Fast Rectangular Matrix Multiplication and Applications

First we study asymptotically fast algorithms for rectangular matrix multiplication. We begin with new algorithms for multiplication of ann×nmatrix by ann×n2matrix in arithmetic timeO(n?),?=3.333953?, which is less by 0.041 than the previous record 3.375477?. Then we present fast multiplication algorithms for matrix pairs of arbitrary dimensions, estimate the asymptotic running time as a function of the dimensions, and optimize the exponents of the complexity estimates. For a large class of input matrix pairs, we improve the known exponents. Finally we show three applications of our results: (a)we decrease from 2.851 to 2.837 the known exponent of the work bounds for fast deterministic (NC) parallel evaluation of the determinant, the characteristic polynomial, and the inverse of ann×nmatrix, as well as for the solution to a nonsingular linear system ofnequations, (b)we asymptotically accelerate the known sequential algorithms for the univariate polynomial composition modxn, yielding the complexity boundO(n1.667) versus the old record ofO(n1.688), and for the univariate polynomial factorization over a finite field, and (c)we improve slightly the known complexity estimates for computing basic solutions to the linear programming problem withmconstraints andnvariables.

[1]  F. Behrend.  On Sets of Integers Which Contain No Three Terms in Arithmetical Progression. , 1946, Proceedings of the National Academy of Sciences of the United States of America.

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

[3]  E. Berlekamp.  Factoring polynomials over large finite fields* , 1970, SYMSAC '71.

[4]  Allan Borodin,et al.  The computational complexity of algebraic and numeric problems , 1975, Elsevier computer science library.

[5]  David P. Dobkin,et al.  On the Number of Multiplications Required for Matrix Multiplication , 1976, SIAM J. Comput..

[6]  H. T. Kung,et al.  Fast Algorithms for Manipulating Formal Power Series , 1978, JACM.

[7]  Victor Y. Pan,et al.  Strassen's algorithm is not optimal trilinear technique of aggregating, uniting and canceling for constructing fast algorithms for matrix operations , 1978, 19th Annual Symposium on Foundations of Computer Science (sfcs 1978).

[8]  Grazia Lotti,et al.  O(n2.7799) Complexity for n*n Approximate Matrix Multiplication , 1979, Inf. Process. Lett..

[9]  Arnold Schönhage,et al.  Partial and Total Matrix Multiplication , 1981, SIAM J. Comput..

[10]  D. Cantor,et al.  A new algorithm for factoring polynomials over finite fields , 1981 .

[11]  Don Coppersmith.  Rapid Multiplication of Rectangular Matrices , 1982, SIAM J. Comput..

[12]  Don Coppersmith,et al.  On the Asymptotic Complexity of Matrix Multiplication , 1982, SIAM J. Comput..

[13]  Victor Y. Pan,et al.  How to Multiply Matrices Faster , 1984, Lecture Notes in Computer Science.

[14]  V. Pan.  How can we speed up matrix multiplication , 1984 .

[15]  Volker Strassen,et al.  The asymptotic spectrum of tensors and the exponent of matrix multiplication , 1986, 27th Annual Symposium on Foundations of Computer Science (sfcs 1986).

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

[17]  V. Strassen.  Relative bilinear complexity and matrix multiplication. , 1987 .

[18]  Richard M. Karp,et al.  A Survey of Parallel Algorithms for Shared-Memory Machines , 1988 .

[19]  V. Strassen.  The asymptotic spectrum of tensors. , 1988 .

[20]  Victor Y. Pan,et al.  Parallel Evaluation of the Determinant and of the Inverse of a Matrix , 1989, Inf. Process. Lett..

[21]  David Eppstein,et al.  Parallel Algorithmic Techniques for Combinatorial Computation , 1988, ICALP.

[22]  Victor Y. Pan,et al.  Processor efficient parallel solution of linear systems over an abstract field , 1991, SPAA '91.

[23]  Victor Y. Pan,et al.  Processor-efficient parallel solution of linear systems. II. The positive characteristic and singular cases , 1992, Proceedings., 33rd Annual Symposium on Foundations of Computer Science.

[24]  H. Niederreiter.  Factorization of polynomials and some linear-algebra problems over finite fields , 1993 .

[25]  V. Strassen.  Algebra and Complexity , 1994 .

[26]  V. Pan,et al.  Polynomial and matrix computations (vol. 1): fundamental algorithms , 1994 .

[27]  Erich Kaltofen,et al.  Subquadratic-time factoring of polynomials over finite fields , 1995, STOC '95.

[28]  H. Niederreiter,et al.  On a new factorization algorithm for polynomials over finite fields , 1995 .

[29]  Victor Y. Pan,et al.  Parallel Computation of Polynomial GCD and Some Related Parallel Computations over Abstract Fields , 1996, Theor. Comput. Sci..

[30]  Don Coppersmith,et al.  Rectangular Matrix Multiplication Revisited , 1997, J. Complex..

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

[32]  Michael Clausen,et al.  Algebraic complexity theory , 1997, Grundlehren der mathematischen Wissenschaften.

[33]  Erich Kaltofen,et al.  Fast polynomial factorization over high algebraic extensions of finite fields , 1997, ISSAC.

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

[35]  Patrice Naudin,et al.  Univariate Polynomial Factorization Over Finite Fields , 1998, Theor. Comput. Sci..

[36]  Peter A. Beling,et al.  Using Fast Matrix Multiplication to Find Basic Solutions , 1998, Theoretical Computer Science.

[37]  I. Kaporin.  A practical algorithm for faster matrix multiplication , 1999 .