Efficient Computation of the Characteristic Polynomial

Jean-Guillaume Dumas (1) , Clément Pernet (1) , Zhendong Wan (2)

1 LMC - IMAG - Laboratoire de Modélisation et Calcul
2 CIS - Department of Computer and Information Sciences [Newark]

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.

Mots clés: characteristic polynomial over finite fields; characteristic polynomial of integer matrices

Domaines: Calcul formel [cs.SC]

Jean-Guillaume Dumas, Clément Pernet, Zhendong Wan. Efficient Computation of the Characteristic Polynomial. International Symposium on Symbolic and Algebraic Computation (ISSAC'15), Jul 2005, Beijing, China. pp.140-147. ⟨hal-00004056v2⟩