Efficient Computation of the Characteristic Polynomial

TL;DR

This paper introduces new algorithms for efficiently computing the characteristic polynomial of dense matrices over finite fields and integers, improving on existing methods in terms of complexity and computational time.

Contribution

It presents novel algorithms based on Krylov iterates, Gaussian elimination, and generalizations of Keller-Gehrig's methods, enhancing efficiency for both finite field and integer matrices.

Findings

New algorithms outperform previous methods in complexity.

Probabilistic approach is effective for sparse and structured matrices.

Early termination and Chinese remaindering improve integer matrix computations.

Abstract

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.