF Return the CPU time + + + + Fundamental Support Routines + + The following fundamental support routines are provided and are + documented in compact form in the relevant chapter introductory + material: + + F06 -- Linear Algebra Support Routines + + F06AAF (DROTG) Generate real plane rotation + + F06EAF (DDOT) Dot product of two real vectors + + F06ECF (DAXPY) Add scalar times real vector to real vector + + F06EDF (DSCAL) Multiply real vector by scalar + + F06EFF (DCOPY) Copy real vector + + F06EGF (DSWAP) Swap two real vectors + + F06EJF (DNRM2) Compute Euclidean norm of real vector + + F06EKF (DASUM) Sum the absolute values of real vector elements + + F06EPF (DROT) Apply real plane rotation + + F06GAF (ZDOTU) Dot product of two complex vectors, unconjugated + + F06GBF (ZDOTC) Dot product of two complex vectors, conjugated + + F06GCF (ZAXPY) Add scalar times complex vector to complex vector + + F06GDF (ZSCAL) Multiply complex vector by complex scalar + + F06GFF (ZCOPY) Copy complex vector + + F06GGF (ZSWAP) Swap two complex vectors + + F06JDF (ZDSCAL) Multiply complex vector by real scalar + + F06JJF (DZNRM2) Compute Euclidean norm of complex vector + + F06JKF (DZASUM) Sum the absolute values of complex vector + elements + + F06JLF (IDAMAX) Index, real vector element with largest absolute + value + + F06JMF (IZAMAX) Index, complex vector element with largest + absolute value + + F06PAF (DGEMV) Matrix-vector product, real rectangular matrix + + F06PBF (DGBMV) Matrix-vector product, real rectangular band + matrix + + F06PCF (DSYMV) Matrix-vector product, real symmetric matrix + + F06PDF (DSBMV) Matrix-vector product, real symmetric band matrix + + F06PEF (DSPMV) Matrix-vector product, real symmetric packed + matrix + + F06PFF (DTRMV) Matrix-vector product, real triangular matrix + + F06PGF (DTBMV) Matrix-vector product, real triangular band + matrix + + F06PHF (DTPMV) Matrix-vector product, real triangular packed + matrix + + F06PJF (DTRSV) System of equations, real triangular matrix + + F06PKF (DTBSV) System of equations, real triangular band matrix + + F06PLF (DTPSV) System of equations, real triangular packed + matrix + + F06PMF (DGER) Rank-1 update, real rectangular matrix + + F06PPF (DSYR) Rank-1 update, real symmetric matrix + + F06PQF (DSPR) Rank-1 update, real symmetric packed matrix + + F06PRF (DSYR2) Rank-2 update, real symmetric matrix + + F06PSF (DSPR2) Rank-2 update, real symmetric packed matrix + + F06SAF (ZGEMV) Matrix-vector product, complex rectangular matrix + + F06SBF (ZGBMV) Matrix-vector product, complex rectangular band + matrix + + F06SCF (ZHEMV) Matrix-vector product, complex Hermitian matrix + + F06SDF (ZHBMV) Matrix-vector product, complex Hermitian band + matrix + + F06SEF (ZHPMV) Matrix-vector product, complex Hermitian packed + matrix + + F06SFF (ZTRMV) Matrix-vector product, complex triangular matrix + + F06SGF (ZTBMV) Matrix-vector product, complex triangular band + matrix + + F06SHF (ZTPMV) Matrix-vector product, complex triangular packed + matrix + + F06SJF (ZTRSV) System of equations, complex triangular matrix + + F06SKF (ZTBSV) System of equations, complex triangular band + matrix + + F06SLF (ZTPSV) System of equations, complex triangular packed + matrix + + F06SMF (ZGERU) Rank-1 update, complex rectangular matrix, + unconjugated vector + + F06SNF (ZGERC) Rank-1 update, complex rectangular matrix, + conjugated vector + + F06SPF (ZHER) Rank-1 update, complex Hermitian matrix + + F06SQF (ZHPR) Rank-1 update, complex Hermitian packed matrix + + F06SRF (ZHER2) Rank-2 update, complex Hermitian matrix + + F06SSF (ZHPR2) Rank-2 update, complex Hermitian packed matrix + + F06YAF (DGEMM) Matrix-matrix product, two real rectangular + matrices + + F06YCF (DSYMM) Matrix-matrix product, one real symmetric matrix, + one real rectangular matrix + + F06YFF (DTRMM) Matrix-matrix product, one real triangular + matrix, one real rectangular matrix + + F06YJF (DTRSM) Solves a system of equations with multiple right- + hand sides, real triangular coefficient matrix + + F06YPF (DSYRK) Rank-k update of a real symmetric matrix + + F06YRF (DSYR2K) Rank-2k update of a real symmetric matrix + + F06ZAF (ZGEMM) Matrix-matrix product, two complex rectangular + matrices + + F06ZCF (ZHEMM) Matrix-matrix product, one complex Hermitian + matrix, one complex rectangular matrix + + F06ZFF (ZTRMM) Matrix-matrix product, one complex triangular + matrix, one complex rectangular matrix + + F06ZJF (ZTRSM) Solves system of equations with multiple right- + hand sides, complex triangular coefficient matrix + + F06ZPF (ZHERK) Rank-k update of a complex Hermitian matrix + + F06ZRF (ZHER2K) Rank-2k update of a complex Hermitian matrix + + F06ZTF (ZSYMM) Matrix-matrix product, one complex symmetric + matrix, one complex rectangular matrix + + F06ZUF (ZSYRK) Rank-k update of a complex symmetric matrix + + F06ZWF (ZSYR2K) Rank-2k update of a complex symmetric matrix + + X01 -- Mathematical Constants + + X01AAF (pi) + + X01ABF Euler's constant, (gamma) + + X02 -- Machine Constants + + X02AHF Largest permissible argument for SIN and COS + + X02AJF Machine precision + + X02AKF Smallest positive model number + + X02ALF Largest positive model number + + X02AMF Safe range of floating-point arithmetic + + X02ANF Safe range of complex floating-point arithmetic + + X02BBF Largest representable integer + + X02BEF Maximum number of decimal digits that can be represented + + X02BHF Parameter of floating-point arithmetic model, b + + X02BJF Parameter of floating-point arithmetic model, p + + X02BKF Parameter of floating-point arithmetic model, e + min + + X02BLF Parameter of floating-point arithmetic model, e + max + + X02DJF Parameter of floating-point arithmetic model, ROUNDS + + + + Routines from the NAG Fortran Library + + A number of routines from the NAG Fortran Library are used in the + Foundation Library as auxiliaries and are not documented here: + + A00AAF + + A02AAF A02ABF A02ACF + + C02AJF + + C05AZF C05NCF C05PCF + + C06FAF C06FBF C06FCF C06FFF C06FJF C06FKF + + C06HAF C06HBF C06HCF C06HDF + + D02CBF D02CHF D02NMF D02NSF D02NVF D02PAF + + D02XAF D02XKF D02YAF D02ZAF + + E02AFF + + E04GBF E04GEF E04YAF + + F01ADF F01AEF F01AFF F01AGF F01AHF F01AJF + + F01AKF F01AMF F01APF F01ATF F01AUF F01AVF + + F01AWF F01AXF F01BCF F01BTF F01CRF F01LBF + + F01LZF F01QAF F01QFF F01QGF F01QJF F01QKF + + F01RFF F01RGF F01RJF F01RKF + + F02AMF F02ANF F02APF F02AQF F02AVF F02AYF + + F02BEF F02SWF F02SXF F02SYF F02SZF F02UWF + + F02UXF F02UYF F02WDF F02WUF F02XUF + + F03AAF F03ABF F03AEF F03AFF + + F04AAF F04AEF F04AFF F04AGF F04AHF F04AJF + + F04AMF F04ANF F04AYF F04LDF F04YAF F04YCF + + F06BAF F06BCF F06BLF F06BMF F06BNF F06CAF + + F06CCF F06CLF F06DBF F06DFF F06FBF F06FCF + + F06FDF F06FGF F06FJF F06FLF F06FPF F06FQF + + F06FRF F06FSF F06HBF F06HGF F06HQF F06HRF + + F06KFF F06KJF F06KLF F06QFF F06QHF F06QKF + + F06QRF F06QSF F06QTF F06QVF F06QWF F06QXF + + F06RAF F06RJF F06TFF F06THF F06TTF F06TXF + + F06VJF F06VKF F06VXF + + F07AGF F07AHF F07AJF F07FGF F07FHF F07FJF + + F07TJF + + G01CEF + + G02BAF G02BUF G02BWF G02DDF + + G13AEF + + M01CBF M01CCF M01DBF M01DCF M01DFF M01ZBF + + P01ABF P01ACF + + S01BAF S07AAF S15ABF + + X03AAF + + X04BAF X04BBF X04CBF X04DBF + +\end{verbatim} +\endscroll +\end{page} + +@ +\subsection{NAG Documentation: introduction} +\label{manpageXXintro} +\index{pages!manpageXXintro!nagaux.ht} +\index{nagaux.ht!pages!manpageXXintro} +\index{manpageXXintro!nagaux.ht!pages} +<>= +\begin{page}{manpageXXintro}{NAG Documentation: introduction} +\beginscroll +\begin{verbatim} + + + + INTRO(3NAG) Foundation Library (12/10/92) INTRO(3NAG) + + + + Introduction Essential Introduction + Essential Introduction to the NAG Foundation Library + + This document is essential reading for any prospective user of + the Library. + + This document appears in both the Handbook and the Reference + Manual for the NAG Foundation Library, but with a different + Section 3 to describe the different forms of routine + documentation in the two publications. + + 1. The Library and its Documentation + + 1.1. Structure of the Library + + 1.2. Structure of the Documentation + + 1.3. On-line Documentation + + 1.4. Implementations of the Library + + 1.5. Library Identification + + 1.6. Fortran Language Standards + + 2. Using the Library + + 2.1. General Advice + + 2.2. Programming Advice + + 2.3. Error handling and the Parameter IFAIL + + 2.4. Input/output in the Library + + 2.5. Auxiliary Routines + + 3. Using the Reference Manual + + 3.1. General Guidance + + 3.2. Structure of Routine Documents + + 3.3. Specifications of Parameters + + 3.3.1. Classification of Parameters + + 3.3.2. Constraints and Suggested Values + + 3.3.3. Array Parameters + + 3.4. Implementation-dependent Information + + 3.5. Example Programs and Results + + 3.6. Summary for New Users + + 4. Relationship between the Foundation Library and other NAG Libraries + + 4.1. NAG Fortran Library + + 4.2. NAG Workstation Library + + 4.3. NAG C Library + + 5. Contact between Users and NAG + + 6. General Information about NAG + + 7. References + + + + 1. The Library and its Documentation + + 1.1. Structure of the Library + + The NAG Foundation Library is a comprehensive collection of + Fortran 77 routines for the solution of numerical and statistical + problems. The word 'routine' is used to denote 'subroutine' or ' + function'. + + The Library is divided into chapters, each devoted to a branch of + numerical analysis or statistics. Each chapter has a three- + character name and a title, e.g. + + D01 -- Quadrature + + Exceptionally one chapter (S) has a one-character name. (The + chapters and their names are based on the ACM modified SHARE + classification index [1].) + + All documented routines in the Library have six-character names, + beginning with the characters of the chapter name, e.g. + + D01AJF + + Note that the second and third characters are digits, not + letters; e.g. 0 is the digit zero, not the letter O. The last + letter of each routine name is always 'F'. + + 1.2. Structure of the Documentation + + There are two types of manual for the NAG Foundation Library: a + Handbook and a Reference Manual. + + The Handbook has the same chapter structure as the Library: each + chapter of routines in the Library has a corresponding chapter + (of the same name) in the Handbook. The chapters occur in + alphanumeric order. General introductory documents and indexes + are placed at the beginning of the Handbook. + + Each chapter in the Handbook contains a Chapter Introduction, + followed by concise summaries of the functionality and parameter + specifications of each routine in the chapter. Exceptionally, in + some chapters (F06, X01, X02) which contain simple support + routines, there are no concise summaries: all the routines are + described together in the Chapter Introduction. + + The Reference Manual provides complete reference documentation + for the NAG Foundation Library. In the Reference Manual, each + chapter consists of the following documents: + + Chapter Introduction, e.g. Introduction -- D01; + + Chapter Contents, e.g. Contents -- D01; + + routine documents, one for each documented routine in the + chapter. + + A routine document has the same name as the routine which it + describes. Within each chapter, routine documents occur in + alphanumeric order. As in the Handbook, chapters F06, X01 and X02 + do not contain separate documentation for individual routines. + + The general introductory documents, indexes and chapter + introductions are the same in the Reference Manual as in the + Handbook. The only exception is that the Essential Introduction + contains a different Section 3 in the two publications, to + describe the different forms of routine documentation. + + 1.3. On-line Documentation + + Extensive on-line documentation is included as an integral part + of the Foundation Library product. This consists of a number of + components: + + -- general introductory material, including the Essential + Introduction + + -- a summary list of all documented routines + + -- a KWIC Index + + -- Chapter Introductions + + -- routine documents + + -- example programs, data and results. + + The material has been derived in a number of forms to cater for + different user requirements, e.g. UNIX man pages, plain text, + RICH TEXT format etc, and the appropriate version is included on + the distribution media. For each implementation of the Foundation + Library the specific documentation (Installers' Note, Users' Note + etc) gives details of what is provided. + + 1.4. Implementations of the Library + + The NAG Foundation Library is available on many different + computer systems. For each distinct system, an implementation of + the Library is prepared by NAG, e.g. the IBM RISC System/6000 + implementation. The implementation is distributed as a tested + compiled library. + + An implementation is usually specific to a range of machines; it + may also be specific to a particular operating system or + compilation system. + + Essentially the same facilities are provided in all + implementations of the Library, but, because of differences in + arithmetic behaviour and in the compilation system, routines + cannot be expected to give identical results on different + systems, especially for sensitive numerical problems. + + The documentation supports all implementations of the Library, + with the help of a few simple conventions, and a small amount of + implementation-dependent information, which is published in a + separate Users' Note for each implementation (see Section 3.4). + + 1.5. Library Identification + + You must know which implementation of the Library you are using + or intend to use. To find out which implementation of the Library + is available on your machine, you can run a program which calls + the NAG Foundation Library routine A00AAF. This routine has no + parameters; it simply outputs text to the advisory message unit + (see Section 2.4). An example of the output is: + + + *** Start of NAG Foundation Library implementation details *** + Implementation title: IBM RISC System/6000 + Precision: FORTRAN double precision + Product Code: FFIB601D + Release: 1 + *** End of NAG Foundation Library implementation details *** + + (The product code can be ignored, except possibly when + communicating with NAG; see Section 4.) + + 1.6. Fortran Language Standards + + All routines in the Library conform to ANSI Standard Fortran 90 + [8]. + + Most of the routines in the Library were originally written to + conform to the earlier Fortran 66 [6] and Fortran 77 [7] + standards, and their calling sequences contain some parameters + which are not strictly necessary in Fortran 90. + + 2. Using the Library + + 2.1. General Advice + + A NAG Foundation Library routine cannot be guaranteed to return + meaningful results, irrespective of the data supplied to it. Care + and thought must be exercised in: + + (a) formulating the problem; + + (b) programming the use of library routines; + + (c) assessing the significance of the results. + + 2.2. Programming Advice + + The NAG Foundation Library and its documentation are designed on + the assumption that users know how to write a calling program in + Fortran. + + When programming a call to a routine, read the routine document + carefully, especially the description of the Parameters. This + states clearly which parameters must have values assigned to them + on entry to the routine, and which return useful values on exit. + See Section 3.3 for further guidance. + + If a call to a Library routine results in an unexpected error + message from the system (or possibly from within the Library), + check the following: + + + Has the NAG routine been called with the correct number of + parameters? + + Do the parameters all have the correct type? + + Have all array parameters been dimensioned correctly? + + Remember that all floating-point parameters must be declared to + be double precision, either with an explicit DOUBLE PRECISION + declaration (or COMPLEX(KIND(1.0D0)) if they are complex), or by + using a suitable IMPLICIT statement. + + Avoid the use of NAG-type names for your own program units or + COMMON blocks: in general, do not use names which contain a + three-character NAG chapter name embedded in them; they may clash + with the names of an auxiliary routine or COMMON block used by + the NAG Library. + + 2.3. Error handling and the Parameter IFAIL + + NAG Foundation Library routines may detect various kinds of + error, failure or warning conditions. Such conditions are handled + in a systematic way by the Library. They fall roughly into three + classes: + + (i) an invalid value of a parameter on entry to a routine; + + (ii) a numerical failure during computation (e.g. approximate + singularity of a matrix, failure of an iteration to + converge); + + (iii) a warning that although the computation has been completed, + the results cannot be guaranteed to be completely reliable. + + All three classes are handled in the same way by the Library, and + are all referred to here simply as 'errors'. + + The error-handling mechanism uses the parameter IFAIL, which is + the last parameter in the calling sequence of most NAG Foundation + Library routines. IFAIL serves two purposes: + + (i) it allows users to specify what action a Library routine + should take if it detects an error; + + (ii) it reports the outcome of a call to a Library routine, + either success (IFAIL = 0) or failure (IFAIL /= 0, with + different values indicating different reasons for the + failure, as explained in Section 6 of the routine document) + . + + For the first purpose IFAIL must be assigned a value before + calling the routine; since IFAIL is reset by the routine, it must + be passed as a variable, not as an integer constant. Allowed + values on entry are: + + IFAIL=0: an error message is output, and execution is + terminated ('hard failure'); + + IFAIL=+1: execution continues without any error message; + + IFAIL=-1: an error message is output, and execution + continues. + + The settings IFAIL =+-1 are referred to as 'soft failure'. + The safest choice is to set IFAIL to 0, but this is not always + convenient: some routines return useful results even though a + failure (in some cases merely a warning) is indicated. However, + if IFAIL is set to +- 1 on entry, it is essential for the program + to test its value on exit from the routine, and to take + appropriate action. + + The specification of IFAIL in Section 5 of a routine document + suggests a suitable setting of IFAIL for that routine. + + 2.4. Input/output in the Library + + Most NAG Foundation Library routines perform no output to an + external file, except possibly to output an error message. All + error messages are written to a logical error message unit. This + unit number (which is set by default to 6 in most + implementations) can be changed by calling the Library routine + X04AAF. + + Some NAG Foundation Library routines may optionally output their + final results, or intermediate results to monitor the course of + computation. All output other than error messages is written to a + logical advisory message unit. This unit number (which is also + set by default to 6 in most implementations) can be changed by + calling the Library routine X04ABF. Although it is logically + distinct from the error message unit, in practice the two unit + numbers may be the same. + + All output from the Library is formatted. + + The only Library routines which perform input from an external + file are a few 'option-setting' routines in Chapter E04: the unit + number is a parameter to the routine, and all input is formatted. + + You must ensure that the relevant Fortran unit numbers are + associated with the desired external files, either by an OPEN + statement in your calling program, or by operating system + commands. + + 2.5. Auxiliary Routines + + In addition to those Library routines which are documented and + are intended to be called by users, the Library also contains + many auxiliary routines. + + In general, you need not be concerned with them at all, although + you may be made aware of their existence if, for example, you + examine a memory map of an executable program which calls NAG + routines. The only exception is that when calling some NAG + Foundation Library routines, you may be required or allowed to + supply the name of an auxiliary routine from the Library as an + external procedure parameter. The routine documents give the + necessary details. In such cases, you only need to supply the + name of the routine; you never need to know details of its + parameter-list. + + NAG auxiliary routines have names which are similar to the name + of the documented routine(s) to which they are related, but with + last letter 'Z', 'Y', and so on, e.g. D01AJZ is an auxiliary + routine called by D01AJF. + + 3. Using the Reference Manual + + 3.1. General Guidance + + The Reference Manual is designed to serve the following + functions: + + -- to give background information about different areas of + numerical and statistical computation; + + -- to advise on the choice of the most suitable NAG + Foundation Library routine or routines to solve a particular + problem; + + -- to give all the information needed to call a NAG + Foundation Library routine correctly from a Fortran program, + and to assess the results. + + At the beginning of the Manual are some general introductory + documents. The following may help you to find the chapter, and + possibly the routine, which you need to solve your problem: + + Contents -- a list of routines in the Library, by + Summary chapter; + + KWIC Index -- a keyword index to chapters and routines. + + Having found a likely chapter or routine, you should read the + corresponding Chapter Introduction, which gives background + information about that area of numerical computation, and + recommendations on the choice of a routine, including indexes, + tables or decision trees. + + When you have chosen a routine, you must consult the routine + document. Each routine document is essentially self-contained (it + may contain references to related documents). It includes a + description of the method, detailed specifications of each + parameter, explanations of each error exit, and remarks on + accuracy. + + Example programs which illustrate the use of each routine are + distributed with the Library in machine-readable form. + + 3.2. Structure of Routine Documents + + All routine documents have the same structure, consisting of nine + numbered sections: + + 1. Purpose + + 2. Specification + + 3. Description + + 4. References + + 5. Parameters (see Section 3.3 below) + + 6. Error Indicators + + 7. Accuracy + + 8. Further Comments + + 9. Example (see Section 3.5 below) + + In a few documents, Section 5 also includes a description of + printed output which may optionally be produced by the routine. + + 3.3. Specifications of Parameters + + Section 5 of each routine document contains the specification of + the parameters, in the order of their appearance in the parameter + list. + + 3.3.1. Classification of Parameters + + Parameters are classified as follows: + + Input : you must assign values to these parameters on or before + entry to the routine, and these values are unchanged on exit from + the routine. + + Output : you need not assign values to these parameters on or + before entry to the routine; the routine may assign values to + them. + + Input/Output : you must assign values to these parameters on or + before entry to the routine, and the routine may then change + these values. + + Workspace: array parameters which are used as workspace by the + routine. You must supply arrays of the correct type and + dimension, but you need not be concerned with their contents. + + External Procedure: a subroutine or function which must be + supplied (e.g. to evaluate an integrand or to print intermediate + output). Usually it must be supplied as part of your calling + program, in which case its specification includes full details of + its parameter-list and specifications of its parameters (all + enclosed in a box). Its parameters are classified in the same way + as those of the Library routine, but because you must write the + procedure rather than call it, the significance of the + classification is different: + + Input : values may be supplied on entry, which your procedure + must not change. + + Output : you may or must assign values to these parameters + before exit from your procedure. + + Input/Output : values may be supplied on entry, and you may + or must assign values to them before exit from your + procedure. + + Occasionally, as mentioned in Section 2.5, the procedure can be + supplied from the NAG Library, and then you only need to know its + name. + + User Workspace: array parameters which are passed by the Library + routine to an external procedure parameter. They are not used by + the routine, but you may use them to pass information between + your calling program and the external procedure. + + 3.3.2. Constraints and Suggested Values + + The word 'Constraint:' or 'Constraints:' in the specification of + an Input parameter introduces a statement of the range of valid + values for that parameter, e.g. + + Constraint: N > 0. + + If the routine is called with an invalid value for the parameter + (e.g. N = 0), the routine will usually take an error exit, + returning a non-zero value of IFAIL (see Section 2.3). + + In newer documents constraints on parameters of type CHARACTER + only list uppercase alphabetic characters, e.g. + + Constraint: STRING = 'A' or 'B'. + + In practice all routines with CHARACTER parameters will permit + the use of lower case characters. + + The phrase 'Suggested Value:' introduces a suggestion for a + reasonable initial setting for an Input parameter (e.g. accuracy + or maximum number of iterations) in case you are unsure what + value to use; you should be prepared to use a different setting + if the suggested value turns out to be unsuitable for your + problem. + + 3.3.3. Array Parameters + + Most array parameters have dimensions which depend on the size of + the problem. In Fortran terminology they have 'adjustable + dimensions': the dimensions occurring in their declarations are + integer variables which are also parameters of the Library + routine. + + For example, a Library routine might have the specification: + + + SUBROUTINE (M, N, A, B, LDB) + INTEGER M, N, A(N), B(LDB,N), LDB + + For a one-dimensional array parameter, such as A in this example, + the specification would begin: + + 3: A(N) -- DOUBLE PRECISION array Input + + You must ensure that the dimension of the array, as declared in + your calling (sub)program, is at least as large as the value you + supply for N. It may be larger; but the routine uses only the + first N elements. + + For a two-dimensional array parameter, such as B in the example, + the specification might be: + + 4: B(LDB,N) -- DOUBLE PRECISION array Input/Output + On entry: the m by n matrix B. + + and the parameter LDB might be described as follows: + + 5: LDB -- INTEGER Input + On entry: the first dimension of the array B as declared in + the (sub)program from which is called. Constraint: + LDB >= M. + + You must supply the first dimension of the array B, as declared + in your calling (sub)program, through the parameter LDB, even + though the number of rows actually used by the routine is + determined by the parameter M. You must ensure that the first + dimension of the array is at least as large as the value you + supply for M. The extra parameter LDB is needed because Fortran + does not allow information about the dimensions of array + parameters to be passed automatically to a routine. + + You must also ensure that the second dimension of the array, as + declared in your calling (sub)program, is at least as large as + the value you supply for N. It may be larger, but the routine + only uses the first N columns. + + A program to call the hypothetical routine used as an example in + this section might include the statements: + + + INTEGER AA(100), BB(100,50) + LDB = 100 + . + . + . + M = 80 + N = 20 + CALL (M,N,AA,BB,LDB) + + Fortran requires that the dimensions which occur in array + declarations, must be greater than zero. Many NAG routines are + designed so that they can be called with a parameter like N in + the above example set to 0 (in which case they would usually exit + immediately without doing anything). If so, the declarations in + the Library routine would use the 'assumed size' array dimension, + and would be given as: + + + INTEGER M, N, A(*), B(LDB,*), LDB + + However, the original declaration of an array in your calling + program must always have constant dimensions, greater than or + equal to 1. + + Consult an expert or a textbook on Fortran, if you have + difficulty in calling NAG routines with array parameters. + + 3.4. Implementation-dependent Information + + In order to support all implementations of the Foundation + Library, the Manual has adopted a convention of using bold + italics to distinguish terms which have different interpretations + in different implementations. + + For example, machine precision denotes the relative precision to + which double precision floating-point numbers are stored in the + computer, e.g. in an implementation with approximately 16 decimal + digits of precision, machine precision has a value of + - 16 + approximately 10 . + + The precise value of machine precision is given by the function + X02AJF. Other functions in Chapter X02 return the values of other + implementation-dependent constants, such as the overflow + threshold, or the largest representable integer. Refer to the X02 + + Chapter Introduction for more details. + + For each implementation of the Library, a separate Users' Note is + provided. This is a short document, revised at each Mark. At most + installations it is available in machine-readable form. It gives + any necessary additional information which applies specifically + to that implementation, in particular: + + -- the interpretation of bold italicised terms; + + -- the values returned by X02 routines; + + -- the default unit numbers for output (see Section 2.4). + + 3.5. Example Programs and Results + + The last section of each routine document describes an example + problem which can be solved by simple use of the routine. The + example programs themselves, together with data and results, are + not printed in the routine document, but are distributed in + machine-readable form with the Library. The programs are designed + so that they can fairly easily be modified, and so serve as the + basis for a simple program to solve a user's own problem. + + The results distributed with each implementation were obtained + using that implementation of the Library; they may not be + identical to the results obtained with other implementations. + + 3.6. Summary for New Users + + If you are unfamiliar with the NAG Foundation Library and are + thinking of using a routine from it, please follow these + instructions: + + (a) read the whole of the Essential Introduction; + + (b) consult the Contents Summary or KWIC Index to choose an + appropriate chapter or routine; + + (c) read the relevant Chapter Introduction; + + (d) choose a routine, and read the routine document. If the + routine does not after all meet your needs, return to steps + (b) or (c); + + (e) read the Users' Note for your implementation; + + (f) consult local documentation, which should be provided by your + local support staff, about access to the NAG Library on your + computing system. + + You should now be in a position to include a call to the routine + in a program, and to attempt to run it. You may of course need to + refer back to the relevant documentation in the case of + difficulties, for advice on assessment of results, and so on. + + As you become familiar with the Library, some of steps (a) to (f) + can be omitted, but it is always essential to: + + -- be familiar with the Chapter Introduction; + + -- read the routine document; + + -- be aware of the Users' Note for your implementation. + + 4. Relationship between the Foundation Library and other NAG Libraries + + 4.1. NAG Fortran Library + + The Foundation Library is a strict subset of the full NAG Fortran + Library (Mark 15 or later). Routines in both libraries have + identical source code (apart from any modifications necessary for + implementation on a specific system) and hence can be called in + exactly the same way, though you should consult the relevant + implementation-specific documentation for details such as values + of machine constants. + + By its very nature, the Foundation Library cannot contain the + same extensive range of routines as the full Fortran Library. If + your application requires a routine which is not in the + Foundation Library, then please consult NAG for information on + relevant material available in the Fortran Library. + + Some routines which occur as user-callable routines in the full + Fortran Library are included as auxiliary routines in the + Foundation Library but they are not documented in this + publication and direct calls to them should only be made if you + are already familiar with their use in the Fortran Library. A + list of all such auxiliary routines is given at the end of the + Foundation Library Contents Summary. + + Whereas the full Fortran Library may be provided in either a + single precision or a double precision version, the Foundation + Library is always provided in double precision. + + 4.2. NAG Workstation Library + + The Foundation Library is a successor product to an earlier, + smaller subset of the full NAG Fortran Library which was called + the NAG Workstation Library. The Foundation Library has greater + functionality than the Workstation Library but is not strictly + upwards compatible, i.e., a number of routines in the earlier + product have been replaced by new material to reflect recent + algorithmic developments. + If you have used the Workstation Library and wish to convert your + programs to call routines from the Foundation Library, please + consult the document 'Converting from the Workstation Library' in + this Manual. + + 4.3. NAG C Library + + NAG has also developed a library of numerical and statistical + software for use by C programmers. This now contains over 200 + user-callable functions and provides similar (but not identical) + coverage to that of the Foundation Library. Please contact NAG + for further details if you have a requirement for similar quality + library code in C. + + 5. Contact between Users and NAG + + If you are using the NAG Foundation Library in a multi-user + environment and require further advice please consult your local + support staff who will be receiving regular information from NAG. + This covers such matters as: + + -- obtaining a copy of the Users' Note for your + implementation; + + -- obtaining information about local access to the Library; + + -- seeking advice about using the Library; + + -- reporting suspected errors in routines or documents; + + -- making suggestions for new routines or features; + + -- purchasing NAG documentation. + + If you are unable to make contact with a local source of support + or are in a single-user environment then please contact NAG + directly at any one of the addresses given at the beginning of + this publication. + + 6. General Information about NAG + + NAG produces and distributes numerical, symbolic, statistical and + graphical software for the solution of problems in a wide range + of applications in such areas as science, engineering, financial + analysis and research. + + For users who write programs and build packages NAG produces sub- + program libraries in a range of computer languages (Ada, C, + Fortran, Pascal, Turbo Pascal). NAG also provides a number of + Fortran programming support products in the NAGWare range -- + Fortran 77 programming tools, Fortran 90 compilers for a number + of machine platforms (including PC-DOS) and VecPar 77 for + restructuring and tuning programs for execution on vector or + parallel computers. + + For users who do not wish to program in the traditional sense but + want the same reliability and qualities offered by our libraries, + NAG provides several powerful mathematical and statistical + packages for interactive use. A major addition to this range of + packages is AXIOM -- the powerful symbolic solver which includes + a Hypertext system and graphical capabilities. + + For further details of any of these products, please contact NAG + at one of the addresses given at the beginning of this + publication. + + References [2], [3], [4], and [5] discuss various aspects of the + design and development of the NAG Library, and NAG's technical + policies and organisation. + + 7. References + + [1] (1960--1976) Collected Algorithms from ACM Index by subject + to algorithms. + + [2] Ford B (1982) Transportable Numerical Software. Lecture + Notes in Computer Science. 142 128--140. + + [3] Ford B, Bentley J, Du Croz J J and Hague S J (1979) The NAG + Library 'machine'. Software Practice and Experience. 9(1) + 65--72. + + [4] Ford B and Pool J C T (1984) The Evolving NAG Library + Service. Sources and Development of Mathematical Software. + (ed W Cowell) Prentice-Hall. 375--397. + + [5] Hague S J, Nugent S M and Ford B (1982) Computer-based + Documentation for the NAG Library. Lecture Notes in Computer + Science. 142 91--127. + + [6] (1966) USA Standard Fortran. Publication X3.9. American + National Standards Institute. + + [7] (1978) American National Standard Fortran. Publication X3.9. + American National Standards Institute. + + [8] (1991) American National Standard Programming Language + Fortran 90. Publication X3.198. + American National Standards Institute. + +\end{verbatim} +\endscroll +\end{page} + +@ +\subsection{NAG Documentation: keyword in context} +\label{manpageXXkwic} +\index{pages!manpageXXkwic!nagaux.ht} +\index{nagaux.ht!pages!manpageXXkwic} +\index{manpageXXkwic!nagaux.ht!pages} +<>= +\begin{page}{manpageXXkwic}{NAG Documentation: keyword in context} +\beginscroll +\begin{verbatim} + + + + KWIC(3NAG) Foundation Library (12/10/92) KWIC(3NAG) + + + + Introduction Keywords in Context + Keywords in Context + + Pre-computed weights and D01BBF + abscissae + for Gaussian quadrature rules, restricted choice of ... + + Sum the F06EKF + absolute + values of real vector elements (DASUM) + + Sum the F06JKF + absolute + values of complex vector elements (DZASUM) + + Index, real vector element with largest F06JLF + absolute + value (IDAMAX) + + Index, complex vector element with largest F06JMF + absolute + value (IZAMAX) + + ODEs, IVP, D02CJF + Adams + method, until function of solution is zero, ... + + 1-D quadrature, D01AJF + adaptive + , finite interval, strategy due to Piessens and de ... + + 1-D quadrature, D01AKF + adaptive + , finite interval, method suitable for oscillating ... + + 1-D quadrature, D01ALF + adaptive + , finite interval, allowing for singularities at ... + + 1-D quadrature, D01AMF + adaptive + , infinite or semi-infinite interval + + 1-D quadrature, D01ANF + adaptive + , finite interval, weight function cos((omega)x) ... + + 1-D quadrature, D01APF + adaptive + , finite interval, weight function with end-point ... + + 1-D quadrature, D01AQF + adaptive + , finite interval, weight function 1/(x-c), ... + + 1-D quadrature, D01ASF + adaptive + , semi-infinite interval, weight function cos((omega)x) + + Multi-dimensional D01FCF + adaptive + quadrature over hyper-rectangle + + Add F06ECF + scalar times real vector to real vector (DAXPY) + + Add F06GCF + scalar times complex vector to complex vector (ZAXPY) + + Return or set unit number for X04ABF + advisory + messages + + Airy S17AGF + function Ai(x) + + Airy S17AHF + function Bi(x) + + Airy S17AJF + function Ai'(x) + + Airy S17AKF + function Bi'(x) + + Airy S17DGF + functions Ai(z) and Ai('z), complex z + + Airy S17DHF + functions Bi(z) and Bi'(z), complex z + + Airy function S17AGF + Ai(x) + + Airy function S17AJF + Ai'(x) + + Airy functions S17DGF + Ai(z) + and Ai'(z), complex z + + Airy functions Ai(z) and S17DGF + Ai'(z) + , complex z + algebraico-logarithmic + type + + Two-way contingency table G01AFF + analysis + 2 + , with (chi) /Fisher's exact test + + Performs principal component G03AAF + analysis + + Performs canonical correlation G03ADF + analysis + + Friedman two-way G08AEF + analysis + of variance on k matched samples + + Kruskal-Wallis one-way G08AFF + analysis + of variance on k samples of unequal size + + L - E02GAF + 1 + approximation + by general linear function + + Approximation E02 + + Approximation S + of special functions + + ARIMA + model + + Univariate time series, estimation, seasonal G13AFF + ARIMA + model + + ARIMA + model + + ARIMA + model + + Safe range of floating-point X02AMF + arithmetic + + Safe range of complex floating-point X02ANF + arithmetic + + Parameter of floating-point X02BHF + arithmetic + model, b + + Parameter of floating-point X02BJF + arithmetic + model, p + + Parameter of floating-point X02BKF + arithmetic + model, e + min + + Parameter of floating-point X02BLF + arithmetic + model, e + max + + Parameter of floating-point X02DJF + arithmetic + model, ROUNDS + + Univariate time series, sample G13ABF + autocorrelation + function + + Univariate time series, partial G13ACF + autocorrelations + from autocorrelations + + Univariate time series, partial autocorrelations from G13ACF + autocorrelations + + Least-squares cubic spline curve fit, E02BEF + automatic + knot placement + + Least-squares surface fit by bicubic splines with E02DCF + automatic + knot placement, data on rectangular grid + + Least-squares surface fit by bicubic splines with E02DDF + automatic + knot placement, scattered data + + B-splines E02 + + Matrix-vector product, real rectangular F06PBF + band + matrix (DGBMV) + + Matrix-vector product, real symmetric F06PDF + band + matrix (DSBMV) + + Matrix-vector product, real triangular F06PGF + band + matrix (DTBMV) + + System of equations, real triangular F06PKF + band + matrix (DTBSV) + + Matrix-vector product, complex rectangular F06SBF + band + matrix (ZGBMV) + + Matrix-vector product, complex Hermitian F06SDF + band + matrix (ZHBMV) + + Matrix-vector product, complex triangular F06SGF + band + matrix (ZTBMV) + + System of equations, complex triangular F06SKF + band + matrix (ZTBSV) + + bandwidth + matrix + + Solution of real symmetric positive-definite variable- F04MCF + bandwidth + simultaneous linear equations (coefficient matrix ... + + Basic F06 + Linear Algebra Subprograms + + ODEs, stiff IVP, D02EJF + BDF + method, until function of solution is zero, + intermediate ... + + Kelvin function S19ABF + bei + x + + Kelvin function S19AAF + ber + x + + Bessel S17ACF + function Y (x) + 0 + + Bessel S17ADF + function Y (x) + 1 + + Bessel S17AEF + function J (x) + 0 + + Bessel S17AFF + function J (x) + 1 + + Bessel S17DCF + functions Y (z), complex z, real (nu)>=0, ... + (nu)+n + + Bessel S17DEF + functions J (z), complex z, real (nu)>=0, ... + (nu)+n + + Modified S18ACF + Bessel + function K (x) + 0 + + Modified S18ADF + Bessel + function K (x) + 1 + + Modified S18AEF + Bessel + function I (x) + 0 + + Modified S18AFF + Bessel + function I (x) + 1 + + Modified S18DCF + Bessel + functions K (z), complex z, real (nu)>=0, ... + (nu)+n + + Modified S18DEF + Bessel + functions I (z), complex z, real (nu)>=0, ... + (nu)+n + + beta + distribution + + Computes deviates for the G01FEF + beta + distribution + + Generates a vector of pseudo-random numbers from a G05FEF + beta + distribution + + Interpolating functions, fitting E01DAF + bicubic + spline, data on rectangular grid + + Least-squares surface fit, E02DAF + bicubic + splines + + Least-squares surface fit by E02DCF + bicubic + splines with automatic knot placement, data on ... + + Least-squares surface fit by E02DDF + bicubic + splines with automatic knot placement, scattered data + + Evaluation of a fitted E02DEF + bicubic + spline at a vector of points + + Evaluation of a fitted E02DFF + bicubic + spline at a mesh of points + + Sort 2-D data into panels for fitting E02ZAF + bicubic + splines + + Fits a generalized linear model with G02GBF + binomial + errors + + binomial + distribution + + Computes probability for the G01HAF + bivariate + Normal distribution + + Airy function S17AHF + Bi(x) + + Airy function S17AKF + Bi'(x) + + Airy functions S17DHF + Bi(z) + and Bi'(z), complex z + + Airy functions Bi(z) and S17DHF + Bi'(z) + , complex z + + BLAS F06 + + Pseudo-random logical G05DZF + (boolean) + value + + ODEs, D02GAF + boundary + value problem, finite difference technique with ... + + ODEs, D02GBF + boundary + value problem, finite difference technique with ... + + ODEs, general nonlinear D02RAF + boundary + value problem, finite difference technique with ... + + bounds + , using function values only + + break-points + + break-points + + Zero of continuous function in given interval, C05ADF + Bus + and Dekker algorithm + + Performs G03ADF + canonical + correlation analysis + + Carlo + method + + Elliptic PDE, Helmholtz equation, 3-D D03FAF + Cartesian + co-ordinates + + Cauchy + principal value (Hilbert transform) + + Pseudo-random real numbers, G05DFF + + Cauchy + distribution + + character + string + + Compare two X05ACF + character + strings representing date and time + + Evaluation of fitted polynomial in one variable from E02AEF + Chebyshev + series form (simplified parameter list) + + Derivative of fitted polynomial in E02AHF + Chebyshev + series form + + Integral of fitted polynomial in E02AJF + Chebyshev + series form + + Evaluation of fitted polynomial in one variable, from E02AKF + Chebyshev + series form + + Check C05ZAF + user's routine for calculating 1st derivatives + + Univariate time series, diagnostic G13ASF + checking + of residuals, following G13AFF + + Cholesky F07FDF + factorization of real symmetric positive-definite ... + + Circular C06EKF + convolution or correlation of two real vectors, no ... + + Cosine integral S13ACF + Ci(x) + + Interpolating functions, method of Renka and E01SAF + Cline + , two variables + + Elliptic PDE, Helmholtz equation, 3-D Cartesian D03FAF + co-ordinates + + coefficient + matrix already factorized by F01MCF) + + coefficient + matrix (DTRSM) + + coefficient + matrix (ZTRSM) + + Kendall/Spearman non-parametric rank correlation G02BNF + coefficients + , no missing values, overwriting input data + + Kendall/Spearman non-parametric rank correlation G02BQF + coefficients + , no missing values, preserving input data + + Operations with orthogonal matrices, form F01QEF + columns + of Q after factorization by F01QCF + + Operations with unitary matrices, form F01REF + columns + of Q after factorization by F01RCF + + Rank M01DJF + columns + of a matrix, real numbers + + Compare X05ACF + two character strings representing date and time + + Complement S15ADF + of error function erfcx + + Unconstrained minimum, pre- E04DGF + conditioned + conjugate gradient algorithm, function of several ... + + Complex C06GBF + conjugate + of Hermitian sequence + + Complex C06GCF + conjugate + of complex sequence + + Complex C06GQF + conjugate + of multiple Hermitian sequences + + Unconstrained minimum, pre-conditioned E04DGF + conjugate + gradient algorithm, function of several variables ... + + Dot product of two complex vectors, F06GBF + conjugated + (ZDOTC) + + Rank-1 update, complex rectangular matrix, F06SNF + conjugated + vector (ZGERC) + + Mathematical X01 + Constants + + Machine X02 + Constants + + constrained + , arbitrary data points + + constraints + , using function values and optionally 1st ... + + Two-way G01AFF + contingency + 2 + table analysis, with (chi) /Fisher's ... + + continuation + facility + + Zero of C05ADF + continuous + function in given interval, Bus and Dekker algorithm + + 2 + continuous + distributions + + Convert C06GSF + Hermitian sequences to general complex sequences + + Convert X05ABF + array of integers representing date and time to ... + + Circular C06EKF + convolution + or correlation of two real vectors, no extra ... + + Copy F06EFF + real vector (DCOPY) + + Copy F06GFF + complex vector (ZCOPY) + + correction + , simple nonlinear problem + + correction + , general linear problem + + correction + , continuation facility + + Circular convolution or C06EKF + correlation + of two real vectors, no extra workspace + + Kendall/Spearman non-parametric rank G02BNF + correlation + coefficients, no missing values, overwriting ... + + Kendall/Spearman non-parametric rank G02BQF + correlation + coefficients, no missing values, preserving input ... + + Computes (optionally weighted) G02BXF + correlation + and covariance matrices + + Performs canonical G03ADF + correlation + analysis + + Multivariate time series, cross- G13BCF + correlations + + cos + ((omega)x) or sin((omega)x) + + cos + ((omega)x) or sin((omega)x) + + Cosine S13ACF + integral Ci(x) + + Covariance E04YCF + matrix for nonlinear least-squares problem + + Computes (optionally weighted) correlation and G02BXF + covariance + matrices + + Return the X05BAF + CPU + time + + Multivariate time series, G13BCF + cross-correlations + + Multivariate time series, smoothed sample G13CDF + cross + spectrum using spectral smoothing by the trapezium ... + + Interpolating functions, E01BAF + cubic + spline interpolant, one variable + + cubic + Hermite, one variable + + Least-squares curve E02BAF + cubic + spline fit (including interpolation) + + Evaluation of fitted E02BBF + cubic + spline, function only + + Evaluation of fitted E02BCF + cubic + spline, function and derivatives + + Evaluation of fitted E02BDF + cubic + spline, definite integral + + Least-squares E02BEF + cubic + spline curve fit, automatic knot placement + + Set up reference vector from supplied G05EXF + cumulative + distribution function or probability distribution ... + + Least-squares E02ADF + curve + fit, by polynomials, arbitrary data points + + Least-squares E02BAF + curve + cubic spline fit (including interpolation) + + Least-squares cubic spline E02BEF + curve + fit, automatic knot placement + + Fresnel integral S20ADF + C(x) + + Daniell) + window + + Daniell) + window + + Return X05AAF + date + and time as an array of integers + + Convert array of integers representing X05ABF + date + and time to character string + + Compare two character strings representing X05ACF + date + and time + + deferred + correction, simple nonlinear problem + + deferred + correction, general linear problem + + deferred + correction, continuation facility + + Interpolated values, interpolant computed by E01BEF, E01BHF + definite + integral, one variable + + Evaluation of fitted cubic spline, E02BDF + definite + integral + + definite + matrix + + T + LDL factorization of real symmetric positive- F01MCF + definite + variable-bandwidth matrix + + definite + + definite + + Solution of real symmetric positive- F04ASF + definite + simultaneous linear equations, one right-hand side ... + + Solution of real symmetric positive- F04FAF + definite + tridiagonal simultaneous linear equations, one ... + + Real sparse symmetric positive- F04MAF + definite + simultaneous linear equations (coefficient matrix ... + + Solution of real symmetric positive- F04MCF + definite + variable-bandwidth simultaneous linear equations ... + + Cholesky factorization of real symmetric positive- F07FDF + definite + matrix (DPOTRF) + + Solution of real symmetric positive- F07FEF + definite + system of linear equations, multiple right-hand ... + + Degenerate S21BAF + symmetrised elliptic integral of 1st kind R ... + C + + Dekker + algorithm + + Computes upper and lower tail and probability G01EEF + density + function probabilities for the beta distribution + + Solution of system of nonlinear equations using 1st C05PBF + derivatives + + Check user's routine for calculating 1st C05ZAF + derivatives + + derivative + , one variable + + Least-squares polynomial fit, values and E02AGF + derivatives + may be constrained, arbitrary data points + + Derivative E02AHF + of fitted polynomial in Chebyshev series form + + Evaluation of fitted cubic spline, function and E02BCF + derivatives + + derivatives + + derivatives + + derivatives + + Computes G01FAF + deviates + for the standard Normal distribution + + Computes G01FBF + deviates + for Student's t-distribution + + Computes G01FCF + deviates + 2 + for the (chi) distribution + + Computes G01FDF + deviates + for the F-distribution + + Computes G01FEF + deviates + for the beta distribution + + Computes G01FFF + deviates + for the gamma distribution + + Univariate time series, G13ASF + diagnostic + checking of residuals, following G13AFF + + ODEs, boundary value problem, finite D02GAF + difference + technique with deferred correction, simple ... + + ODEs, boundary value problem, finite D02GBF + difference + technique with deferred correction, general linear ... + + difference + technique with deferred correction, continuation ... + + Elliptic PDE, solution of finite D03EDF + difference + equations by a multigrid technique + + Univariate time series, seasonal and non-seasonal G13AAF + differencing + + Single 1-D real C06EAF + discrete + Fourier transform, no extra workspace + + Single 1-D Hermitian C06EBF + discrete + Fourier transform, no extra workspace + + Single 1-D complex C06ECF + discrete + Fourier transform, no extra workspace + + Multiple 1-D real C06FPF + discrete + Fourier transforms + + Multiple 1-D Hermitian C06FQF + discrete + Fourier transforms + + Multiple 1-D complex C06FRF + discrete + Fourier transforms + + 2-D complex C06FUF + discrete + Fourier transform + + Discretize D03EEF + a 2nd order elliptic PDE on a rectangle + + Computes probabilities for the standard Normal G01EAF + distribution + + Computes probabilities for Student's t- G01EBF + distribution + + 2 + Computes probabilities for (chi) G01ECF + distribution + + Computes probabilities for F- G01EDF + distribution + + distribution + + Computes probabilities for the gamma G01EFF + distribution + + Computes deviates for the standard Normal G01FAF + distribution + + Computes deviates for Student's t- G01FBF + distribution + + 2 + Computes deviates for the (chi) G01FCF + distribution + + Computes deviates for the F- G01FDF + distribution + + Computes deviates for the beta G01FEF + distribution + + Computes deviates for the gamma G01FFF + distribution + + Computes probability for the bivariate Normal G01HAF + distribution + + Pseudo-random real numbers, uniform G05CAF + distribution + over (0,1) + + Pseudo-random real numbers, Normal G05DDF + distribution + + Pseudo-random real numbers, Cauchy G05DFF + distribution + + Pseudo-random real numbers, Weibull G05DPF + distribution + + Pseudo-random integer from uniform G05DYF + distribution + + Set up reference vector for multivariate Normal G05EAF + distribution + + distribution + + distribution + + Set up reference vector from supplied cumulative G05EXF + distribution + function or probability distribution function + + distribution + function + + Generates a vector of random numbers from a uniform G05FAF + distribution + + distribution + + Generates a vector of random numbers from a Normal G05FDF + distribution + + distribution + + distribution + + 2 + (chi) goodness of fit test, for standard continuous G08CGF + distributions + + Inverse G01F + distributions + + Doncker + , allowing for badly-behaved integrands + + Dot F06EAF + product of two real vectors (DDOT) + + Dot F06GAF + product of two complex vectors, unconjugated (ZDOTU) + + Dot F06GBF + product of two complex vectors, conjugated (ZDOTC) + + eigenfunction + , user-specified break-points + + All eigenvalues of generalized real F02ADF + eigenproblem + of the form Ax=(lambda)Bx where A and B are ... + + All eigenvalues and eigenvectors of generalized real F02AEF + eigenproblem + of the form Ax=(lambda)Bx where A and B are ... + + eigenproblem + by QZ algorithm, real matrices + + eigenproblem + + eigenvalue + and eigenfunction, user-specified break-points + + All F02AAF + eigenvalues + of real symmetric matrix + + All F02ABF + eigenvalues + and eigenvectors of real symmetric matrix + + All F02ADF + eigenvalues + of generalized real eigenproblem of the form + Ax=(lambda)Bx + + All F02AEF + eigenvalues + and eigenvectors of generalized real ... + + All F02AFF + eigenvalues + of real matrix + + All F02AGF + eigenvalues + and eigenvectors of real matrix + + All F02AJF + eigenvalues + of complex matrix + + All F02AKF + eigenvalues + and eigenvectors of complex matrix + + All F02AWF + eigenvalues + of complex Hermitian matrix + + All F02AXF + eigenvalues + and eigenvectors of complex Hermitian matrix + + Selected F02BBF + eigenvalues + and eigenvectors of real symmetric matrix + + All F02BJF + eigenvalues + and optionally eigenvectors of generalized ... + + Selected F02FJF + eigenvalues + and eigenvectors of sparse symmetric eigenproblem + + All eigenvalues and F02ABF + eigenvectors + of real symmetric matrix + + All eigenvalues and F02AEF + eigenvectors + of generalized real eigenproblem of the form ... + + All eigenvalues and F02AGF + eigenvectors + of real matrix + + All eigenvalues and F02AKF + eigenvectors + of complex matrix + + All eigenvalues and F02AXF + eigenvectors + of complex Hermitian matrix + + Selected eigenvalues and F02BBF + eigenvectors + of real symmetric matrix + + All eigenvalues and optionally F02BJF + eigenvectors + of generalized eigenproblem by QZ algorithm, ... + + Selected eigenvalues and F02FJF + eigenvectors + of sparse symmetric eigenproblem + + Elliptic D03EDF + PDE, solution of finite difference equations by a ... + + Discretize a 2nd order D03EEF + elliptic + PDE on a rectangle + + Elliptic D03FAF + PDE, Helmholtz equation, 3-D Cartesian co-ordinates + + Degenerate symmetrised S21BAF + elliptic + integral of 1st kind R (x,y) + C + + Symmetrised S21BBF + elliptic + integral of 1st kind R (x,y,z) + F + + Symmetrised S21BCF + elliptic + integral of 2nd kind R (x,y,z) + D + + Symmetrised S21BDF + elliptic + integral of 3rd kind R (x,y,z,r) + J + + end-point + singularities of algebraico-logarithmic type + + error + + Fits a generalized linear model with binomial G02GBF + errors + + Fits a generalized linear model with Poisson G02GCF + errors + + Complement of S15ADF + error + function erfc x + + Error S15AEF + function erf x + + Return or set unit number for X04AAF + error + messages + + Computes G02DNF + estimable + function of a general linear regression model and ... + + Univariate time series, preliminary G13ADF + estimation + , seasonal ARIMA model + + Univariate time series, G13AFF + estimation + , seasonal ARIMA model + + Multivariate time series, preliminary G13BDF + estimation + of transfer function model + + Multivariate time series, G13BEF + estimation + of multi-input model + + Compute F06EJF + Euclidean + norm of real vector (DNRM2) + + Compute F06JJF + Euclidean + norm of complex vector (DZNRM2) + + Evaluation E02AEF + of fitted polynomial in one variable from ... + + Evaluation E02AKF + of fitted polynomial in one variable, from ... + + Evaluation E02BBF + of fitted cubic spline, function only + + Evaluation E02BCF + of fitted cubic spline, function and derivatives + + Evaluation E02BDF + of fitted cubic spline, definite integral + + Evaluation E02DEF + of a fitted bicubic spline at a vector of points + + Evaluation E02DFF + of a fitted bicubic spline at a mesh of points + + 2 + exact + test + + Computes the G08AJF + exact + probabilities for the Mann-Whitney U statistic, no ... + + Computes the G08AKF + exact + probabilities for the Mann-Whitney U statistic, ties .. + + exponential + distribution + + Complex S01EAF + exponential + z + , e + + Exponential S13AAF + integral E (x) + 1 + + Computes a five-point summary (median, hinges and G01ALF + extremes) + + Computes probabilities for G01EDF + F + -distribution + + Computes deviates for the G01FDF + F + -distribution + + LU F01BRF + factorization + of real sparse matrix + + LU F01BSF + factorization + of real sparse matrix with known sparsity pattern + + T + LL F01MAF + factorization + of real sparse symmetric positive-definite matrix + + T + LDL F01MCF + factorization + of real symmetric positive-definite ... + + QR F01QCF + factorization + of real m by n matrix (m>=n) + + T + factorization + by F01QCF + + factorization + by F01QCF + + QR F01RCF + factorization + of complex m by n matrix (m>=n) + + H + factorization + by F01RCF + + factorization + by F01RCF + + LU F07ADF + factorization + of real m by n matrix (DGETRF) + + Cholesky F07FDF + factorization + of real symmetric positive-definite matrix ... + + Multivariate time series, G13BAF + filtering + (pre-whitening) by an ARIMA model + + 1-D quadrature, adaptive, D01AJF + finite + interval, strategy due to Piessens and de Doncker, ... + + 1-D quadrature, adaptive, D01AKF + finite + interval, method suitable for oscillating functions + + 1-D quadrature, adaptive, D01ALF + finite + interval, allowing for singularities at user-specified + + 1-D quadrature, adaptive, D01ANF + finite + interval, weight function cos((omega)x) or sin... + + 1-D quadrature, adaptive, D01APF + finite + interval, weight function with end-point ... + + 1-D quadrature, adaptive, D01AQF + finite + interval, weight function 1/(x-c), Cauchy ... + + ODEs, boundary value problem, D02GAF + finite + difference technique with deferred correction, simple . + + ODEs, boundary value problem, D02GBF + finite + difference technique with deferred correction, general + + finite/infinite + range, eigenvalue and eigenfunction, ... + + ODEs, general nonlinear boundary value problem, D02RAF + finite + difference technique with deferred correction, ... + + Elliptic PDE, solution of D03EDF + finite + difference equations by a multigrid technique + + 2 + Fisher's + exact test + + Interpolating functions, E01DAF + fitting + bicubic spline, data on rectangular grid + + Least-squares curve E02ADF + fit + , by polynomials, arbitrary data points + + Evaluation of E02AEF + fitted + polynomial in one variable from Chebyshev series form . + + Least-squares polynomial E02AGF + fit + , values and derivatives may be constrained, arbitrary + + Derivative of E02AHF + fitted + polynomial in Chebyshev series form + + Integral of E02AJF + fitted + polynomial in Chebyshev series form + + Evaluation of E02AKF + fitted + polynomial in one variable, from Chebyshev series form + + Least-squares curve cubic spline E02BAF + fit + (including interpolation) + + Evaluation of E02BBF + fitted + cubic spline, function only + + Evaluation of E02BCF + fitted + cubic spline, function and derivatives + + Evaluation of E02BDF + fitted + cubic spline, definite integral + + Least-squares cubic spline curve E02BEF + fit + , automatic knot placement + + Least-squares surface E02DAF + fit + , bicubic splines + + Least-squares surface E02DCF + fit + by bicubic splines with automatic knot placement, data + on ... + + Least-squares surface E02DDF + fit + by bicubic splines with automatic knot placement, ... + + Evaluation of a E02DEF + fitted + bicubic spline at a vector of points + + Evaluation of a E02DFF + fitted + bicubic spline at a mesh of points + + Sort 2-D data into panels for E02ZAF + fitting + bicubic splines + + Fits G02DAF + a general (multiple) linear regression model + + Fits G02DGF + a general linear regression model for new dependent ... + + Fits G02GBF + a generalized linear model with binomial errors + + Fits G02GCF + a generalized linear model with Poisson errors + + 2 + Performs the (chi) goodness of G08CGF + fit + test, for standard continuous distributions + + Goodness of G08 + fit + tests + + Computes a G01ALF + five-point + summary (median, hinges and extremes) + + Safe range of X02AMF + floating-point + arithmetic + + Safe range of complex X02ANF + floating-point + arithmetic + + Parameter of X02BHF + floating-point + arithmetic model, b + + Parameter of X02BJF + floating-point + arithmetic model, p + + Parameter of X02BKF + floating-point + arithmetic model, e + min + + Parameter of X02BLF + floating-point + arithmetic model, e + max + + Parameter of X02DJF + floating-point + arithmetic model, ROUNDS + + Univariate time series, update state set for G13AGF + forecasting + + Univariate time series, G13AHF + forecasting + from state set + + Univariate time series, state set and G13AJF + forecasts + , from fully specified seasonal ARIMA model + + Multivariate time series, state set and G13BJF + forecasts + from fully specified multi-input model + + Single 1-D real discrete C06EAF + Fourier + transform, no extra workspace + + Single 1-D Hermitian discrete C06EBF + Fourier + transform, no extra workspace + + Single 1-D complex discrete C06ECF + Fourier + transform, no extra workspace + + Multiple 1-D real discrete C06FPF + Fourier + transforms + + Multiple 1-D Hermitian discrete C06FQF + Fourier + transforms + + Multiple 1-D complex discrete C06FRF + Fourier + transforms + + 2-D complex discrete C06FUF + Fourier + transform + + frequency + table + + Frequency G01AEF + table from raw data + + frequency + (Daniell) window + + frequency + (Daniell) window + + Fresnel S20ACF + integral S(x) + + Fresnel S20ADF + integral C(x) + + Friedman G08AEF + two-way analysis of variance on k matched samples + + Computes probabilities for the G01EFF + gamma + distribution + + Computes deviates for the G01FFF + gamma + distribution + + Generates a vector of pseudo-random numbers from a G05FFF + gamma + distribution + + Gamma S14AAF + function + + Log S14ABF + Gamma + function + + Incomplete S14BAF + gamma + functions P(a,x) and Q(a,x) + + Unconstrained minimum of a sum of squares, combined E04FDF + Gauss-Newton + and modified Newton algorithm using function ... + + Unconstrained minimum of a sum of squares, combined E04GCF + Gauss-Newton + and quasi-Newton algorithm, using 1st derivatives + + Pre-computed weights and abscissae for D01BBF + Gaussian + quadrature rules, restricted choice of rule + + All eigenvalues of F02ADF + generalized + real eigenproblem of the form Ax=(lambda)Bx where ... + + All eigenvalues and eigenvectors of F02AEF + generalized + real eigenproblem of the form Ax=(lambda)Bx where ... + + All eigenvalues and optionally eigenvectors of F02BJF + generalized + eigenproblem by QZ algorithm, real matrices + + Fits a G02GBF + generalized + linear model with binomial errors + + Fits a G02GCF + generalized + linear model with Poisson errors + + Computes orthogonal rotations for loading matrix, G03BAF + generalized + orthomax criterion + + Generate F06AAF + real plane rotation (DROTG) + + Initialise random number G05CBF + generating + routines to give repeatable sequence + + Initialise random number G05CCF + generating + routines to give non-repeatable sequence + + Save state of random number G05CFF + generating + routines + + Restore state of random number G05CGF + generating + routines + + Set up reference vector for G05ECF + generating + pseudo-random integers, Poisson distribution + + Set up reference vector for G05EDF + generating + pseudo-random integers, binomial distribution + + Generates G05FAF + a vector of random numbers from a uniform ... + + Generates G05FBF + a vector of random numbers from an (negative) ... + + Generates G05FDF + a vector of random numbers from a Normal distribution + + Generates G05FEF + a vector of pseudo-random numbers from a beta ... + + Generates G05FFF + a vector of pseudo-random numbers from a gamma ... + + Generates G05HDF + a realisation of a multivariate time series from a ... + + Gill-Miller + method + + 2 + Performs the (chi) G08CGF + goodness + of fit test, for standard continuous distributions + + Goodness G08 + of fit tests + + Unconstrained minimum, pre-conditioned conjugate E04DGF + gradient + algorithm, function of several variables using 1st ... + + Hankel S17DLF + (j) + functions H (z), j=1,2, ... + (nu)+n + + Elliptic PDE, D03FAF + Helmholtz + equation, 3-D Cartesian co-ordinates + + Hermite + , one variable + + Single 1-D C06EBF + Hermitian + discrete Fourier transform, no extra workspace + + Multiple 1-D C06FQF + Hermitian + discrete Fourier transforms + + Complex conjugate of C06GBF + Hermitian + sequence + + Complex conjugate of multiple C06GQF + Hermitian + sequences + + Convert C06GSF + Hermitian + sequences to general complex sequences + + All eigenvalues of complex F02AWF + Hermitian + matrix + + All eigenvalues and eigenvectors of complex F02AXF + Hermitian + matrix + + Matrix-vector product, complex F06SCF + Hermitian + matrix (ZHEMV) + + Matrix-vector product, complex F06SDF + Hermitian + band matrix (ZHBMV) + + Matrix-vector product, complex F06SEF + Hermitian + packed matrix (ZHPMV) + + Rank-1 update, complex F06SPF + Hermitian + matrix (ZHER) + + Rank-1 update, complex F06SQF + Hermitian + packed matrix (ZHPR) + + Rank-2 update, complex F06SRF + Hermitian + matrix (ZHER2) + + Rank-2 update, complex F06SSF + Hermitian + packed matrix (ZHPR2) + + Matrix-matrix product, one complex F06ZCF + Hermitian + matrix, one complex rectangular matrix (ZHEMM) + + Rank-k update of a complex F06ZPF + Hermitian + matrix (ZHERK) + + Rank-2k update of a complex F06ZRF + Hermitian + matrix (ZHER2K) + + Hilbert + transform) + + Computes a five-point summary (median, G01ALF + hinges + and extremes) + + Multi-dimensional adaptive quadrature over D01FCF + hyper-rectangle + + Multi-dimensional quadrature over D01GBF + hyper-rectangle + , Monte Carlo method + + Incomplete S14BAF + gamma functions P(a,x) and Q(a,x) + + Index F06JLF + , real vector element with largest absolute value + (IDAMAX) + + Index F06JMF + , complex vector element with largest absolute value .. + + 1-D quadrature, adaptive, D01AMF + infinite + or semi-infinite interval + + 1-D quadrature, adaptive, infinite or semi- D01AMF + infinite + interval + + 1-D quadrature, adaptive, semi- D01ASF + infinite + interval, weight function cos((omega)x) or ... + + infinite + range, eigenvalue and eigenfunction, user-specified ... + + Calculates standardized residuals and G02FAF + influence + statistics + + Initialise G05CBF + random number generating routines to give ... + + Initialise G05CCF + random number generating routines to give ... + + input + data + + input + data + + Multivariate time series, estimation of multi- G13BEF + input + model + + input + model + + Input/output X04 + utilities + + Pseudo-random G05DYF + integer + from uniform distribution + + Set up reference vector for generating pseudo-random G05ECF + integers + , Poisson distribution + + Set up reference vector for generating pseudo-random G05EDF + integers + , binomial distribution + + Pseudo-random permutation of an G05EHF + integer + vector + + Pseudo-random sample from an G05EJF + integer + vector + + Pseudo-random G05EYF + integer + from reference vector + + Largest representable X02BBF + integer + + Return date and time as an array of X05AAF + integers + + Convert array of X05ABF + integers + representing date and time to character string + + integral + , one variable + + Integral E02AJF + of fitted polynomial in Chebyshev series form + + Evaluation of fitted cubic spline, definite E02BDF + integral + + Exponential S13AAF + integral + E (x) + 1 + + Cosine S13ACF + integral + Ci(x) + + Sine S13ADF + integral + Si(x) + + Fresnel S20ACF + integral + S(x) + + Fresnel S20ADF + integral + C(x) + + Degenerate symmetrised elliptic S21BAF + integral + of 1st kind R (x,y) + C + + Symmetrised elliptic S21BBF + integral + of 1st kind R (x,y,z) + F + + Symmetrised elliptic S21BCF + integral + of 2nd kind R (x,y,z) + D + + Symmetrised elliptic S21BDF + integral + of 3rd kind R (x,y,z,r) + J + + 1-D quadrature, D01GAF + integration + of function defined by data values, Gill-Miller ... + + Numerical D01 + integration + + Interpolating E01BAF + functions, cubic spline interpolant, one variable + + Interpolating functions, cubic spline E01BAF + interpolant + , one variable + + Interpolating E01BEF + functions, monotonicity-preserving, piecewise ... + + Interpolated E01BFF + values, interpolant computed by E01BEF, function ... + + Interpolated values, E01BFF + interpolant + computed by E01BEF, function only, one variable + + Interpolated E01BGF + values, interpolant computed by E01BEF, function ... + + Interpolated values, E01BGF + interpolant + computed by E01BEF, function and 1st derivative, ... + + Interpolated E01BHF + values, interpolant computed by E01BEF, definite ... + + Interpolated values, E01BHF + interpolant + computed by E01BEF, definite integral, one variable + + Interpolating E01DAF + functions, fitting bicubic spline, data on ... + + Interpolating E01SAF + functions, method of Renka and Cline, two ... + + Interpolating E01SEF + functions, modified Shepard's method, two ... + + Least-squares curve cubic spline fit (including E02BAF + interpolation) + + Inverse G01F + distributions + + Invert M01ZAF + a permutation + + iterative + refinement + + iterative + refinement + + ODEs, D02BBF + IVP + , Runge-Kutta-Merson method, over a range, intermediate + + ODEs, D02BHF + IVP + , Runge-Kutta-Merson method, until function of solution + is ... + + ODEs, D02CJF + IVP + , Adams method, until function of solution is zero, ... + + ODEs, stiff D02EJF + IVP + , BDF method, until function of solution is zero, ... + + Kelvin function S19ADF + kei + x + + Kelvin S19AAF + function ber x + + Kelvin S19ABF + function bei x + + Kelvin S19ACF + function ker x + + Kelvin S19ADF + function kei x + + Kendall/Spearman G02BNF + non-parametric rank correlation ... + + Kendall/Spearman G02BQF + non-parametric rank correlation ... + + Kelvin function S19ACF + ker + x + + Least-squares cubic spline curve fit, automatic E02BEF + knot + placement + + knot + placement, data on rectangular grid + + knot + placement, scattered data + + Kruskal-Wallis G08AFF + one-way analysis of variance on k samples of ... + + Mean, variance, skewness, G01AAF + kurtosis + etc, one variable, from raw data + + Mean, variance, skewness, G01ADF + kurtosis + etc, one variable, from frequency table + + All zeros of complex polynomial, modified C02AFF + Laguerre + method + + All zeros of real polynomial, modified C02AGF + Laguerre + method + + Index, real vector element with F06JLF + largest + absolute value (IDAMAX) + + Index, complex vector element with F06JMF + largest + absolute value (IZAMAX) + + Largest X02ALF + positive model number + + Largest X02BBF + representable integer + + LDL F01MCF + T + factorization of real symmetric positive-definite ... + + Constructs a stem and G01ARF + leaf + plot + + Least-squares E02ADF + curve fit, by polynomials, arbitrary data points + + Least-squares E02AGF + polynomial fit, values and derivatives may be ... + + Least-squares E02BAF + curve cubic spline fit (including interpolation) + + Least-squares E02BEF + cubic spline curve fit, automatic knot placement + + Least-squares E02DAF + surface fit, bicubic splines + + Least-squares E02DCF + surface fit by bicubic splines with automatic ... + + Least-squares E02DDF + surface fit by bicubic splines with automatic ... + + Covariance matrix for nonlinear E04YCF + least-squares + problem + + Least-squares F04JGF + (if rank=n) or minimal least-squares (if ... + + Least-squares (if rank=n) or minimal F04JGF + least-squares + (if rank=n + + Rank-1 F06PMF + update, real rectangular matrix (DGER) + + Rank-1 F06PPF + update, real symmetric matrix (DSYR) + + Rank-1 F06PQF + update, real symmetric packed matrix (DSPR) + + Rank-2 F06PRF + update, real symmetric matrix (DSYR2) + + Rank-2 F06PSF + update, real symmetric packed matrix (DSPR2) + + Rank-1 F06SMF + update, complex rectangular matrix, unconjugated ... + + Rank-1 F06SNF + update, complex rectangular matrix, conjugated vector . + + Rank-1 F06SPF + update, complex Hermitian matrix (ZHER) + + Rank-1 F06SQF + update, complex Hermitian packed matrix (ZHPR) + + Rank-2 F06SRF + update, complex Hermitian matrix (ZHER2) + + Rank-2 F06SSF + update, complex Hermitian packed matrix (ZHPR2) + + Rank-k F06YPF + update of a real symmetric matrix (DSYRK) + + Rank-2k F06YRF + update of a real symmetric matrix (DSYR2K) + + Rank-k F06ZPF + update of a complex Hermitian matrix (ZHERK) + + Rank-2k F06ZRF + update of a complex Hermitian matrix (ZHER2K) + + Rank-k F06ZUF + update of a complex symmetric matrix (ZSYRK) + + Rank-2k F06ZWF + update of a complex symmetric matrix (ZHER2K) + + Kendall/Spearman non-parametric G02BNF + rank + correlation coefficients, no missing values, + overwriting ... + + Kendall/Spearman non-parametric G02BQF + rank + correlation coefficients, no missing values, preserving + + rank + test + + Rank M01DAF + a vector, real numbers + + Rank M01DEF + rows of a matrix, real numbers + + Rank M01DJF + columns of a matrix, real numbers + + Rearrange a vector according to given M01EAF + ranks + , real numbers + + Generates a G05HDF + realisation + of a multivariate time series from a VARMA model + + Rearrange M01EAF + a vector according to given ranks, real numbers + + Multi-dimensional adaptive quadrature over hyper- D01FCF + rectangle + + Multi-dimensional quadrature over hyper- D01GBF + rectangle + , Monte Carlo method + + Discretize a 2nd order elliptic PDE on a D03EEF + rectangle + + rectangular + grid + + rectangular + grid + + Matrix-vector product, real F06PAF + rectangular + matrix (DGEMV) + + Matrix-vector product, real F06PBF + rectangular + band matrix (DGBMV) + + Rank-1 update, real F06PMF + rectangular + matrix (DGER) + + Matrix-vector product, complex F06SAF + rectangular + matrix (ZGEMV) + + Matrix-vector product, complex F06SBF + rectangular + band matrix (ZGBMV) + + Rank-1 update, complex F06SMF + rectangular + matrix, unconjugated vector (ZGERU) + + Rank-1 update, complex F06SNF + rectangular + matrix, conjugated vector (ZGERC) + + Matrix-matrix product, two real F06YAF + rectangular + matrices (DGEMM) + + rectangular + matrix (DSYMM) + + rectangular + matrix (DTRMM) + + Matrix-matrix product, two complex F06ZAF + rectangular + matrices (ZGEMM) + + rectangular + matrix (ZHEMM) + + rectangular + matrix (ZTRMM) + + rectangular + matrix (ZSYMM) + + Set up G05EAF + reference + vector for multivariate Normal distribution + + Set up G05ECF + reference + vector for generating pseudo-random integers, ... + + Set up G05EDF + reference + vector for generating pseudo-random integers, ... + + Set up G05EXF + reference + vector from supplied cumulative distribution ... + + Pseudo-random integer from G05EYF + reference + vector + + Pseudo-random multivariate Normal vector from G05EZF + reference + vector + + refinement + + refinement + + Simple linear G02CAF + regression + with constant term, no missing values + + Fits a general (multiple) linear G02DAF + regression + model + + Fits a general linear G02DGF + regression + model for new dependent variable + + Computes estimable function of a general linear G02DNF + regression + model and its standard error + + Nonlinear E04 + regression + + 2nd order Sturm-Liouville problem, D02KEF + regular/singular + system, finite/infinite range, eigenvalue ... + + Interpolating functions, method of E01SAF + Renka + and Cline, two variables + + Calculates standardized G02FAF + residuals + and influence statistics + Univariate time series, diagnostic checking of G13ASF + residuals + , following G13AFF + + right-hand + sides + + Solution of real simultaneous linear equations, one F04ARF + right-hand + side + + right-hand + side using iterative refinement + + Solution of real simultaneous linear equations, one F04ATF + right-hand + side using iterative refinement + + right-hand + side + + Solves a system of equations with multiple F06YJF + right-hand + sides, real triangular coefficient matrix (DTRSM) + + Solves system of equations with multiple F06ZJF + right-hand + sides, complex triangular coefficient matrix (ZTRSM) + + Solution of real system of linear equations, multiple F07AEF + right-hand + sides, matrix already factorized by F07ADF (DGETRS) + + right-hand + sides, matrix already factorized by F07FDF (DPOTRS) + + Generate real plane F06AAF + rotation + (DROTG) + + Apply real plane F06EPF + rotation + (DROT) + + Computes orthogonal G03BAF + rotations + for loading matrix, generalized orthomax criterion + + rules + , restricted choice of rule + + rule + ODEs, IVP, D02BBF + Runge-Kutta-Merson + method, over a range, intermediate output + + ODEs, IVP, D02BHF + Runge-Kutta-Merson + method, until function of solution is zero ... + + Safe X02AMF + range of floating-point arithmetic + + Safe X02ANF + range of complex floating-point arithmetic + + Pseudo-random G05EJF + sample + from an integer vector + + Sign test on two paired G08AAF + samples + + Median test on two G08ACF + samples + of unequal size + + Friedman two-way analysis of variance on k matched G08AEF + samples + + Kruskal-Wallis one-way analysis of variance on k G08AFF + samples + of unequal size + + Performs the Wilcoxon one- G08AGF + sample + (matched pairs) signed rank test + + Performs the Mann-Whitney U test on two independent G08AHF + samples + + sample + + sample + + Univariate time series, G13ABF + sample + autocorrelation function + + Univariate time series, smoothed G13CBF + sample + spectrum using spectral smoothing by the trapezium ... + + Multivariate time series, smoothed G13CDF + sample + cross spectrum using spectral smoothing by the ... + + Add F06ECF + scalar + times real vector to real vector (DAXPY) + + Multiply real vector by F06EDF + scalar + (DSCAL) + + Add F06GCF + scalar + times complex vector to complex vector (ZAXPY) + + Multiply complex vector by complex F06GDF + scalar + (ZSCAL) + + Multiply complex vector by real F06JDF + scalar + (ZDSCAL) + + scattered + data + + Univariate time series, G13AAF + seasonal + and non-seasonal differencing + + Univariate time series, seasonal and non- G13AAF + seasonal + differencing + + Univariate time series, preliminary estimation, G13ADF + seasonal + ARIMA model + + Univariate time series, estimation, G13AFF + seasonal + ARIMA model + + seasonal + ARIMA model + + 1-D quadrature, adaptive, infinite or D01AMF + semi-infinite + interval + + 1-D quadrature, adaptive, D01ASF + semi-infinite + interval, weight function cos((omega)x) ... + Complex conjugate of Hermitian C06GBF + sequence + + Complex conjugate of complex C06GCF + sequence + + Complex conjugate of multiple Hermitian C06GQF + sequences + + Convert Hermitian C06GSF + sequences + to general complex sequences + + Convert Hermitian sequences to general complex C06GSF + sequences + + sequence + + sequence + + Minimum, function of several variables, E04UCF + sequential + QP method, nonlinear constraints, using function ... + + Interpolating functions, modified E01SEF + Shepard's + method, two variables + + Sign G08AAF + test on two paired samples + + Performs the Wilcoxon one-sample (matched pairs) G08AGF + signed + rank test + + Solution of complex F04ADF + simultaneous + linear equations with multiple right-hand sides + + Solution of real F04ARF + simultaneous + linear equations, one right-hand side + + Solution of real symmetric positive-definite F04ASF + simultaneous + linear equations, one right-hand side using ... + + Solution of real F04ATF + simultaneous + linear equations, one right-hand side using ... + + Solution of real sparse F04AXF + simultaneous + linear equations (coefficient matrix already ... + + simultaneous + linear equations, one right-hand side + + Real sparse symmetric positive-definite F04MAF + simultaneous + linear equations (coefficient matrix already ... + + Real sparse symmetric F04MBF + simultaneous + linear equations + + simultaneous + linear equations (coefficient matrix already ... + + sin + ((omega)x) + + sin + ((omega)x) + + Sine S13ADF + integral Si(x) + + 2nd order Sturm-Liouville problem, regular/ D02KEF + singular + system, finite/infinite range, eigenvalue and ... + + singularities + at user-specified break-points + + singularities + of algebraico-logarithmic type + + Mean, variance, G01AAF + skewness + , kurtosis etc, one variable, from raw data + + Mean, variance, G01ADF + skewness + , kurtosis etc, one variable, from frequency table + + Smallest X02AKF + positive model number + + Univariate time series, G13CBF + smoothed + sample spectrum using spectral smoothing by the ... + + smoothing + by the trapezium frequency (Daniell) window + + Multivariate time series, G13CDF + smoothed + sample cross spectrum using spectral smoothing by ... + + smoothing + by the trapezium frequency (Daniell) window + + Sort E02ZAF + 2-D data into panels for fitting bicubic splines + + Sort M01CAF + a vector, real numbers + + LU factorization of real F01BRF + sparse + matrix + + LU factorization of real F01BSF + sparse + matrix with known sparsity pattern + + LU factorization of real sparse matrix with known F01BSF + sparsity + pattern + + T + LL factorization of real F01MAF + sparse + symmetric positive-definite matrix + + Selected eigenvalues and eigenvectors of F02FJF + sparse + symmetric eigenproblem + + Solution of real F04AXF + sparse + simultaneous linear equations (coefficient matrix ... + + Real F04MAF + sparse + symmetric positive-definite simultaneous linear ... + + Real F04MBF + sparse + symmetric simultaneous linear equations + + Sparse F04QAF + linear least-squares problem, m real equations in ... + + Kendall/ G02BNF + Spearman + non-parametric rank correlation coefficients, no ... + + Kendall/ G02BQF + Spearman + non-parametric rank correlation coefficients, no ... + + Approximation of S + special + functions + + Univariate time series, smoothed sample G13CBF + spectrum + using spectral smoothing by the trapezium frequency ... + + spectral + smoothing by the trapezium frequency (Daniell) window + + Multivariate time series, smoothed sample cross G13CDF + spectrum + using spectral smoothing by the trapezium frequency ... + + spectral + smoothing by the trapezium frequency (Daniell) window + + Interpolating functions, cubic E01BAF + spline + interpolant, one variable + + Interpolating functions, fitting bicubic E01DAF + spline + , data on rectangular grid + + Least-squares curve cubic E02BAF + spline + fit (including interpolation) + + Evaluation of fitted cubic E02BBF + spline + , function only + + Evaluation of fitted cubic E02BCF + spline + , function and derivatives + + Evaluation of fitted cubic E02BDF + spline + , definite integral + + Least-squares cubic E02BEF + spline + curve fit, automatic knot placement + Least-squares surface fit, bicubic E02DAF + splines + + Least-squares surface fit by bicubic E02DCF + splines + with automatic knot placement, data on rectangular grid + + Least-squares surface fit by bicubic E02DDF + splines + with automatic knot placement, scattered data + + Evaluation of a fitted bicubic E02DEF + spline + at a vector of points + + Evaluation of a fitted bicubic E02DFF + spline + at a mesh of points + + Sort 2-D data into panels for fitting bicubic E02ZAF + splines + + B- E02 + splines + + Least- E02ADF + squares + curve fit, by polynomials, arbitrary data points + + Least- E02AGF + squares + polynomial fit, values and derivatives may be ... + + Least- E02BAF + squares + curve cubic spline fit (including interpolation) + + Least- E02BEF + squares + cubic spline curve fit, automatic knot placement + + Least- E02DAF + squares + surface fit, bicubic splines + + Least- E02DCF + squares + surface fit by bicubic splines with automatic knot ... + + Least- E02DDF + squares + surface fit by bicubic splines with automatic knot ... + Unconstrained minimum of a sum of E04FDF + squares + , combined Gauss-Newton and modified Newton algorithm . + + Unconstrained minimum of a sum of E04GCF + squares + , combined Gauss-Newton and quasi-Newton algorithm, ... + + Covariance matrix for nonlinear least- E04YCF + squares + problem + + Least- F04JGF + squares + (if rank=n) or minimal least-squares (if ... + + Least-squares (if rank=n) or minimal least- F04JGF + squares + (if rank>= +\begin{page}{manpageXXconvert}{NAG Documentation: conversion} +\beginscroll +\begin{verbatim} + + + + CONVERSION(3NAG) Foundation Library (12/10/92) CONVERSION(3NAG) + + + + Introduction Converting from the Workstation Library + Converting from the Workstation Library + + The NAG Foundation Library is a successor product to an earlier, + smaller subset of the full NAG Fortran Library which was called + the NAG Workstation Library. The Foundation Library has been + designed to be upwards compatible, in terms of functionality, + with the Workstation Library. However some routines that were + present in the Workstation Library have been replaced by more up- + to-date routines from the NAG Fortran Library, which provide + improved algorithms or software design. + + The list below gives the names of those routines which were + available in the Workstation Library, but are not included in the + Foundation Library. For each such routine, it also gives the name + of the routine in the Foundation Library which best covers the + same functionality. + + Workstation Foundation + Library Library + + C02AEF C02AGF + + D02CBF D02CJF + + D02CHF D02CJF + + D02EBF D02EJF + + D02EHF D02EJF + + D02HAF D02GAF + + D02HBF D02RAF + + D02SAF D02RAF + + E02DBF E02DEF + + E04VDF E04UCF + + E04ZCF E04UCF (see Note 1) + + F01BTF F07ADF + + F01BXF F07FDF + + F02WAF F02WEF + + F04AYF F07AEF + + F04AZF F07FEF + + F04YAF G02DAF + + G01ABF G02BXF (with M = 2) + + G01BAF G01EBF + + G01BBF G01EDF + + G01BCF G01ECF + + G01BDF G01EEF + + G01CAF G01FBF + + G01CBF G01FDF + + G01CCF G01FCF + + G01CDF G01FEF + + G01CEF G01FAF + + G02BAF G02BXF + + G02BGF G02BXF + + G02CEF G02DAF (see Note 2) + + G02CGF G02DAF + + G02CJF G02DAF + + G05DBF G05FBF + + G05DCF G05CAF (see Note 3) + + G05DEF G05FFF + + G05DHF G05FFF (see Note 4) + + G05EGF G05HDF + + G05EWF G05HDF + + G08ABF G08AGF + + G08ADF G08AHF + + M01AKF M01DAF + + M01APF M01CAF + + S15ABF G01EAF + + S15ACF G01EAF + + X02AAF X02AJF + + X02ACF X02ALF + + Notes: + + 1. E04ZCF checks user-supplied routines for evaluating the + first derivatives of the objective function and constraint + functions supplied to E04VDF. This functionality is now + provided by E04UCF, using the optional parameters Verify + Objective Gradients and Verify Constraint Gradients. + + 2. G02CEF selects variables to be included in a linear + regression performed by G02CGF. This functionality is now + provided by the parameter ISX of G02DAF. + + 3. A call to G05DCF can be replaced by a simple transformation + of the result of a call to G05CAF. The statement + X = G05DCF(A,B) + can be replaced by the statements + + X = G05CAF(X) + X = A + B*LOG(X/(1.0D0-X)) + + 2 + 4. G05DHF generates random numbers from a (chi) distribution + with N degrees of freedom. This can be achieved by calling + G05FFF with the values DBLE(N)/2.0D0 and 2.0D0 for the + parameters A and B respectively. + +\end{verbatim} +\endscroll +\end{page} + +@ +\section{nagc.ht} +\subsection{ Zeros of Polynomials} +\label{manpageXXc02} +\index{pages!manpageXXc02!nagc.ht} +\index{nagc.ht!pages!manpageXXc02} +\index{manpageXXc02!nagc.ht!pages} +<>= +\begin{page}{manpageXXc02}{NAG Documentation: c02} +\beginscroll +\begin{verbatim} + + + + C02(3NAG) Foundation Library (12/10/92) C02(3NAG) + + + + C02 -- Zeros of Polynomials Introduction -- C02 + Chapter C02 + Zeros of Polynomials + + 1. Scope of the Chapter + + This chapter is concerned with computing the zeros of a + polynomial with real or complex coefficients. + + 2. Background to the Problems + + Let f(z) be a polynomial of degree n with complex coefficients + a : + i + + n n-1 n-2 + f(z)==a z +a z +a z +...+a z+a , a /=0. + 0 1 2 n-1 n 0 + + A complex number z is called a zero of f(z) (or equivalently a + 1 + root of the equation f(z)=0), if: + + f(z )=0. + 1 + + If z is a zero, then f(z) can be divided by a factor (z-z ): + 1 1 + + f(z)=(z-z )f (z) (1) + 1 1 + + where f (z) is a polynomial of degree n-1. By the Fundamental + 1 + Theorem of Algebra, a polynomial f(z) always has a zero, and so + the process of dividing out factors (z-z ) can be continued until + i + we have a complete factorization of f(z) + + f(z)==a (z-z )(z-z )...(z-z ). + 0 1 2 n + + Here the complex numbers z ,z ,...,z are the zeros of f(z); they + 1 2 n + may not all be distinct, so it is sometimes more convenient to + write: + + m m m + 1 2 k + f(z)==a (z-z ) (z-z ) ...(z-z ) , k<=n, + 0 1 2 k + + with distinct zeros z ,z ,...,z and multiplicities m >=1. If + 1 2 k i + m =1, z is called a single zero, if m >1, z is called a + i i i i + multiple or repeated zero; a multiple zero is also a zero of the + derivative of f(z). + + If the coefficients of f(z) are all real, then the zeros of f(z) + are either real or else occur as pairs of conjugate complex + numbers x+iy and x-iy. A pair of complex conjugate zeros are the + 2 + zeros of a quadratic factor of f(z), (z +rz+s), with real + coefficients r and s. + + Mathematicians are accustomed to thinking of polynomials as + pleasantly simple functions to work with. However the problem of + numerically computing the zeros of an arbitrary polynomial is far + from simple. A great variety of algorithms have been proposed, of + which a number have been widely used in practice; for a fairly + comprehensive survey, see Householder [1]. All general algorithms + are iterative. Most converge to one zero at a time; the + corresponding factor can then be divided out as in equation (1) + above - this process is called deflation or, loosely, dividing + out the zero - and the algorithm can be applied again to the + polynomial f (z). A pair of complex conjugate zeros can be + 1 + divided out together - this corresponds to dividing f(z) by a + quadratic factor. + + Whatever the theoretical basis of the algorithm, a number of + practical problems arise: for a thorough discussion of some of + them see Peters and Wilkinson [2] and Wilkinson [3]. The most + elementary point is that, even if z is mathematically an exact + 1 + zero of f(z), because of the fundamental limitations of computer + arithmetic the computed value of f(z ) will not necessarily be + 1 + exactly 0.0. In practice there is usually a small region of + values of z about the exact zero at which the computed value of + f(z) becomes swamped by rounding errors. Moreover in many + algorithms this inaccuracy in the computed value of f(z) results + in a similar inaccuracy in the computed step from one iterate to + the next. This limits the precision with which any zero can be + computed. Deflation is another potential cause of trouble, since, + in the notation of equation (1), the computed coefficients of + f (z) will not be completely accurate, especially if z is not an + 1 1 + exact zero of f(z); so the zeros of the computed f (z) will + 1 + deviate from the zeros of f(z). + + A zero is called ill-conditioned if it is sensitive to small + changes in the coefficients of the polynomial. An ill-conditioned + zero is likewise sensitive to the computational inaccuracies just + mentioned. Conversely a zero is called well-conditioned if it is + comparatively insensitive to such perturbations. Roughly speaking + a zero which is well separated from other zeros is well- + conditioned, while zeros which are close together are ill- + conditioned, but in talking about 'closeness' the decisive factor + is not the absolute distance between neighbouring zeros but their + ratio: if the ratio is close to 1 the zeros are ill-conditioned. + In particular, multiple zeros are ill-conditioned. A multiple + zero is usually split into a cluster of zeros by perturbations in + the polynomial or computational inaccuracies. + + 2.1. References + + [1] Householder A S (1970) The Numerical Treatment of a Single + Nonlinear Equation. McGraw-Hill. + + [2] Peters G and Wilkinson J H (1971) Practical Problems Arising + in the Solution of Polynomial Equations. J. Inst. Maths + Applics. 8 16--35. + + [3] Wilkinson J H (1963) Rounding Errors in Algebraic Processes, + Chapter 2. HMSO. + + 3. Recommendations on Choice and Use of Routines + + 3.1. Discussion + + Two routines are available: C02AFF for polynomials with complex + coefficients and C02AGF for polynomials with real coefficients. + + C02AFF and C02AGF both use a variant of Laguerre's Method due to + BT Smith to calculate each zero until the degree of the deflated + polynomial is less than 3, whereupon the remaining zeros are + obtained using the 'standard' closed formulae for a quadratic or + linear equation. + + The accuracy of the roots will depend on how ill-conditioned they + are. Peters and Wilkinson [2] describe techniques for estimating + the errors in the zeros after they have been computed. + + 3.2. Index + + Zeros of a complex polynomial C02AFF + Zeros of a real polynomial C02AGF + + + C02 -- Zeros of Polynomials Contents -- C02 + Chapter C02 + + Zeros of Polynomials + + C02AFF All zeros of complex polynomial, modified Laguerre method + + C02AGF All zeros of real polynomial, modified Laguerre method + +\end{verbatim} +\endscroll +\end{page} + +@ +\subsection{ Roots of a complex polynomial equation} +\label{manpageXXc02aff} +\index{pages!manpageXXc02aff!nagc.ht} +\index{nagc.ht!pages!manpageXXc02aff} +\index{manpageXXc02aff!nagc.ht!pages} +<>= +\begin{page}{manpageXXc02aff}{NAG Documentation: c02aff} +\beginscroll +\begin{verbatim} + + + + C02AFF(3NAG) Foundation Library (12/10/92) C02AFF(3NAG) + + + + C02 -- Zeros of Polynomials C02AFF + C02AFF -- NAG Foundation Library Routine Document + + Note: Before using this routine, please read the Users' Note for + your implementation to check implementation-dependent details. + The symbol (*) after a NAG routine name denotes a routine that is + not included in the Foundation Library. + + 1. Purpose + + C02AFF finds all the roots of a complex polynomial equation, + using a variant of Laguerre's Method. + + 2. Specification + + SUBROUTINE C02AFF (A, N, SCALE, Z, W, IFAIL) + INTEGER N, IFAIL + DOUBLE PRECISION A(2,N+1), Z(2,N), W(4*(N+1)) + LOGICAL SCALE + + 3. Description + + The routine attempts to find all the roots of the nth degree + complex polynomial equation + + n n-1 n-2 + P(z)=a z +a z +a z +...+a z+a =0. + 0 1 2 n-1 n + + The roots are located using a modified form of Laguerre's Method, + originally proposed by Smith [2]. + + The method of Laguerre [3] can be described by the iterative + scheme + + -n*P(z ) + k + L(z )=z -z = ----------------, + k k+1 k + P'(z )+- /H(z ) + k \/ k + + 2 + where H(z )=(n-1)*[(n-1)*(P'(z )) -n*P(z )P''(z )], and z is + k k k k 0 + specified. + + The sign in the denominator is chosen so that the modulus of the + Laguerre step at z , viz. |L(z )|, is as small as possible. The + k k + method can be shown to be cubically convergent for isolated roots + (real or complex) and linearly convergent for multiple roots. + The routine generates a sequence of iterates z , z , z ,..., such + 1 2 3 + that |P(z )|<|P(z )| and ensures that z +L(z ) 'roughly' + k+1 k k+1 k+1 + lies inside a circular region of radius |F| about z known to + k + contain a zero of P(z); that is, |L(z )|<=|F|, where F denotes + k+1 + the Fejer bound (see Marden [1]) at the point z . Following Smith + k + [2], F is taken to be min(B,1.445*n*R), where B is an upper bound + for the magnitude of the smallest zero given by + + 1/n + B=1.0001*min(\/n*L(z ),|r |,|a /a | ), + k 1 n 0 + + r is the zero X of smaller magnitude of the quadratic equation + 1 + + 2 + 2(P''(z )/(2*n*(n-1)))X +2(P'(z )/n)X+P(z )=0 + k k k + + and the Cauchy lower bound R for the smallest zero is computed + (using Newton's Method) as the positive root of the polynomial + equation + + n n-1 n-2 + |a |z +|a |z +|a |z +...+|a |z-|a |=0. + 0 1 2 n-1 n + + Starting from the origin, successive iterates are generated + according to the rule z =z +L(z ) for k = 1,2,3,... and L(z ) + k+1 k k k + is 'adjusted' so that |P(z )|<|P(z )| and |L(z )|<=|F|. The + k+1 k k+1 + iterative procedure terminates if P(z ) is smaller in absolute + k+1 + value than the bound on the rounding error in P(z ) and the + k+1 + current iterate z =z is taken to be a zero of P(z). The + p k+1 + ~ + deflated polynomial P(z)=P(z)/(z-z ) of degree n-1 is then + p + formed, and the above procedure is repeated on the deflated + polynomial until n<3, whereupon the remaining roots are obtained + via the 'standard' closed formulae for a linear (n = 1) or + quadratic (n = 2) equation. + + To obtain the roots of a quadratic polynomial, C02AHF(*) can be + used. + + 4. References + + [1] Marden M (1966) Geometry of Polynomials. Mathematical + Surveys. 3 Am. Math. Soc., Providence, RI. + + [2] Smith B T (1967) ZERPOL: A Zero Finding Algorithm for + Polynomials Using Laguerre's Method. Technical Report. + Department of Computer Science, University of Toronto, + Canada. + + [3] Wilkinson J H (1965) The Algebraic Eigenvalue Problem. + Clarendon Press. + + 5. Parameters + + 1: A(2,N+1) -- DOUBLE PRECISION array Input + On entry: if A is declared with bounds (2,0:N), then A(1,i) + and A(2,i) must contain the real and imaginary parts of a + i + n-i + (i.e., the coefficient of z ), for i=0,1,...,n. + Constraint: A(1,0) /= 0.0 or A(2,0) /= 0.0. + + 2: N -- INTEGER Input + On entry: the degree of the polynomial, n. Constraint: N >= + 1. + + 3: SCALE -- LOGICAL Input + On entry: indicates whether or not the polynomial is to be + scaled. See Section 8 for advice on when it may be + preferable to set SCALE = .FALSE. and for a description of + the scaling strategy. Suggested value: SCALE = .TRUE.. + + 4: Z(2,N) -- DOUBLE PRECISION array Output + On exit: the real and imaginary parts of the roots are + stored in Z(1,i) and Z(2,i) respectively, for i=1,2,...,n. + + 5: W(4*(N+1)) -- DOUBLE PRECISION array Workspace + + 6: IFAIL -- INTEGER Input/Output + On entry: IFAIL must be set to 0, -1 or 1. For users not + familiar with this parameter (described in the Essential + Introduction) the recommended value is 0. + + On exit: IFAIL = 0 unless the routine detects an error (see + Section 6). + + 6. Error Indicators and Warnings + + Errors detected by the routine: + + If on entry IFAIL = 0 or -1, explanatory error messages are + output on the current error message unit (as defined by X04AAF). + + IFAIL= 1 + On entry A(1,0) = 0.0 and A(2,0) = 0.0, + + or N < 1. + + IFAIL= 2 + The iterative procedure has failed to converge. This error + is very unlikely to occur. If it does, please contact NAG + immediately, as some basic assumption for the arithmetic has + been violated. See also Section 8. + + IFAIL= 3 + Either overflow or underflow prevents the evaluation of P(z) + near some of its zeros. This error is very unlikely to + occur. If it does, please contact NAG immediately. See also + Section 8. + + 7. Accuracy + + All roots are evaluated as accurately as possible, but because of + the inherent nature of the problem complete accuracy cannot be + guaranteed. + + 8. Further Comments + + If SCALE = .TRUE., then a scaling factor for the coefficients is + chosen as a power of the base B of the machine so that the + EMAX-P + largest coefficient in magnitude approaches THRESH = B . + Users should note that no scaling is performed if the largest + coefficient in magnitude exceeds THRESH, even if SCALE = .TRUE.. + (For definition of B, EMAX and P see the Chapter Introduction + X02.) + + However, with SCALE = .TRUE., overflow may be encountered when + the input coefficients a ,a ,a ,...,a vary widely in magnitude, + 0 1 2 n + (4*P) + particularly on those machines for which B overflows. In + such cases, SCALE should be set to .FALSE. and the coefficients + scaled so that the largest coefficient in magnitude does not + (EMAX-2*P) + exceed B . + + Even so, the scaling strategy used in C02AFF is sometimes + insufficient to avoid overflow and/or underflow conditions. In + such cases, the user is recommended to scale the independent + variable (z) so that the disparity between the largest and + smallest coefficient in magnitude is reduced. That is, use the + routine to locate the zeros of the polynomial d*P(cz) for some + suitable values of c and d. For example, if the original + -100 100 20 -10 + polynomial was P(z)=2 i+2 z , then choosing c=2 and + 100 20 + d=2 , for instance, would yield the scaled polynomial i+z , + which is well-behaved relative to overflow and underflow and has + 10 + zeros which are 2 times those of P(z). + + If the routine fails with IFAIL = 2 or 3, then the real and + imaginary parts of any roots obtained before the failure occurred + are stored in Z in the reverse order in which they were found. + Let n denote the number of roots found before the failure + R + occurred. Then Z(1,n) and Z(2,n) contain the real and imaginary + parts of the 1st root found, Z(1,n-1) and Z(2,n-1) contain the + real and imaginary parts of the 2nd root found, ..., Z(1,n ) and + R + Z(2,n ) contain the real and imaginary parts of the n th root + R R + found. After the failure has occurred, the remaining 2*(n-n ) + R + elements of Z contain a large negative number (equal to + + + -1/(X02AMF().\/2)). + + 9. Example + + 5 4 3 2 + To find the roots of the polynomial a z +a z +a z +a z +a z+a =0, + 0 1 2 3 4 5 + where a =(5.0+6.0i), a =(30.0+20.0i), a =-(0.2+6.0i), + 0 1 2 + a =(50.0+100000.0i), a =-(2.0-40.0i) and a =(10.0+1.0i). + 3 4 5 + + The example program is not reproduced here. The source code for + all example programs is distributed with the NAG Foundation + Library software and should be available on-line. + +\end{verbatim} +\endscroll +\end{page} + +@ +\subsection{ Roots of a real polynomial equation} +\label{manpageXXc02agf} +\index{pages!manpageXXc02agf!nagc.ht} +\index{nagc.ht!pages!manpageXXc02agf} +\index{manpageXXc02agf!nagc.ht!pages} +<>= +\begin{page}{manpageXXc02agf}{NAG Documentation: c02agf} +\beginscroll +\begin{verbatim} + + + + C02AGF(3NAG) Foundation Library (12/10/92) C02AGF(3NAG) + + + + C02 -- Zeros of Polynomials C02AGF + C02AGF -- NAG Foundation Library Routine Document + + Note: Before using this routine, please read the Users' Note for + your implementation to check implementation-dependent details. + The symbol (*) after a NAG routine name denotes a routine that is + not included in the Foundation Library. + + 1. Purpose + + C02AGF finds all the roots of a real polynomial equation, using a + variant of Laguerre's Method. + + 2. Specification + + SUBROUTINE C02AGF (A, N, SCALE, Z, W, IFAIL) + INTEGER N, IFAIL + DOUBLE PRECISION A(N+1), Z(2,N), W(2*(N+1)) + LOGICAL SCALE + + 3. Description + + The routine attempts to find all the roots of the nth degree real + polynomial equation + + n n-1 n-2 + P(z)=a z +a z +a z +...+a z+a =0. + 0 1 2 n-1 n + + The roots are located using a modified form of Laguerre's Method, + originally proposed by Smith [2]. + + The method of Laguerre [3] can be described by the iterative + scheme + + -n*P(z ) + k + L(z )=z -z = ----------------, + k k+1 k + P'(z )+- /H(z ) + k \/ k + + 2 + where H(z )=(n-1)*[(n-1)*(P'(z )) -n*P(z )P''(z )], and z is + k k k k 0 + specified. + + The sign in the denominator is chosen so that the modulus of the + Laguerre step at z , viz. |L(z )|, is as small as possible. The + k k + method can be shown to be cubically convergent for isolated roots + (real or complex) and linearly convergent for multiple roots. + The routine generates a sequence of iterates z , z , z ,..., such + 1 2 3 + that |P(z +1)|<|P(z )| and ensures that z +L(z ) 'roughly' + k k k+1 k+1 + lies inside a circular region of radius |F| about z known to + k + contain a zero of P(z); that is, |L(z )|<=|F|, where F denotes + k+1 + the Fejer bound (see Marden [1]) at the point z . Following Smith + k + [2], F is taken to be min(B,1.445*n*R), where B is an upper bound + for the magnitude of the smallest zero given by + + 1/n + B=1.0001*min(\/n*L(z ),|r |,|a /a | ), + k 1 n 0 + + r is the zero X of smaller magnitude of the quadratic equation + 1 + + 2 + 2(P''(z )/(2*n*(n-1)))X +2(P'(z )/n)X+P(z )=0 + k k k + + and the Cauchy lower bound R for the smallest zero is computed + (using Newton's Method) as the positive root of the polynomial + equation + + n n-1 n-2 + |a |z +|a |z +|a |z +...+|a |z-|a |=0. + 0 1 2 n-1 n + + Starting from the origin, successive iterates are generated + according to the rule z =z +L(z ) for k=1,2,3,... and L(z ) is + k+1 k k k + k+1 k k+1 + iterative procedure terminates if P(z ) is smaller in absolute + k+1 + value than the bound on the rounding error in P(z ) and the + k+1 + current iterate z =z is taken to be a zero of P(z) (as is its + p k-1 + + + conjugate z if z is complex). The deflated polynomial + p p + ~ + P(z)=P(z)/(z-z ) of degree n-1 if z is real + p p + ~ + (P(z)=P(z)/((z-z )(z-z )) of degree n-2 if z is complex) is then + p p p + formed, and the above procedure is repeated on the deflated + polynomial until n<3, whereupon the remaining roots are obtained + via the 'standard' closed formulae for a linear (n = 1) or + quadratic (n = 2) equation. + + To obtain the roots of a quadratic polynomial, C02AJF(*) can be + used. + + 4. References + + [1] Marden M (1966) Geometry of Polynomials. Mathematical + Surveys. 3 Am. Math. Soc., Providence, RI. + + [2] Smith B T (1967) ZERPOL: A Zero Finding Algorithm for + Polynomials Using Laguerre's Method. Technical Report. + Department of Computer Science, University of Toronto, + Canada. + + [3] Wilkinson J H (1965) The Algebraic Eigenvalue Problem. + Clarendon Press. + + 5. Parameters + + 1: A(N+1) -- DOUBLE PRECISION array Input + On entry: if A is declared with bounds (0:N), then A(i) + n-i + must contain a (i.e., the coefficient of z ), for + i + i=0,1,...,n. Constraint: A(0) /= 0.0. + + 2: N -- INTEGER Input + On entry: the degree of the polynomial, n. Constraint: N >= + 1. + + 3: SCALE -- LOGICAL Input + On entry: indicates whether or not the polynomial is to be + scaled. See Section 8 for advice on when it may be + preferable to set SCALE = .FALSE. and for a description of + the scaling strategy. Suggested value: SCALE = .TRUE.. + + 4: Z(2,N) -- DOUBLE PRECISION array Output + On exit: the real and imaginary parts of the roots are + stored in Z(1,i) and Z(2,i) respectively, for i=1,2,...,n. + Complex conjugate pairs of roots are stored in consecutive + pairs of elements of Z; that is, Z(1,i+1) = Z(1,i) and + Z(2,i+1)=-Z(2,i). + + 5: W(2*(N+1)) -- DOUBLE PRECISION array Workspace + + 6: IFAIL -- INTEGER Input/Output + On entry: IFAIL must be set to 0, -1 or 1. For users not + familiar with this parameter (described in the Essential + Introduction) the recommended value is 0. + On exit: IFAIL = 0 unless the routine detects an error (see + Section 6). + + 6. Error Indicators and Warnings + + Errors detected by the routine: + + If on entry IFAIL = 0 or -1, explanatory error messages are + output on the current error message unit (as defined by X04AAF). + + IFAIL= 1 + On entry A(0) = 0.0, + + or N < 1. + + IFAIL= 2 + The iterative procedure has failed to converge. This error + is very unlikely to occur. If it does, please contact NAG + immediately, as some basic assumption for the arithmetic has + been violated. See also Section 8. + + IFAIL= 3 + Either overflow or underflow prevents the evaluation of P(z) + near some of its zeros. This error is very unlikely to + occur. If it does, please contact NAG immediately. See also + Section 8. + + 7. Accuracy + + All roots are evaluated as accurately as possible, but because of + the inherent nature of the problem complete accuracy cannot be + guaranteed. + + 8. Further Comments + + If SCALE = .TRUE., then a scaling factor for the coefficients is + chosen as a power of the base B of the machine so that the + EMAX-P + largest coefficient in magnitude approaches THRESH = B . + Users should note that no scaling is performed if the largest + coefficient in magnitude exceeds THRESH, even if SCALE = .TRUE.. + (For definition of B, EMAX and P see the Chapter Introduction + X02.) + + However, with SCALE = .TRUE., overflow may be encountered when + the input coefficients a ,a ,a ,...,a vary widely in magnitude, + 0 1 2 n + (4*P) + particularly on those machines for which B overflows. In + such cases, SCALE should be set to .FALSE. and the coefficients + scaled so that the largest coefficient in magnitude does not + (EMAX-2*P) + exceed B . + + Even so, the scaling strategy used in C02AGF is sometimes + insufficient to avoid overflow and/or underflow conditions. In + such cases, the user is recommended to scale the independent + variable (z) so that the disparity between the largest and + smallest coefficient in magnitude is reduced. That is, use the + routine to locate the zeros of the polynomial d*P(cz) for some + suitable values of c and d. For example, if the original + -100 100 20 -10 + polynomial was P(z)=2 +2 z , then choosing c=2 and + 100 20 + d=2 , for instance, would yield the scaled polynomial 1+z , + which is well-behaved relative to overflow and underflow and has + 10 + zeros which are 2 times those of P(z). + + If the routine fails with IFAIL = 2 or 3, then the real and + imaginary parts of any roots obtained before the failure occurred + are stored in Z in the reverse order in which they were found. + Let n denote the number of roots found before the failure + R + occurred. Then Z(1,n) and Z(2,n) contain the real and imaginary + parts of the 1st root found, Z(1,n-1) and Z(2,n-1) contain the + real and imaginary parts of the 2nd root found, ..., Z(1,n ) and + R + Z(2,n ) contain the real and imaginary parts of the n th root + R R + found. After the failure has occurred, the remaining 2*(n-n ) + R + elements of Z contain a large negative number (equal to + + + -1/(X02AMF().\/2)). + + 9. Example + + To find the roots of the 5th degree polynomial + 5 4 3 2 + z +2z +3z +4z +5z+6=0. + + The example program is not reproduced here. The source code for + all example programs is distributed with the NAG Foundation + Library software and should be available on-line. + +\end{verbatim} +\endscroll +\end{page} + +@ +\subsection{ Roots of One or More Transcendental Equations} +\label{manpageXXc05} +\index{pages!manpageXXc05!nagc.ht} +\index{nagc.ht!pages!manpageXXc05} +\index{manpageXXc05!nagc.ht!pages} +<>= +\begin{page}{manpageXXc05}{NAG Documentation: c05} +\beginscroll +\begin{verbatim} + + + + C05(3NAG) Foundation Library (12/10/92) C05(3NAG) + + + + C05 -- Roots of One or More Transcendental Equations + Introduction -- C05 + Chapter C05 + Roots of One or More Transcendental Equations + + 1. Scope of the Chapter + + This chapter is concerned with the calculation of real zeros of + continuous real functions of one or more variables. (Complex + equations must be expressed in terms of the equivalent larger + system of real equations.) + + 2. Background to the Problems + + The chapter divides naturally into two parts. + + 2.1. A Single Equation + + The first deals with the real zeros of a real function of a + single variable f(x). + + At present, there is only one routine with a simple calling + sequence. This routine assumes that the user can determine an + initial interval [a,b] within which the desired zero lies, that + is f(a)*f(b)<0, and outside which all other zeros lie. The + routine then systematically subdivides the interval to produce a + final interval containing the zero. This final interval has a + length bounded by the user's specified error requirements; the + end of the interval where the function has smallest magnitude is + returned as the zero. This routine is guaranteed to converge to a + simple zero of the function. (Here we define a simple zero as a + zero corresponding to a sign-change of the function.) The + algorithm used is due to Bus and Dekker. + + 2.2. Systems of Equations + + The routines in the second part of this chapter are designed to + solve a set of nonlinear equations in n unknowns + + + T + f (x)=0, i=1,2,...,n, x=(x ,x ,...,x ) (1) + i 1 2 n + + where T stands for transpose. + + It is assumed that the functions are continuous and + differentiable so that the matrix of first partial derivatives of + the functions, the Jacobian matrix J (x)=ddf /ddx evaluated at + ij i j + the point x, exists, though it may not be possible to calculate + it directly. + + The functions f must be independent, otherwise there will be an + i + infinity of solutions and the methods will fail. However, even + when the functions are independent the solutions may not be + unique. Since the methods are iterative, an initial guess at the + solution has to be supplied, and the solution located will + usually be the one closest to this initial guess. + + 2.3. References + + [1] Gill P E and Murray W (1976) Algorithms for the Solution of + the Nonlinear Least-squares Problem. NAC 71 National + Physical Laboratory. + + [2] More J J, Garbow B S and Hillstrom K E (1974) User Guide for + Minpack-1. ANL-80-74 Argonne National Laboratory. + + [3] Ortega J M and Rheinboldt W C (1970) Iterative Solution of + Nonlinear Equations in Several Variables. Academic Press. + + [4] Rabinowitz P (1970) Numerical Methods for Nonlinear + Algebraic Equations. Gordon and Breach. + + 3. Recommendations on Choice and Use of Routines + + 3.1. Zeros of Functions of One Variable + + There is only one routine (C05ADF) for solving a single nonlinear + equation. This routine is designed for solving problems where the + function f(x) whose zero is to be calculated, can be coded as a + user-supplied routine. + + C05ADF may only be used when the user can supply an interval + [a,b] containing the zero, that is f(a)*f(b)<0. + + 3.2. Solution of Sets of Nonlinear Equations + + The solution of a set of nonlinear equations + + f (x ,x ,...,x )=0, i=1,2,...,n (2) + i 1 2 n + + can be regarded as a special case of the problem of finding a + minimum of a sum of squares + + m + / 2 + s(x)= | [f (x ,x ,...,x )] (m>=n). (3) + / i 1 2 n + i=1 + + So the routines in Chapter E04 of the Library are relevant as + well as the special nonlinear equations routines. + + There are two routines (C05NBF and C05PBF) for solving a set of + nonlinear equations. These routines require the f (and possibly + i + their derivatives) to be calculated in user-supplied routines. + These should be set up carefully so the Library routines can work + as efficiently as possible. + + The main decision which has to be made by the user is whether to + ddf + i + supply the derivatives ----. It is advisable to do so if + ddx + j + possible, since the results obtained by algorithms which use + derivatives are generally more reliable than those obtained by + algorithms which do not use derivatives. + + C05PBF requires the user to provide the derivatives, whilst + C05NBF does not. C05NBF and C05PBF are easy-to-use routines. A + routine, C05ZAF, is provided for use in conjunction with C05PBF + to check the user-provided derivatives for consistency with the + functions themselves. The user is strongly advised to make use of + this routine whenever C05PBF is used. + + Firstly, the calculation of the functions and their derivatives + should be ordered so that cancellation errors are avoided. This + is particularly important in a routine that uses these quantities + to build up estimates of higher derivatives. + + Secondly, scaling of the variables has a considerable effect on + the efficiency of a routine. The problem should be designed so + that the elements of x are of similar magnitude. The same comment + applies to the functions, all the f should be of comparable + i + size. + + The accuracy is usually determined by the accuracy parameters of + the routines, but the following points may be useful: + + (i) Greater accuracy in the solution may be requested by + choosing smaller input values for the accuracy parameters. + However, if unreasonable accuracy is demanded, rounding + errors may become important and cause a failure. + + (ii) Some idea of the accuracies of the x may be obtained by + i + monitoring the progress of the routine to see how many + figures remain unchanged during the last few iterations. + + (iii) An approximation to the error in the solution x, given by e + where e is the solution to the set of linear equations + + J(x)e=-f(x) + + T + where f(x)=(f (x),f (x),...,f (x)) (see Chapter F04). + 1 2 n + + (iv) If the functions f (x) are changed by small amounts + i + (epsilon) , for i=1,2,...,n, then the corresponding change + i + in the solution x is given approximately by (sigma), where + (sigma) is the solution of the set of linear equations + + J(x)(sigma)=-(epsilon), (see Chapter F04). + + Thus one can estimate the sensitivity of x to any + uncertainties in the specification of f (x), for + i + i=1,2,...,n. + + 3.3. Index + + Zeros of functions of one variable: + Bus and Dekker algorithm C05ADF + Zeros of functions of several variables: + easy-to-use C05NBF + easy-to-use, derivatives required C05PBF + Checking Routine: + Checks user-supplied Jacobian C05ZAF + + + C05 -- Roots of One or More Transcendental Equations + Contents -- C05 + Chapter C05 + + Roots of One or More Transcendental Equations + + C05ADF Zero of continuous function in given interval, Bus and + Dekker algorithm + + C05NBF Solution of system of nonlinear equations using function + values only + + C05PBF Solution of system of nonlinear equations using 1st + derivatives + + C05ZAF Check user's routine for calculating 1st derivatives + +\end{verbatim} +\endscroll +\end{page} + +@ +\subsection{ Zero of a continuous function in a given interval} +\label{manpageXXc05adf} +\index{pages!manpageXXc05adf!nagc.ht} +\index{nagc.ht!pages!manpageXXc05adf} +\index{manpageXXc05adf!nagc.ht!pages} +<>= +\begin{page}{manpageXXc05adf}{NAG Documentation: c05adf} +\beginscroll +\begin{verbatim} + + + + C05ADF(3NAG) Foundation Library (12/10/92) C05ADF(3NAG) + + + + C05 -- Roots of One or More Transcendental Equations C05ADF + C05ADF -- NAG Foundation Library Routine Document + + Note: Before using this routine, please read the Users' Note for + your implementation to check implementation-dependent details. + The symbol (*) after a NAG routine name denotes a routine that is + not included in the Foundation Library. + + 1. Purpose + + C05ADF locates a zero of a continuous function in a given + interval by a combination of the methods of linear interpolation, + extrapolation and bisection. + + 2. Specification + + SUBROUTINE C05ADF (A, B, EPS, ETA, F, X, IFAIL) + INTEGER IFAIL + DOUBLE PRECISION A, B, EPS, ETA, F, X + EXTERNAL F + + 3. Description + + The routine attempts to obtain an approximation to a simple zero + of the function f(x) given an initial interval [a,b] such that + f(a)*f(b)<=0. The zero is found by calls to C05AZF(*) whose + specification should be consulted for details of the method used. + + The approximation x to the zero (alpha) is determined so that one + or both of the following criteria are satisfied: + + (i) |x-(alpha)|