Black box Frobenius decompositions over small fields

A new randomized algorithm is presented for computation of the Frobenius form and transition matrix for an n × n matrix over a field. Using standard matrix and polynomial arithmetic, the algorithm has an asymptotic expected complexity that matches the worst case complexity of the best known deterministic algorithmic for this problem, recently given by Storjohann and Villard [16]. The new algorithm is based on the evaluation of Krylov spaces, rather than an climination technique, and may therefore be superior when applied to sparse or structured matrices with a small number of invariant factors.

[1]  Allan K. Steel,et al.  Algorithm for the Computation of Canonical Forms of Matrices over Fields , 1997, J. Symb. Comput..

[2]  Mark Giesbrecht,et al.  Nearly Optimal Algorithms for Canonical Matrix Forms , 1995, SIAM J. Comput..

[3]  I. Blake,et al.  Computational aspects of discrete logarithms , 1996 .

[4]  Patrick Ozello,et al.  Calcul exact des formes de Jordan et de Frobenius d'une matrice. (Exact computation of the Jordan and Frobenius forms of a matrix) , 1987 .

[5]  Douglas H. Wiedemann.  Solving sparse linear equations over finite fields , 1986, IEEE Trans. Inf. Theory.

[6]  Wayne Eberly,et al.  Asymptotically Efficient Algorithms for the Frobenius Form , 2002 .

[7]  Joachim von zur Gathen,et al.  Modern Computer Algebra , 1998 .

[8]  G. Villard.  Computing the Frobenius Normal Form of a Sparse Matrix , 2000 .

[9]  Erich Kaltofen,et al.  Challenges of Symbolic Computation: My Favorite Open Problems , 2000, J. Symb. Comput..

[10]  Heinz Lüneburg,et al.  On the rational normal form of endomorphisms : a primer to constructive algebra , 1987 .

[11]  James L. Massey,et al.  Step-by-step decoding of the Bose-Chaudhuri- Hocquenghem codes , 1965, IEEE Trans. Inf. Theory.

[12]  Daniel Augot,et al.  On the Computation of Minimal Polynomials, Cyclic Vectors, and Frobenius Forms , 1997 .

[13]  V. Strassen.  Gaussian elimination is not optimal , 1969 .

[14]  Don Coppersmith,et al.  Matrix multiplication via arithmetic progressions , 1987, STOC.

[15]  Arne Storjohann,et al.  An O(n3) algorithm for the Frobenius normal form , 1998, ISSAC '98.

[16]  Elwyn R. Berlekamp,et al.  Algebraic coding theory , 1984, McGraw-Hill series in systems science.

[17]  James L. Massey,et al.  Shift-register synthesis and BCH decoding , 1969, IEEE Trans. Inf. Theory.