Triangular x-basis decompositions and derandomization of linear algebra algorithms over K[x]

Deterministic algorithms are given for some computational problems that take as input a nonsingular polynomial matrix A over K[x], K an abstract field, including solving a linear system involving A and computing a row reduced form of A. The fastest known algorithms for linear system solving based on the technique of high-order lifting by Storjohann (2003), and for row reduction based on the fast minimal approximant basis computation algorithm by Giorgi et al. (2003), use randomization to find either a linear or small degree polynomial that is relatively prime to detA. We derandomize these algorithms by first computing a factorization of A=UH, with x not dividing detU and x-1 not dividing detH. A partial linearization technique, that is applicable also to other problems, is developed to transform a system involving H, which may have some columns of large degrees, to an equivalent system that has degrees reduced to that of the average column degree.

[1]  Arne Storjohann,et al.  High-order lifting and integrality certification , 2003, J. Symb. Comput..

[2]  Arne Storjohann High-order lifting , 2002, ISSAC '02.

[3]  George Labahn,et al.  Recursiveness in matrix rational interpolation problems , 1997 .

[4]  Numerische Mathematik Exact Solution of Linear Equations Using P-Adie Expansions* , 2005 .

[5]  George Labahn,et al.  Normal forms for general polynomial matrices , 2006, J. Symb. Comput..

[6]  Soumojit Sarkar,et al.  Normalization of row reduced matrices , 2011, ISSAC '11.

[7]  Arne Storjohann,et al.  The shifted number system for fast linear algebra on integer matrices , 2005, J. Complex..

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

[9]  Claude-Pierre Jeannerod,et al.  On the complexity of polynomial matrix computations , 2003, ISSAC '03.

[10]  Robert T. Moenck,et al.  Approximate algorithms to derive exact solutions to systems of linear equations , 1979, EUROSAM.

[11]  B. Beckermann,et al.  A Uniform Approach for the Fast Computation of Matrix-Type Padé Approximants , 1994, SIAM J. Matrix Anal. Appl..

[12]  Thomas Kailath,et al.  Linear Systems , 1980 .

[13]  A. Storjohann Algorithms for matrix canonical forms , 2000 .

[14]  Joachim von zur Gathen,et al.  Modern Computer Algebra (3. ed.) , 2003 .

[15]  Irving Kaplansky,et al.  Elementary divisors and modules , 1949 .

[16]  Arne Storjohann,et al.  On lattice reduction for polynomial matrices , 2000 .

[17]  Mark Giesbrecht,et al.  Fast computation of the Smith normal form of an integer matrix , 1995, ISSAC '95.

[18]  Claude-Pierre Jeannerod,et al.  LSP matrix decomposition revisited LSP Matrix Decomposition Revisited LSP Matrix Decomposition Revisited , 2006 .

[19]  Claude-Pierre Jeannerod,et al.  Asymptotically fast polynomial matrix algorithms for multivariable systems , 2005, ArXiv.

[20]  Arne Storjohann,et al.  Certified dense linear system solving , 2004, J. Symb. Comput..

[21]  Arne Storjohann,et al.  Fast Algorithms for for Linear Algebra Modulo N , 1998, ESA.

[22]  George Labahn,et al.  Shifted normal forms of polynomial matrices , 1999, ISSAC '99.

[23]  Jia Yu,et al.  A local construction of the Smith normal form of a matrix polynomial , 2008, J. Symb. Comput..

[24]  Gilles Villard,et al.  Generalized Subresultants for Computing the Smith Normal Form of Polynomial Matrices , 1995, J. Symb. Comput..

[25]  Oscar H. Ibarra,et al.  A Generalization of the Fast LUP Matrix Decomposition Algorithm and Applications , 1982, J. Algorithms.

[26]  John A. Howell,et al.  Spans in the module (Zm ) s , 1986 .

[27]  James Lee Hafner,et al.  Asymptotically fast triangulation of matrices over rings , 1991, SODA '90.

[28]  Erich Kaltofen,et al.  Parallel algorithms for matrix normal forms , 1990 .

[29]  Arne Storjohann Notes on computing minimal approximant bases , 2006, Challenges in Symbolic Computation Software.

[30]  Arne Storjohann,et al.  Rational solutions of singular linear systems , 2000, ISSAC.