On Efficient Sparse Integer Matrix Smith Normal Form Computations

Abstract

We present a new algorithm to compute the Integer Smith normal form of large sparse matrices. We reduce the computation of the Smith form to independent, and therefore parallel, computations modulo powers of word-size primes. Consequently, the algorithm does not suffer from coefficient growth. We have implemented several variants of this algorithm (elimination and/or black box techniques) since practical performance depends strongly on the memory available. Our method has proven useful in algebraic topology for the computation of the homology of some large simplicial complexes.

Computation of cubical homology, cohomology, and (co)homological operations via chain contraction

“…However, the boundary matrices derived from cellular complexes are very sparse, that is, have a very small number entries in their columns and rows, which can be used for designing more efficient algorithms (e.g. [15]). Indeed, simple reduction techniques motivated by geometric interpretation of the data may dramatically reduce the matrices in size before coming to a point where no more reductions of this type can be applied and thus the general algorithm for the computation of the SNF must be used.…”

The “Seven Dwarfs” of Symbolic Computation

“…We observe additional trends today: Strassen's fast matrix multiplication algorithm and cache-efficient BLAS libraries improve performance of exact linear algebra [Dumas et al 2008]; characteristic polynomials and integer Smith normal forms of sparse integer matrices [Dumas et al 2001;Giesbrecht 2001] are important invariants, for instance in computing the so-called bar code of a persistent topology of data; and structured exact linear problem solvers such as the matrix Berlekamp/Massey algorithm [Kaltofen and Yuhasz 2013] form a fundamental ingredient in sparse solvers.…”

Sheafhom: Software for Sparse Integer Matrices

“…Some authors approach sparse matrices over Z by working modulo prime numbers p. One can reduce the matrix mod p for several different p and reconstruct the integer result at the end. For an example, see [7]. For a similar idea with dense matrices, see [3].…”