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Efficient Computation of the Characteristic Polynomial

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.
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https://hal.archives-ouvertes.fr/hal-00004056
Contributor : Jean-Guillaume Dumas Connect in order to contact the contributor
Submitted on : Tuesday, February 8, 2005 - 6:19:03 PM
Last modification on : Saturday, April 23, 2022 - 5:56:03 PM
Long-term archiving on: : Friday, September 17, 2010 - 6:32:37 PM

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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⟩

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