Abstract
Symmetric polynomials over F p in n indeterminates x 1,..., x n are expressible as rational functions of the first n power sums s j =x jl + ...+ x jn with exponents j not divisible by p. There exist fairly simple regular specializations of these power sums by elements from F p so that all denominators of such rational expressions remain nonzero. This leads to a new class of fast parallel algorithms for the determinant or the characteristic polynomial of n×n matrices over any field of characteristic p with arithmetic circuit depth θ(logn)2.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
S.J. Berkowitz, On computing the determinant in small parallel time using a small number of processors, Inf. Proc. Letters 18 (1984), 147–150.
A. Borodin, J. von zur Gathen, and J.E. Hopcroft, Fast parallel matrix and GCD computations, Inf. Control 52(1982), 241–256.
A.L. Chistov, Fast parallel calculation of the rank of matrices overa field of arbitrary characteristic, Proc. FCT '85, Lect. Notes Comp. Sci. 199 (1985), 63–69.
L. Csanky, Fast parallel matrix inversion algorithms, SIAM J. Comput. 5 (1976), 618–623.
S. Kakeya, On fundamental systems of symmetric functions, Jap. J. Math. 2 (1925), 69–80.
S. Kakeya, On fundamental systems of symmetric functions 2, Jap. J. Math. 4 (1927), 77–85.
U.J.J. Leverrier, Sur les variations séculaires des éléments elliptiques des sept planètes principales: Mercure, Vénus, la Terre, Mars, Jupiter, Saturne, et Uranus, J. Math. Pures Appl. 5 (1840), 220–254.
K. Nakamura, On the representation of symmetric fonctions by power-sums which form the fundamental system, Jap. J. Math. 4 (1927), 87–92.
F.P. Preparata, and D.V. Sarwate, An improved parallel processor bound in fast matrix inversion, Inf. Proc. Letters 7 (1978), 148–150.
K.T. Vahlen, über Fundamentalsysteme für symmetrische Funktionen, Acta Mathematica 23 (1900), 91–120.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1993 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Schönhage, A. (1993). Fast parallel computation of characteristic polynomials by Leverrier's power sum method adapted to fields of finite characteristic. In: Lingas, A., Karlsson, R., Carlsson, S. (eds) Automata, Languages and Programming. ICALP 1993. Lecture Notes in Computer Science, vol 700. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-56939-1_90
Download citation
DOI: https://doi.org/10.1007/3-540-56939-1_90
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-56939-8
Online ISBN: 978-3-540-47826-3
eBook Packages: Springer Book Archive