On computing the determinant in small parallel time using a small number of processors
Information Processing Letters
Fast algorithms for the characteristic polynomial
Theoretical Computer Science
On computing determinants of matrices without divisions
ISSAC '92 Papers from the international symposium on Symbolic and algebraic computation
Field testing for cosmic ray soft errors in semiconductor memories
IBM Journal of Research and Development - Special issue: terrestrial cosmic rays and soft errors
Modern computer algebra
On efficient sparse integer matrix Smith normal form computations
Journal of Symbolic Computation - Special issue on computer algebra and mechanized reasoning: selected St. Andrews' ISSAC/Calculemus 2000 contributions
Computing rational forms of integer matrices
Journal of Symbolic Computation
Finite field linear algebra subroutines
Proceedings of the 2002 international symposium on Symbolic and algebraic computation
LU based algorithms for characteristic polynomial over a finite field
ACM SIGSAM Bulletin
FFPACK: finite field linear algebra package
ISSAC '04 Proceedings of the 2004 international symposium on Symbolic and algebraic computation
Reliable Krylov-based algorithms for matrix null space and rank
ISSAC '04 Proceedings of the 2004 international symposium on Symbolic and algebraic computation
On the complexity of computing determinants
Computational Complexity
Faster algorithms for the characteristic polynomial
Proceedings of the 2007 international symposium on Symbolic and algebraic computation
Some recent progress in exact linear algebra and related questions
Proceedings of the 2007 international symposium on Symbolic and algebraic computation
Dense Linear Algebra over Word-Size Prime Fields: the FFLAS and FFPACK Packages
ACM Transactions on Mathematical Software (TOMS)
On finding multiplicities of characteristic polynomial factors of black-box matrices
Proceedings of the 2009 international symposium on Symbolic and algebraic computation
Generic design of Chinese remaindering schemes
Proceedings of the 4th International Workshop on Parallel and Symbolic Computation
Output-sensitive decoding for redundant residue systems
Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation
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We deal with the computation of the characteristic polynomial of dense matrices over word size finite fields and over the integers. We first present two algorithms for 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.