Generic Linear Algebra and Quotient Rings in Maple
نویسندگان
چکیده
The algorithms for linear algebra in the Magma and Axiom computer algebra systems work over an arbitrary ring. For example, the implementation of Gaussian elimination for reducing a matrix to (reduced) row Echelon form works over any field that the user constructs. In contrast, Maple’s facilities for linear algebra in its LinearAlgebra package only work for specific rings. If the input matrix contains general expressions, the algorithms may work incorrectly. Motivated by a need to do linear algebra over quotient rings and finite fields in Maple, we have designed a simple to use facility that permits the Maple user to define a field, Euclidean domain, integral domain or ring so that our “generic” algorithms for linear algebra are immediately available. These algorithms comprise a package called GenericLinearAlgebra which is being integrated into Maple 11. We have also implemented a package for computing in quotient rings. This package, called QuotientRings, exports all the necessary operations so that one can immediately do linear algebra over a quotient ring, for example, the trigonometric polynomial ring Q[s, c]/〈s + c − 1〉. The package also includes a new algorithm for simplifying fractions over a quotient ring to canonical form that we discuss.
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