Abstract

A Las Vegas type probabilistic algorithm is presented for finding the Frobenius canonical form of an n x n matrix T over any field K. The algorithm requires O~(MM(n)) = MM(n) (log n) ^ O(1) operations in K, where O(MM(n)) operations in K are sufficient to multiply two n x n matrices over K. This nearly matches the lower bound of \Omega(MM(n)) operations in K for this problem, and improves on the O(n^4) operations in K required by the previously best known algorithms. We also demonstrate a fast parallel implementation of our algorithm for the Frobenius form, which is processor-efficient on a PRAM. As an application we give an algorithm to evaluate a polynomial g(x) in K[x] at T which requires only O~(MM(n)) operations in K when deg g < n^2. Other applications include sequential and parallel algorithms for computing the minimal and characteristic polynomials of a matrix, the rational Jordan form of a matrix, for testing whether two matrices are similar, and for matrix powering, which are substantially faster than those previously known.

Citations

2172	The design and analysis of computer algorithms - Aho, Hopcroft, et al. - 1974
693	Matrix multiplication via arithmetical progressions - Coppersmith, Winograd - 1990
268	Parallel algorithms for sharedmemory machines - Karp, Ramachandran - 1990
153	Theory of Matrices, Vols - Gantmacher - 1959
143	Solving sparse linear equations over finite fields - Wiedemann - 1986
131	Factoring polynomials over large finite fields - Berlekamp - 1970
127	On fast multiplication of polynomials over arbitrary algebras - Cantor, Kaltofen - 1991
109	The complexity of partial derivatives - Baur, Strassen - 1983
93	The computational complexity of algebraic and numeric problems - Borodin, Munro - 1975
91	Polinomial algorithms for computing the smith and hermite normal forms of an integer matrix - Kannan, Bachem - 1979
85	Efficient Parallel Solution of Linear Systems - Pan, Reif - 1985
60	Computational Methods of Linear Algebra - Faddeev, Faddeeva - 1963
41	Triangular Factorization and Inversion by Fast Matrix Multiplication - Bunch, Hopcroft - 1974
40	Fast construction of irreducible polynomials over finite fields - Shoup
39	On the number of nonscalar multiplications necessary to evaluate polynomials - Paterson, Stockmeyer, et al. - 1973
28	Fast parallel computation of Hermite and Smith forms of polynomial matrices - Kaltofen, Krishnamoorthy, et al. - 1987
28	Fast algorithms for the characteristic polynomial.’ Theoret - Keller-Gehrig - 1985
22	Calcul exact des formes de Jordan et de Frobenius d’une matrice - Ozello - 1987
12	Fast polynomial transform algorithms for digital convolution - Nussbaumer - 1980
9	Polynomial-time algorithms for solving systems of linear equations over polynomials - Kannan - 1985
8	Efficient parallel independent subsets and matrix factorizations - Eberly - 1991
3	onhage, Schnelle Multiplikation von Polynomen uber Korpern der Charakteristik 2 - Sch - 1977
3	Schnelle multiplikation groer zahlen - onhage, A, et al. - 1971
2	The numerical solution of the secular equation - Danilevsky - 1937
2	Efficient solution of sparse linear systems - Kaltofen - 1992
2	Calcul formel et parall'elisme : 'etude de la forme normale de Smith - Villard
1	Matrix extensions of the RSA algorithm - Chuang, Dunham - 1990
1	Uber den Rang einer Matrix, Sitzungsberichte der Koniglich Preu��ischen Akademie der Wissenschaften zu - Frobenius - 1911
1	parallel solution of linear systems II: The general case - Processor-efficient - 1992
1	uneburg, On Rational Normal Form of Endomorphisms: a Primer to Constructive Algebra - L - 1987
1	The general term of a recurring series - Ranum - 1911