ABSTRACT
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.
- J. Abdeljaoued and G. I. Malaschonok. Efficient algorithms for computing the characteristic polynomial in a domain. J. of Pure and Applied Algebra, 156:127--145, 2001.Google ScholarCross Ref
- W. Baur and V. Strassen. The complexity of partial derivatives. Theor. Computer Science, 22(3):317--330, 1983.Google ScholarCross Ref
- S. J. Berkowitz. On computing the determinant in small parallel time using a small number of processors. Inf. Process. Lett., 18(3):147--150, 1984. Google ScholarDigital Library
- J.-G. Dumas, T. Gautier, and C. Pernet. Finite field linear algebra subroutines. In T. Mora, editor, ISSAC'2002. ACM Press, New York, July 2002. Google ScholarDigital Library
- J.-G. Dumas, P. Giorgi, and C. Pernet. FFPACK: Finite field linear algebra package. In Gutierrez 2004:ISSAC. Google ScholarDigital Library
- J.-G. Dumas, B. D. Saunders, and G. Villard. On efficient sparse integer matrix Smith normal form computations. J. Symb. Comp., 32(1/2):71--99, July--Aug. 2001. Google ScholarDigital Library
- W. Eberly. Reliable krylov-based algorithms for matrix null space and rank. In Gutierrez 2004:ISSAC. Google ScholarDigital Library
- J. v. Gathen and J. Gerhard. Modern Computer Algebra. Cambridge University Press. 1999. Google ScholarDigital Library
- M. Giesbrecht and A. Storjohann. Rational forms of integer matrices. J. Symb. Comp., 34(3):157--172, 2002. Google ScholarDigital Library
- J. Gutierrez, editor. ISSAC'2004. Santander, Spain. ACM Press, New York, July 2004.Google Scholar
- A. Householder. The Theory of Matrices in Numerical Analysis. Blaisdell, Waltham, Mass., 1964.Google Scholar
- O. H. Ibarra, S. Moran, and R. Hui. A generalization of the fast LUP matrix decomposition algorithm and applications. J. of Algorithms, 3(1):45--56, Mar. 1982.Google ScholarCross Ref
- E. Kaltofen. On computing determinants of matrices without divisions. In P. S. Wang, editor, ISSAC'92. ACM Press, New York, July 1992. Google ScholarDigital Library
- E. Kaltofen and G. Villard. On the complexity of computing determinants. Comp. Complexity, 13:91--130, 2004. Google ScholarDigital Library
- W. Keller-Gehrig. Fast algorithms for the characteristic polynomial. Theoretical computer science, 36:309--317, 1985. Google ScholarDigital Library
- H. Lombardi and J. Abdeljaoued. Méthodes matricielles - Introduction à la complexité algébrique. Springer, 2004.Google Scholar
- T. J. O'Gorman, J. M. Ross, A. H. Taber, J. F. Ziegler, H. P. Muhlfeld, C. J. Montrose, H. W. Curtis, and J. L. Walsh. Field testing for cosmic ray soft errors in semiconductor memories. IBM J. of R&D, 40(1):41--50, Jan. 1996. Google ScholarDigital Library
- C. Pernet. Calcul du polynôme caractéristique sur des corps finis. Master's thesis, U. J. Fourier, June 2003. www-lmc.imag.fr/lmc-mosaic/Clement.Pernet.Google Scholar
- C. Pernet and Z. Wan. L U based algorithms for characteristic polynomial over a finite field. SIGSAM Bull., 37(3):83--84, 2003. Poster available at www-lmc.imag.fr/lmc-mosaic/Clement.Pernet. Google ScholarDigital Library
- A. Steel. Personnal communication. 2005Google Scholar
- A. Storjohann. Algorithms for Matrix Canonical Forms. PhD thesis, Institut für Wissenschaftliches Rechnen, ETH-Zentrum, Zürich, Switzerland, Nov. 2000.Google Scholar
- A. Storjohann. Computing the frobenius form of a sparse integer matrix. Manuscript, Apr. 2000.Google Scholar
- G. Villard. Computing the Frobenius normal form of a sparse matrix. In V. G. Ganzha, E. W. Mayr, and E. V. Vorozhtsov, editors, CASC'00, Oct. 2000.Google Scholar
Index Terms
- Efficient computation of the characteristic polynomial
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