Faster inversion and other black box matrix computations using efficient block projections

Efficient block projections of non-singular matrices have recently been used by the authors in [10] to obtain an efficient algorithm to find rational solutions for sparse systems of linear equations. In particular a bound ofO~(n2.5) machine operations is presented for this computation assuming that the input matrix can be multiplied by a vector with constant-sized entries using O~(n) machine operations. Somewhat more general bounds for black-box matrix computations are also derived. Unfortunately, the correctness of this algorithm depends on the existence of efficient block projections of non-singular matrices, and this was only conjectured. In this paper we establish the correctness of the algorithm from [10] by proving the existence of efficient block projections for arbitrary non-singular matrices over sufficiently large fields. We further demonstrate the usefulness of these projections by incorporating them into existing black-box matrix algorithms to derive improved bounds for the cost of several matrix problems. We consider, in particular, matrices that can be multiplied by a vector using O~(n) field operations: We show how to compute the inverse of any such non-singular matrix over any field using an expected number of O~(n2.27) operations in that field. A basis for the null space of such a matrix, and a certification of its rank, are obtained at the same cost. An application of this technique to Kaltofen and Villard's Baby-Steps/Giant-Steps algorithms for the determinant and Smith Form of an integer matrix is also sketched, yielding algorithms requiring O~(n2.66) machine operations. More general bounds involving the number of black-box matrix operations to be used are also obtained. The derived algorithms are all probabilistic of the Las Vegas type. They are assumed to be able to generate random elements - bits or field elements - at unit cost, and always output the correct answer in the expected time given.

[1]  M. G. Bruin,et al.  A uniform approach for the fast computation of Matrix-type Padé approximants , 1996 .

[2]  Erich Kaltofen,et al.  On Wiedemann's Method of Solving Sparse Linear Systems , 1991, AAECC.

[3]  Wayne Eberly Processor-efficient parallel matrix inversion over abstract fields: two extensions , 1997, PASCO '97.

[4]  Jean-Guillaume Dumas,et al.  Integer Smith form via the valence: experience with large sparse matrices from homology , 2000, ISSAC.

[5]  J. Dixon Exact solution of linear equations usingP-adic expansions , 1982 .

[6]  Erich Kaltofen,et al.  Analysis of Coppersmith's Block Wiedemann Algorithm for the Parallel Solution of Sparse Linear Systems , 1993, AAECC.

[7]  Erich Kaltofen,et al.  On the complexity of computing determinants , 2001, computational complexity.

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

[9]  Mark Giesbrecht,et al.  Fast computation of the Smith form of a sparse integer matrix , 2002, computational complexity.

[10]  Erich Kaltofen,et al.  On fast multiplication of polynomials over arbitrary algebras , 1991, Acta Informatica.

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

[12]  Mark Giesbrecht,et al.  Efficient parallel solution of sparse systems of linear diophantine equations , 1997, PASCO '97.

[13]  Erich Kaltofen,et al.  LINBOX: A GENERIC LIBRARY FOR EXACT LINEAR ALGEBRA , 2002 .

[14]  Victor Y. Pan,et al.  Improved algorithms for computing determinants and resultants , 2005, J. Complex..

[15]  Victor Y. Pan,et al.  Fast Rectangular Matrix Multiplication and Applications , 1998, J. Complex..

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

[17]  Jacob T. Schwartz,et al.  Fast Probabilistic Algorithms for Verification of Polynomial Identities , 1980, J. ACM.

[18]  George Labahn,et al.  The Inverses of Block Hankel and Block Toeplitz Matrices , 1990, SIAM J. Comput..

[19]  Richard Zippel,et al.  Probabilistic algorithms for sparse polynomials , 1979, EUROSAM.

[20]  V. Pan,et al.  TR-2004013: Toeplitz and Hankel Meet Hensel and Newton: Nearly Optimal Algorithms and Their Practical Acceleration with Saturated Initialization , 2004 .

[21]  Gilles Villard,et al.  Solving sparse rational linear systems , 2006, ISSAC '06.

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

[23]  Éric Schost,et al.  Polynomial evaluation and interpolation on special sets of points , 2005, J. Complex..

[24]  B. D. Saunders,et al.  Efficient matrix preconditioners for black box linear algebra , 2002 .