Nearly Optimal Algorithms for Canonical Matrix Forms

Abstract

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 su cient 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 in K required by the previously best known algorithms. We also demonstrate a fast parallel implementation of our algorithm for the Frobenius form, which is processor-e cient on a PRAM. As an application we give an algorithm to evaluate a polynomial g 2 K x] at T which requires only O~(MM(n)) operations in K when degg n 2. Other applications include sequential and parallel algorithms for computing the minimal and characteristic polynomials of a matrix, the rational Jordan form of a matrix, for testing whether two matrices are similar, and for matrix powering, which are substantially faster than those previously known.

On the complexity of polynomial matrix computations

“…To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. computed in O(n ω ) operations in K. Under the model of computation trees over K, we know that ω is also the exponent of the problems of computing the determinant, the matrix inverse, the rank, the characteristic polynomial (we refer to the survey in [5,Chap.16]) or the Frobenius normal form [8,15]. On an algebraic Ram, all these problems can be solved with O˜(n ω ) operations in K, hence the corresponding algorithms are optimal up to logarithmic terms.…”

Algorithms for matrix groups

“…Characteristic and minimal polynomials can be computed in the claimed time using the Las Vegas algorithms of [2,69] and [47] respectively. Neunhöffer & Praeger [87] describe Monte Carlo and deterministic algorithms to construct the minimal polynomial; these have complexity O(d 3 ) and O(d 4 ) respectively and are implemented in GAP.…”

Computation of Homology Groups and Generators

“…The best known deterministic algorithm has been proposed by Storjohann [Sto96]. Stochastic algorithms have for example been proposed by Giesbrecht et al [Gie95]. They are generally more efficient than deterministic ones on sparse matrices, but are quite equivalent on dense matrices.…”