Fast computation of Smith forms of sparse matrices over local rings

• Authors: Mustafa Elsheikh;Mark Giesbrecht;Andy Novocin;B. David Saunders
• Venue: Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation
• Year: 2012

We present algorithms to compute the Smith Normal Form of matrices over two families of local rings. The algorithms use the black-box model which is suitable for sparse and structured matrices. The algorithms depend on a number of tools, such as matrix rank computation over finite fields, for which the best-known time- and memory-efficient algorithms are probabilistic. For an n x n matrix A over the ring F[z]/(fe), where fe is a power of an irreducible polynomial f ∈ F[z] of degree d, our algorithm requires O(ηde2n) operations in F, where our black-box is assumed to require O(η) operations in F to compute a matrix-vector product by a vector over F[z]/(fe) (and η is assumed greater than nde). The algorithm only requires additional storage for O(nde) elements of F. In particular, if η = O(nde), then our algorithm requires only O(n2d2e3) operations in F, which is an improvement on known dense methods for small d and e. For the ring Z/peZ, where p is a prime, we give an algorithm which is time- and memory-efficient when the number of nontrivial invariant factors is small. We describe a method for dimension reduction while preserving the invariant factors. The time complexity is essentially linear in μnre log p, where μ is the number of operations in Z/pZ to evaluate the black-box (assumed greater than n) and r is the total number of non-zero invariant factors. To avoid the practical cost of conditioning, we give a Monte Carlo certificate, which at low cost, provides either a high probability of success or a proof of failure. The quest for a time- and memory-efficient solution without restrictions on the number of nontrivial invariant factors remains open. We offer a conjecture which may contribute toward that end.