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

Smith Normal Form over Local Rings and Related Problems

The Smith normal form is a diagonalization of matrices with many applications in diophantine analysis, graph theory, system control theory, simplicial homology, and more recently, in topological analysis of big data. Efficient computation of Smith normal form is a well-studied area for matrices with integer and polynomial entries. Existing successful algorithms typically rely on elimination for dense matrices and iterative Krylov space methods for sparse matrices. Our interest lies in computing Smith normal form for sparse matrices over local rings, where traditional iterative methods face challenges due to the lack of unique minimal polynomials. We explore different approaches to tackling this problem for two local rings: the integers modulo a prime power, and the polynomials modulo a power of an irreducible polynomial. Over local polynomial rings, we find success in linearization into larger dimension matrices over the base field. Effectively we transform the problem of computing the Smith normal form into a small number of rank problems over the base field. The latter problem has existing efficient algorithms for sparse and dense matrices. The problem is harder over local integer rings. We take the approach of hybrid sparse-dense algorithms. We also tackle a restricted version of the problem where we detect only the first non-trivial invariant factor. We also give an algorithm to find the first few invariant factors using iterative rank-1 updates. This method becomes dense when applied to finding all the invariant factors. We digress slightly into the related problem of preconditioning. We show that linear- time preconditioners are suitable for computing Smith normal form, and computing nullspace samples. For the latter problem we design an algorithm for computing uniform samples from the nullspace. On a separate track, we focus on the properties of the Smith normal form decomposition. We relate the invariant factors to the eigenvalues. Our ultimate goal is to extend the applications of numerical algorithms for computing eigenvalues to computing the invariant factors of symbolic matrices