Sheafhom: Software for Sparse Integer Matrices

Abstract

Sheafhom is a free software package for large-scale computations in the category of finitely-generated modules over the integers and related rings. Its front end is a language for problems in algebraic topology and geometry. These problems come down to sparse systems of linear equations over the integers. Sheafhom's back end solves the sparse systems, with emphasis on avoiding fill-in and integer explosion. We survey and compare algorithms for integer sparse matrices, and we present implementation techniques in Common Lisp. The final section finds the quotient of the free abelian group on 26 letters by sets of words in the dictionary, in the spirit of [8].