Parallel algorithms for matrix normal forms

Parallel Algorithms for Matrix Normal Forms

Here we offer a new randomized parallel algorithm that determines the Smith normal form of a matrix with entries being univariate polynomials with coefficients in an arbitrary field. The algorithm has two important advantages over our previous one: the multipliers relating the Smith form to the input matrix are computed, and the algorithm is probabilistic of Las Vegas type, i.e., always finds t...

Output-sensitive modular algorithms for polynomial matrix normal forms

We give modular algorithms to compute row-reduced forms, weak Popov forms, and Popov forms of polynomial matrices, as well as the corresponding unimodular transformation matrices. Our algorithms improve on existing fraction-free algorithms. In each case we define lucky homomorphisms, determine the appropriate normalization, as well as bound the number of homomorphic images required. The algorit...

Nearly Optimal Algorithms for Canonical Matrix Forms

A Las Vegas type probabilistic algorithm is presented for nding the Frobenius canonical form of an n n matrix T over any eld K. The algorithm requires O~(MM(n)) = MM(n) (logn) O(1) operations in K, where O(MM(n)) operations in K are suucient to multiply two n n matrices over K. This nearly matches the lower bound of (MM(n)) operations in K for this problem, and improves on the O(n 4) operations...

Polynomial Algorithms for Computing the Smith and Hermite Normal Forms of an Integer Matrix

Combinatorial Algorithms for Parallel Sparse Matrix Distributions

Combinatorial algorithms have long played a crucial enabling role in parallel computing. An important problem in sparse matrix computations is how to distribute the sparse matrix and vectors among processors. We review graph, bipartite graph, and hypergraph models for both 1d (row or column) distributions and 2d distributions. A valuable tool is hypergraph partitioning. We present results using...