To write functions that implement the operations of a domain, you want to choose the most computationally efficient data structure to represent the elements of your domain.
A classic problem in computer algebra is the optimal choice for an internal representation of polynomials. If you create a polynomial, say , how does Axiom hold this value internally? There are many ways. Axiom has nearly a dozen different representations of polynomials, one to suit almost any purpose. Algorithms for solving polynomial equations work most efficiently with polynomials represented one way, whereas those for factoring polynomials are most efficient using another. One often-used representation is a list of terms, each term consisting of exponent-coefficient records written in the order of decreasing exponents. For example, the polynomial is represented by the list .
What is the optimal data structure for a matrix? It depends on the application. For large sparse matrices, a linked-list structure of records holding only the non-zero elements may be optimal. If the elements can be defined by a simple formula , then a compiled function for may be optimal. Some programmers prefer to represent ordinary matrices as vectors of vectors. Others prefer to represent matrices by one big linear array where elements are accessed with linearly computable indexes.
While all these simultaneous structures tend to be confusing, Axiom provides a helpful organizational tool for such a purpose: categories. PolynomialCategory, for example, provides a uniform user interface across all polynomial types. Each kind of polynomial implements functions for all these operations, each in its own way. If you use only the top-level operations in PolynomialCategory you usually do not care what kind of polynomial implementation is used.
Within a given domain, however, you define (at most) one representation.You can make that representation a Union type, however. See ugTypesUnions for examples of unions. If you want to have multiple representations (that is, several domains, each with its own representation), use a category to describe the Exports, then define separate domains for each representation.