A local construction of the Smith normal form of a matrix polynomial

We present an algorithm for computing a Smith form with multipliers of a regular matrix polynomial over a field. This algorithm differs from previous ones in that it computes a local Smith form for each irreducible factor in the determinant separately and then combines them into a global Smith form, whereas other algorithms apply a sequence of unimodular row and column operations to the original matrix. The performance of the algorithm in exact arithmetic is reported for several test cases.Comment: 26 pages, 6 figures; introduction expanded, 10 references added, two additional tests performe

Fast Computation of Smith Forms of Sparse Matrices Over Local Rings

We present algorithms to compute the Smith Normal Form of matrices over two families of local rings. The algorithms use the \emph{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 \nxn matrix $A$ over the ring \Fzfe, where $f^e$ is a power of an irreducible polynomial f \in \Fz of degree $d$, our algorithm requires \bigO(\eta de^2n) operations in \F, where our black-box is assumed to require \bigO(\eta) operations in \F to compute a matrix-vector product by a vector over \Fzfe (and $\eta$ is assumed greater than \Pden). The algorithm only requires additional storage for \bigO(\Pden) elements of \F. In particular, if \eta=\softO(\Pden), then our algorithm requires only \softO(n^2d^2e^3) operations in \F, which is an improvement on known dense methods for small $d$ and $e$. For the ring \ZZ/p^e\ZZ, 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 $\mu n r e \log p,$ where $\mu$ is the number of operations in \ZZ/p\ZZ 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.Comment: Preliminary version to appear at ISSAC 201

Solving Sparse Integer Linear Systems

We propose a new algorithm to solve sparse linear systems of equations over the integers. This algorithm is based on a $p$-adic lifting technique combined with the use of block matrices with structured blocks. It achieves a sub-cubic complexity in terms of machine operations subject to a conjecture on the effectiveness of certain sparse projections. A LinBox-based implementation of this algorithm is demonstrated, and emphasizes the practical benefits of this new method over the previous state of the art

Efficient Computation of the Characteristic Polynomial

This article deals with the computation of the characteristic polynomial of dense matrices over small finite fields and over the integers. We first present two algorithms for the finite fields: one is based on Krylov iterates and Gaussian elimination. We compare it to an improvement of the second algorithm of Keller-Gehrig. Then we show that a generalization of Keller-Gehrig's third algorithm could improve both complexity and computational time. We use these results as a basis for the computation of the characteristic polynomial of integer matrices. We first use early termination and Chinese remaindering for dense matrices. Then a probabilistic approach, based on integer minimal polynomial and Hensel factorization, is particularly well suited to sparse and/or structured matrices

Generic design of Chinese remaindering schemes

We propose a generic design for Chinese remainder algorithms. A Chinese remainder computation consists in reconstructing an integer value from its residues modulo non coprime integers. We also propose an efficient linear data structure, a radix ladder, for the intermediate storage and computations. Our design is structured into three main modules: a black box residue computation in charge of computing each residue; a Chinese remaindering controller in charge of launching the computation and of the termination decision; an integer builder in charge of the reconstruction computation. We then show that this design enables many different forms of Chinese remaindering (e.g. deterministic, early terminated, distributed, etc.), easy comparisons between these forms and e.g. user-transparent parallelism at different parallel grains