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Unformatted text preview: On the Complexity of Polynomial Matrix Computations Pascal Giorgi, ClaudePierre Jeannerod and Gilles Villard CNRS, INRIA, Laboratoire LIP ENSL, 46, Alle dItalie 69364 Lyon Cedex 07, France http://www.enslyon.fr/ { pgiorgi,cpjeanne,gvillard } ABSTRACT We study the link between the complexity of polynomial matrix multiplication and the complexity of solving other basic linear algebra problems on polynomial matrices. By polynomial matrices we mean n n matrices in K [ x ] of de gree bounded by d , with K a commutative field. Under the straightline program model we show that multiplica tion is reducible to the problem of computing the coefficient of degree d of the determinant. Conversely, we propose al gorithms for minimal approximant computation and column reduction that are based on polynomial matrix multiplica tion; for the determinant, the straightline program we give also relies on matrix product over K [ x ] and provides an alter native to the determinant algorithm of [16, 17]. We further show that all these problems can be solved in particular in O ( n d ) operations in K . Here the soft O notation O in dicates some missing log( nd ) factors and is the exponent of matrix multiplication over K . Categories and Subject Descriptors F.2.1 [ Analysis of Algorithms and Problem Complex ity ]: Numerical Algorithms and Problems computations on matrices, computations on polynomials. General Terms Algorithms. Keywords Matrix polynomial, minimal basis, column reduced form, matrix gcd, determinant, polynomial matrix multiplication. 1. INTRODUCTION The link between matrix multiplication and other basic linear algebra problems is well known under the algebraic complexity model. For K a commutative field, we will as sume that the product of two n n matrices over K can be Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. ISSAC03, August 36, 2003, Philadelphia, Pennsylvania, USA. Copyright 2003 ACM 1581136412/03/0008 ... $ 5.00. computed in O ( n ) operations in K . Under the model of computation trees over K , we know that is also the ex ponent of the problems of computing the determinant, the matrix inverse, the rank, the characteristic polynomial (we refer to the survey in [5, Chap.16]) or the Frobenius normal form [8, 15]. On an algebraic Ram , all these problems can be solved with O ( n ) operations in K , hence the correspond ing algorithms are optimal up to logarithmic terms. Here and in the rest of the paper, for any exponent e 1 , O ( n e 1 ) denotes O ( n e 1 log e 2 n ) for any exponent e 2 ....
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