diff --git a/books/Makefile.pamphlet b/books/Makefile.pamphlet index d5e906d..28b6176 100644 --- a/books/Makefile.pamphlet +++ b/books/Makefile.pamphlet @@ -11,8 +11,14 @@ as pdf files in the final output tree. \eject \tableofcontents \eject +Note that we need to set the SHELL variable here because the +default shell appears to be /bin/sh which does not follow the +semantics of echo used by /bin/bash. Thus, while trying to +write the backslash-newpage lines the backslash-n gets interpreted +as a newline. \section{The Makefile} <<*>>= +SHELL=bash PDF=${AXIOM}/doc IN=${SPD}/books LATEX=latex diff --git a/books/bookvol10.3.pamphlet b/books/bookvol10.3.pamphlet index 919003f..3510eef 100644 --- a/books/bookvol10.3.pamphlet +++ b/books/bookvol10.3.pamphlet @@ -12999,6 +12999,83 @@ DifferentialSparseMultivariatePolynomial(R, S, V): @ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section{domain DPMM DirectProductMatrixModule} +\pagehead{DirectProductMatrixModule}{DPMM} +\pagepic{ps/v103directproductmatrixmodule.ps}{DPMM}{1.00} +See also:\\ +\refto{OppositeMonogenicLinearOperator}{OMLO} +\refto{OrdinaryDifferentialRing}{ODR} +\refto{DirectProductModule}{DPMO} +<>= +)abbrev domain DPMM DirectProductMatrixModule +++ Author: Stephen M. Watt +++ Date Created: 1986 +++ Date Last Updated: June 4, 1991 +++ Basic Operations: +++ Related Domains: +++ Also See: +++ AMS Classifications: +++ Keywords: equation +++ Examples: +++ References: +++ Description: +++ This constructor provides a direct product type with a +++ left matrix-module view. + +DirectProductMatrixModule(n, R, M, S): DPcategory == DPcapsule where + n: PositiveInteger + R: Ring + RowCol ==> DirectProduct(n,R) + M: SquareMatrixCategory(n,R,RowCol,RowCol) + S: LeftModule(R) + + DPcategory == Join(DirectProductCategory(n,S), LeftModule(R), LeftModule(M)) + + DPcapsule == DirectProduct(n, S) add + Rep := Vector(S) + r:R * x:$ == [r*x.i for i in 1..n] + m:M * x:$ == [ +/[m(i,j)*x.j for j in 1..n] for i in 1..n] + +@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section{domain DPMO DirectProductModule} +\pagehead{DirectProductModule}{DPMO} +\pagepic{ps/v103directproductmodule.ps}{DPMO}{1.00} +See also:\\ +\refto{OppositeMonogenicLinearOperator}{OMLO} +\refto{OrdinaryDifferentialRing}{ODR} +\refto{DirectProductMatrixModule}{DPMM} +<>= +)abbrev domain DPMO DirectProductModule +++ Author: Stephen M. Watt +++ Date Created: 1986 +++ Date Last Updated: June 4, 1991 +++ Basic Operations: +++ Related Domains: +++ Also See: +++ AMS Classifications: +++ Keywords: equation +++ Examples: +++ References: +++ Description: +++ This constructor provides a direct product of R-modules +++ with an R-module view. + +DirectProductModule(n, R, S): DPcategory == DPcapsule where + n: NonNegativeInteger + R: Ring + S: LeftModule(R) + + DPcategory == Join(DirectProductCategory(n,S), LeftModule(R)) + -- with if S has Algebra(R) then Algebra(R) + -- + + DPcapsule == DirectProduct(n,S) add + Rep := Vector(S) + r:R * x:$ == [r * x.i for i in 1..n] + +@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{domain DMP DistributedMultivariatePolynomial} <>= -- gdpoly.spad.pamphlet DistributedMultivariatePolynomial.input @@ -31820,6 +31897,1600 @@ Library(): TableAggregate(String, Any) with @ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section{domain LSQM LieSquareMatrix} +\pagehead{LieSquareMatrix}{LSQM} +\pagepic{ps/v103liesquarematrix.ps}{LSQM}{1.00} +See also:\\ +\refto{AssociatedLieAlgebra}{LIE} +\refto{AssociatedJordanAlgebra}{JORDAN} +<>= +)abbrev domain LSQM LieSquareMatrix +++ Author: J. Grabmeier +++ Date Created: 07 March 1991 +++ Date Last Updated: 08 March 1991 +++ Basic Operations: +++ Related Constructors: +++ Also See: +++ AMS Classifications: +++ Keywords: +++ References: +++ Description: +++ LieSquareMatrix(n,R) implements the Lie algebra of the n by n +++ matrices over the commutative ring R. +++ The Lie bracket (commutator) of the algebra is given by +++ \spad{a*b := (a *$SQMATRIX(n,R) b - b *$SQMATRIX(n,R) a)}, +++ where \spadfun{*$SQMATRIX(n,R)} is the usual matrix multiplication. +LieSquareMatrix(n,R): Exports == Implementation where + + n : PositiveInteger + R : CommutativeRing + + Row ==> DirectProduct(n,R) + Col ==> DirectProduct(n,R) + + Exports ==> Join(SquareMatrixCategory(n,R,Row,Col), CoercibleTo Matrix R,_ + FramedNonAssociativeAlgebra R) --with + + Implementation ==> AssociatedLieAlgebra (R,SquareMatrix(n, R)) add + + Rep := AssociatedLieAlgebra (R,SquareMatrix(n, R)) + -- local functions + n2 : PositiveInteger := n*n + + convDM : DirectProduct(n2,R) -> % + conv : DirectProduct(n2,R) -> SquareMatrix(n,R) + --++ converts n2-vector to (n,n)-matrix row by row + conv v == + cond : Matrix(R) := new(n,n,0$R)$Matrix(R) + z : Integer := 0 + for i in 1..n repeat + for j in 1..n repeat + z := z+1 + setelt(cond,i,j,v.z) + squareMatrix(cond)$SquareMatrix(n, R) + + + coordinates(a:%,b:Vector(%)):Vector(R) == + -- only valid for b canonicalBasis + res : Vector R := new(n2,0$R) + z : Integer := 0 + for i in 1..n repeat + for j in 1..n repeat + z := z+1 + res.z := elt(a,i,j)$% + res + + + convDM v == + sq := conv v + coerce(sq)$Rep :: % + + basis() == + n2 : PositiveInteger := n*n + ldp : List DirectProduct(n2,R) := + [unitVector(i::PositiveInteger)$DirectProduct(n2,R) for i in 1..n2] + res:Vector % := vector map(convDM,_ + ldp)$ListFunctions2(DirectProduct(n2,R), %) + + someBasis() == basis() + rank() == n*n + + +-- transpose: % -> % +-- ++ computes the transpose of a matrix +-- squareMatrix: Matrix R -> % +-- ++ converts a Matrix to a LieSquareMatrix +-- coerce: % -> Matrix R +-- ++ converts a LieSquareMatrix to a Matrix +-- symdecomp : % -> Record(sym:%,antisym:%) +-- if R has commutative("*") then +-- minorsVect: -> Vector(Union(R,"uncomputed")) --range: 1..2**n-1 +-- if R has commutative("*") then central +-- if R has commutative("*") and R has unitsKnown then unitsKnown + +@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section{domain LODO LinearOrdinaryDifferentialOperator} +<>= +-- lodo.spad.pamphlet LinearOrdinaryDifferentialOperator.input +)spool LinearOrdinaryDifferentialOperator.output +)set message test on +)set message auto off +)clear all +--S 1 of 16 +Dx: LODO(EXPR INT, f +-> D(f, x)) +--R +--R Type: Void +--E 1 + +--S 2 of 16 +Dx := D() +--R +--R +--R (2) D +--IType: LinearOrdinaryDifferentialOperator(Expression Integer,theMap LAMBDA-CLOSURE(NIL,NIL,NIL,G1404 envArg,SPADCALL(G1404,QUOTE x,ELT(*1;anonymousFunction;0;frame0;internal;MV,0)))) +--E 2 + +--S 3 of 16 +Dop:= Dx^3 + G/x^2*Dx + H/x^3 - 1 +--R +--R +--R 3 +--R 3 G - x + H +--R (3) D + -- D + -------- +--R 2 3 +--R x x +--IType: LinearOrdinaryDifferentialOperator(Expression Integer,theMap LAMBDA-CLOSURE(NIL,NIL,NIL,G1404 envArg,SPADCALL(G1404,QUOTE x,ELT(*1;anonymousFunction;0;frame0;internal;MV,0)))) +--E 3 + +--S 4 of 16 +n == 3 +--R +--R Type: Void +--E 4 + +--S 5 of 16 +phi == reduce(+,[subscript(s,[i])*exp(x)/x^i for i in 0..n]) +--R +--R Type: Void +--E 5 + +--S 6 of 16 +phi1 == Dop(phi) / exp x +--R +--R Type: Void +--E 6 + +--S 7 of 16 +phi2 == phi1 *x**(n+3) +--R +--R Type: Void +--E 7 + +--S 8 of 16 +phi3 == retract(phi2)@(POLY INT) +--R +--R Type: Void +--E 8 + +--S 9 of 16 +pans == phi3 ::UP(x,POLY INT) +--R +--R Type: Void +--E 9 + +--S 10 of 16 +pans1 == [coefficient(pans, (n+3-i) :: NNI) for i in 2..n+1] +--R +--R Type: Void +--E 10 + +--S 11 of 16 +leq == solve(pans1,[subscript(s,[i]) for i in 1..n]) +--R +--R Type: Void +--E 11 + +--S 12 of 16 +leq +--R +--R Compiling body of rule n to compute value of type PositiveInteger +--R Compiling body of rule phi to compute value of type Expression +--R Integer +--R Compiling body of rule phi1 to compute value of type Expression +--R Integer +--R Compiling body of rule phi2 to compute value of type Expression +--R Integer +--R Compiling body of rule phi3 to compute value of type Polynomial +--R Integer +--R Compiling body of rule pans to compute value of type +--R UnivariatePolynomial(x,Polynomial Integer) +--R Compiling body of rule pans1 to compute value of type List +--R Polynomial Integer +--R Compiling body of rule leq to compute value of type List List +--R Equation Fraction Polynomial Integer +--I Compiling function G3349 with type Integer -> Boolean +--R +--R (12) +--R 2 3 2 +--R s G 3s H + s G + 6s G (9s G + 54s )H + s G + 18s G + 72s G +--R 0 0 0 0 0 0 0 0 0 +--R [[s = ---,s = ------------------,s = --------------------------------------]] +--R 1 3 2 18 3 162 +--R Type: List List Equation Fraction Polynomial Integer +--E 12 + +--S 13 of 16 +n==4 +--R +--R Compiled code for n has been cleared. +--R Compiled code for leq has been cleared. +--R Compiled code for pans1 has been cleared. +--R Compiled code for phi2 has been cleared. +--R Compiled code for phi has been cleared. +--R Compiled code for phi3 has been cleared. +--R Compiled code for phi1 has been cleared. +--R Compiled code for pans has been cleared. +--R 1 old definition(s) deleted for function or rule n +--R Type: Void +--E 13 + +--S 14 of 16 +leq +--R +--R Compiling body of rule n to compute value of type PositiveInteger +--R Compiling body of rule phi to compute value of type Expression +--R Integer +--R Compiling body of rule phi1 to compute value of type Expression +--R Integer +--R Compiling body of rule phi2 to compute value of type Expression +--R Integer +--R Compiling body of rule phi3 to compute value of type Polynomial +--R Integer +--R Compiling body of rule pans to compute value of type +--R UnivariatePolynomial(x,Polynomial Integer) +--R Compiling body of rule pans1 to compute value of type List +--R Polynomial Integer +--R Compiling body of rule leq to compute value of type List List +--R Equation Fraction Polynomial Integer +--R +--R (14) +--R [ +--R 2 +--R s G 3s H + s G + 6s G +--R 0 0 0 0 +--R [s = ---, s = ------------------, +--R 1 3 2 18 +--R 3 2 +--R (9s G + 54s )H + s G + 18s G + 72s G +--R 0 0 0 0 0 +--R s = --------------------------------------, +--R 3 162 +--R +--R s = +--R 4 +--R 2 2 4 3 2 +--R 27s H + (18s G + 378s G + 1296s )H + s G + 36s G + 396s G +--R 0 0 0 0 0 0 0 +--R + +--R 1296s G +--R 0 +--R / +--R 1944 +--R ] +--R ] +--R Type: List List Equation Fraction Polynomial Integer +--E 14 + +--S 15 of 16 +n==7 +--R +--R Compiled code for n has been cleared. +--R Compiled code for leq has been cleared. +--R Compiled code for pans1 has been cleared. +--R Compiled code for phi2 has been cleared. +--R Compiled code for phi has been cleared. +--R Compiled code for phi3 has been cleared. +--R Compiled code for phi1 has been cleared. +--R Compiled code for pans has been cleared. +--R 1 old definition(s) deleted for function or rule n +--R Type: Void +--E 15 + +--S 16 of 16 +leq +--R +--R Compiling body of rule n to compute value of type PositiveInteger +--R Compiling body of rule phi to compute value of type Expression +--R Integer +--R Compiling body of rule phi1 to compute value of type Expression +--R Integer +--R Compiling body of rule phi2 to compute value of type Expression +--R Integer +--R Compiling body of rule phi3 to compute value of type Polynomial +--R Integer +--R Compiling body of rule pans to compute value of type +--R UnivariatePolynomial(x,Polynomial Integer) +--R Compiling body of rule pans1 to compute value of type List +--R Polynomial Integer +--R Compiling body of rule leq to compute value of type List List +--R Equation Fraction Polynomial Integer +--R +--R (16) +--R [ +--R 2 +--R s G 3s H + s G + 6s G +--R 0 0 0 0 +--R [s = ---, s = ------------------, +--R 1 3 2 18 +--R 3 2 +--R (9s G + 54s )H + s G + 18s G + 72s G +--R 0 0 0 0 0 +--R s = --------------------------------------, +--R 3 162 +--R +--R s = +--R 4 +--R 2 2 4 3 2 +--R 27s H + (18s G + 378s G + 1296s )H + s G + 36s G + 396s G +--R 0 0 0 0 0 0 0 +--R + +--R 1296s G +--R 0 +--R / +--R 1944 +--R , +--R +--R s = +--R 5 +--R 2 3 2 +--R (135s G + 2268s )H + (30s G + 1350s G + 16416s G + 38880s )H +--R 0 0 0 0 0 0 +--R + +--R 5 4 3 2 +--R s G + 60s G + 1188s G + 9504s G + 25920s G +--R 0 0 0 0 0 +--R / +--R 29160 +--R , +--R +--R s = +--R 6 +--R 3 2 2 +--R 405s H + (405s G + 18468s G + 174960s )H +--R 0 0 0 0 +--R + +--R 4 3 2 6 +--R (45s G + 3510s G + 88776s G + 777600s G + 1166400s )H + s G +--R 0 0 0 0 0 0 +--R + +--R 5 4 3 2 +--R 90s G + 2628s G + 27864s G + 90720s G +--R 0 0 0 0 +--R / +--R 524880 +--R , +--R +--R s = +--R 7 +--R 3 +--R (2835s G + 91854s )H +--R 0 0 +--R + +--R 3 2 2 +--R (945s G + 81648s G + 2082996s G + 14171760s )H +--R 0 0 0 0 +--R + +--R 5 4 3 2 +--R (63s G + 7560s G + 317520s G + 5554008s G + 34058880s G)H +--R 0 0 0 0 0 +--R + +--R 7 6 5 4 3 2 +--R s G + 126s G + 4788s G + 25272s G - 1744416s G - 26827200s G +--R 0 0 0 0 0 0 +--R + +--R - 97977600s G +--R 0 +--R / +--R 11022480 +--R ] +--R ] +--R Type: List List Equation Fraction Polynomial Integer +--E 16 +)spool + +Dx: LODO(EXPR INT, f +-> D(f, x)) +Dx := D() +Dop:= Dx^3 + G/x^2*Dx + H/x^3 - 1 +n == 3 +phi == reduce(+,[subscript(s,[i])*exp(x)/x^i for i in 0..n]) +phi1 == Dop(phi) / exp x +phi2 == phi1 *x**(n+3) +phi3 == retract(phi2)@(POLY INT) +pans == phi3 ::UP(x,POLY INT) +pans1 == [coefficient(pans, (n+3-i) :: NNI) for i in 2..n+1] +leq == solve(pans1,[subscript(s,[i]) for i in 1..n]) +leq +n==4 +leq +n==7 +leq +)spool +)lisp (bye) +@ +<>= +==================================================================== +LinearOrdinaryDifferentialOperator examples +==================================================================== + +LinearOrdinaryDifferentialOperator(A, diff) is the domain of linear +ordinary differential operators with coefficients in a ring A with a +given derivation. + +==================================================================== +Differential Operators with Series Coefficients +==================================================================== + +Problem: + Find the first few coefficients of exp(x)/x^i of Dop phi where + + Dop := D^3 + G/x^2 * D + H/x^3 - 1 + phi := sum(s[i]*exp(x)/x^i, i = 0..) + +Solution: + +Define the differential. + + Dx: LODO(EXPR INT, f +-> D(f, x)) + Type: Void + + Dx := D() + D + Type: LinearOrdinaryDifferentialOperator(Expression Integer, + theMap LAMBDA-CLOSURE(NIL,NIL,NIL,G1404 envArg, + SPADCALL(G1404,QUOTE x, + ELT(*1;anonymousFunction;0;frame0;internal;MV,0)))) + +Now define the differential operator Dop. + + Dop:= Dx^3 + G/x^2*Dx + H/x^3 - 1 + 3 + 3 G - x + H + D + -- D + -------- + 2 3 + x x + Type: LinearOrdinaryDifferentialOperator(Expression Integer, + theMap LAMBDA-CLOSURE(NIL,NIL,NIL,G1404 envArg, + SPADCALL(G1404,QUOTE x, + ELT(*1;anonymousFunction;0;frame0;internal;MV,0)))) + + n == 3 + Type: Void + + phi == reduce(+,[subscript(s,[i])*exp(x)/x^i for i in 0..n]) + Type: Void + + phi1 == Dop(phi) / exp x + Type: Void + + phi2 == phi1 *x**(n+3) + Type: Void + + phi3 == retract(phi2)@(POLY INT) + Type: Void + + pans == phi3 ::UP(x,POLY INT) + Type: Void + + pans1 == [coefficient(pans, (n+3-i) :: NNI) for i in 2..n+1] + Type: Void + + leq == solve(pans1,[subscript(s,[i]) for i in 1..n]) + Type: Void + +Evaluate this for several values of n. + + leq + 2 3 2 + s G 3s H + s G + 6s G (9s G + 54s )H + s G + 18s G + 72s G + 0 0 0 0 0 0 0 0 0 + [[s = ---,s = ------------------,s = --------------------------------------]] + 1 3 2 18 3 162 + Type: List List Equation Fraction Polynomial Integer + + n==4 + Type: Void + + leq + [ + 2 + s G 3s H + s G + 6s G + 0 0 0 0 + [s = ---, s = ------------------, + 1 3 2 18 + 3 2 + (9s G + 54s )H + s G + 18s G + 72s G + 0 0 0 0 0 + s = --------------------------------------, + 3 162 + + s = + 4 + 2 2 4 3 2 + 27s H + (18s G + 378s G + 1296s )H + s G + 36s G + 396s G + 0 0 0 0 0 0 0 + + + 1296s G + 0 + / + 1944 + ] + ] + Type: List List Equation Fraction Polynomial Integer + + n==7 + Type: Void + + leq + [ + 2 + s G 3s H + s G + 6s G + 0 0 0 0 + [s = ---, s = ------------------, + 1 3 2 18 + 3 2 + (9s G + 54s )H + s G + 18s G + 72s G + 0 0 0 0 0 + s = --------------------------------------, + 3 162 + + s = + 4 + 2 2 4 3 2 + 27s H + (18s G + 378s G + 1296s )H + s G + 36s G + 396s G + 0 0 0 0 0 0 0 + + + 1296s G + 0 + / + 1944 + , + + s = + 5 + 2 3 2 + (135s G + 2268s )H + (30s G + 1350s G + 16416s G + 38880s )H + 0 0 0 0 0 0 + + + 5 4 3 2 + s G + 60s G + 1188s G + 9504s G + 25920s G + 0 0 0 0 0 + / + 29160 + , + + s = + 6 + 3 2 2 + 405s H + (405s G + 18468s G + 174960s )H + 0 0 0 0 + + + 4 3 2 6 + (45s G + 3510s G + 88776s G + 777600s G + 1166400s )H + s G + 0 0 0 0 0 0 + + + 5 4 3 2 + 90s G + 2628s G + 27864s G + 90720s G + 0 0 0 0 + / + 524880 + , + + s = + 7 + 3 + (2835s G + 91854s )H + 0 0 + + + 3 2 2 + (945s G + 81648s G + 2082996s G + 14171760s )H + 0 0 0 0 + + + 5 4 3 2 + (63s G + 7560s G + 317520s G + 5554008s G + 34058880s G)H + 0 0 0 0 0 + + + 7 6 5 4 3 2 + s G + 126s G + 4788s G + 25272s G - 1744416s G - 26827200s G + 0 0 0 0 0 0 + + + - 97977600s G + 0 + / + 11022480 + ] + ] + Type: List List Equation Fraction Polynomial Integer + +See Also: +o )show LinearOrdinaryDifferentialOperator +o $AXIOM/doc/src/algebra/lodo.spad.dvi + +@ +\pagehead{LinearOrdinaryDifferentialOperator}{LODO} +\pagepic{ps/v103linearordinarydifferentialoperator.ps}{LODO}{1.00} +<>= +)abbrev domain LODO LinearOrdinaryDifferentialOperator +++ Author: Manuel Bronstein +++ Date Created: 9 December 1993 +++ Date Last Updated: 15 April 1994 +++ Keywords: differential operator +++ Description: +++ \spad{LinearOrdinaryDifferentialOperator} defines a ring of +++ differential operators with coefficients in a ring A with a given +++ derivation. +++ Multiplication of operators corresponds to functional composition: +++ \spad{(L1 * L2).(f) = L1 L2 f} +LinearOrdinaryDifferentialOperator(A:Ring, diff: A -> A): + LinearOrdinaryDifferentialOperatorCategory A + == SparseUnivariateSkewPolynomial(A, 1, diff) add + Rep := SparseUnivariateSkewPolynomial(A, 1, diff) + + outputD := "D"@String :: Symbol :: OutputForm + + coerce(l:%):OutputForm == outputForm(l, outputD) + elt(p:%, a:A):A == apply(p, 0, a) + + if A has Field then + import LinearOrdinaryDifferentialOperatorsOps(A, %) + + symmetricProduct(a, b) == symmetricProduct(a, b, diff) + symmetricPower(a, n) == symmetricPower(a, n, diff) + directSum(a, b) == directSum(a, b, diff) + +@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section{domain LODO1 LinearOrdinaryDifferentialOperator1} +<>= +-- lodo.spad.pamphlet LinearOrdinaryDifferentialOperator1.input +)spool LinearOrdinaryDifferentialOperator1.output +)set message test on +)set message auto off +)clear all +--S 1 of 20 +RFZ := Fraction UnivariatePolynomial('x, Integer) +--R +--R +--R (1) Fraction UnivariatePolynomial(x,Integer) +--R Type: Domain +--E 1 + +--S 2 of 20 +x : RFZ := 'x +--R +--R +--R (2) x +--R Type: Fraction UnivariatePolynomial(x,Integer) +--E 2 + +--S 3 of 20 +Dx : LODO1 RFZ := D() +--R +--R +--R (3) D +--RType: LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer) +--E 3 + +--S 4 of 20 +b : LODO1 RFZ := 3*x**2*Dx**2 + 2*Dx + 1/x +--R +--R +--R 2 2 1 +--R (4) 3x D + 2D + - +--R x +--RType: LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer) +--E 4 + +--S 5 of 20 +a : LODO1 RFZ := b*(5*x*Dx + 7) +--R +--R +--R 3 3 2 2 7 +--R (5) 15x D + (51x + 10x)D + 29D + - +--R x +--RType: LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer) +--E 5 + +--S 6 of 20 +p := x**2 + 1/x**2 +--R +--R +--R 4 +--R x + 1 +--R (6) ------ +--R 2 +--R x +--R Type: Fraction UnivariatePolynomial(x,Integer) +--E 6 + +--S 7 of 20 +(a*b - b*a) p +--R +--R +--R 4 +--R - 75x + 540x - 75 +--R (7) ------------------ +--R 4 +--R x +--R Type: Fraction UnivariatePolynomial(x,Integer) +--E 7 + +--S 8 of 20 +ld := leftDivide(a,b) +--R +--R +--R (8) [quotient= 5x D + 7,remainder= 0] +--RType: Record(quotient: LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer),remainder: LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer)) +--E 8 + +--S 9 of 20 +a = b * ld.quotient + ld.remainder +--R +--R +--R 3 3 2 2 7 3 3 2 2 7 +--R (9) 15x D + (51x + 10x)D + 29D + -= 15x D + (51x + 10x)D + 29D + - +--R x x +--RType: Equation LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer) +--E 9 + +--S 10 of 20 +rd := rightDivide(a,b) +--R +--R +--R 5 +--R (10) [quotient= 5x D + 7,remainder= 10D + -] +--R x +--RType: Record(quotient: LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer),remainder: LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer)) +--E 10 + +--S 11 of 20 +a = rd.quotient * b + rd.remainder +--R +--R +--R 3 3 2 2 7 3 3 2 2 7 +--R (11) 15x D + (51x + 10x)D + 29D + -= 15x D + (51x + 10x)D + 29D + - +--R x x +--RType: Equation LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer) +--E 11 + +--S 12 of 20 +rightQuotient(a,b) +--R +--R +--R (12) 5x D + 7 +--RType: LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer) +--E 12 + +--S 13 of 20 +rightRemainder(a,b) +--R +--R +--R 5 +--R (13) 10D + - +--R x +--RType: LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer) +--E 13 + +--S 14 of 20 +leftExactQuotient(a,b) +--R +--R +--R (14) 5x D + 7 +--RType: Union(LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer),...) +--E 14 + +--S 15 of 20 +e := leftGcd(a,b) +--R +--R +--R 2 2 1 +--R (15) 3x D + 2D + - +--R x +--RType: LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer) +--E 15 + +--S 16 of 20 +leftRemainder(a, e) +--R +--R +--R (16) 0 +--RType: LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer) +--E 16 + +--S 17 of 20 +rightRemainder(a, e) +--R +--R +--R 5 +--R (17) 10D + - +--R x +--RType: LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer) +--E 17 + +--S 18 of 20 +f := rightLcm(a,b) +--R +--R +--R 3 3 2 2 7 +--R (18) 15x D + (51x + 10x)D + 29D + - +--R x +--RType: LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer) +--E 18 + +--S 19 of 20 +rightRemainder(f, b) +--R +--R +--R 5 +--R (19) 10D + - +--R x +--RType: LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer) +--E 19 + +--S 20 of 20 +leftRemainder(f, b) +--R +--R +--R (20) 0 +--RType: LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer) +--E 20 +)spool +)lisp (bye) +@ + +<>= +==================================================================== +LinearOrdinaryDifferentialOperator1 example +==================================================================== + +LinearOrdinaryDifferentialOperator1(A) is the domain of linear +ordinary differential operators with coefficients in the differential +ring A. + +==================================================================== +Differential Operators with Rational Function Coefficients +==================================================================== + +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. + +We begin by defining RFZ to be the rational functions in x with +integer coefficients and Dx to be the differential operator for d/dx. + + RFZ := Fraction UnivariatePolynomial('x, Integer) + Fraction UnivariatePolynomial(x,Integer) + Type: Domain + + x : RFZ := 'x + x + Type: Fraction UnivariatePolynomial(x,Integer) + + Dx : LODO1 RFZ := D() + D + Type: LinearOrdinaryDifferentialOperator1 + Fraction UnivariatePolynomial(x,Integer) + +Operators are created using the usual arithmetic operations. + + b : LODO1 RFZ := 3*x**2*Dx**2 + 2*Dx + 1/x + 2 2 1 + 3x D + 2D + - + x + Type: LinearOrdinaryDifferentialOperator1 + Fraction UnivariatePolynomial(x,Integer) + + a : LODO1 RFZ := b*(5*x*Dx + 7) + 3 3 2 2 7 + 15x D + (51x + 10x)D + 29D + - + x + Type: LinearOrdinaryDifferentialOperator1 + Fraction UnivariatePolynomial(x,Integer) + +Operator multiplication corresponds to functional composition. + + p := x**2 + 1/x**2 + 4 + x + 1 + ------ + 2 + x + Type: Fraction UnivariatePolynomial(x,Integer) + +Since operator coefficients depend on x, the multiplication is not +commutative. + + (a*b - b*a) p + 4 + - 75x + 540x - 75 + ------------------ + 4 + x + Type: Fraction UnivariatePolynomial(x,Integer) + +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 leftDivide and rightDivide are quotient-remainder pairs +satisfying: + + leftDivide(a,b) = [q, r] such that a = b*q + r + rightDivide(a,b) = [q, r] such that a = q*b + r + +In both cases, the degree of the remainder, r, is less than the degree +of b. + + ld := leftDivide(a,b) + [quotient= 5x D + 7,remainder= 0] + Type: Record(quotient: LinearOrdinaryDifferentialOperator1 + Fraction UnivariatePolynomial(x,Integer), + remainder: LinearOrdinaryDifferentialOperator1 + Fraction UnivariatePolynomial(x,Integer)) + + a = b * ld.quotient + ld.remainder + 3 3 2 2 7 3 3 2 2 7 + 15x D + (51x + 10x)D + 29D + -= 15x D + (51x + 10x)D + 29D + - + Type: Equation LinearOrdinaryDifferentialOperator1 + Fraction UnivariatePolynomial(x,Integer) + +The operations of left and right division are so-called because the +quotient is obtained by dividing a on that side by b. + + rd := rightDivide(a,b) + [quotient= 5x D + 7,remainder= 10D + -] + Type: Record(quotient: LinearOrdinaryDifferentialOperator1 + Fraction UnivariatePolynomial(x,Integer), + remainder: LinearOrdinaryDifferentialOperator1 + Fraction UnivariatePolynomial(x,Integer)) + + a = rd.quotient * b + rd.remainder + 3 3 2 2 7 3 3 2 2 7 + 15x D + (51x + 10x)D + 29D + -= 15x D + (51x + 10x)D + 29D + - + Type: Equation LinearOrdinaryDifferentialOperator1 + Fraction UnivariatePolynomial(x,Integer) + +Operations rightQuotient and rightRemainder are available if only one +of the quotient or remainder are of interest to you. This is the +quotient from right division. + + rightQuotient(a,b) + 5x D + 7 + Type: LinearOrdinaryDifferentialOperator1 + Fraction UnivariatePolynomial(x,Integer) + +This is the remainder from right division. The corresponding "left" +functions, leftQuotient and leftRemainder are also available. + + rightRemainder(a,b) + 5 + 10D + - + x + Type: LinearOrdinaryDifferentialOperator1 + Fraction UnivariatePolynomial(x,Integer) + +For exact division, operations leftExactQuotient and rightExactQuotient +are supplied. These return the quotient but only if the remainder is zero. +The call rightExactQuotient(a,b) would yield an error. + + leftExactQuotient(a,b) + 5x D + 7 + Type: Union(LinearOrdinaryDifferentialOperator1 + Fraction UnivariatePolynomial(x,Integer),...) + +The division operations allow the computation of left and right greatest +common divisors, leftGcd and rightGcd via remainder sequences, and +consequently the computation of left and right least common multiples, +rightLcm and leftLcm. + + e := leftGcd(a,b) + 2 2 1 + 3x D + 2D + - + x + Type: LinearOrdinaryDifferentialOperator1 + Fraction UnivariatePolynomial(x,Integer) + +Note that a greatest common divisor doesn't necessarily divide a and b +on both sides. Here the left greatest common divisor does not divide +a on the right. + + leftRemainder(a, e) + 0 + Type: LinearOrdinaryDifferentialOperator1 + Fraction UnivariatePolynomial(x,Integer) + + rightRemainder(a, e) + 5 + 10D + - + x + Type: LinearOrdinaryDifferentialOperator1 + Fraction UnivariatePolynomial(x,Integer) + +Similarly, a least common multiple is not necessarily divisible from +both sides. + + f := rightLcm(a,b) + 3 3 2 2 7 + 15x D + (51x + 10x)D + 29D + - + Type: LinearOrdinaryDifferentialOperator1 + Fraction UnivariatePolynomial(x,Integer) + + rightRemainder(f, b) + 5 + 10D + - + x + Type: LinearOrdinaryDifferentialOperator1 + Fraction UnivariatePolynomial(x,Integer) + + leftRemainder(f, b) + 0 + Type: LinearOrdinaryDifferentialOperator1 + Fraction UnivariatePolynomial(x,Integer) + +See Also: +o )show LinearOrdinaryDifferentialOperator1 +o $AXIOM/doc/src/algebra/lodo.spad.dvi + +@ +\pagehead{LinearOrdinaryDifferentialOperator1}{LODO1} +\pagepic{ps/v103linearordinarydifferentialoperator1.ps}{LODO1}{1.00} +<>= +)abbrev domain LODO1 LinearOrdinaryDifferentialOperator1 +++ Author: Manuel Bronstein +++ Date Created: 9 December 1993 +++ Date Last Updated: 31 January 1994 +++ Keywords: differential operator +++ Description: +++ \spad{LinearOrdinaryDifferentialOperator1} defines a ring of +++ differential operators with coefficients in a differential ring A. +++ Multiplication of operators corresponds to functional composition: +++ \spad{(L1 * L2).(f) = L1 L2 f} +LinearOrdinaryDifferentialOperator1(A:DifferentialRing) == + LinearOrdinaryDifferentialOperator(A, differentiate$A) + +@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section{domain LODO2 LinearOrdinaryDifferentialOperator2} +<>= +-- lodo.spad.pamphlet LinearOrdinaryDifferentialOperator2.input +)spool LinearOrdinaryDifferentialOperator2.output +)set message test on +)set message auto off +)clear all +--S 1 of 26 +Q := Fraction Integer +--R +--R +--R (1) Fraction Integer +--R Type: Domain +--E 1 + +--S 2 of 26 +PQ := UnivariatePolynomial('x, Q) +--R +--R +--R (2) UnivariatePolynomial(x,Fraction Integer) +--R Type: Domain +--E 2 + +--S 3 of 26 +x: PQ := 'x +--R +--R +--R (3) x +--R Type: UnivariatePolynomial(x,Fraction Integer) +--E 3 + +--S 4 of 26 +Dx: LODO2(Q, PQ) := D() +--R +--R +--R (4) D +--RType: LinearOrdinaryDifferentialOperator2(Fraction Integer,UnivariatePolynomial(x,Fraction Integer)) +--E 4 + +--S 5 of 26 +a := Dx + 1 +--R +--R +--R (5) D + 1 +--RType: LinearOrdinaryDifferentialOperator2(Fraction Integer,UnivariatePolynomial(x,Fraction Integer)) +--E 5 + +--S 6 of 26 +b := a + 1/2*Dx**2 - 1/2 +--R +--R +--R 1 2 1 +--R (6) - D + D + - +--R 2 2 +--RType: LinearOrdinaryDifferentialOperator2(Fraction Integer,UnivariatePolynomial(x,Fraction Integer)) +--E 6 + +--S 7 of 26 +p := 4*x**2 + 2/3 +--R +--R +--R 2 2 +--R (7) 4x + - +--R 3 +--R Type: UnivariatePolynomial(x,Fraction Integer) +--E 7 + +--S 8 of 26 +a p +--R +--R +--R 2 2 +--R (8) 4x + 8x + - +--R 3 +--R Type: UnivariatePolynomial(x,Fraction Integer) +--E 8 + +--S 9 of 26 +(a * b) p = a b p +--R +--R +--R 2 37 2 37 +--R (9) 2x + 12x + --= 2x + 12x + -- +--R 3 3 +--R Type: Equation UnivariatePolynomial(x,Fraction Integer) +--E 9 + +--S 10 of 26 +c := (1/9)*b*(a + b)^2 +--R +--R +--R 1 6 5 5 13 4 19 3 79 2 7 1 +--R (10) -- D + -- D + -- D + -- D + -- D + -- D + - +--R 72 36 24 18 72 12 8 +--RType: LinearOrdinaryDifferentialOperator2(Fraction Integer,UnivariatePolynomial(x,Fraction Integer)) +--E 10 + +--S 11 of 26 +(a**2 - 3/4*b + c) (p + 1) +--R +--R +--R 2 44 541 +--R (11) 3x + -- x + --- +--R 3 36 +--R Type: UnivariatePolynomial(x,Fraction Integer) +--E 11 +)clear all +--S 12 of 26 +PZ := UnivariatePolynomial(x,Integer) +--R +--R +--R (1) UnivariatePolynomial(x,Integer) +--R Type: Domain +--E 12 + +--S 13 of 26 +x:PZ := 'x +--R +--R +--R (2) x +--R Type: UnivariatePolynomial(x,Integer) +--E 13 + +--S 14 of 26 +Mat := SquareMatrix(3,PZ) +--R +--R +--R (3) SquareMatrix(3,UnivariatePolynomial(x,Integer)) +--R Type: Domain +--E 14 + +--S 15 of 26 +Vect := DPMM(3, PZ, Mat, PZ) +--R +--R +--R (4) +--R DirectProductMatrixModule(3,UnivariatePolynomial(x,Integer),SquareMatrix(3,Un +--R ivariatePolynomial(x,Integer)),UnivariatePolynomial(x,Integer)) +--R Type: Domain +--E 15 + +--S 16 of 26 +Modo := LODO2(Mat, Vect) +--R +--R +--R (5) +--R LinearOrdinaryDifferentialOperator2(SquareMatrix(3,UnivariatePolynomial(x,Int +--R eger)),DirectProductMatrixModule(3,UnivariatePolynomial(x,Integer),SquareMatr +--R ix(3,UnivariatePolynomial(x,Integer)),UnivariatePolynomial(x,Integer))) +--R Type: Domain +--E 16 + +--S 17 of 26 +m:Mat := matrix [ [x^2,1,0],[1,x^4,0],[0,0,4*x^2] ] +--R +--R +--R + 2 + +--R |x 1 0 | +--R | | +--R (6) | 4 | +--R |1 x 0 | +--R | | +--R | 2| +--R +0 0 4x + +--R Type: SquareMatrix(3,UnivariatePolynomial(x,Integer)) +--E 17 + +--S 18 of 26 +p:Vect := directProduct [3*x^2+1,2*x,7*x^3+2*x] +--R +--R +--R 2 3 +--R (7) [3x + 1,2x,7x + 2x] +--RType: DirectProductMatrixModule(3,UnivariatePolynomial(x,Integer),SquareMatrix(3,UnivariatePolynomial(x,Integer)),UnivariatePolynomial(x,Integer)) +--E 18 + +--S 19 of 26 +q: Vect := m * p +--R +--R +--R 4 2 5 2 5 3 +--R (8) [3x + x + 2x,2x + 3x + 1,28x + 8x ] +--RType: DirectProductMatrixModule(3,UnivariatePolynomial(x,Integer),SquareMatrix(3,UnivariatePolynomial(x,Integer)),UnivariatePolynomial(x,Integer)) +--E 19 + +--S 20 of 26 +Dx : Modo := D() +--R +--R +--R (9) D +--RType: LinearOrdinaryDifferentialOperator2(SquareMatrix(3,UnivariatePolynomial(x,Integer)),DirectProductMatrixModule(3,UnivariatePolynomial(x,Integer),SquareMatrix(3,UnivariatePolynomial(x,Integer)),UnivariatePolynomial(x,Integer))) +--E 20 + +--S 21 of 26 +a : Modo := Dx + m +--R +--R +--R + 2 + +--R |x 1 0 | +--R | | +--R (10) D + | 4 | +--R |1 x 0 | +--R | | +--R | 2| +--R +0 0 4x + +--RType: LinearOrdinaryDifferentialOperator2(SquareMatrix(3,UnivariatePolynomial(x,Integer)),DirectProductMatrixModule(3,UnivariatePolynomial(x,Integer),SquareMatrix(3,UnivariatePolynomial(x,Integer)),UnivariatePolynomial(x,Integer))) +--E 21 + +--S 22 of 26 +b : Modo := m*Dx + 1 +--R +--R +--R + 2 + +--R |x 1 0 | +1 0 0+ +--R | | | | +--R (11) | 4 |D + |0 1 0| +--R |1 x 0 | | | +--R | | +0 0 1+ +--R | 2| +--R +0 0 4x + +--RType: LinearOrdinaryDifferentialOperator2(SquareMatrix(3,UnivariatePolynomial(x,Integer)),DirectProductMatrixModule(3,UnivariatePolynomial(x,Integer),SquareMatrix(3,UnivariatePolynomial(x,Integer)),UnivariatePolynomial(x,Integer))) +--E 22 + +--S 23 of 26 +c := a*b +--R +--R +--R (12) +--R + 2 + + 4 4 2 + + 2 + +--R |x 1 0 | |x + 2x + 2 x + x 0 | |x 1 0 | +--R | | 2 | | | | +--R | 4 |D + | 4 2 8 3 |D + | 4 | +--R |1 x 0 | | x + x x + 4x + 2 0 | |1 x 0 | +--R | | | | | | +--R | 2| | 4 | | 2| +--R +0 0 4x + + 0 0 16x + 8x + 1+ +0 0 4x + +--RType: LinearOrdinaryDifferentialOperator2(SquareMatrix(3,UnivariatePolynomial(x,Integer)),DirectProductMatrixModule(3,UnivariatePolynomial(x,Integer),SquareMatrix(3,UnivariatePolynomial(x,Integer)),UnivariatePolynomial(x,Integer))) +--E 23 + +--S 24 of 26 +a p +--R +--R +--R 4 2 5 2 5 3 2 +--R (13) [3x + x + 8x,2x + 3x + 3,28x + 8x + 21x + 2] +--RType: DirectProductMatrixModule(3,UnivariatePolynomial(x,Integer),SquareMatrix(3,UnivariatePolynomial(x,Integer)),UnivariatePolynomial(x,Integer)) +--E 24 + +--S 25 of 26 +b p +--R +--R +--R 3 2 4 4 3 2 +--R (14) [6x + 3x + 3,2x + 8x,84x + 7x + 8x + 2x] +--RType: DirectProductMatrixModule(3,UnivariatePolynomial(x,Integer),SquareMatrix(3,UnivariatePolynomial(x,Integer)),UnivariatePolynomial(x,Integer)) +--E 25 + +--S 26 of 26 +(a + b + c) (p + q) +--R +--R +--R (15) +--R 8 7 6 5 4 3 2 +--R [10x + 12x + 16x + 30x + 85x + 94x + 40x + 40x + 17, +--R 12 9 8 7 6 5 4 3 2 +--R 10x + 10x + 12x + 92x + 6x + 32x + 72x + 28x + 49x + 32x + 19, +--R 8 7 6 5 4 3 2 +--R 2240x + 224x + 1280x + 3508x + 492x + 751x + 98x + 18x + 4] +--RType: DirectProductMatrixModule(3,UnivariatePolynomial(x,Integer),SquareMatrix(3,UnivariatePolynomial(x,Integer)),UnivariatePolynomial(x,Integer)) +--E 26 +)spool +)lisp (bye) +@ +<>= +==================================================================== +LinearOrdinaryDifferentialOperator2 +==================================================================== + +LinearOrdinaryDifferentialOperator2(A, M) is the domain of linear +ordinary differential operators with coefficients in the differential +ring A and operating on M, an A-module. This includes the cases of +operators which are polynomials in 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. + +==================================================================== +Differential Operators with Constant Coefficients +==================================================================== + +This example shows differential operators with rational number +coefficients operating on univariate polynomials. + +We begin by making type assignments so we can conveniently refer +to univariate polynomials in x over the rationals. + + Q := Fraction Integer + Fraction Integer + Type: Domain + + PQ := UnivariatePolynomial('x, Q) + UnivariatePolynomial(x,Fraction Integer) + Type: Domain + + x: PQ := 'x + x + Type: UnivariatePolynomial(x,Fraction Integer) + +Now we assign Dx to be the differential operator D corresponding to d/dx. + + Dx: LODO2(Q, PQ) := D() + D + Type: LinearOrdinaryDifferentialOperator2(Fraction Integer, + UnivariatePolynomial(x,Fraction Integer)) + +New operators are created as polynomials in D(). + + a := Dx + 1 + D + 1 + Type: LinearOrdinaryDifferentialOperator2(Fraction Integer, + UnivariatePolynomial(x,Fraction Integer)) + + b := a + 1/2*Dx**2 - 1/2 + 1 2 1 + - D + D + - + 2 2 + Type: LinearOrdinaryDifferentialOperator2(Fraction Integer, + UnivariatePolynomial(x,Fraction Integer)) + +To apply the operator a to the value p the usual function call syntax +is used. + + p := 4*x**2 + 2/3 + 2 2 + 4x + - + 3 + Type: UnivariatePolynomial(x,Fraction Integer) + + a p + 2 2 + 4x + 8x + - + 3 + Type: UnivariatePolynomial(x,Fraction Integer) + +Operator multiplication is defined by the identity (a*b)p = a(b(p)) + + (a * b) p = a b p + 2 37 2 37 + 2x + 12x + --= 2x + 12x + -- + 3 3 + Type: Equation UnivariatePolynomial(x,Fraction Integer) + +Exponentiation follows from multiplication. + + c := (1/9)*b*(a + b)^2 + 1 6 5 5 13 4 19 3 79 2 7 1 + -- D + -- D + -- D + -- D + -- D + -- D + - + 72 36 24 18 72 12 8 + Type: LinearOrdinaryDifferentialOperator2(Fraction Integer, + UnivariatePolynomial(x,Fraction Integer)) + +Finally, note that operator expressions may be applied directly. + + (a**2 - 3/4*b + c) (p + 1) + 2 44 541 + 3x + -- x + --- + 3 36 + Type: UnivariatePolynomial(x,Fraction Integer) + +==================================================================== +Differential Operators with Matrix Coefficients Operating on Vectors} +==================================================================== + +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. + +In this section, the operators have three by three matrix coefficients +with polynomial entries. + + PZ := UnivariatePolynomial(x,Integer) + UnivariatePolynomial(x,Integer) + Type: Domain + + x:PZ := 'x + x + Type: UnivariatePolynomial(x,Integer) + + Mat := SquareMatrix(3,PZ) + SquareMatrix(3,UnivariatePolynomial(x,Integer)) + Type: Domain + +The operators act on the vectors considered as a Mat-module. + + Vect := DPMM(3, PZ, Mat, PZ) + DirectProductMatrixModule(3,UnivariatePolynomial(x,Integer),SquareMatrix(3,Un + ivariatePolynomial(x,Integer)),UnivariatePolynomial(x,Integer)) + Type: Domain + + Modo := LODO2(Mat, Vect) + LinearOrdinaryDifferentialOperator2(SquareMatrix(3,UnivariatePolynomial(x,Int + eger)),DirectProductMatrixModule(3,UnivariatePolynomial(x,Integer),SquareMatr + ix(3,UnivariatePolynomial(x,Integer)),UnivariatePolynomial(x,Integer))) + Type: Domain + +The matrix m is used as a coefficient and the vectors p and q are +operated upon. + + m:Mat := matrix [ [x^2,1,0],[1,x^4,0],[0,0,4*x^2] ] + + 2 + + |x 1 0 | + | | + | 4 | + |1 x 0 | + | | + | 2| + +0 0 4x + + Type: SquareMatrix(3,UnivariatePolynomial(x,Integer)) + + p:Vect := directProduct [3*x^2+1,2*x,7*x^3+2*x] + 2 3 + [3x + 1,2x,7x + 2x] + Type: DirectProductMatrixModule(3,UnivariatePolynomial(x,Integer), + SquareMatrix(3,UnivariatePolynomial(x,Integer)), + UnivariatePolynomial(x,Integer)) + + q: Vect := m * p + 4 2 5 2 5 3 + [3x + x + 2x,2x + 3x + 1,28x + 8x ] + Type: DirectProductMatrixModule(3,UnivariatePolynomial(x,Integer), + SquareMatrix(3,UnivariatePolynomial(x,Integer)), + UnivariatePolynomial(x,Integer)) + +Now form a few operators. + + Dx : Modo := D() + D + Type: LinearOrdinaryDifferentialOperator2( + SquareMatrix(3,UnivariatePolynomial(x,Integer)), + DirectProductMatrixModule(3,UnivariatePolynomial(x,Integer), + SquareMatrix(3,UnivariatePolynomial(x,Integer)), + UnivariatePolynomial(x,Integer))) + + a : Modo := Dx + m + + 2 + + |x 1 0 | + | | + D + | 4 | + |1 x 0 | + | | + | 2| + +0 0 4x + + Type: LinearOrdinaryDifferentialOperator2( + SquareMatrix(3,UnivariatePolynomial(x,Integer)), + DirectProductMatrixModule(3,UnivariatePolynomial(x,Integer), + SquareMatrix(3,UnivariatePolynomial(x,Integer)), + UnivariatePolynomial(x,Integer))) + + b : Modo := m*Dx + 1 + + 2 + + |x 1 0 | +1 0 0+ + | | | | + | 4 |D + |0 1 0| + |1 x 0 | | | + | | +0 0 1+ + | 2| + +0 0 4x + + Type: LinearOrdinaryDifferentialOperator2( + SquareMatrix(3,UnivariatePolynomial(x,Integer)), + DirectProductMatrixModule(3,UnivariatePolynomial(x,Integer), + SquareMatrix(3,UnivariatePolynomial(x,Integer)), + UnivariatePolynomial(x,Integer))) + + c := a*b + + 2 + + 4 4 2 + + 2 + + |x 1 0 | |x + 2x + 2 x + x 0 | |x 1 0 | + | | 2 | | | | + | 4 |D + | 4 2 8 3 |D + | 4 | + |1 x 0 | | x + x x + 4x + 2 0 | |1 x 0 | + | | | | | | + | 2| | 4 | | 2| + +0 0 4x + + 0 0 16x + 8x + 1+ +0 0 4x + + Type: LinearOrdinaryDifferentialOperator2( + SquareMatrix(3,UnivariatePolynomial(x,Integer)), + DirectProductMatrixModule(3,UnivariatePolynomial(x,Integer), + SquareMatrix(3,UnivariatePolynomial(x,Integer)), + UnivariatePolynomial(x,Integer))) + +These operators can be applied to vector values. + + a p + 4 2 5 2 5 3 2 + [3x + x + 8x,2x + 3x + 3,28x + 8x + 21x + 2] + Type: DirectProductMatrixModule(3,UnivariatePolynomial(x,Integer), + SquareMatrix(3,UnivariatePolynomial(x,Integer)), + UnivariatePolynomial(x,Integer)) + + b p + 3 2 4 4 3 2 + [6x + 3x + 3,2x + 8x,84x + 7x + 8x + 2x] + Type: DirectProductMatrixModule(3,UnivariatePolynomial(x,Integer), + SquareMatrix(3,UnivariatePolynomial(x,Integer)), + UnivariatePolynomial(x,Integer)) + + (a + b + c) (p + q) + 8 7 6 5 4 3 2 + [10x + 12x + 16x + 30x + 85x + 94x + 40x + 40x + 17, + 12 9 8 7 6 5 4 3 2 + 10x + 10x + 12x + 92x + 6x + 32x + 72x + 28x + 49x + 32x + 19, + 8 7 6 5 4 3 2 + 2240x + 224x + 1280x + 3508x + 492x + 751x + 98x + 18x + 4] + Type: DirectProductMatrixModule(3,UnivariatePolynomial(x,Integer), + SquareMatrix(3,UnivariatePolynomial(x,Integer)), + UnivariatePolynomial(x,Integer)) + +See Also: +o )show LinearOrdinaryDifferentialOperator2 +o $AXIOM/doc/src/algebra/lodo.spad.dvi + +@ +\pagehead{LinearOrdinaryDifferentialOperator2}{LODO2} +\pagepic{ps/v103linearordinarydifferentialoperator2.ps}{LODO2}{1.00} +<>= +)abbrev domain LODO2 LinearOrdinaryDifferentialOperator2 +++ Author: Stephen M. Watt, Manuel Bronstein +++ Date Created: 1986 +++ Date Last Updated: 1 February 1994 +++ Keywords: differential operator +++ Description: +++ \spad{LinearOrdinaryDifferentialOperator2} defines a ring of +++ differential operators with coefficients in a differential ring A +++ and acting on an A-module M. +++ Multiplication of operators corresponds to functional composition: +++ \spad{(L1 * L2).(f) = L1 L2 f} +LinearOrdinaryDifferentialOperator2(A, M): Exports == Implementation where + A: DifferentialRing + M: LeftModule A with + differentiate: $ -> $ + ++ differentiate(x) returns the derivative of x + + Exports ==> Join(LinearOrdinaryDifferentialOperatorCategory A, Eltable(M, M)) + + Implementation ==> LinearOrdinaryDifferentialOperator(A, differentiate$A) add + elt(p:%, m:M):M == + apply(p, differentiate, m)$ApplyUnivariateSkewPolynomial(A, M, %) + +@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{domain LIST List} <>= -- list.spad.pamphlet List.input @@ -32521,99 +34192,6 @@ List(S:Type): Exports == Implementation where @ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\section{domain LSQM LieSquareMatrix} -\pagehead{LieSquareMatrix}{LSQM} -\pagepic{ps/v103liesquarematrix.ps}{LSQM}{1.00} -See also:\\ -\refto{AssociatedLieAlgebra}{LIE} -\refto{AssociatedJordanAlgebra}{JORDAN} -<>= -)abbrev domain LSQM LieSquareMatrix -++ Author: J. Grabmeier -++ Date Created: 07 March 1991 -++ Date Last Updated: 08 March 1991 -++ Basic Operations: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: -++ Description: -++ LieSquareMatrix(n,R) implements the Lie algebra of the n by n -++ matrices over the commutative ring R. -++ The Lie bracket (commutator) of the algebra is given by -++ \spad{a*b := (a *$SQMATRIX(n,R) b - b *$SQMATRIX(n,R) a)}, -++ where \spadfun{*$SQMATRIX(n,R)} is the usual matrix multiplication. -LieSquareMatrix(n,R): Exports == Implementation where - - n : PositiveInteger - R : CommutativeRing - - Row ==> DirectProduct(n,R) - Col ==> DirectProduct(n,R) - - Exports ==> Join(SquareMatrixCategory(n,R,Row,Col), CoercibleTo Matrix R,_ - FramedNonAssociativeAlgebra R) --with - - Implementation ==> AssociatedLieAlgebra (R,SquareMatrix(n, R)) add - - Rep := AssociatedLieAlgebra (R,SquareMatrix(n, R)) - -- local functions - n2 : PositiveInteger := n*n - - convDM : DirectProduct(n2,R) -> % - conv : DirectProduct(n2,R) -> SquareMatrix(n,R) - --++ converts n2-vector to (n,n)-matrix row by row - conv v == - cond : Matrix(R) := new(n,n,0$R)$Matrix(R) - z : Integer := 0 - for i in 1..n repeat - for j in 1..n repeat - z := z+1 - setelt(cond,i,j,v.z) - squareMatrix(cond)$SquareMatrix(n, R) - - - coordinates(a:%,b:Vector(%)):Vector(R) == - -- only valid for b canonicalBasis - res : Vector R := new(n2,0$R) - z : Integer := 0 - for i in 1..n repeat - for j in 1..n repeat - z := z+1 - res.z := elt(a,i,j)$% - res - - - convDM v == - sq := conv v - coerce(sq)$Rep :: % - - basis() == - n2 : PositiveInteger := n*n - ldp : List DirectProduct(n2,R) := - [unitVector(i::PositiveInteger)$DirectProduct(n2,R) for i in 1..n2] - res:Vector % := vector map(convDM,_ - ldp)$ListFunctions2(DirectProduct(n2,R), %) - - someBasis() == basis() - rank() == n*n - - --- transpose: % -> % --- ++ computes the transpose of a matrix --- squareMatrix: Matrix R -> % --- ++ converts a Matrix to a LieSquareMatrix --- coerce: % -> Matrix R --- ++ converts a LieSquareMatrix to a Matrix --- symdecomp : % -> Record(sym:%,antisym:%) --- if R has commutative("*") then --- minorsVect: -> Vector(Union(R,"uncomputed")) --range: 1..2**n-1 --- if R has commutative("*") then central --- if R has commutative("*") and R has unitsKnown then unitsKnown - -@ -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{domain LMOPS ListMonoidOps} \pagehead{ListMonoidOps}{LMOPS} \pagepic{ps/v103listmonoidops.ps}{LMOPS}{1.00} @@ -32778,6 +34356,155 @@ ListMonoidOps(S, E, un): Exports == Implementation where @ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section{domain LMDICT ListMultiDictionary} +\pagehead{ListMultiDictionary}{LMDICT} +\pagepic{ps/v103listmultidictionary.ps}{LMDICT}{1.00} +<>= +)abbrev domain LMDICT ListMultiDictionary +++ Author: MBM Nov/87, MB Oct/89 +++ Date Created: +++ Date Last Updated: 13 June 1994 Frederic Lehobey +++ Basic Operations: +++ Related Constructors: +++ Also See: +++ AMS Classifications: +++ Keywords: +++ References: +++ Description: The \spadtype{ListMultiDictionary} domain implements a +++ dictionary with duplicates +++ allowed. The representation is a list with duplicates represented +++ explicitly. Hence most operations will be relatively inefficient +++ when the number of entries in the dictionary becomes large. +-- The operations \spadfun{pick}, \spadfun{count} and \spadfun{delete} can be used to iterate +-- over the objects in the dictionary. +-- [FDLL : those functions have not been implemented in the parent Categories] +++ If the objects in the +++ dictionary belong to an ordered set, the entries are maintained in +++ ascending order. + +NNI ==> NonNegativeInteger +D ==> Record(entry:S, count:NonNegativeInteger) + +ListMultiDictionary(S:SetCategory): MultiDictionary(S) with + finiteAggregate + duplicates?: % -> Boolean + ++ duplicates?(d) tests if dictionary d has duplicate entries. + substitute : (S, S, %) -> % + ++ substitute(x,y,d) replace x's with y's in dictionary d. + == add + Rep := Reference List S + + sub: (S, S, S) -> S + + coerce(s:%):OutputForm == + prefix("dictionary"::OutputForm, [x::OutputForm for x in parts s]) + + #s == # parts s + copy s == dictionary copy parts s + empty? s == empty? parts s + bag l == dictionary l + dictionary() == dictionary empty() + + empty():% == ref empty() + + dictionary(ls:List S):% == + empty? ls => empty() + lmd := empty() + for x in ls repeat insert_!(x,lmd) + lmd + + if S has ConvertibleTo InputForm then + convert(lmd:%):InputForm == + convert [convert("dictionary"::Symbol)@InputForm, + convert(parts lmd)@InputForm] + + map(f, s) == dictionary map(f, parts s) + map_!(f, s) == dictionary map_!(f, parts s) + parts s == deref s + sub(x, y, z) == (z = x => y; z) + insert_!(x, s, n) == (for i in 1..n repeat insert_!(x, s); s) + substitute(x, y, s) == dictionary map(sub(x, y, #1), parts s) + removeDuplicates_! s == dictionary removeDuplicates_! parts s + + inspect s == + empty? s => error "empty dictionary" + first parts s + + extract_! s == + empty? s => error "empty dictionary" + x := first(p := parts s) + setref(s, rest p) + x + + duplicates? s == + empty?(p := parts s) => false + q := rest p + while not empty? q repeat + first p = first q => return true + p := q + q := rest q + false + + remove_!(p: S->Boolean, lmd:%):% == + for x in removeDuplicates parts lmd | p(x) repeat remove_!(x,lmd) + lmd + + select_!(p: S->Boolean, lmd:%):% == remove_!(not p(#1), lmd) + + duplicates(lmd:%):List D == + ld: List D := empty() + for x in removeDuplicates parts lmd | (n := count(x, lmd)) > + 1$NonNegativeInteger repeat + ld := cons([x, n], ld) + ld + + if S has OrderedSet then + s = t == parts s = parts t + + remove_!(x:S, s:%) == + p := deref s + while not empty? p and x = first p repeat p := rest p + setref(s, p) + empty? p => s + q := rest p + while not empty? q and x > first q repeat (p := q; q := rest q) + while not empty? q and x = first q repeat q := rest q + p.rest := q + s + + insert_!(x, s) == + p := deref s + empty? p or x < first p => + setref(s, concat(x, p)) + s + q := rest p + while not empty? q and x > first q repeat (p := q; q := rest q) + p.rest := concat(x, q) + s + + else + remove_!(x:S, s:%) == (setref(s, remove_!(x, parts s)); s) + + s = t == + a := copy s + while not empty? a repeat + x := inspect a + count(x, s) ^= count(x, t) => return false + remove_!(x, a) + true + + insert_!(x, s) == + p := deref s + while not empty? p repeat + x = first p => + p.rest := concat(x, rest p) + s + p := rest p + setref(s, concat(x, deref s)) + s + +@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{domain LA LocalAlgebra} \pagehead{LocalAlgebra}{LA} \pagepic{ps/v103localalgebra.ps}{LA}{1.00} @@ -34153,6 +35880,51 @@ OnePointCompletion(R:SetCategory): Exports == Implementation where @ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section{domain OMLO OppositeMonogenicLinearOperator} +\pagehead{OppositeMonogenicLinearOperator}{OMLO} +\pagepic{ps/v103oppositemonogeniclinearoperator.ps}{OMLO}{1.00} +See also:\\ +\refto{OrdinaryDifferentialRing}{ODR} +\refto{DirectProductModule}{DPMO} +\refto{DirectProductMatrixModule}{DPMM} +<>= +)abbrev domain OMLO OppositeMonogenicLinearOperator +++ Author: Stephen M. Watt +++ Date Created: 1986 +++ Date Last Updated: May 30, 1991 +++ Basic Operations: +++ Related Domains: MonogenicLinearOperator +++ Also See: +++ AMS Classifications: +++ Keywords: opposite ring +++ Examples: +++ References: +++ Description: +++ This constructor creates the \spadtype{MonogenicLinearOperator} domain +++ which is ``opposite'' in the ring sense to P. +++ That is, as sets \spad{P = $} but \spad{a * b} in \spad{$} is equal to +++ \spad{b * a} in P. + +OppositeMonogenicLinearOperator(P, R): OPRcat == OPRdef where + P: MonogenicLinearOperator(R) + R: Ring + + OPRcat == MonogenicLinearOperator(R) with + if P has DifferentialRing then DifferentialRing + op: P -> $ ++ op(p) creates a value in $ equal to p in P. + po: $ -> P ++ po(q) creates a value in P equal to q in $. + + OPRdef == P add + Rep := P + x, y: $ + a: P + op a == a: $ + po x == x: P + x*y == (y:P) *$P (x:P) + coerce(x): OutputForm == prefix(op::OutputForm, [coerce(x:P)$P]) + +@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{domain ORDCOMP OrderedCompletion} <>= "ORDCOMP" -> "SETCAT" @@ -35070,6 +36842,53 @@ OrderlyDifferentialVariable(S:OrderedSet):DifferentialVariableCategory(S) @ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section{domain ODR OrdinaryDifferentialRing} +\pagehead{OrdinaryDifferentialRing}{ODR} +\pagepic{ps/v103ordinarydifferentialring.ps}{ODR}{1.00} +See also:\\ +\refto{OppositeMonogenicLinearOperator}{OMLO} +\refto{DirectProductModule}{DPMO} +\refto{DirectProductMatrixModule}{DPMM} +<>= +)abbrev domain ODR OrdinaryDifferentialRing +++ Author: Stephen M. Watt +++ Date Created: 1986 +++ Date Last Updated: June 3, 1991 +++ Basic Operations: +++ Related Domains: +++ Also See: +++ AMS Classifications: +++ Keywords: differential ring +++ Examples: +++ References: +++ Description: +++ This constructor produces an ordinary differential ring from +++ a partial differential ring by specifying a variable. + +OrdinaryDifferentialRing(Kernels,R,var): DRcategory == DRcapsule where + Kernels:SetCategory + R: PartialDifferentialRing(Kernels) + var : Kernels + DRcategory == Join(BiModule($,$), DifferentialRing) with + if R has Field then Field + coerce: R -> $ + ++ coerce(r) views r as a value in the ordinary differential ring. + coerce: $ -> R + ++ coerce(p) views p as a valie in the partial differential ring. + DRcapsule == R add + n: Integer + Rep := R + coerce(u:R):$ == u::Rep::$ + coerce(p:$):R == p::Rep::R + differentiate p == differentiate(p, var) + + if R has Field then + p / q == ((p::R) /$R (q::R))::$ + p ** n == ((p::R) **$R n)::$ + inv(p) == (inv(p::R)$R)::$ + +@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{domain OSI OrdSetInts} \pagehead{OrdSetInts}{OSI} \pagepic{ps/v103ordsetints.ps}{OSI}{1.00} @@ -38259,6 +40078,156 @@ ScriptFormulaFormat(): public == private where @ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\section{domain SETMN SetOfMIntegersInOneToN} +\pagehead{SetOfMIntegersInOneToN}{SETMN} +\pagepic{ps/v103setofmintegersinoneton.ps}{SETMN}{1.00} +<>= +)abbrev domain SETMN SetOfMIntegersInOneToN +++ Author: Manuel Bronstein +++ Date Created: 10 January 1994 +++ Date Last Updated: 10 January 1994 +++ Description: +++ \spadtype{SetOfMIntegersInOneToN} implements the subsets of M integers +++ in the interval \spad{[1..n]} +SetOfMIntegersInOneToN(m, n): Exports == Implementation where + PI ==> PositiveInteger + N ==> NonNegativeInteger + U ==> Union(%, "failed") + n,m: PI + + Exports ==> Finite with + incrementKthElement: (%, PI) -> U + ++ incrementKthElement(S,k) increments the k^{th} element of S, + ++ and returns "failed" if the result is not a set of M integers + ++ in \spad{1..n} any more. + replaceKthElement: (%, PI, PI) -> U + ++ replaceKthElement(S,k,p) replaces the k^{th} element of S by p, + ++ and returns "failed" if the result is not a set of M integers + ++ in \spad{1..n} any more. + elements: % -> List PI + ++ elements(S) returns the list of the elements of S in increasing order. + setOfMinN: List PI -> % + ++ setOfMinN([a_1,...,a_m]) returns the set {a_1,...,a_m}. + ++ Error if {a_1,...,a_m} is not a set of M integers in \spad{1..n}. + enumerate: () -> Vector % + ++ enumerate() returns a vector of all the sets of M integers in + ++ \spad{1..n}. + member?: (PI, %) -> Boolean + ++ member?(p, s) returns true is p is in s, false otherwise. + delta: (%, PI, PI) -> N + ++ delta(S,k,p) returns the number of elements of S which are strictly + ++ between p and the k^{th} element of S. + + Implementation ==> add + Rep := Record(bits:Bits, pos:N) + + reallyEnumerate: () -> Vector % + enum: (N, N, PI) -> List Bits + + all:Reference Vector % := ref empty() + sz:Reference N := ref 0 + + s1 = s2 == s1.bits =$Bits s2.bits + coerce(s:%):OutputForm == brace [i::OutputForm for i in elements s] + random() == index((1 + (random()$Integer rem size()))::PI) + reallyEnumerate() == [[b, i] for b in enum(m, n, n) for i in 1..] + member?(p, s) == s.bits.p + + enumerate() == + if empty? all() then all() := reallyEnumerate() + all() + +-- enumerates the sets of p integers in 1..q, returns them as sets in 1..n +-- must have p <= q + enum(p, q, n) == + zero? p or zero? q => empty() + p = q => + b := new(n, false)$Bits + for i in 1..p repeat b.i := true + [b] + q1 := (q - 1)::N + l := enum((p - 1)::N, q1, n) + if empty? l then l := [new(n, false)$Bits] + for s in l repeat s.q := true + concat_!(enum(p, q1, n), l) + + size() == + if zero? sz() then + sz() := binomial(n, m)$IntegerCombinatoricFunctions(Integer) :: N + sz() + + lookup s == + if empty? all() then all() := reallyEnumerate() + if zero?(s.pos) then s.pos := position(s, all()) :: N + s.pos :: PI + + index p == + p > size() => error "index: argument too large" + if empty? all() then all() := reallyEnumerate() + all().p + + setOfMinN l == + s := new(n, false)$Bits + count:N := 0 + for i in l repeat + count := count + 1 + count > m or zero? i or i > n or s.i => + error "setOfMinN: improper set of integers" + s.i := true + count < m => error "setOfMinN: improper set of integers" + [s, 0] + + elements s == + b := s.bits + l:List PI := empty() + found:N := 0 + i:PI := 1 + while found < m repeat + if b.i then + l := concat(i, l) + found := found + 1 + i := i + 1 + reverse_! l + + incrementKthElement(s, k) == + b := s.bits + found:N := 0 + i:N := 1 + while found < k repeat + if b.i then found := found + 1 + i := i + 1 + i > n or b.i => "failed" + newb := copy b + newb.i := true + newb.((i-1)::N) := false + [newb, 0] + + delta(s, k, p) == + b := s.bits + count:N := found:N := 0 + i:PI := 1 + while found < k repeat + if b.i then + found := found + 1 + if i > p and found < k then count := count + 1 + i := i + 1 + count + + replaceKthElement(s, k, p) == + b := s.bits + found:N := 0 + i:PI := 1 + while found < k repeat + if b.i then found := found + 1 + if found < k then i := i + 1 + b.p and i ^= p => "failed" + newb := copy b + newb.p := true + newb.i := false + [newb, (i = p => s.pos; 0)] + +@ +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{domain SDPOL SequentialDifferentialPolynomial} \pagehead{SequentialDifferentialPolynomial}{SDPOL} \pagepic{ps/v103sequentialdifferentialpolynomial.ps}{SDPOL}{1.00} @@ -43412,6 +45381,8 @@ Note that this code is not included in the generated catdef.spad file. <> <> <> +<> +<> <> <> <> @@ -43515,8 +45486,12 @@ Note that this code is not included in the generated catdef.spad file. <> <> <> +<> +<> +<> <> <> +<> <> <> @@ -43535,10 +45510,12 @@ Note that this code is not included in the generated catdef.spad file. <> <> <> +<> <> <> <> <> +<> <> <> @@ -43562,6 +45539,7 @@ Note that this code is not included in the generated catdef.spad file. <> <> <> +<> <> <> <> diff --git a/books/ps/v103directproductmatrixmodule.ps b/books/ps/v103directproductmatrixmodule.ps new file mode 100644 index 0000000..131df11 --- /dev/null +++ b/books/ps/v103directproductmatrixmodule.ps @@ -0,0 +1,248 @@ +%!PS-Adobe-2.0 +%%Creator: dot version 2.8 (Thu Sep 14 20:34:11 UTC 2006) +%%For: (root) root +%%Title: pic +%%Pages: (atend) +%%BoundingBox: 36 36 222 80 +%%EndComments +save +%%BeginProlog +/DotDict 200 dict def +DotDict begin + +/setupLatin1 { +mark +/EncodingVector 256 array def + EncodingVector 0 + +ISOLatin1Encoding 0 255 getinterval putinterval +EncodingVector 45 /hyphen put + +% Set up ISO Latin 1 character encoding +/starnetISO { + dup dup findfont dup length dict begin + { 1 index /FID ne { def }{ pop pop } ifelse + } forall + /Encoding EncodingVector 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src/algebra/Makefile remove mts.spad +20081210 tpd src/algebra/mts.spad removed, moved domains to bookvol10.3 +20081210 tpd src/algebra/Makefile remove mset.spad +20081210 tpd src/algebra/mset.spad removed, moved domains to bookvol10.3 +20081210 tpd src/algebra/mring.spad moved domains to bookvol10.3 +20081210 tpd src/algebra/Makefile remove moebius.spad +20081210 tpd src/algebra/moebius.spad removed, moved domains to bookvol10.3 +20081210 tpd src/algebra/Makefile remove modring.spad +20081210 tpd src/algebra/modring.spad removed, moved domains to bookvol10.3 +20081210 tpd src/algebra/Makefile remove modmon.spad +20081210 tpd src/algebra/modmon.spad removed, moved domains to bookvol10.3 +20081210 tpd src/algebra/Makefile remove monmonom.spad +20081210 tpd src/algebra/modmonom.spad removed, moved domains to bookvol10.3 +20081210 tpd src/algebra/mkfunc.spad move domains to bookvol10.3 +20081210 tpd src/algebra/Makefile remove misc.spad +20081210 tpd src/algebra/misc.spad removed, moved domains to bookvol10.3 +20081210 tpd src/input/Makefile remove matrix.spad +20081210 tpd src/algebra/Makefile remove matrix.spad +20081210 tpd src/algebra/matrix.spad removed, moved domains to bookvol10.3 +20081210 tpd books/ps/v103setofmintegersinoneton.ps added +20081210 tpd books/ps/v103ordinarydifferentialring.ps added +20081210 tpd books/ps/v103oppositemonogeniclinearoperator.ps added +20081210 tpd books/ps/v103listmultidictionary.ps added +20081210 tpd books/ps/v103linearordinarydifferentialoperator2.ps added +20081210 tpd books/ps/v103linearordinarydifferentialoperator1.ps added +20081210 tpd books/ps/v103linearordinarydifferentialoperator.ps added +20081210 tpd books/ps/v103directproductmodule.ps added +20081210 tpd books/ps/v103directproductmatrixmodule.ps added +20081210 tpd src/algebra/Makefile convert lodo help files +20081210 tpd src/algebra/lodo.spad move domains to bookvol10.3 +20081210 tpd src/algebra/lodop.spad move domains to bookvol10.3 +20081210 tpd src/algebra/lodof.spad move domain to bookvol10.3 +20081210 tpd src/algebra/Makefile remove lmdict.spad +20081210 tpd src/algebra/lmdict.spad removed, move domain to bookvol10.3 20081209 tpd books/ps/v103univariatelaurentseriesconstructor.ps added 20081209 tpd books/ps/v103univariatelaurentseries.ps added 20081209 tpd books/ps/v103splithomogeneousdirectproduct.ps added diff --git a/src/algebra/Makefile.pamphlet b/src/algebra/Makefile.pamphlet index f7366c5..8c74041 100644 --- a/src/algebra/Makefile.pamphlet +++ b/src/algebra/Makefile.pamphlet @@ -368,8 +368,6 @@ drawpak.spad.pamphlet (DRAWCX) mappkg.spad.pamphlet (MAPHACK1 MAPHACK2 MAPHACK3 MAPHACK4 MAPPKG1 MAPPKG2 MAPPKG3 MAPPKG4) mesh.spad.pamphlet (MESH) -moebius.spad.pamphlet (MOEBIUS) -mring.spad.pamphlet (MRING MRF2) opalg.spad.pamphlet (MODOP OP) partperm.spad.pamphlet (PARTPERM) pgrobner.spad.pamphlet (PGROEB) @@ -538,9 +536,6 @@ matfuns.spad.pamphlet (IMATLIN MATCAT2 RMCAT2 IMATQF MATLIN) mfinfact.spad.pamphlet (MFINFACT) mlift.spad.pamphlet (MLIST) moddfact.spad.pamphlet (MDDFACT) -modmon.spad.pamphlet (MODMON) -modring.spad.pamphlet (MODRING EMR MODFIELD) -mts.spad.pamphlet (SMTS TS) multsqfr.spad.pamphlet (MULTSQFR) newpoint.spad.pamphlet (PTCAT POINT COMPPROP SUBSPACE PTPACK PTFUNC2) numtheor.spad.pamphlet (INTHEORY PNTHEORY) @@ -756,12 +751,8 @@ intaux.spad.pamphlet (IR IR2) intclos.spad.pamphlet (TRIMAT IBATOOL FFINTBAS WFFINTBS NFINTBAS) integer.spad.pamphlet (INTSLPE INT NNI PI ROMAN) kl.spad.pamphlet (CACHSET SCACHE MKCHSET KERNEL KERNEL2) -lmdict.spad.pamphlet (LMDICT) -matrix.spad.pamphlet (IMATRIX MATRIX RMATRIX SQMATRIX) -misc.spad.pamphlet (SAOS) mkfunc.spad.pamphlet (INFORM INFORM1 MKFUNC MKUCFUNC MKBCFUNC MKFLCFN) modgcd.spad.pamphlet (INMODGCD) -mset.spad.pamphlet (MSET) multpoly.spad.pamphlet (POLY POLY2 MPOLY SMP INDE) newdata.spad.pamphlet (IPRNTPK TBCMPPK SPLNODE SPLTREE) omerror.spad.pamphlet (OMERRK OMERR) @@ -1181,17 +1172,15 @@ SPADFILES= \ ${OUTSRC}/laplace.spad ${OUTSRC}/laurent.spad ${OUTSRC}/leadcdet.spad \ ${OUTSRC}/limitps.spad ${OUTSRC}/lindep.spad \ ${OUTSRC}/lingrob.spad ${OUTSRC}/liouv.spad ${OUTSRC}/listgcd.spad \ - ${OUTSRC}/list.spad ${OUTSRC}/lmdict.spad ${OUTSRC}/lodof.spad \ + ${OUTSRC}/list.spad ${OUTSRC}/lodof.spad \ ${OUTSRC}/lodop.spad ${OUTSRC}/lodo.spad \ ${OUTSRC}/manip.spad ${OUTSRC}/mappkg.spad \ ${OUTSRC}/matfuns.spad ${OUTSRC}/mathml.spad \ - ${OUTSRC}/matrix.spad ${OUTSRC}/matstor.spad \ - ${OUTSRC}/mesh.spad ${OUTSRC}/mfinfact.spad ${OUTSRC}/misc.spad \ + ${OUTSRC}/matstor.spad \ + ${OUTSRC}/mesh.spad ${OUTSRC}/mfinfact.spad \ ${OUTSRC}/mkfunc.spad ${OUTSRC}/mkrecord.spad \ ${OUTSRC}/mlift.spad ${OUTSRC}/moddfact.spad ${OUTSRC}/modgcd.spad \ - ${OUTSRC}/modmonom.spad ${OUTSRC}/modmon.spad ${OUTSRC}/modring.spad \ - ${OUTSRC}/moebius.spad ${OUTSRC}/mring.spad ${OUTSRC}/mset.spad \ - ${OUTSRC}/mts.spad ${OUTSRC}/multfact.spad ${OUTSRC}/multpoly.spad \ + ${OUTSRC}/multfact.spad ${OUTSRC}/multpoly.spad \ ${OUTSRC}/multsqfr.spad \ ${OUTSRC}/naalg.spad \ ${OUTSRC}/newdata.spad ${OUTSRC}/newpoint.spad \ @@ -1337,17 +1326,15 @@ DOCFILES= \ ${DOC}/laplace.spad.dvi ${DOC}/laurent.spad.dvi ${DOC}/leadcdet.spad.dvi \ ${DOC}/limitps.spad.dvi ${DOC}/lindep.spad.dvi \ ${DOC}/lingrob.spad.dvi ${DOC}/liouv.spad.dvi ${DOC}/listgcd.spad.dvi \ - ${DOC}/list.spad.dvi ${DOC}/lmdict.spad.dvi ${DOC}/lodof.spad.dvi \ + ${DOC}/list.spad.dvi ${DOC}/lodof.spad.dvi \ ${DOC}/lodop.spad.dvi ${DOC}/lodo.spad.dvi \ ${DOC}/manip.spad.dvi ${DOC}/mappkg.spad.dvi \ ${DOC}/matfuns.spad.dvi ${DOC}/mathml.spad.dvi \ - ${DOC}/matrix.spad.dvi ${DOC}/matstor.spad.dvi \ - ${DOC}/mesh.spad.dvi ${DOC}/mfinfact.spad.dvi ${DOC}/misc.spad.dvi \ + ${DOC}/matstor.spad.dvi \ + ${DOC}/mesh.spad.dvi ${DOC}/mfinfact.spad.dvi \ ${DOC}/mkfunc.spad.dvi ${DOC}/mkrecord.spad.dvi ${DOC}/mlift.spad.jhd.dvi \ ${DOC}/mlift.spad.dvi ${DOC}/moddfact.spad.dvi ${DOC}/modgcd.spad.dvi \ - ${DOC}/modmonom.spad.dvi ${DOC}/modmon.spad.dvi ${DOC}/modring.spad.dvi \ - ${DOC}/moebius.spad.dvi ${DOC}/mring.spad.dvi ${DOC}/mset.spad.dvi \ - ${DOC}/mts.spad.dvi ${DOC}/multfact.spad.dvi ${DOC}/multpoly.spad.dvi \ + ${DOC}/multfact.spad.dvi ${DOC}/multpoly.spad.dvi \ ${DOC}/multsqfr.spad.dvi \ ${DOC}/naalg.spad.dvi ${DOC}/ndftip.as.dvi \ ${DOC}/nepip.as.dvi ${DOC}/newdata.spad.dvi ${DOC}/newpoint.spad.dvi \ @@ -1943,12 +1930,13 @@ which is essentially a list of all of the algebra help files. Each item in this list refers to a stanza that looks like: \begin{verbatim} -${HELP}/AssociationList.help: ${IN}/list.spad.pamphlet - @echo 7000 create AssociationList.help from ${IN}/list.spad.pamphlet - @${TANGLE} -R"AssociationList.help" ${IN}/list.spad.pamphlet \ +${HELP}/AssociationList.help: ${BOOKS}/bookvol10.3.pamphlet + @echo 7000 create AssociationList.help from \ + ${BOOKS}/bookvol10.3.pamphlet + @${TANGLE} -R"AssociationList.help" ${BOOKS}/bookvol10.3.pamphlet \ >${HELP}/AssociationList.help @cp ${HELP}/AssociationList.help ${HELP}/ALIST.help - @${TANGLE} -R"AssociationList.input" ${IN}/list.spad.pamphlet \ + @${TANGLE} -R"AssociationList.input" ${BOOKS}/bookvol10.3.pamphlet \ >${INPUT}/AssociationList.input \end{verbatim} @@ -2148,12 +2136,13 @@ all: ${REGRESS} @echo algebra test cases complete. @ <>= -${HELP}/AssociationList.help: ${IN}/list.spad.pamphlet - @echo 7000 create AssociationList.help from ${IN}/list.spad.pamphlet - @${TANGLE} -R"AssociationList.help" ${IN}/list.spad.pamphlet \ +${HELP}/AssociationList.help: ${BOOKS}/bookvol10.3.pamphlet + @echo 7000 create AssociationList.help from \ + ${BOOKS}/bookvol10.3.pamphlet + @${TANGLE} -R"AssociationList.help" ${BOOKS}/bookvol10.3.pamphlet \ >${HELP}/AssociationList.help @cp ${HELP}/AssociationList.help ${HELP}/ALIST.help - @${TANGLE} -R"AssociationList.input" ${IN}/list.spad.pamphlet \ + @${TANGLE} -R"AssociationList.input" ${BOOKS}/bookvol10.3.pamphlet \ >${INPUT}/AssociationList.input ${HELP}/BalancedBinaryTree.help: ${IN}/tree.spad.pamphlet @@ -2231,12 +2220,12 @@ ${HELP}/CliffordAlgebra.help: ${BOOKS}/bookvol10.3.pamphlet @${TANGLE} -R"CliffordAlgebra.input" ${BOOKS}/bookvol10.3.pamphlet \ >${INPUT}/CliffordAlgebra.input -${HELP}/Complex.help: ${IN}/gaussian.spad.pamphlet - @echo 7010 create Complex.help from ${IN}/gaussian.spad.pamphlet - @${TANGLE} -R"Complex.help" ${IN}/gaussian.spad.pamphlet \ +${HELP}/Complex.help: ${BOOKS}/bookvol10.3.pamphlet + @echo 7010 create Complex.help from ${BOOKS}/bookvol10.3.pamphlet + @${TANGLE} -R"Complex.help" ${BOOKS}/bookvol10.3.pamphlet \ >${HELP}/Complex.help @-cp ${HELP}/Complex.help ${HELP}/COMPLEX.help - @${TANGLE} -R"Complex.input" ${IN}/gaussian.spad.pamphlet \ + @${TANGLE} -R"Complex.input" ${BOOKS}/bookvol10.3.pamphlet \ >${INPUT}/Complex.input ${HELP}/ContinuedFraction.help: ${BOOKS}/bookvol10.3.pamphlet @@ -2447,12 +2436,12 @@ ${HELP}/HomogeneousDistributedMultivariatePolynomial.help: \ ${BOOKS}/bookvol10.3.pamphlet \ >${INPUT}/HomogeneousDistributedMultivariatePolynomial.input -${HELP}/Integer.help: ${IN}/integer.spad.pamphlet - @echo 7033 create Integer.help from ${IN}/integer.spad.pamphlet - @${TANGLE} -R"Integer.help" ${IN}/integer.spad.pamphlet \ +${HELP}/Integer.help: ${BOOKS}/bookvol10.3.pamphlet + @echo 7033 create Integer.help from ${BOOKS}/bookvol10.3.pamphlet + @${TANGLE} -R"Integer.help" ${BOOKS}/bookvol10.3.pamphlet \ >${HELP}/Integer.help @cp ${HELP}/Integer.help ${HELP}/INT.help - @${TANGLE} -R"Integer.input" ${IN}/integer.spad.pamphlet \ + @${TANGLE} -R"Integer.input" ${BOOKS}/bookvol10.3.pamphlet \ >${INPUT}/Integer.input ${HELP}/IntegerLinearDependence.help: ${IN}/lindep.spad.pamphlet @@ -2478,12 +2467,12 @@ ${HELP}/IntegerNumberTheoryFunctions.help: ${IN}/numtheor.spad.pamphlet ${IN}/numtheor.spad.pamphlet \ >${INPUT}/IntegerNumberTheoryFunctions.input -${HELP}/Kernel.help: ${IN}/kl.spad.pamphlet - @echo 7036 create Kernel.help from ${IN}/kl.spad.pamphlet - @${TANGLE} -R"Kernel.help" ${IN}/kl.spad.pamphlet \ +${HELP}/Kernel.help: ${BOOKS}/bookvol10.3.pamphlet + @echo 7036 create Kernel.help from ${BOOKS}/bookvol10.3.pamphlet + @${TANGLE} -R"Kernel.help" ${BOOKS}/bookvol10.3.pamphlet \ >${HELP}/Kernel.help @-cp ${HELP}/Kernel.help ${HELP}/KERNEL.help - @${TANGLE} -R"Kernel.input" ${IN}/kl.spad.pamphlet \ + @${TANGLE} -R"Kernel.input" ${BOOKS}/bookvol10.3.pamphlet \ >${INPUT}/Kernel.input ${HELP}/KeyedAccessFile.help: ${BOOKS}/bookvol10.3.pamphlet @@ -2528,45 +2517,47 @@ ${HELP}/LiePolynomial.help: ${IN}/xlpoly.spad.pamphlet @${TANGLE} -R"LiePolynomial.input" ${IN}/xlpoly.spad.pamphlet \ >${INPUT}/LiePolynomial.input -${HELP}/LinearOrdinaryDifferentialOperator.help: ${IN}/lodo.spad.pamphlet +${HELP}/LinearOrdinaryDifferentialOperator.help: ${BOOKS}/bookvol10.3.pamphlet @echo 7042 create LinearOrdinaryDifferentialOperator.help \ - from ${IN}/lodo.spad.pamphlet + from ${BOOKS}/bookvol10.3.pamphlet @${TANGLE} -R"LinearOrdinaryDifferentialOperator.help" \ - ${IN}/lodo.spad.pamphlet \ + ${BOOKS}/bookvol10.3.pamphlet \ >${HELP}/LinearOrdinaryDifferentialOperator.help @cp ${HELP}/LinearOrdinaryDifferentialOperator.help ${HELP}/LODO.help @${TANGLE} -R"LinearOrdinaryDifferentialOperator.input" \ - ${IN}/lodo.spad.pamphlet \ + ${BOOKS}/bookvol10.3.pamphlet \ >${INPUT}/LinearOrdinaryDifferentialOperator.input -${HELP}/LinearOrdinaryDifferentialOperator1.help: ${IN}/lodo.spad.pamphlet +${HELP}/LinearOrdinaryDifferentialOperator1.help: \ + ${BOOKS}/bookvol10.3.pamphlet @echo 7043 create LinearOrdinaryDifferentialOperator1.help \ - from ${IN}/lodo.spad.pamphlet + from ${BOOKS}/bookvol10.3.pamphlet @${TANGLE} -R"LinearOrdinaryDifferentialOperator1.help" \ - ${IN}/lodo.spad.pamphlet \ + ${BOOKS}/bookvol10.3.pamphlet \ >${HELP}/LinearOrdinaryDifferentialOperator1.help @cp ${HELP}/LinearOrdinaryDifferentialOperator1.help ${HELP}/LODO1.help @${TANGLE} -R"LinearOrdinaryDifferentialOperator1.input" \ - ${IN}/lodo.spad.pamphlet \ + ${BOOKS}/bookvol10.3.pamphlet \ >${INPUT}/LinearOrdinaryDifferentialOperator1.input -${HELP}/LinearOrdinaryDifferentialOperator2.help: ${IN}/lodo.spad.pamphlet +${HELP}/LinearOrdinaryDifferentialOperator2.help: \ + ${BOOKS}/bookvol10.3.pamphlet @echo 7044 create LinearOrdinaryDifferentialOperator2.help \ - from ${IN}/lodo.spad.pamphlet + from ${BOOKS}/bookvol10.3.pamphlet @${TANGLE} -R"LinearOrdinaryDifferentialOperator2.help" \ - ${IN}/lodo.spad.pamphlet \ + ${BOOKS}/bookvol10.3.pamphlet \ >${HELP}/LinearOrdinaryDifferentialOperator2.help @cp ${HELP}/LinearOrdinaryDifferentialOperator2.help ${HELP}/LODO2.help @${TANGLE} -R"LinearOrdinaryDifferentialOperator2.input" \ - ${IN}/lodo.spad.pamphlet \ + ${BOOKS}/bookvol10.3.pamphlet \ >${INPUT}/LinearOrdinaryDifferentialOperator2.input -${HELP}/List.help: ${IN}/list.spad.pamphlet - @echo 7045 create List.help from ${IN}/list.spad.pamphlet - @${TANGLE} -R"List.help" ${IN}/list.spad.pamphlet \ +${HELP}/List.help: ${BOOKS}/bookvol10.3.pamphlet + @echo 7045 create List.help from ${BOOKS}/bookvol10.3.pamphlet + @${TANGLE} -R"List.help" ${BOOKS}/bookvol10.3.pamphlet \ >${HELP}/List.help @-cp ${HELP}/List.help ${HELP}/LIST.help - @${TANGLE} -R"List.input" ${IN}/list.spad.pamphlet \ + @${TANGLE} -R"List.input" ${BOOKS}/bookvol10.3.pamphlet \ >${INPUT}/List.input ${HELP}/LyndonWord.help: ${IN}/xlpoly.spad.pamphlet @@ -2617,31 +2608,31 @@ ${HELP}/MappingPackage3.help: ${IN}/mappkg.spad.pamphlet @${TANGLE} -R"MappingPackage3.input" ${IN}/mappkg.spad.pamphlet \ >${INPUT}/MappingPackage3.input -${HELP}/Matrix.help: ${IN}/matrix.spad.pamphlet - @echo 7052 create Matrix.help from ${IN}/matrix.spad.pamphlet - @${TANGLE} -R"Matrix.help" ${IN}/matrix.spad.pamphlet \ +${HELP}/Matrix.help: ${BOOKS}/bookvol10.3.pamphlet + @echo 7052 create Matrix.help from ${BOOKS}/bookvol10.3.pamphlet + @${TANGLE} -R"Matrix.help" ${BOOKS}/bookvol10.3.pamphlet \ >${HELP}/Matrix.help @-cp ${HELP}/Matrix.help ${HELP}/MATRIX.help - @${TANGLE} -R"Matrix.input" ${IN}/matrix.spad.pamphlet \ + @${TANGLE} -R"Matrix.input" ${BOOKS}/bookvol10.3.pamphlet \ >${INPUT}/Matrix.input -${HELP}/Multiset.help: ${IN}/mset.spad.pamphlet - @echo 7053 create Multiset.help from ${IN}/mset.spad.pamphlet - @${TANGLE} -R"Multiset.help" ${IN}/mset.spad.pamphlet \ +${HELP}/Multiset.help: ${BOOKS}/bookvol10.3.pamphlet + @echo 7053 create Multiset.help from ${BOOKS}/bookvol10.3.pamphlet + @${TANGLE} -R"Multiset.help" ${BOOKS}/bookvol10.3.pamphlet \ >${HELP}/Multiset.help @cp ${HELP}/Multiset.help ${HELP}/MSET.help - @${TANGLE} -R"Multiset.input" ${IN}/mset.spad.pamphlet \ + @${TANGLE} -R"Multiset.input" ${BOOKS}/bookvol10.3.pamphlet \ >${INPUT}/Multiset.input -${HELP}/MultivariatePolynomial.help: ${IN}/multpoly.spad.pamphlet +${HELP}/MultivariatePolynomial.help: ${BOOKS}/bookvol10.3.pamphlet @echo 7054 create MultivariatePolynomial.help from \ - ${IN}/multpoly.spad.pamphlet + ${BOOKS}/bookvol10.3.pamphlet @${TANGLE} -R"MultivariatePolynomial.help" \ - ${IN}/multpoly.spad.pamphlet \ + ${BOOKS}/bookvol10.3.pamphlet \ >${HELP}/MultivariatePolynomial.help @cp ${HELP}/MultivariatePolynomial.help ${HELP}/MPOLY.help @${TANGLE} -R"MultivariatePolynomial.input" \ - ${IN}/multpoly.spad.pamphlet \ + ${BOOKS}/bookvol10.3.pamphlet \ >${INPUT}/MultivariatePolynomial.input ${HELP}/None.help: ${BOOKS}/bookvol10.3.pamphlet @@ -2739,12 +2730,12 @@ ${HELP}/Plot.help: ${IN}/plot.spad.pamphlet @-cp ${HELP}/Plot.help ${HELP}/PLOT.help @${TANGLE} -R"Plot.input" ${IN}/plot.spad.pamphlet >${INPUT}/Plot.input -${HELP}/Polynomial.help: ${IN}/multpoly.spad.pamphlet - @echo 7065 create Polynomial.help from ${IN}/multpoly.spad.pamphlet - @${TANGLE} -R"Polynomial.help" ${IN}/multpoly.spad.pamphlet \ +${HELP}/Polynomial.help: ${BOOKS}/bookvol10.3.pamphlet + @echo 7065 create Polynomial.help from ${BOOKS}/bookvol10.3.pamphlet + @${TANGLE} -R"Polynomial.help" ${BOOKS}/bookvol10.3.pamphlet \ >${HELP}/Polynomial.help @cp ${HELP}/Polynomial.help ${HELP}/POLY.help - @${TANGLE} -R"Polynomial.input" ${IN}/multpoly.spad.pamphlet \ + @${TANGLE} -R"Polynomial.input" ${BOOKS}/bookvol10.3.pamphlet \ >${INPUT}/Polynomial.input ${HELP}/Quaternion.help: ${IN}/quat.spad.pamphlet @@ -2788,12 +2779,12 @@ ${HELP}/RegularTriangularSet.help: ${IN}/regset.spad.pamphlet @${TANGLE} -R"RegularTriangularSet.input" ${IN}/regset.spad.pamphlet \ >${INPUT}/RegularTriangularSet.input -${HELP}/RomanNumeral.help: ${IN}/integer.spad.pamphlet - @echo 7071 create RomanNumeral.help from ${IN}/integer.spad.pamphlet - @${TANGLE} -R"RomanNumeral.help" ${IN}/integer.spad.pamphlet \ +${HELP}/RomanNumeral.help: ${BOOKS}/bookvol10.3.pamphlet + @echo 7071 create RomanNumeral.help from ${BOOKS}/bookvol10.3.pamphlet + @${TANGLE} -R"RomanNumeral.help" ${BOOKS}/bookvol10.3.pamphlet \ >${HELP}/RomanNumeral.help @cp ${HELP}/RomanNumeral.help ${HELP}/ROMAN.help - @${TANGLE} -R"RomanNumeral.input" ${IN}/integer.spad.pamphlet \ + @${TANGLE} -R"RomanNumeral.input" ${BOOKS}/bookvol10.3.pamphlet \ >${INPUT}/RomanNumeral.input ${HELP}/Segment.help: ${IN}/seg.spad.pamphlet @@ -2828,12 +2819,12 @@ ${HELP}/SparseTable.help: ${IN}/table.spad.pamphlet @${TANGLE} -R"SparseTable.input" ${IN}/table.spad.pamphlet \ >${INPUT}/SparseTable.input -${HELP}/SquareMatrix.help: ${IN}/matrix.spad.pamphlet - @echo 7076 create SquareMatrix.help from ${IN}/matrix.spad.pamphlet - @${TANGLE} -R"SquareMatrix.help" ${IN}/matrix.spad.pamphlet \ +${HELP}/SquareMatrix.help: ${BOOKS}/bookvol10.3.pamphlet + @echo 7076 create SquareMatrix.help from ${BOOKS}/bookvol10.3.pamphlet + @${TANGLE} -R"SquareMatrix.help" ${BOOKS}/bookvol10.3.pamphlet \ >${HELP}/SquareMatrix.help @cp ${HELP}/SquareMatrix.help ${HELP}/SQMATRIX.help - @${TANGLE} -R"SquareMatrix.input" ${IN}/matrix.spad.pamphlet \ + @${TANGLE} -R"SquareMatrix.input" ${BOOKS}/bookvol10.3.pamphlet \ >${INPUT}/SquareMatrix.input ${HELP}/SquareFreeRegularTriangularSet.help: ${IN}/sregset.spad.pamphlet diff --git a/src/algebra/lmdict.spad.pamphlet b/src/algebra/lmdict.spad.pamphlet deleted file mode 100644 index cba196e..0000000 --- a/src/algebra/lmdict.spad.pamphlet +++ /dev/null @@ -1,200 +0,0 @@ -\documentclass{article} -\usepackage{axiom} -\begin{document} -\title{\$SPAD/src/algebra lmdict.spad} -\author{Michael Monagan, Frederic Lehobey} -\maketitle -\begin{abstract} -\end{abstract} -\eject -\tableofcontents -\eject -\section{domain LMDICT ListMultiDictionary} -<>= -)abbrev domain LMDICT ListMultiDictionary -++ Author: MBM Nov/87, MB Oct/89 -++ Date Created: -++ Date Last Updated: 13 June 1994 Frederic Lehobey -++ Basic Operations: -++ Related Constructors: -++ Also See: -++ AMS Classifications: -++ Keywords: -++ References: -++ Description: The \spadtype{ListMultiDictionary} domain implements a -++ dictionary with duplicates -++ allowed. The representation is a list with duplicates represented -++ explicitly. Hence most operations will be relatively inefficient -++ when the number of entries in the dictionary becomes large. --- The operations \spadfun{pick}, \spadfun{count} and \spadfun{delete} can be used to iterate --- over the objects in the dictionary. --- [FDLL : those functions have not been implemented in the parent Categories] -++ If the objects in the -++ dictionary belong to an ordered set, the entries are maintained in -++ ascending order. - -NNI ==> NonNegativeInteger -D ==> Record(entry:S, count:NonNegativeInteger) - -ListMultiDictionary(S:SetCategory): MultiDictionary(S) with - finiteAggregate - duplicates?: % -> Boolean - ++ duplicates?(d) tests if dictionary d has duplicate entries. - substitute : (S, S, %) -> % - ++ substitute(x,y,d) replace x's with y's in dictionary d. - == add - Rep := Reference List S - - sub: (S, S, S) -> S - - coerce(s:%):OutputForm == - prefix("dictionary"::OutputForm, [x::OutputForm for x in parts s]) - - #s == # parts s - copy s == dictionary copy parts s - empty? s == empty? parts s - bag l == dictionary l - dictionary() == dictionary empty() - - empty():% == ref empty() - - dictionary(ls:List S):% == - empty? ls => empty() - lmd := empty() - for x in ls repeat insert_!(x,lmd) - lmd - - if S has ConvertibleTo InputForm then - convert(lmd:%):InputForm == - convert [convert("dictionary"::Symbol)@InputForm, - convert(parts lmd)@InputForm] - - map(f, s) == dictionary map(f, parts s) - map_!(f, s) == dictionary map_!(f, parts s) - parts s == deref s - sub(x, y, z) == (z = x => y; z) - insert_!(x, s, n) == (for i in 1..n repeat insert_!(x, s); s) - substitute(x, y, s) == dictionary map(sub(x, y, #1), parts s) - removeDuplicates_! s == dictionary removeDuplicates_! parts s - - inspect s == - empty? s => error "empty dictionary" - first parts s - - extract_! s == - empty? s => error "empty dictionary" - x := first(p := parts s) - setref(s, rest p) - x - - duplicates? s == - empty?(p := parts s) => false - q := rest p - while not empty? q repeat - first p = first q => return true - p := q - q := rest q - false - - remove_!(p: S->Boolean, lmd:%):% == - for x in removeDuplicates parts lmd | p(x) repeat remove_!(x,lmd) - lmd - - select_!(p: S->Boolean, lmd:%):% == remove_!(not p(#1), lmd) - - duplicates(lmd:%):List D == - ld: List D := empty() - for x in removeDuplicates parts lmd | (n := count(x, lmd)) > - 1$NonNegativeInteger repeat - ld := cons([x, n], ld) - ld - - if S has OrderedSet then - s = t == parts s = parts t - - remove_!(x:S, s:%) == - p := deref s - while not empty? p and x = first p repeat p := rest p - setref(s, p) - empty? p => s - q := rest p - while not empty? q and x > first q repeat (p := q; q := rest q) - while not empty? q and x = first q repeat q := rest q - p.rest := q - s - - insert_!(x, s) == - p := deref s - empty? p or x < first p => - setref(s, concat(x, p)) - s - q := rest p - while not empty? q and x > first q repeat (p := q; q := rest q) - p.rest := concat(x, q) - s - - else - remove_!(x:S, s:%) == (setref(s, remove_!(x, parts s)); s) - - s = t == - a := copy s - while not empty? a repeat - x := inspect a - count(x, s) ^= count(x, t) => return false - remove_!(x, a) - true - - insert_!(x, s) == - p := deref s - while not empty? p repeat - x = first p => - p.rest := concat(x, rest p) - s - p := rest p - setref(s, concat(x, deref s)) - s - -@ -\section{License} -<>= ---Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd. ---All rights reserved. --- ---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. -@ -<<*>>= -<> - -<> -@ -\eject -\begin{thebibliography}{99} -\bibitem{1} nothing -\end{thebibliography} -\end{document} diff --git a/src/algebra/lodo.spad.pamphlet b/src/algebra/lodo.spad.pamphlet index fd3136a..939f355 100644 --- a/src/algebra/lodo.spad.pamphlet +++ b/src/algebra/lodo.spad.pamphlet @@ -113,1498 +113,6 @@ LinearOrdinaryDifferentialOperatorsOps(A, L): Exports == Implementation where p @ -\section{domain LODO LinearOrdinaryDifferentialOperator} -<>= --- lodo.spad.pamphlet LinearOrdinaryDifferentialOperator.input -)spool LinearOrdinaryDifferentialOperator.output -)set message test on -)set message auto off -)clear all ---S 1 of 16 -Dx: LODO(EXPR INT, f +-> D(f, x)) ---R ---R Type: Void ---E 1 - ---S 2 of 16 -Dx := D() ---R ---R ---R (2) D ---IType: LinearOrdinaryDifferentialOperator(Expression Integer,theMap LAMBDA-CLOSURE(NIL,NIL,NIL,G1404 envArg,SPADCALL(G1404,QUOTE x,ELT(*1;anonymousFunction;0;frame0;internal;MV,0)))) ---E 2 - ---S 3 of 16 -Dop:= Dx^3 + G/x^2*Dx + H/x^3 - 1 ---R ---R ---R 3 ---R 3 G - x + H ---R (3) D + -- D + -------- ---R 2 3 ---R x x ---IType: LinearOrdinaryDifferentialOperator(Expression Integer,theMap LAMBDA-CLOSURE(NIL,NIL,NIL,G1404 envArg,SPADCALL(G1404,QUOTE x,ELT(*1;anonymousFunction;0;frame0;internal;MV,0)))) ---E 3 - ---S 4 of 16 -n == 3 ---R ---R Type: Void ---E 4 - ---S 5 of 16 -phi == reduce(+,[subscript(s,[i])*exp(x)/x^i for i in 0..n]) ---R ---R Type: Void ---E 5 - ---S 6 of 16 -phi1 == Dop(phi) / exp x ---R ---R Type: Void ---E 6 - ---S 7 of 16 -phi2 == phi1 *x**(n+3) ---R ---R Type: Void ---E 7 - ---S 8 of 16 -phi3 == retract(phi2)@(POLY INT) ---R ---R Type: Void ---E 8 - ---S 9 of 16 -pans == phi3 ::UP(x,POLY INT) ---R ---R Type: Void ---E 9 - ---S 10 of 16 -pans1 == [coefficient(pans, (n+3-i) :: NNI) for i in 2..n+1] ---R ---R Type: Void ---E 10 - ---S 11 of 16 -leq == solve(pans1,[subscript(s,[i]) for i in 1..n]) ---R ---R Type: Void ---E 11 - ---S 12 of 16 -leq ---R ---R Compiling body of rule n to compute value of type PositiveInteger ---R Compiling body of rule phi to compute value of type Expression ---R Integer ---R Compiling body of rule phi1 to compute value of type Expression ---R Integer ---R Compiling body of rule phi2 to compute value of type Expression ---R Integer ---R Compiling body of rule phi3 to compute value of type Polynomial ---R Integer ---R Compiling body of rule pans to compute value of type ---R UnivariatePolynomial(x,Polynomial Integer) ---R Compiling body of rule pans1 to compute value of type List ---R Polynomial Integer ---R Compiling body of rule leq to compute value of type List List ---R Equation Fraction Polynomial Integer ---I Compiling function G3349 with type Integer -> Boolean ---R ---R (12) ---R 2 3 2 ---R s G 3s H + s G + 6s G (9s G + 54s )H + s G + 18s G + 72s G ---R 0 0 0 0 0 0 0 0 0 ---R [[s = ---,s = ------------------,s = --------------------------------------]] ---R 1 3 2 18 3 162 ---R Type: List List Equation Fraction Polynomial Integer ---E 12 - ---S 13 of 16 -n==4 ---R ---R Compiled code for n has been cleared. ---R Compiled code for leq has been cleared. ---R Compiled code for pans1 has been cleared. ---R Compiled code for phi2 has been cleared. ---R Compiled code for phi has been cleared. ---R Compiled code for phi3 has been cleared. ---R Compiled code for phi1 has been cleared. ---R Compiled code for pans has been cleared. ---R 1 old definition(s) deleted for function or rule n ---R Type: Void ---E 13 - ---S 14 of 16 -leq ---R ---R Compiling body of rule n to compute value of type PositiveInteger ---R Compiling body of rule phi to compute value of type Expression ---R Integer ---R Compiling body of rule phi1 to compute value of type Expression ---R Integer ---R Compiling body of rule phi2 to compute value of type Expression ---R Integer ---R Compiling body of rule phi3 to compute value of type Polynomial ---R Integer ---R Compiling body of rule pans to compute value of type ---R UnivariatePolynomial(x,Polynomial Integer) ---R Compiling body of rule pans1 to compute value of type List ---R Polynomial Integer ---R Compiling body of rule leq to compute value of type List List ---R Equation Fraction Polynomial Integer ---R ---R (14) ---R [ ---R 2 ---R s G 3s H + s G + 6s G ---R 0 0 0 0 ---R [s = ---, s = ------------------, ---R 1 3 2 18 ---R 3 2 ---R (9s G + 54s )H + s G + 18s G + 72s G ---R 0 0 0 0 0 ---R s = --------------------------------------, ---R 3 162 ---R ---R s = ---R 4 ---R 2 2 4 3 2 ---R 27s H + (18s G + 378s G + 1296s )H + s G + 36s G + 396s G ---R 0 0 0 0 0 0 0 ---R + ---R 1296s G ---R 0 ---R / ---R 1944 ---R ] ---R ] ---R Type: List List Equation Fraction Polynomial Integer ---E 14 - ---S 15 of 16 -n==7 ---R ---R Compiled code for n has been cleared. ---R Compiled code for leq has been cleared. ---R Compiled code for pans1 has been cleared. ---R Compiled code for phi2 has been cleared. ---R Compiled code for phi has been cleared. ---R Compiled code for phi3 has been cleared. ---R Compiled code for phi1 has been cleared. ---R Compiled code for pans has been cleared. ---R 1 old definition(s) deleted for function or rule n ---R Type: Void ---E 15 - ---S 16 of 16 -leq ---R ---R Compiling body of rule n to compute value of type PositiveInteger ---R Compiling body of rule phi to compute value of type Expression ---R Integer ---R Compiling body of rule phi1 to compute value of type Expression ---R Integer ---R Compiling body of rule phi2 to compute value of type Expression ---R Integer ---R Compiling body of rule phi3 to compute value of type Polynomial ---R Integer ---R Compiling body of rule pans to compute value of type ---R UnivariatePolynomial(x,Polynomial Integer) ---R Compiling body of rule pans1 to compute value of type List ---R Polynomial Integer ---R Compiling body of rule leq to compute value of type List List ---R Equation Fraction Polynomial Integer ---R ---R (16) ---R [ ---R 2 ---R s G 3s H + s G + 6s G ---R 0 0 0 0 ---R [s = ---, s = ------------------, ---R 1 3 2 18 ---R 3 2 ---R (9s G + 54s )H + s G + 18s G + 72s G ---R 0 0 0 0 0 ---R s = --------------------------------------, ---R 3 162 ---R ---R s = ---R 4 ---R 2 2 4 3 2 ---R 27s H + (18s G + 378s G + 1296s )H + s G + 36s G + 396s G ---R 0 0 0 0 0 0 0 ---R + ---R 1296s G ---R 0 ---R / ---R 1944 ---R , ---R ---R s = ---R 5 ---R 2 3 2 ---R (135s G + 2268s )H + (30s G + 1350s G + 16416s G + 38880s )H ---R 0 0 0 0 0 0 ---R + ---R 5 4 3 2 ---R s G + 60s G + 1188s G + 9504s G + 25920s G ---R 0 0 0 0 0 ---R / ---R 29160 ---R , ---R ---R s = ---R 6 ---R 3 2 2 ---R 405s H + (405s G + 18468s G + 174960s )H ---R 0 0 0 0 ---R + ---R 4 3 2 6 ---R (45s G + 3510s G + 88776s G + 777600s G + 1166400s )H + s G ---R 0 0 0 0 0 0 ---R + ---R 5 4 3 2 ---R 90s G + 2628s G + 27864s G + 90720s G ---R 0 0 0 0 ---R / ---R 524880 ---R , ---R ---R s = ---R 7 ---R 3 ---R (2835s G + 91854s )H ---R 0 0 ---R + ---R 3 2 2 ---R (945s G + 81648s G + 2082996s G + 14171760s )H ---R 0 0 0 0 ---R + ---R 5 4 3 2 ---R (63s G + 7560s G + 317520s G + 5554008s G + 34058880s G)H ---R 0 0 0 0 0 ---R + ---R 7 6 5 4 3 2 ---R s G + 126s G + 4788s G + 25272s G - 1744416s G - 26827200s G ---R 0 0 0 0 0 0 ---R + ---R - 97977600s G ---R 0 ---R / ---R 11022480 ---R ] ---R ] ---R Type: List List Equation Fraction Polynomial Integer ---E 16 -)spool - -Dx: LODO(EXPR INT, f +-> D(f, x)) -Dx := D() -Dop:= Dx^3 + G/x^2*Dx + H/x^3 - 1 -n == 3 -phi == reduce(+,[subscript(s,[i])*exp(x)/x^i for i in 0..n]) -phi1 == Dop(phi) / exp x -phi2 == phi1 *x**(n+3) -phi3 == retract(phi2)@(POLY INT) -pans == phi3 ::UP(x,POLY INT) -pans1 == [coefficient(pans, (n+3-i) :: NNI) for i in 2..n+1] -leq == solve(pans1,[subscript(s,[i]) for i in 1..n]) -leq -n==4 -leq -n==7 -leq -)spool -)lisp (bye) -@ -<>= -==================================================================== -LinearOrdinaryDifferentialOperator examples -==================================================================== - -LinearOrdinaryDifferentialOperator(A, diff) is the domain of linear -ordinary differential operators with coefficients in a ring A with a -given derivation. - -==================================================================== -Differential Operators with Series Coefficients -==================================================================== - -Problem: - Find the first few coefficients of exp(x)/x^i of Dop phi where - - Dop := D^3 + G/x^2 * D + H/x^3 - 1 - phi := sum(s[i]*exp(x)/x^i, i = 0..) - -Solution: - -Define the differential. - - Dx: LODO(EXPR INT, f +-> D(f, x)) - Type: Void - - Dx := D() - D - Type: LinearOrdinaryDifferentialOperator(Expression Integer, - theMap LAMBDA-CLOSURE(NIL,NIL,NIL,G1404 envArg, - SPADCALL(G1404,QUOTE x, - ELT(*1;anonymousFunction;0;frame0;internal;MV,0)))) - -Now define the differential operator Dop. - - Dop:= Dx^3 + G/x^2*Dx + H/x^3 - 1 - 3 - 3 G - x + H - D + -- D + -------- - 2 3 - x x - Type: LinearOrdinaryDifferentialOperator(Expression Integer, - theMap LAMBDA-CLOSURE(NIL,NIL,NIL,G1404 envArg, - SPADCALL(G1404,QUOTE x, - ELT(*1;anonymousFunction;0;frame0;internal;MV,0)))) - - n == 3 - Type: Void - - phi == reduce(+,[subscript(s,[i])*exp(x)/x^i for i in 0..n]) - Type: Void - - phi1 == Dop(phi) / exp x - Type: Void - - phi2 == phi1 *x**(n+3) - Type: Void - - phi3 == retract(phi2)@(POLY INT) - Type: Void - - pans == phi3 ::UP(x,POLY INT) - Type: Void - - pans1 == [coefficient(pans, (n+3-i) :: NNI) for i in 2..n+1] - Type: Void - - leq == solve(pans1,[subscript(s,[i]) for i in 1..n]) - Type: Void - -Evaluate this for several values of n. - - leq - 2 3 2 - s G 3s H + s G + 6s G (9s G + 54s )H + s G + 18s G + 72s G - 0 0 0 0 0 0 0 0 0 - [[s = ---,s = ------------------,s = --------------------------------------]] - 1 3 2 18 3 162 - Type: List List Equation Fraction Polynomial Integer - - n==4 - Type: Void - - leq - [ - 2 - s G 3s H + s G + 6s G - 0 0 0 0 - [s = ---, s = ------------------, - 1 3 2 18 - 3 2 - (9s G + 54s )H + s G + 18s G + 72s G - 0 0 0 0 0 - s = --------------------------------------, - 3 162 - - s = - 4 - 2 2 4 3 2 - 27s H + (18s G + 378s G + 1296s )H + s G + 36s G + 396s G - 0 0 0 0 0 0 0 - + - 1296s G - 0 - / - 1944 - ] - ] - Type: List List Equation Fraction Polynomial Integer - - n==7 - Type: Void - - leq - [ - 2 - s G 3s H + s G + 6s G - 0 0 0 0 - [s = ---, s = ------------------, - 1 3 2 18 - 3 2 - (9s G + 54s )H + s G + 18s G + 72s G - 0 0 0 0 0 - s = --------------------------------------, - 3 162 - - s = - 4 - 2 2 4 3 2 - 27s H + (18s G + 378s G + 1296s )H + s G + 36s G + 396s G - 0 0 0 0 0 0 0 - + - 1296s G - 0 - / - 1944 - , - - s = - 5 - 2 3 2 - (135s G + 2268s )H + (30s G + 1350s G + 16416s G + 38880s )H - 0 0 0 0 0 0 - + - 5 4 3 2 - s G + 60s G + 1188s G + 9504s G + 25920s G - 0 0 0 0 0 - / - 29160 - , - - s = - 6 - 3 2 2 - 405s H + (405s G + 18468s G + 174960s )H - 0 0 0 0 - + - 4 3 2 6 - (45s G + 3510s G + 88776s G + 777600s G + 1166400s )H + s G - 0 0 0 0 0 0 - + - 5 4 3 2 - 90s G + 2628s G + 27864s G + 90720s G - 0 0 0 0 - / - 524880 - , - - s = - 7 - 3 - (2835s G + 91854s )H - 0 0 - + - 3 2 2 - (945s G + 81648s G + 2082996s G + 14171760s )H - 0 0 0 0 - + - 5 4 3 2 - (63s G + 7560s G + 317520s G + 5554008s G + 34058880s G)H - 0 0 0 0 0 - + - 7 6 5 4 3 2 - s G + 126s G + 4788s G + 25272s G - 1744416s G - 26827200s G - 0 0 0 0 0 0 - + - - 97977600s G - 0 - / - 11022480 - ] - ] - Type: List List Equation Fraction Polynomial Integer - -See Also: -o )show LinearOrdinaryDifferentialOperator -o $AXIOM/doc/src/algebra/lodo.spad.dvi - -@ -<>= -)abbrev domain LODO LinearOrdinaryDifferentialOperator -++ Author: Manuel Bronstein -++ Date Created: 9 December 1993 -++ Date Last Updated: 15 April 1994 -++ Keywords: differential operator -++ Description: -++ \spad{LinearOrdinaryDifferentialOperator} defines a ring of -++ differential operators with coefficients in a ring A with a given -++ derivation. -++ Multiplication of operators corresponds to functional composition: -++ \spad{(L1 * L2).(f) = L1 L2 f} -LinearOrdinaryDifferentialOperator(A:Ring, diff: A -> A): - LinearOrdinaryDifferentialOperatorCategory A - == SparseUnivariateSkewPolynomial(A, 1, diff) add - Rep := SparseUnivariateSkewPolynomial(A, 1, diff) - - outputD := "D"@String :: Symbol :: OutputForm - - coerce(l:%):OutputForm == outputForm(l, outputD) - elt(p:%, a:A):A == apply(p, 0, a) - - if A has Field then - import LinearOrdinaryDifferentialOperatorsOps(A, %) - - symmetricProduct(a, b) == symmetricProduct(a, b, diff) - symmetricPower(a, n) == symmetricPower(a, n, diff) - directSum(a, b) == directSum(a, b, diff) - -@ -\section{domain LODO1 LinearOrdinaryDifferentialOperator1} -<>= --- lodo.spad.pamphlet LinearOrdinaryDifferentialOperator1.input -)spool LinearOrdinaryDifferentialOperator1.output -)set message test on -)set message auto off -)clear all ---S 1 of 20 -RFZ := Fraction UnivariatePolynomial('x, Integer) ---R ---R ---R (1) Fraction UnivariatePolynomial(x,Integer) ---R Type: Domain ---E 1 - ---S 2 of 20 -x : RFZ := 'x ---R ---R ---R (2) x ---R Type: Fraction UnivariatePolynomial(x,Integer) ---E 2 - ---S 3 of 20 -Dx : LODO1 RFZ := D() ---R ---R ---R (3) D ---RType: LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer) ---E 3 - ---S 4 of 20 -b : LODO1 RFZ := 3*x**2*Dx**2 + 2*Dx + 1/x ---R ---R ---R 2 2 1 ---R (4) 3x D + 2D + - ---R x ---RType: LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer) ---E 4 - ---S 5 of 20 -a : LODO1 RFZ := b*(5*x*Dx + 7) ---R ---R ---R 3 3 2 2 7 ---R (5) 15x D + (51x + 10x)D + 29D + - ---R x ---RType: LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer) ---E 5 - ---S 6 of 20 -p := x**2 + 1/x**2 ---R ---R ---R 4 ---R x + 1 ---R (6) ------ ---R 2 ---R x ---R Type: Fraction UnivariatePolynomial(x,Integer) ---E 6 - ---S 7 of 20 -(a*b - b*a) p ---R ---R ---R 4 ---R - 75x + 540x - 75 ---R (7) ------------------ ---R 4 ---R x ---R Type: Fraction UnivariatePolynomial(x,Integer) ---E 7 - ---S 8 of 20 -ld := leftDivide(a,b) ---R ---R ---R (8) [quotient= 5x D + 7,remainder= 0] ---RType: Record(quotient: LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer),remainder: LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer)) ---E 8 - ---S 9 of 20 -a = b * ld.quotient + ld.remainder ---R ---R ---R 3 3 2 2 7 3 3 2 2 7 ---R (9) 15x D + (51x + 10x)D + 29D + -= 15x D + (51x + 10x)D + 29D + - ---R x x ---RType: Equation LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer) ---E 9 - ---S 10 of 20 -rd := rightDivide(a,b) ---R ---R ---R 5 ---R (10) [quotient= 5x D + 7,remainder= 10D + -] ---R x ---RType: Record(quotient: LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer),remainder: LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer)) ---E 10 - ---S 11 of 20 -a = rd.quotient * b + rd.remainder ---R ---R ---R 3 3 2 2 7 3 3 2 2 7 ---R (11) 15x D + (51x + 10x)D + 29D + -= 15x D + (51x + 10x)D + 29D + - ---R x x ---RType: Equation LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer) ---E 11 - ---S 12 of 20 -rightQuotient(a,b) ---R ---R ---R (12) 5x D + 7 ---RType: LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer) ---E 12 - ---S 13 of 20 -rightRemainder(a,b) ---R ---R ---R 5 ---R (13) 10D + - ---R x ---RType: LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer) ---E 13 - ---S 14 of 20 -leftExactQuotient(a,b) ---R ---R ---R (14) 5x D + 7 ---RType: Union(LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer),...) ---E 14 - ---S 15 of 20 -e := leftGcd(a,b) ---R ---R ---R 2 2 1 ---R (15) 3x D + 2D + - ---R x ---RType: LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer) ---E 15 - ---S 16 of 20 -leftRemainder(a, e) ---R ---R ---R (16) 0 ---RType: LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer) ---E 16 - ---S 17 of 20 -rightRemainder(a, e) ---R ---R ---R 5 ---R (17) 10D + - ---R x ---RType: LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer) ---E 17 - ---S 18 of 20 -f := rightLcm(a,b) ---R ---R ---R 3 3 2 2 7 ---R (18) 15x D + (51x + 10x)D + 29D + - ---R x ---RType: LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer) ---E 18 - ---S 19 of 20 -rightRemainder(f, b) ---R ---R ---R 5 ---R (19) 10D + - ---R x ---RType: LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer) ---E 19 - ---S 20 of 20 -leftRemainder(f, b) ---R ---R ---R (20) 0 ---RType: LinearOrdinaryDifferentialOperator1 Fraction UnivariatePolynomial(x,Integer) ---E 20 -)spool -)lisp (bye) -@ - -<>= -==================================================================== -LinearOrdinaryDifferentialOperator1 example -==================================================================== - -LinearOrdinaryDifferentialOperator1(A) is the domain of linear -ordinary differential operators with coefficients in the differential -ring A. - -==================================================================== -Differential Operators with Rational Function Coefficients -==================================================================== - -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. - -We begin by defining RFZ to be the rational functions in x with -integer coefficients and Dx to be the differential operator for d/dx. - - RFZ := Fraction UnivariatePolynomial('x, Integer) - Fraction UnivariatePolynomial(x,Integer) - Type: Domain - - x : RFZ := 'x - x - Type: Fraction UnivariatePolynomial(x,Integer) - - Dx : LODO1 RFZ := D() - D - Type: LinearOrdinaryDifferentialOperator1 - Fraction UnivariatePolynomial(x,Integer) - -Operators are created using the usual arithmetic operations. - - b : LODO1 RFZ := 3*x**2*Dx**2 + 2*Dx + 1/x - 2 2 1 - 3x D + 2D + - - x - Type: LinearOrdinaryDifferentialOperator1 - Fraction UnivariatePolynomial(x,Integer) - - a : LODO1 RFZ := b*(5*x*Dx + 7) - 3 3 2 2 7 - 15x D + (51x + 10x)D + 29D + - - x - Type: LinearOrdinaryDifferentialOperator1 - Fraction UnivariatePolynomial(x,Integer) - -Operator multiplication corresponds to functional composition. - - p := x**2 + 1/x**2 - 4 - x + 1 - ------ - 2 - x - Type: Fraction UnivariatePolynomial(x,Integer) - -Since operator coefficients depend on x, the multiplication is not -commutative. - - (a*b - b*a) p - 4 - - 75x + 540x - 75 - ------------------ - 4 - x - Type: Fraction UnivariatePolynomial(x,Integer) - -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 leftDivide and rightDivide are quotient-remainder pairs -satisfying: - - leftDivide(a,b) = [q, r] such that a = b*q + r - rightDivide(a,b) = [q, r] such that a = q*b + r - -In both cases, the degree of the remainder, r, is less than the degree -of b. - - ld := leftDivide(a,b) - [quotient= 5x D + 7,remainder= 0] - Type: Record(quotient: LinearOrdinaryDifferentialOperator1 - Fraction UnivariatePolynomial(x,Integer), - remainder: LinearOrdinaryDifferentialOperator1 - Fraction UnivariatePolynomial(x,Integer)) - - a = b * ld.quotient + ld.remainder - 3 3 2 2 7 3 3 2 2 7 - 15x D + (51x + 10x)D + 29D + -= 15x D + (51x + 10x)D + 29D + - - Type: Equation LinearOrdinaryDifferentialOperator1 - Fraction UnivariatePolynomial(x,Integer) - -The operations of left and right division are so-called because the -quotient is obtained by dividing a on that side by b. - - rd := rightDivide(a,b) - [quotient= 5x D + 7,remainder= 10D + -] - Type: Record(quotient: LinearOrdinaryDifferentialOperator1 - Fraction UnivariatePolynomial(x,Integer), - remainder: LinearOrdinaryDifferentialOperator1 - Fraction UnivariatePolynomial(x,Integer)) - - a = rd.quotient * b + rd.remainder - 3 3 2 2 7 3 3 2 2 7 - 15x D + (51x + 10x)D + 29D + -= 15x D + (51x + 10x)D + 29D + - - Type: Equation LinearOrdinaryDifferentialOperator1 - Fraction UnivariatePolynomial(x,Integer) - -Operations rightQuotient and rightRemainder are available if only one -of the quotient or remainder are of interest to you. This is the -quotient from right division. - - rightQuotient(a,b) - 5x D + 7 - Type: LinearOrdinaryDifferentialOperator1 - Fraction UnivariatePolynomial(x,Integer) - -This is the remainder from right division. The corresponding "left" -functions, leftQuotient and leftRemainder are also available. - - rightRemainder(a,b) - 5 - 10D + - - x - Type: LinearOrdinaryDifferentialOperator1 - Fraction UnivariatePolynomial(x,Integer) - -For exact division, operations leftExactQuotient and rightExactQuotient -are supplied. These return the quotient but only if the remainder is zero. -The call rightExactQuotient(a,b) would yield an error. - - leftExactQuotient(a,b) - 5x D + 7 - Type: Union(LinearOrdinaryDifferentialOperator1 - Fraction UnivariatePolynomial(x,Integer),...) - -The division operations allow the computation of left and right greatest -common divisors, leftGcd and rightGcd via remainder sequences, and -consequently the computation of left and right least common multiples, -rightLcm and leftLcm. - - e := leftGcd(a,b) - 2 2 1 - 3x D + 2D + - - x - Type: LinearOrdinaryDifferentialOperator1 - Fraction UnivariatePolynomial(x,Integer) - -Note that a greatest common divisor doesn't necessarily divide a and b -on both sides. Here the left greatest common divisor does not divide -a on the right. - - leftRemainder(a, e) - 0 - Type: LinearOrdinaryDifferentialOperator1 - Fraction UnivariatePolynomial(x,Integer) - - rightRemainder(a, e) - 5 - 10D + - - x - Type: LinearOrdinaryDifferentialOperator1 - Fraction UnivariatePolynomial(x,Integer) - -Similarly, a least common multiple is not necessarily divisible from -both sides. - - f := rightLcm(a,b) - 3 3 2 2 7 - 15x D + (51x + 10x)D + 29D + - - Type: LinearOrdinaryDifferentialOperator1 - Fraction UnivariatePolynomial(x,Integer) - - rightRemainder(f, b) - 5 - 10D + - - x - Type: LinearOrdinaryDifferentialOperator1 - Fraction UnivariatePolynomial(x,Integer) - - leftRemainder(f, b) - 0 - Type: LinearOrdinaryDifferentialOperator1 - Fraction UnivariatePolynomial(x,Integer) - -See Also: -o )show LinearOrdinaryDifferentialOperator1 -o $AXIOM/doc/src/algebra/lodo.spad.dvi - -@ -<>= -)abbrev domain LODO1 LinearOrdinaryDifferentialOperator1 -++ Author: Manuel Bronstein -++ Date Created: 9 December 1993 -++ Date Last Updated: 31 January 1994 -++ Keywords: differential operator -++ Description: -++ \spad{LinearOrdinaryDifferentialOperator1} defines a ring of -++ differential operators with coefficients in a differential ring A. -++ Multiplication of operators corresponds to functional composition: -++ \spad{(L1 * L2).(f) = L1 L2 f} -LinearOrdinaryDifferentialOperator1(A:DifferentialRing) == - LinearOrdinaryDifferentialOperator(A, differentiate$A) - -@ -\section{domain LODO2 LinearOrdinaryDifferentialOperator2} -<>= --- lodo.spad.pamphlet LinearOrdinaryDifferentialOperator2.input -)spool LinearOrdinaryDifferentialOperator2.output -)set message test on -)set message auto off -)clear all ---S 1 of 26 -Q := Fraction Integer ---R ---R ---R (1) Fraction Integer ---R Type: Domain ---E 1 - ---S 2 of 26 -PQ := UnivariatePolynomial('x, Q) ---R ---R ---R (2) UnivariatePolynomial(x,Fraction Integer) ---R Type: Domain ---E 2 - ---S 3 of 26 -x: PQ := 'x ---R ---R ---R (3) x ---R Type: UnivariatePolynomial(x,Fraction Integer) ---E 3 - ---S 4 of 26 -Dx: LODO2(Q, PQ) := D() ---R ---R ---R (4) D ---RType: LinearOrdinaryDifferentialOperator2(Fraction Integer,UnivariatePolynomial(x,Fraction Integer)) ---E 4 - ---S 5 of 26 -a := Dx + 1 ---R ---R ---R (5) D + 1 ---RType: LinearOrdinaryDifferentialOperator2(Fraction Integer,UnivariatePolynomial(x,Fraction Integer)) ---E 5 - ---S 6 of 26 -b := a + 1/2*Dx**2 - 1/2 ---R ---R ---R 1 2 1 ---R (6) - D + D + - ---R 2 2 ---RType: LinearOrdinaryDifferentialOperator2(Fraction Integer,UnivariatePolynomial(x,Fraction Integer)) ---E 6 - ---S 7 of 26 -p := 4*x**2 + 2/3 ---R ---R ---R 2 2 ---R (7) 4x + - ---R 3 ---R Type: UnivariatePolynomial(x,Fraction Integer) ---E 7 - ---S 8 of 26 -a p ---R ---R ---R 2 2 ---R (8) 4x + 8x + - ---R 3 ---R Type: UnivariatePolynomial(x,Fraction Integer) ---E 8 - ---S 9 of 26 -(a * b) p = a b p ---R ---R ---R 2 37 2 37 ---R (9) 2x + 12x + --= 2x + 12x + -- ---R 3 3 ---R Type: Equation UnivariatePolynomial(x,Fraction Integer) ---E 9 - ---S 10 of 26 -c := (1/9)*b*(a + b)^2 ---R ---R ---R 1 6 5 5 13 4 19 3 79 2 7 1 ---R (10) -- D + -- D + -- D + -- D + -- D + -- D + - ---R 72 36 24 18 72 12 8 ---RType: LinearOrdinaryDifferentialOperator2(Fraction Integer,UnivariatePolynomial(x,Fraction Integer)) ---E 10 - ---S 11 of 26 -(a**2 - 3/4*b + c) (p + 1) ---R ---R ---R 2 44 541 ---R (11) 3x + -- x + --- ---R 3 36 ---R Type: UnivariatePolynomial(x,Fraction Integer) ---E 11 -)clear all ---S 12 of 26 -PZ := UnivariatePolynomial(x,Integer) ---R ---R ---R (1) UnivariatePolynomial(x,Integer) ---R Type: Domain ---E 12 - ---S 13 of 26 -x:PZ := 'x ---R ---R ---R (2) x ---R Type: UnivariatePolynomial(x,Integer) ---E 13 - ---S 14 of 26 -Mat := SquareMatrix(3,PZ) ---R ---R ---R (3) SquareMatrix(3,UnivariatePolynomial(x,Integer)) ---R Type: Domain ---E 14 - ---S 15 of 26 -Vect := DPMM(3, PZ, Mat, PZ) ---R ---R ---R (4) ---R DirectProductMatrixModule(3,UnivariatePolynomial(x,Integer),SquareMatrix(3,Un ---R ivariatePolynomial(x,Integer)),UnivariatePolynomial(x,Integer)) ---R Type: Domain ---E 15 - ---S 16 of 26 -Modo := LODO2(Mat, Vect) ---R ---R ---R (5) ---R LinearOrdinaryDifferentialOperator2(SquareMatrix(3,UnivariatePolynomial(x,Int ---R eger)),DirectProductMatrixModule(3,UnivariatePolynomial(x,Integer),SquareMatr ---R ix(3,UnivariatePolynomial(x,Integer)),UnivariatePolynomial(x,Integer))) ---R Type: Domain ---E 16 - ---S 17 of 26 -m:Mat := matrix [ [x^2,1,0],[1,x^4,0],[0,0,4*x^2] ] ---R ---R ---R + 2 + ---R |x 1 0 | ---R | | ---R (6) | 4 | ---R |1 x 0 | ---R | | ---R | 2| ---R +0 0 4x + ---R Type: SquareMatrix(3,UnivariatePolynomial(x,Integer)) ---E 17 - ---S 18 of 26 -p:Vect := directProduct [3*x^2+1,2*x,7*x^3+2*x] ---R ---R ---R 2 3 ---R (7) [3x + 1,2x,7x + 2x] ---RType: DirectProductMatrixModule(3,UnivariatePolynomial(x,Integer),SquareMatrix(3,UnivariatePolynomial(x,Integer)),UnivariatePolynomial(x,Integer)) ---E 18 - ---S 19 of 26 -q: Vect := m * p ---R ---R ---R 4 2 5 2 5 3 ---R (8) [3x + x + 2x,2x + 3x + 1,28x + 8x ] ---RType: DirectProductMatrixModule(3,UnivariatePolynomial(x,Integer),SquareMatrix(3,UnivariatePolynomial(x,Integer)),UnivariatePolynomial(x,Integer)) ---E 19 - ---S 20 of 26 -Dx : Modo := D() ---R ---R ---R (9) D ---RType: LinearOrdinaryDifferentialOperator2(SquareMatrix(3,UnivariatePolynomial(x,Integer)),DirectProductMatrixModule(3,UnivariatePolynomial(x,Integer),SquareMatrix(3,UnivariatePolynomial(x,Integer)),UnivariatePolynomial(x,Integer))) ---E 20 - ---S 21 of 26 -a : Modo := Dx + m ---R ---R ---R + 2 + ---R |x 1 0 | ---R | | ---R (10) D + | 4 | ---R |1 x 0 | ---R | | ---R | 2| ---R +0 0 4x + ---RType: LinearOrdinaryDifferentialOperator2(SquareMatrix(3,UnivariatePolynomial(x,Integer)),DirectProductMatrixModule(3,UnivariatePolynomial(x,Integer),SquareMatrix(3,UnivariatePolynomial(x,Integer)),UnivariatePolynomial(x,Integer))) ---E 21 - ---S 22 of 26 -b : Modo := m*Dx + 1 ---R ---R ---R + 2 + ---R |x 1 0 | +1 0 0+ ---R | | | | ---R (11) | 4 |D + |0 1 0| ---R |1 x 0 | | | ---R | | +0 0 1+ ---R | 2| ---R +0 0 4x + ---RType: LinearOrdinaryDifferentialOperator2(SquareMatrix(3,UnivariatePolynomial(x,Integer)),DirectProductMatrixModule(3,UnivariatePolynomial(x,Integer),SquareMatrix(3,UnivariatePolynomial(x,Integer)),UnivariatePolynomial(x,Integer))) ---E 22 - ---S 23 of 26 -c := a*b ---R ---R ---R (12) ---R + 2 + + 4 4 2 + + 2 + ---R |x 1 0 | |x + 2x + 2 x + x 0 | |x 1 0 | ---R | | 2 | | | | ---R | 4 |D + | 4 2 8 3 |D + | 4 | ---R |1 x 0 | | x + x x + 4x + 2 0 | |1 x 0 | ---R | | | | | | ---R | 2| | 4 | | 2| ---R +0 0 4x + + 0 0 16x + 8x + 1+ +0 0 4x + ---RType: LinearOrdinaryDifferentialOperator2(SquareMatrix(3,UnivariatePolynomial(x,Integer)),DirectProductMatrixModule(3,UnivariatePolynomial(x,Integer),SquareMatrix(3,UnivariatePolynomial(x,Integer)),UnivariatePolynomial(x,Integer))) ---E 23 - ---S 24 of 26 -a p ---R ---R ---R 4 2 5 2 5 3 2 ---R (13) [3x + x + 8x,2x + 3x + 3,28x + 8x + 21x + 2] ---RType: DirectProductMatrixModule(3,UnivariatePolynomial(x,Integer),SquareMatrix(3,UnivariatePolynomial(x,Integer)),UnivariatePolynomial(x,Integer)) ---E 24 - ---S 25 of 26 -b p ---R ---R ---R 3 2 4 4 3 2 ---R (14) [6x + 3x + 3,2x + 8x,84x + 7x + 8x + 2x] ---RType: DirectProductMatrixModule(3,UnivariatePolynomial(x,Integer),SquareMatrix(3,UnivariatePolynomial(x,Integer)),UnivariatePolynomial(x,Integer)) ---E 25 - ---S 26 of 26 -(a + b + c) (p + q) ---R ---R ---R (15) ---R 8 7 6 5 4 3 2 ---R [10x + 12x + 16x + 30x + 85x + 94x + 40x + 40x + 17, ---R 12 9 8 7 6 5 4 3 2 ---R 10x + 10x + 12x + 92x + 6x + 32x + 72x + 28x + 49x + 32x + 19, ---R 8 7 6 5 4 3 2 ---R 2240x + 224x + 1280x + 3508x + 492x + 751x + 98x + 18x + 4] ---RType: DirectProductMatrixModule(3,UnivariatePolynomial(x,Integer),SquareMatrix(3,UnivariatePolynomial(x,Integer)),UnivariatePolynomial(x,Integer)) ---E 26 -)spool -)lisp (bye) -@ -<>= -==================================================================== -LinearOrdinaryDifferentialOperator2 -==================================================================== - -LinearOrdinaryDifferentialOperator2(A, M) is the domain of linear -ordinary differential operators with coefficients in the differential -ring A and operating on M, an A-module. This includes the cases of -operators which are polynomials in 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. - -==================================================================== -Differential Operators with Constant Coefficients -==================================================================== - -This example shows differential operators with rational number -coefficients operating on univariate polynomials. - -We begin by making type assignments so we can conveniently refer -to univariate polynomials in x over the rationals. - - Q := Fraction Integer - Fraction Integer - Type: Domain - - PQ := UnivariatePolynomial('x, Q) - UnivariatePolynomial(x,Fraction Integer) - Type: Domain - - x: PQ := 'x - x - Type: UnivariatePolynomial(x,Fraction Integer) - -Now we assign Dx to be the differential operator D corresponding to d/dx. - - Dx: LODO2(Q, PQ) := D() - D - Type: LinearOrdinaryDifferentialOperator2(Fraction Integer, - UnivariatePolynomial(x,Fraction Integer)) - -New operators are created as polynomials in D(). - - a := Dx + 1 - D + 1 - Type: LinearOrdinaryDifferentialOperator2(Fraction Integer, - UnivariatePolynomial(x,Fraction Integer)) - - b := a + 1/2*Dx**2 - 1/2 - 1 2 1 - - D + D + - - 2 2 - Type: LinearOrdinaryDifferentialOperator2(Fraction Integer, - UnivariatePolynomial(x,Fraction Integer)) - -To apply the operator a to the value p the usual function call syntax -is used. - - p := 4*x**2 + 2/3 - 2 2 - 4x + - - 3 - Type: UnivariatePolynomial(x,Fraction Integer) - - a p - 2 2 - 4x + 8x + - - 3 - Type: UnivariatePolynomial(x,Fraction Integer) - -Operator multiplication is defined by the identity (a*b)p = a(b(p)) - - (a * b) p = a b p - 2 37 2 37 - 2x + 12x + --= 2x + 12x + -- - 3 3 - Type: Equation UnivariatePolynomial(x,Fraction Integer) - -Exponentiation follows from multiplication. - - c := (1/9)*b*(a + b)^2 - 1 6 5 5 13 4 19 3 79 2 7 1 - -- D + -- D + -- D + -- D + -- D + -- D + - - 72 36 24 18 72 12 8 - Type: LinearOrdinaryDifferentialOperator2(Fraction Integer, - UnivariatePolynomial(x,Fraction Integer)) - -Finally, note that operator expressions may be applied directly. - - (a**2 - 3/4*b + c) (p + 1) - 2 44 541 - 3x + -- x + --- - 3 36 - Type: UnivariatePolynomial(x,Fraction Integer) - -==================================================================== -Differential Operators with Matrix Coefficients Operating on Vectors} -==================================================================== - -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. - -In this section, the operators have three by three matrix coefficients -with polynomial entries. - - PZ := UnivariatePolynomial(x,Integer) - UnivariatePolynomial(x,Integer) - Type: Domain - - x:PZ := 'x - x - Type: UnivariatePolynomial(x,Integer) - - Mat := SquareMatrix(3,PZ) - SquareMatrix(3,UnivariatePolynomial(x,Integer)) - Type: Domain - -The operators act on the vectors considered as a Mat-module. - - Vect := DPMM(3, PZ, Mat, PZ) - DirectProductMatrixModule(3,UnivariatePolynomial(x,Integer),SquareMatrix(3,Un - ivariatePolynomial(x,Integer)),UnivariatePolynomial(x,Integer)) - Type: Domain - - Modo := LODO2(Mat, Vect) - LinearOrdinaryDifferentialOperator2(SquareMatrix(3,UnivariatePolynomial(x,Int - eger)),DirectProductMatrixModule(3,UnivariatePolynomial(x,Integer),SquareMatr - ix(3,UnivariatePolynomial(x,Integer)),UnivariatePolynomial(x,Integer))) - Type: Domain - -The matrix m is used as a coefficient and the vectors p and q are -operated upon. - - m:Mat := matrix [ [x^2,1,0],[1,x^4,0],[0,0,4*x^2] ] - + 2 + - |x 1 0 | - | | - | 4 | - |1 x 0 | - | | - | 2| - +0 0 4x + - Type: SquareMatrix(3,UnivariatePolynomial(x,Integer)) - - p:Vect := directProduct [3*x^2+1,2*x,7*x^3+2*x] - 2 3 - [3x + 1,2x,7x + 2x] - Type: DirectProductMatrixModule(3,UnivariatePolynomial(x,Integer), - SquareMatrix(3,UnivariatePolynomial(x,Integer)), - UnivariatePolynomial(x,Integer)) - - q: Vect := m * p - 4 2 5 2 5 3 - [3x + x + 2x,2x + 3x + 1,28x + 8x ] - Type: DirectProductMatrixModule(3,UnivariatePolynomial(x,Integer), - SquareMatrix(3,UnivariatePolynomial(x,Integer)), - UnivariatePolynomial(x,Integer)) - -Now form a few operators. - - Dx : Modo := D() - D - Type: LinearOrdinaryDifferentialOperator2( - SquareMatrix(3,UnivariatePolynomial(x,Integer)), - DirectProductMatrixModule(3,UnivariatePolynomial(x,Integer), - SquareMatrix(3,UnivariatePolynomial(x,Integer)), - UnivariatePolynomial(x,Integer))) - - a : Modo := Dx + m - + 2 + - |x 1 0 | - | | - D + | 4 | - |1 x 0 | - | | - | 2| - +0 0 4x + - Type: LinearOrdinaryDifferentialOperator2( - SquareMatrix(3,UnivariatePolynomial(x,Integer)), - DirectProductMatrixModule(3,UnivariatePolynomial(x,Integer), - SquareMatrix(3,UnivariatePolynomial(x,Integer)), - UnivariatePolynomial(x,Integer))) - - b : Modo := m*Dx + 1 - + 2 + - |x 1 0 | +1 0 0+ - | | | | - | 4 |D + |0 1 0| - |1 x 0 | | | - | | +0 0 1+ - | 2| - +0 0 4x + - Type: LinearOrdinaryDifferentialOperator2( - SquareMatrix(3,UnivariatePolynomial(x,Integer)), - DirectProductMatrixModule(3,UnivariatePolynomial(x,Integer), - SquareMatrix(3,UnivariatePolynomial(x,Integer)), - UnivariatePolynomial(x,Integer))) - - c := a*b - + 2 + + 4 4 2 + + 2 + - |x 1 0 | |x + 2x + 2 x + x 0 | |x 1 0 | - | | 2 | | | | - | 4 |D + | 4 2 8 3 |D + | 4 | - |1 x 0 | | x + x x + 4x + 2 0 | |1 x 0 | - | | | | | | - | 2| | 4 | | 2| - +0 0 4x + + 0 0 16x + 8x + 1+ +0 0 4x + - Type: LinearOrdinaryDifferentialOperator2( - SquareMatrix(3,UnivariatePolynomial(x,Integer)), - DirectProductMatrixModule(3,UnivariatePolynomial(x,Integer), - SquareMatrix(3,UnivariatePolynomial(x,Integer)), - UnivariatePolynomial(x,Integer))) - -These operators can be applied to vector values. - - a p - 4 2 5 2 5 3 2 - [3x + x + 8x,2x + 3x + 3,28x + 8x + 21x + 2] - Type: DirectProductMatrixModule(3,UnivariatePolynomial(x,Integer), - SquareMatrix(3,UnivariatePolynomial(x,Integer)), - UnivariatePolynomial(x,Integer)) - - b p - 3 2 4 4 3 2 - [6x + 3x + 3,2x + 8x,84x + 7x + 8x + 2x] - Type: DirectProductMatrixModule(3,UnivariatePolynomial(x,Integer), - SquareMatrix(3,UnivariatePolynomial(x,Integer)), - UnivariatePolynomial(x,Integer)) - - (a + b + c) (p + q) - 8 7 6 5 4 3 2 - [10x + 12x + 16x + 30x + 85x + 94x + 40x + 40x + 17, - 12 9 8 7 6 5 4 3 2 - 10x + 10x + 12x + 92x + 6x + 32x + 72x + 28x + 49x + 32x + 19, - 8 7 6 5 4 3 2 - 2240x + 224x + 1280x + 3508x + 492x + 751x + 98x + 18x + 4] - Type: DirectProductMatrixModule(3,UnivariatePolynomial(x,Integer), - SquareMatrix(3,UnivariatePolynomial(x,Integer)), - UnivariatePolynomial(x,Integer)) - -See Also: -o )show LinearOrdinaryDifferentialOperator2 -o $AXIOM/doc/src/algebra/lodo.spad.dvi - -@ -<>= -)abbrev domain LODO2 LinearOrdinaryDifferentialOperator2 -++ Author: Stephen M. Watt, Manuel Bronstein -++ Date Created: 1986 -++ Date Last Updated: 1 February 1994 -++ Keywords: differential operator -++ Description: -++ \spad{LinearOrdinaryDifferentialOperator2} defines a ring of -++ differential operators with coefficients in a differential ring A -++ and acting on an A-module M. -++ Multiplication of operators corresponds to functional composition: -++ \spad{(L1 * L2).(f) = L1 L2 f} -LinearOrdinaryDifferentialOperator2(A, M): Exports == Implementation where - A: DifferentialRing - M: LeftModule A with - differentiate: $ -> $ - ++ differentiate(x) returns the derivative of x - - Exports ==> Join(LinearOrdinaryDifferentialOperatorCategory A, Eltable(M, M)) - - Implementation ==> LinearOrdinaryDifferentialOperator(A, differentiate$A) add - elt(p:%, m:M):M == - apply(p, differentiate, m)$ApplyUnivariateSkewPolynomial(A, M, %) - -@ \section{License} <>= --Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd. @@ -1642,9 +150,6 @@ LinearOrdinaryDifferentialOperator2(A, M): Exports == Implementation where <> <> -<> -<> -<> @ \eject \begin{thebibliography}{99} diff --git a/src/algebra/lodof.spad.pamphlet b/src/algebra/lodof.spad.pamphlet index fd72c15..daaedcb 100644 --- a/src/algebra/lodof.spad.pamphlet +++ b/src/algebra/lodof.spad.pamphlet @@ -9,153 +9,6 @@ \eject \tableofcontents \eject -\section{domain SETMN SetOfMIntegersInOneToN} -<>= -)abbrev domain SETMN SetOfMIntegersInOneToN -++ Author: Manuel Bronstein -++ Date Created: 10 January 1994 -++ Date Last Updated: 10 January 1994 -++ Description: -++ \spadtype{SetOfMIntegersInOneToN} implements the subsets of M integers -++ in the interval \spad{[1..n]} -SetOfMIntegersInOneToN(m, n): Exports == Implementation where - PI ==> PositiveInteger - N ==> NonNegativeInteger - U ==> Union(%, "failed") - n,m: PI - - Exports ==> Finite with - incrementKthElement: (%, PI) -> U - ++ incrementKthElement(S,k) increments the k^{th} element of S, - ++ and returns "failed" if the result is not a set of M integers - ++ in \spad{1..n} any more. - replaceKthElement: (%, PI, PI) -> U - ++ replaceKthElement(S,k,p) replaces the k^{th} element of S by p, - ++ and returns "failed" if the result is not a set of M integers - ++ in \spad{1..n} any more. - elements: % -> List PI - ++ elements(S) returns the list of the elements of S in increasing order. - setOfMinN: List PI -> % - ++ setOfMinN([a_1,...,a_m]) returns the set {a_1,...,a_m}. - ++ Error if {a_1,...,a_m} is not a set of M integers in \spad{1..n}. - enumerate: () -> Vector % - ++ enumerate() returns a vector of all the sets of M integers in - ++ \spad{1..n}. - member?: (PI, %) -> Boolean - ++ member?(p, s) returns true is p is in s, false otherwise. - delta: (%, PI, PI) -> N - ++ delta(S,k,p) returns the number of elements of S which are strictly - ++ between p and the k^{th} element of S. - - Implementation ==> add - Rep := Record(bits:Bits, pos:N) - - reallyEnumerate: () -> Vector % - enum: (N, N, PI) -> List Bits - - all:Reference Vector % := ref empty() - sz:Reference N := ref 0 - - s1 = s2 == s1.bits =$Bits s2.bits - coerce(s:%):OutputForm == brace [i::OutputForm for i in elements s] - random() == index((1 + (random()$Integer rem size()))::PI) - reallyEnumerate() == [[b, i] for b in enum(m, n, n) for i in 1..] - member?(p, s) == s.bits.p - - enumerate() == - if empty? all() then all() := reallyEnumerate() - all() - --- enumerates the sets of p integers in 1..q, returns them as sets in 1..n --- must have p <= q - enum(p, q, n) == - zero? p or zero? q => empty() - p = q => - b := new(n, false)$Bits - for i in 1..p repeat b.i := true - [b] - q1 := (q - 1)::N - l := enum((p - 1)::N, q1, n) - if empty? l then l := [new(n, false)$Bits] - for s in l repeat s.q := true - concat_!(enum(p, q1, n), l) - - size() == - if zero? sz() then - sz() := binomial(n, m)$IntegerCombinatoricFunctions(Integer) :: N - sz() - - lookup s == - if empty? all() then all() := reallyEnumerate() - if zero?(s.pos) then s.pos := position(s, all()) :: N - s.pos :: PI - - index p == - p > size() => error "index: argument too large" - if empty? all() then all() := reallyEnumerate() - all().p - - setOfMinN l == - s := new(n, false)$Bits - count:N := 0 - for i in l repeat - count := count + 1 - count > m or zero? i or i > n or s.i => - error "setOfMinN: improper set of integers" - s.i := true - count < m => error "setOfMinN: improper set of integers" - [s, 0] - - elements s == - b := s.bits - l:List PI := empty() - found:N := 0 - i:PI := 1 - while found < m repeat - if b.i then - l := concat(i, l) - found := found + 1 - i := i + 1 - reverse_! l - - incrementKthElement(s, k) == - b := s.bits - found:N := 0 - i:N := 1 - while found < k repeat - if b.i then found := found + 1 - i := i + 1 - i > n or b.i => "failed" - newb := copy b - newb.i := true - newb.((i-1)::N) := false - [newb, 0] - - delta(s, k, p) == - b := s.bits - count:N := found:N := 0 - i:PI := 1 - while found < k repeat - if b.i then - found := found + 1 - if i > p and found < k then count := count + 1 - i := i + 1 - count - - replaceKthElement(s, k, p) == - b := s.bits - found:N := 0 - i:PI := 1 - while found < k repeat - if b.i then found := found + 1 - if found < k then i := i + 1 - b.p and i ^= p => "failed" - newb := copy b - newb.p := true - newb.i := false - [newb, (i = p => s.pos; 0)] - -@ \section{package PREASSOC PrecomputedAssociatedEquations} <>= )abbrev package PREASSOC PrecomputedAssociatedEquations @@ -521,7 +374,6 @@ LinearOrdinaryDifferentialOperatorFactorizer(F, UP): Exports == Impl where -- oderf.spad odealg.spad nlode.spad nlinsol.spad riccati.spad -- kovacic.spad lodof.spad odeef.spad -<> <> <> <> diff --git a/src/algebra/lodop.spad.pamphlet b/src/algebra/lodop.spad.pamphlet index 37d3567..61ce08e 100644 --- a/src/algebra/lodop.spad.pamphlet +++ b/src/algebra/lodop.spad.pamphlet @@ -9,44 +9,6 @@ \eject \tableofcontents \eject -\section{domain OMLO OppositeMonogenicLinearOperator} -<>= -)abbrev domain OMLO OppositeMonogenicLinearOperator -++ Author: Stephen M. Watt -++ Date Created: 1986 -++ Date Last Updated: May 30, 1991 -++ Basic Operations: -++ Related Domains: MonogenicLinearOperator -++ Also See: -++ AMS Classifications: -++ Keywords: opposite ring -++ Examples: -++ References: -++ Description: -++ This constructor creates the \spadtype{MonogenicLinearOperator} domain -++ which is ``opposite'' in the ring sense to P. -++ That is, as sets \spad{P = $} but \spad{a * b} in \spad{$} is equal to -++ \spad{b * a} in P. - -OppositeMonogenicLinearOperator(P, R): OPRcat == OPRdef where - P: MonogenicLinearOperator(R) - R: Ring - - OPRcat == MonogenicLinearOperator(R) with - if P has DifferentialRing then DifferentialRing - op: P -> $ ++ op(p) creates a value in $ equal to p in P. - po: $ -> P ++ po(q) creates a value in P equal to q in $. - - OPRdef == P add - Rep := P - x, y: $ - a: P - op a == a: $ - po x == x: P - x*y == (y:P) *$P (x:P) - coerce(x): OutputForm == prefix(op::OutputForm, [coerce(x:P)$P]) - -@ \section{package NCODIV NonCommutativeOperatorDivision} <>= )abbrev package NCODIV NonCommutativeOperatorDivision @@ -143,109 +105,6 @@ NonCommutativeOperatorDivision(P, F): PDcat == PDdef where b0*u @ -\section{domain ODR OrdinaryDifferentialRing} -<>= -)abbrev domain ODR OrdinaryDifferentialRing -++ Author: Stephen M. Watt -++ Date Created: 1986 -++ Date Last Updated: June 3, 1991 -++ Basic Operations: -++ Related Domains: -++ Also See: -++ AMS Classifications: -++ Keywords: differential ring -++ Examples: -++ References: -++ Description: -++ This constructor produces an ordinary differential ring from -++ a partial differential ring by specifying a variable. - -OrdinaryDifferentialRing(Kernels,R,var): DRcategory == DRcapsule where - Kernels:SetCategory - R: PartialDifferentialRing(Kernels) - var : Kernels - DRcategory == Join(BiModule($,$), DifferentialRing) with - if R has Field then Field - coerce: R -> $ - ++ coerce(r) views r as a value in the ordinary differential ring. - coerce: $ -> R - ++ coerce(p) views p as a valie in the partial differential ring. - DRcapsule == R add - n: Integer - Rep := R - coerce(u:R):$ == u::Rep::$ - coerce(p:$):R == p::Rep::R - differentiate p == differentiate(p, var) - - if R has Field then - p / q == ((p::R) /$R (q::R))::$ - p ** n == ((p::R) **$R n)::$ - inv(p) == (inv(p::R)$R)::$ - -@ -\section{domain DPMO DirectProductModule} -<>= -)abbrev domain DPMO DirectProductModule -++ Author: Stephen M. Watt -++ Date Created: 1986 -++ Date Last Updated: June 4, 1991 -++ Basic Operations: -++ Related Domains: -++ Also See: -++ AMS Classifications: -++ Keywords: equation -++ Examples: -++ References: -++ Description: -++ This constructor provides a direct product of R-modules -++ with an R-module view. - -DirectProductModule(n, R, S): DPcategory == DPcapsule where - n: NonNegativeInteger - R: Ring - S: LeftModule(R) - - DPcategory == Join(DirectProductCategory(n,S), LeftModule(R)) - -- with if S has Algebra(R) then Algebra(R) - -- - - DPcapsule == DirectProduct(n,S) add - Rep := Vector(S) - r:R * x:$ == [r * x.i for i in 1..n] - -@ -\section{domain DPMM DirectProductMatrixModule} -<>= -)abbrev domain DPMM DirectProductMatrixModule -++ Author: Stephen M. Watt -++ Date Created: 1986 -++ Date Last Updated: June 4, 1991 -++ Basic Operations: -++ Related Domains: -++ Also See: -++ AMS Classifications: -++ Keywords: equation -++ Examples: -++ References: -++ Description: -++ This constructor provides a direct product type with a -++ left matrix-module view. - -DirectProductMatrixModule(n, R, M, S): DPcategory == DPcapsule where - n: PositiveInteger - R: Ring - RowCol ==> DirectProduct(n,R) - M: SquareMatrixCategory(n,R,RowCol,RowCol) - S: LeftModule(R) - - DPcategory == Join(DirectProductCategory(n,S), LeftModule(R), LeftModule(M)) - - DPcapsule == DirectProduct(n, S) add - Rep := Vector(S) - r:R * x:$ == [r*x.i for i in 1..n] - m:M * x:$ == [ +/[m(i,j)*x.j for j in 1..n] for i in 1..n] - -@ \section{License} <>= --Copyright (c) 1991-2002, The Numerical ALgorithms Group Ltd. @@ -282,11 +141,7 @@ DirectProductMatrixModule(n, R, M, S): DPcategory == DPcapsule where <<*>>= <> -<> <> -<> -<> -<> @ \eject \begin{thebibliography}{99} diff --git a/src/algebra/matrix.spad.pamphlet b/src/algebra/matrix.spad.pamphlet deleted file mode 100644 index 590cd3e..0000000 --- a/src/algebra/matrix.spad.pamphlet +++ /dev/null @@ -1,1586 +0,0 @@ -\documentclass{article} -\usepackage{axiom} -\begin{document} -\title{\$SPAD/src/algebra matrix.spad} -\author{Johannes Grabmeier, Oswald Gschnitzer, Clifton J. Williamson} -\maketitle -\begin{abstract} -\end{abstract} -\eject -\tableofcontents -\eject -\section{domain IMATRIX IndexedMatrix} -<>= -)abbrev domain IMATRIX IndexedMatrix -++ Author: Grabmeier, Gschnitzer, Williamson -++ Date Created: 1987 -++ Date Last Updated: July 1990 -++ Basic Operations: -++ Related Domains: Matrix, RectangularMatrix, SquareMatrix, -++ StorageEfficientMatrixOperations -++ Also See: -++ AMS Classifications: -++ Keywords: matrix, linear algebra -++ Examples: -++ References: -++ Description: -++ An \spad{IndexedMatrix} is a matrix where the minimal row and column -++ indices are parameters of the type. The domains Row and Col -++ are both IndexedVectors. -++ The index of the 'first' row may be obtained by calling the -++ function \spadfun{minRowIndex}. The index of the 'first' column may -++ be obtained by calling the function \spadfun{minColIndex}. The index of -++ the first element of a 'Row' is the same as the index of the -++ first column in a matrix and vice versa. -IndexedMatrix(R,mnRow,mnCol): Exports == Implementation where - R : Ring - mnRow, mnCol : Integer - Row ==> IndexedVector(R,mnCol) - Col ==> IndexedVector(R,mnRow) - MATLIN ==> MatrixLinearAlgebraFunctions(R,Row,Col,$) - - Exports ==> MatrixCategory(R,Row,Col) - - Implementation ==> - InnerIndexedTwoDimensionalArray(R,mnRow,mnCol,Row,Col) add - - swapRows_!(x,i1,i2) == - (i1 < minRowIndex(x)) or (i1 > maxRowIndex(x)) or _ - (i2 < minRowIndex(x)) or (i2 > maxRowIndex(x)) => - error "swapRows!: index out of range" - i1 = i2 => x - minRow := minRowIndex x - xx := x pretend PrimitiveArray(PrimitiveArray(R)) - n1 := i1 - minRow; n2 := i2 - minRow - row1 := qelt(xx,n1) - qsetelt_!(xx,n1,qelt(xx,n2)) - qsetelt_!(xx,n2,row1) - xx pretend $ - - if R has commutative("*") then - - determinant x == determinant(x)$MATLIN - minordet x == minordet(x)$MATLIN - - if R has EuclideanDomain then - - rowEchelon x == rowEchelon(x)$MATLIN - - if R has IntegralDomain then - - rank x == rank(x)$MATLIN - nullity x == nullity(x)$MATLIN - nullSpace x == nullSpace(x)$MATLIN - - if R has Field then - - inverse x == inverse(x)$MATLIN - -@ -\section{domain MATRIX Matrix} -<>= --- matrix.spad.pamphlet Matrix.input -)spool Matrix.output -)set message test on -)set message auto off -)clear all ---S 1 of 38 -m : Matrix(Integer) := new(3,3,0) ---R ---R ---R +0 0 0+ ---R | | ---R (1) |0 0 0| ---R | | ---R +0 0 0+ ---R Type: Matrix Integer ---E 1 - ---S 2 of 38 -setelt(m,2,3,5) ---R ---R ---R (2) 5 ---R Type: PositiveInteger ---E 2 - ---S 3 of 38 -m(1,2) := 10 ---R ---R ---R (3) 10 ---R Type: PositiveInteger ---E 3 - ---S 4 of 38 -m ---R ---R ---R +0 10 0+ ---R | | ---R (4) |0 0 5| ---R | | ---R +0 0 0+ ---R Type: Matrix Integer ---E 4 - ---S 5 of 38 -matrix [ [1,2,3,4],[0,9,8,7] ] ---R ---R ---R +1 2 3 4+ ---R (5) | | ---R +0 9 8 7+ ---R Type: Matrix Integer ---E 5 - ---S 6 of 38 -dm := diagonalMatrix [1,x**2,x**3,x**4,x**5] ---R ---R ---R +1 0 0 0 0 + ---R | | ---R | 2 | ---R |0 x 0 0 0 | ---R | | ---R | 3 | ---R (6) |0 0 x 0 0 | ---R | | ---R | 4 | ---R |0 0 0 x 0 | ---R | | ---R | 5| ---R +0 0 0 0 x + ---R Type: Matrix Polynomial Integer ---E 6 - ---S 7 of 38 -setRow!(dm,5,vector [1,1,1,1,1]) ---R ---R ---R +1 0 0 0 0+ ---R | | ---R | 2 | ---R |0 x 0 0 0| ---R | | ---R (7) | 3 | ---R |0 0 x 0 0| ---R | | ---R | 4 | ---R |0 0 0 x 0| ---R | | ---R +1 1 1 1 1+ ---R Type: Matrix Polynomial Integer ---E 7 - ---S 8 of 38 -setColumn!(dm,2,vector [y,y,y,y,y]) ---R ---R ---R +1 y 0 0 0+ ---R | | ---R |0 y 0 0 0| ---R | | ---R | 3 | ---R (8) |0 y x 0 0| ---R | | ---R | 4 | ---R |0 y 0 x 0| ---R | | ---R +1 y 1 1 1+ ---R Type: Matrix Polynomial Integer ---E 8 - ---S 9 of 38 -cdm := copy(dm) ---R ---R ---R +1 y 0 0 0+ ---R | | ---R |0 y 0 0 0| ---R | | ---R | 3 | ---R (9) |0 y x 0 0| ---R | | ---R | 4 | ---R |0 y 0 x 0| ---R | | ---R +1 y 1 1 1+ ---R Type: Matrix Polynomial Integer ---E 9 - ---S 10 of 38 -setelt(dm,4,1,1-x**7) ---R ---R ---R 7 ---R (10) - x + 1 ---R Type: Polynomial Integer ---E 10 - ---S 11 of 38 -[dm,cdm] ---R ---R ---R + 1 y 0 0 0+ +1 y 0 0 0+ ---R | | | | ---R | 0 y 0 0 0| |0 y 0 0 0| ---R | | | | ---R | 3 | | 3 | ---R (11) [| 0 y x 0 0|,|0 y x 0 0|] ---R | | | | ---R | 7 4 | | 4 | ---R |- x + 1 y 0 x 0| |0 y 0 x 0| ---R | | | | ---R + 1 y 1 1 1+ +1 y 1 1 1+ ---R Type: List Matrix Polynomial Integer ---E 11 - ---S 12 of 38 -subMatrix(dm,2,3,2,4) ---R ---R ---R +y 0 0+ ---R (12) | | ---R | 3 | ---R +y x 0+ ---R Type: Matrix Polynomial Integer ---E 12 - ---S 13 of 38 -d := diagonalMatrix [1.2,-1.3,1.4,-1.5] ---R ---R ---R +1.2 0.0 0.0 0.0 + ---R | | ---R |0.0 - 1.3 0.0 0.0 | ---R (13) | | ---R |0.0 0.0 1.4 0.0 | ---R | | ---R +0.0 0.0 0.0 - 1.5+ ---R Type: Matrix Float ---E 13 - ---S 14 of 38 -e := matrix [ [6.7,9.11],[-31.33,67.19] ] ---R ---R ---R + 6.7 9.11 + ---R (14) | | ---R +- 31.33 67.19+ ---R Type: Matrix Float ---E 14 - ---S 15 of 38 -setsubMatrix!(d,1,2,e) ---R ---R ---R +1.2 6.7 9.11 0.0 + ---R | | ---R |0.0 - 31.33 67.19 0.0 | ---R (15) | | ---R |0.0 0.0 1.4 0.0 | ---R | | ---R +0.0 0.0 0.0 - 1.5+ ---R Type: Matrix Float ---E 15 - ---S 16 of 38 -d ---R ---R ---R +1.2 6.7 9.11 0.0 + ---R | | ---R |0.0 - 31.33 67.19 0.0 | ---R (16) | | ---R |0.0 0.0 1.4 0.0 | ---R | | ---R +0.0 0.0 0.0 - 1.5+ ---R Type: Matrix Float ---E 16 - ---S 17 of 38 -a := matrix [ [1/2,1/3,1/4],[1/5,1/6,1/7] ] ---R ---R ---R +1 1 1+ ---R |- - -| ---R |2 3 4| ---R (17) | | ---R |1 1 1| ---R |- - -| ---R +5 6 7+ ---R Type: Matrix Fraction Integer ---E 17 - ---S 18 of 38 -b := matrix [ [3/5,3/7,3/11],[3/13,3/17,3/19] ] ---R ---R ---R +3 3 3+ ---R |- - --| ---R |5 7 11| ---R (18) | | ---R | 3 3 3| ---R |-- -- --| ---R +13 17 19+ ---R Type: Matrix Fraction Integer ---E 18 - ---S 19 of 38 -horizConcat(a,b) ---R ---R ---R +1 1 1 3 3 3+ ---R |- - - - - --| ---R |2 3 4 5 7 11| ---R (19) | | ---R |1 1 1 3 3 3| ---R |- - - -- -- --| ---R +5 6 7 13 17 19+ ---R Type: Matrix Fraction Integer ---E 19 - ---S 20 of 38 -vab := vertConcat(a,b) ---R ---R ---R +1 1 1 + ---R |- - - | ---R |2 3 4 | ---R | | ---R |1 1 1 | ---R |- - - | ---R |5 6 7 | ---R (20) | | ---R |3 3 3| ---R |- - --| ---R |5 7 11| ---R | | ---R | 3 3 3| ---R |-- -- --| ---R +13 17 19+ ---R Type: Matrix Fraction Integer ---E 20 - ---S 21 of 38 -transpose vab ---R ---R ---R +1 1 3 3+ ---R |- - - --| ---R |2 5 5 13| ---R | | ---R |1 1 3 3| ---R (21) |- - - --| ---R |3 6 7 17| ---R | | ---R |1 1 3 3| ---R |- - -- --| ---R +4 7 11 19+ ---R Type: Matrix Fraction Integer ---E 21 - ---S 22 of 38 -m := matrix [ [1,2],[3,4] ] ---R ---R ---R +1 2+ ---R (22) | | ---R +3 4+ ---R Type: Matrix Integer ---E 22 - ---S 23 of 38 -4 * m * (-5) ---R ---R ---R +- 20 - 40+ ---R (23) | | ---R +- 60 - 80+ ---R Type: Matrix Integer ---E 23 - ---S 24 of 38 -n := matrix([ [1,0,-2],[-3,5,1] ]) ---R ---R ---R + 1 0 - 2+ ---R (24) | | ---R +- 3 5 1 + ---R Type: Matrix Integer ---E 24 - ---S 25 of 38 -m * n ---R ---R ---R +- 5 10 0 + ---R (25) | | ---R +- 9 20 - 2+ ---R Type: Matrix Integer ---E 25 - ---S 26 of 38 -vec := column(n,3) ---R ---R ---R (26) [- 2,1] ---R Type: Vector Integer ---E 26 - ---S 27 of 38 -vec * m ---R ---R ---R (27) [1,0] ---R Type: Vector Integer ---E 27 - ---S 28 of 38 -m * vec ---R ---R ---R (28) [0,- 2] ---R Type: Vector Integer ---E 28 - ---S 29 of 38 -hilb := matrix([ [1/(i + j) for i in 1..3] for j in 1..3]) ---R ---R ---R +1 1 1+ ---R |- - -| ---R |2 3 4| ---R | | ---R |1 1 1| ---R (29) |- - -| ---R |3 4 5| ---R | | ---R |1 1 1| ---R |- - -| ---R +4 5 6+ ---R Type: Matrix Fraction Integer ---E 29 - ---S 30 of 38 -inverse(hilb) ---R ---R ---R + 72 - 240 180 + ---R | | ---R (30) |- 240 900 - 720| ---R | | ---R + 180 - 720 600 + ---R Type: Union(Matrix Fraction Integer,...) ---E 30 - ---S 31 of 38 -mm := matrix([ [1,2,3,4], [5,6,7,8], [9,10,11,12], [13,14,15,16] ]) ---R ---R ---R +1 2 3 4 + ---R | | ---R |5 6 7 8 | ---R (31) | | ---R |9 10 11 12| ---R | | ---R +13 14 15 16+ ---R Type: Matrix Integer ---E 31 - ---S 32 of 38 -inverse(mm) ---R ---R ---R (32) "failed" ---R Type: Union("failed",...) ---E 32 - ---S 33 of 38 -determinant(mm) ---R ---R ---R (33) 0 ---R Type: NonNegativeInteger ---E 33 - ---S 34 of 38 -trace(mm) ---R ---R ---R (34) 34 ---R Type: PositiveInteger ---E 34 - ---S 35 of 38 -rank(mm) ---R ---R ---R (35) 2 ---R Type: PositiveInteger ---E 35 - ---S 36 of 38 -nullity(mm) ---R ---R ---R (36) 2 ---R Type: PositiveInteger ---E 36 - ---S 37 of 38 -nullSpace(mm) ---R ---R ---R (37) [[1,- 2,1,0],[2,- 3,0,1]] ---R Type: List Vector Integer ---E 37 - ---S 38 of 38 -rowEchelon(mm) ---R ---R ---R +1 2 3 4 + ---R | | ---R |0 4 8 12| ---R (38) | | ---R |0 0 0 0 | ---R | | ---R +0 0 0 0 + ---R Type: Matrix Integer ---E 38 -)spool -)lisp (bye) -@ -<>= -==================================================================== -Matrix examples -==================================================================== - -The Matrix domain provides arithmetic operations on matrices -and standard functions from linear algebra. -This domain is similar to the TwoDimensionalArray domain, except -that the entries for Matrix must belong to a Ring. - -==================================================================== -Creating Matrices -==================================================================== - -There are many ways to create a matrix from a collection of values or -from existing matrices. - -If the matrix has almost all items equal to the same value, use new to -create a matrix filled with that value and then reset the entries that -are different. - - m : Matrix(Integer) := new(3,3,0) - +0 0 0+ - | | - |0 0 0| - | | - +0 0 0+ - Type: Matrix Integer - -To change the entry in the second row, third column to 5, use setelt. - - setelt(m,2,3,5) - 5 - Type: PositiveInteger - -An alternative syntax is to use assignment. - - m(1,2) := 10 - 10 - Type: PositiveInteger - -The matrix was destructively modified. - - m - +0 10 0+ - | | - |0 0 5| - | | - +0 0 0+ - Type: Matrix Integer - -If you already have the matrix entries as a list of lists, use matrix. - - matrix [ [1,2,3,4],[0,9,8,7] ] - +1 2 3 4+ - | | - +0 9 8 7+ - Type: Matrix Integer - -If the matrix is diagonal, use diagonalMatrix. - - dm := diagonalMatrix [1,x**2,x**3,x**4,x**5] - +1 0 0 0 0 + - | | - | 2 | - |0 x 0 0 0 | - | | - | 3 | - |0 0 x 0 0 | - | | - | 4 | - |0 0 0 x 0 | - | | - | 5| - +0 0 0 0 x + - Type: Matrix Polynomial Integer - -Use setRow and setColumn to change a row or column of a matrix. - - setRow!(dm,5,vector [1,1,1,1,1]) - +1 0 0 0 0+ - | | - | 2 | - |0 x 0 0 0| - | | - | 3 | - |0 0 x 0 0| - | | - | 4 | - |0 0 0 x 0| - | | - +1 1 1 1 1+ - Type: Matrix Polynomial Integer - - setColumn!(dm,2,vector [y,y,y,y,y]) - +1 y 0 0 0+ - | | - |0 y 0 0 0| - | | - | 3 | - |0 y x 0 0| - | | - | 4 | - |0 y 0 x 0| - | | - +1 y 1 1 1+ - Type: Matrix Polynomial Integer - -Use copy to make a copy of a matrix. - - cdm := copy(dm) - +1 y 0 0 0+ - | | - |0 y 0 0 0| - | | - | 3 | - |0 y x 0 0| - | | - | 4 | - |0 y 0 x 0| - | | - +1 y 1 1 1+ - Type: Matrix Polynomial Integer - -This is useful if you intend to modify a matrix destructively but -want a copy of the original. - - setelt(dm,4,1,1-x**7) - 7 - - x + 1 - Type: Polynomial Integer - - [dm,cdm] - + 1 y 0 0 0+ +1 y 0 0 0+ - | | | | - | 0 y 0 0 0| |0 y 0 0 0| - | | | | - | 3 | | 3 | - [| 0 y x 0 0|,|0 y x 0 0|] - | | | | - | 7 4 | | 4 | - |- x + 1 y 0 x 0| |0 y 0 x 0| - | | | | - + 1 y 1 1 1+ +1 y 1 1 1+ - Type: List Matrix Polynomial Integer - -Use subMatrix to extract part of an existing matrix. The syntax is -subMatrix(m, firstrow, lastrow, firstcol, lastcol). - - subMatrix(dm,2,3,2,4) - +y 0 0+ - | | - | 3 | - +y x 0+ - Type: Matrix Polynomial Integer - -To change a submatrix, use setsubMatrix. - - d := diagonalMatrix [1.2,-1.3,1.4,-1.5] - +1.2 0.0 0.0 0.0 + - | | - |0.0 - 1.3 0.0 0.0 | - | | - |0.0 0.0 1.4 0.0 | - | | - +0.0 0.0 0.0 - 1.5+ - Type: Matrix Float - -If e is too big to fit where you specify, an error message is -displayed. Use subMatrix to extract part of e, if necessary. - - e := matrix [ [6.7,9.11],[-31.33,67.19] ] - + 6.7 9.11 + - | | - +- 31.33 67.19+ - Type: Matrix Float - -This changes the submatrix of d whose upper left corner is at the -first row and second column and whose size is that of e. - - setsubMatrix!(d,1,2,e) - +1.2 6.7 9.11 0.0 + - | | - |0.0 - 31.33 67.19 0.0 | - | | - |0.0 0.0 1.4 0.0 | - | | - +0.0 0.0 0.0 - 1.5+ - Type: Matrix Float - - d - +1.2 6.7 9.11 0.0 + - | | - |0.0 - 31.33 67.19 0.0 | - | | - |0.0 0.0 1.4 0.0 | - | | - +0.0 0.0 0.0 - 1.5+ - Type: Matrix Float - -Matrices can be joined either horizontally or vertically to make -new matrices. - - a := matrix [ [1/2,1/3,1/4],[1/5,1/6,1/7] ] - +1 1 1+ - |- - -| - |2 3 4| - | | - |1 1 1| - |- - -| - +5 6 7+ - Type: Matrix Fraction Integer - - b := matrix [ [3/5,3/7,3/11],[3/13,3/17,3/19] ] - +3 3 3+ - |- - --| - |5 7 11| - | | - | 3 3 3| - |-- -- --| - +13 17 19+ - Type: Matrix Fraction Integer - -Use horizConcat to append them side to side. The two matrices must -have the same number of rows. - - horizConcat(a,b) - +1 1 1 3 3 3+ - |- - - - - --| - |2 3 4 5 7 11| - | | - |1 1 1 3 3 3| - |- - - -- -- --| - +5 6 7 13 17 19+ - Type: Matrix Fraction Integer - -Use vertConcat to stack one upon the other. The two matrices must -have the same number of columns. - - vab := vertConcat(a,b) - +1 1 1 + - |- - - | - |2 3 4 | - | | - |1 1 1 | - |- - - | - |5 6 7 | - | | - |3 3 3| - |- - --| - |5 7 11| - | | - | 3 3 3| - |-- -- --| - +13 17 19+ - Type: Matrix Fraction Integer - -The operation transpose is used to create a new matrix by reflection -across the main diagonal. - - transpose vab - +1 1 3 3+ - |- - - --| - |2 5 5 13| - | | - |1 1 3 3| - |- - - --| - |3 6 7 17| - | | - |1 1 3 3| - |- - -- --| - +4 7 11 19+ - Type: Matrix Fraction Integer - -==================================================================== -Operations on Matrices -==================================================================== - -Axiom provides both left and right scalar multiplication. - - m := matrix [ [1,2],[3,4] ] - +1 2+ - | | - +3 4+ - Type: Matrix Integer - - 4 * m * (-5) - +- 20 - 40+ - | | - +- 60 - 80+ - Type: Matrix Integer - -You can add, subtract, and multiply matrices provided, of course, that -the matrices have compatible dimensions. If not, an error message is -displayed. - - n := matrix([ [1,0,-2],[-3,5,1] ]) - + 1 0 - 2+ - | | - +- 3 5 1 + - Type: Matrix Integer - -This following product is defined but n * m is not. - - m * n - +- 5 10 0 + - | | - +- 9 20 - 2+ - Type: Matrix Integer - -The operations nrows and ncols return the number of rows and columns -of a matrix. You can extract a row or a column of a matrix using the -operations row and column. The object returned is a Vector. - -Here is the third column of the matrix n. - - vec := column(n,3) - [- 2,1] - Type: Vector Integer - -You can multiply a matrix on the left by a "row vector" and on the right -by a "column vector". - - vec * m - [1,0] - Type: Vector Integer - -Of course, the dimensions of the vector and the matrix must be compatible -or an error message is returned. - - m * vec - [0,- 2] - Type: Vector Integer - -The operation inverse computes the inverse of a matrix if the matrix -is invertible, and returns "failed" if not. - -This Hilbert matrix is invertible. - - hilb := matrix([ [1/(i + j) for i in 1..3] for j in 1..3]) - +1 1 1+ - |- - -| - |2 3 4| - | | - |1 1 1| - |- - -| - |3 4 5| - | | - |1 1 1| - |- - -| - +4 5 6+ - Type: Matrix Fraction Integer - - inverse(hilb) - + 72 - 240 180 + - | | - |- 240 900 - 720| - | | - + 180 - 720 600 + - Type: Union(Matrix Fraction Integer,...) - -This matrix is not invertible. - - mm := matrix([ [1,2,3,4], [5,6,7,8], [9,10,11,12], [13,14,15,16] ]) - +1 2 3 4 + - | | - |5 6 7 8 | - | | - |9 10 11 12| - | | - +13 14 15 16+ - Type: Matrix Integer - - inverse(mm) - "failed" - Type: Union("failed",...) - -The operation determinant computes the determinant of a matrix -provided that the entries of the matrix belong to a CommutativeRing. - -The above matrix mm is not invertible and, hence, must have determinant 0. - - determinant(mm) - 0 - Type: NonNegativeInteger - -The operation trace computes the trace of a square matrix. - - trace(mm) - 34 - Type: PositiveInteger - -The operation rank computes the rank of a matrix: the maximal number -of linearly independent rows or columns. - - rank(mm) - 2 - Type: PositiveInteger - -The operation nullity computes the nullity of a matrix: the dimension -of its null space. - - nullity(mm) - 2 - Type: PositiveInteger - -The operation nullSpace 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. - - nullSpace(mm) - [[1,- 2,1,0],[2,- 3,0,1]] - Type: List Vector Integer - -The operation rowEchelon returns the row echelon form of a matrix. It -is easy to see that the rank of this matrix is two and that its -nullity is also two. - - rowEchelon(mm) - +1 2 3 4 + - | | - |0 4 8 12| - | | - |0 0 0 0 | - | | - +0 0 0 0 + - Type: Matrix Integer - -See Also -o )help OneDimensionalArray -o )help TwoDimensionalArray -o )help Vector -o )help Permanent -o )show Matrix -o $AXIOM/doc/src/algebra/matrix.spad.dvi - -@ -<>= -)abbrev domain MATRIX Matrix -++ Author: Grabmeier, Gschnitzer, Williamson -++ Date Created: 1987 -++ Date Last Updated: July 1990 -++ Basic Operations: -++ Related Domains: IndexedMatrix, RectangularMatrix, SquareMatrix -++ Also See: -++ AMS Classifications: -++ Keywords: matrix, linear algebra -++ Examples: -++ References: -++ Description: -++ \spadtype{Matrix} is a matrix domain where 1-based indexing is used -++ for both rows and columns. -Matrix(R): Exports == Implementation where - R : Ring - Row ==> Vector R - Col ==> Vector R - mnRow ==> 1 - mnCol ==> 1 - MATLIN ==> MatrixLinearAlgebraFunctions(R,Row,Col,$) - MATSTOR ==> StorageEfficientMatrixOperations(R) - - Exports ==> MatrixCategory(R,Row,Col) with - diagonalMatrix: Vector R -> $ - ++ \spad{diagonalMatrix(v)} returns a diagonal matrix where the elements - ++ of v appear on the diagonal. - - if R has ConvertibleTo InputForm then ConvertibleTo InputForm - - if R has Field then - inverse: $ -> Union($,"failed") - ++ \spad{inverse(m)} returns the inverse of the matrix m. - ++ If the matrix is not invertible, "failed" is returned. - ++ Error: if the matrix is not square. --- matrix: Vector Vector R -> $ --- ++ \spad{matrix(v)} converts the vector of vectors v to a matrix, where --- ++ the vector of vectors is viewed as a vector of the rows of the --- ++ matrix --- diagonalMatrix: Vector $ -> $ --- ++ \spad{diagonalMatrix([m1,...,mk])} creates a block diagonal matrix --- ++ M with block matrices {\em m1},...,{\em mk} down the diagonal, --- ++ with 0 block matrices elsewhere. --- vectorOfVectors: $ -> Vector Vector R --- ++ \spad{vectorOfVectors(m)} returns the rows of the matrix m as a --- ++ vector of vectors - - Implementation ==> - InnerIndexedTwoDimensionalArray(R,mnRow,mnCol,Row,Col) add - minr ==> minRowIndex - maxr ==> maxRowIndex - minc ==> minColIndex - maxc ==> maxColIndex - mini ==> minIndex - maxi ==> maxIndex - - minRowIndex x == mnRow - minColIndex x == mnCol - - swapRows_!(x,i1,i2) == - (i1 < minRowIndex(x)) or (i1 > maxRowIndex(x)) or _ - (i2 < minRowIndex(x)) or (i2 > maxRowIndex(x)) => - error "swapRows!: index out of range" - i1 = i2 => x - minRow := minRowIndex x - xx := x pretend PrimitiveArray(PrimitiveArray(R)) - n1 := i1 - minRow; n2 := i2 - minRow - row1 := qelt(xx,n1) - qsetelt_!(xx,n1,qelt(xx,n2)) - qsetelt_!(xx,n2,row1) - xx pretend $ - - positivePower:($,Integer,NonNegativeInteger) -> $ - positivePower(x,n,nn) == --- one? n => x - (n = 1) => x - -- no need to allocate space for 3 additional matrices - n = 2 => x * x - n = 3 => x * x * x - n = 4 => (y := x * x; y * y) - a := new(nn,nn,0) pretend Matrix(R) - b := new(nn,nn,0) pretend Matrix(R) - c := new(nn,nn,0) pretend Matrix(R) - xx := x pretend Matrix(R) - power_!(a,b,c,xx,n :: NonNegativeInteger)$MATSTOR pretend $ - - x:$ ** n:NonNegativeInteger == - not((nn := nrows x) = ncols x) => - error "**: matrix must be square" - zero? n => scalarMatrix(nn,1) - positivePower(x,n,nn) - - if R has commutative("*") then - - determinant x