NDLI: On the complexity of polynomial matrix computations

Content Provider	ACM Digital Library
Author	Jeannerod, Claude-Pierre Giorgi, Pascal Villard, Gilles
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 ntimes n matrices in K[x] of degree bounded by d, with K a commutative field. Under the straight-line program model we show that multiplication is reducible to the problem of computing the coefficient of degree d of the determinant. Conversely, we propose algorithms for minimal approximant computation and column reduction that are based on polynomial matrix multiplication; for the determinant, the straight-line program we give also relies on matrix product over K[x] and provides an alternative to the determinant algorithm of [16, 17]. We further show that all these problems can be solved in particular in O $}(^{ω})$ operations in K. Here the "soft O" notation O indicates some missing log (nd) factors and ω is the exponent of matrix multiplication over K.
Starting Page	135
Ending Page	142
Page Count	8
File Format	PDF
ISBN	1581136412
DOI	10.1145/860854.860889
Language	English
Publisher	Association for Computing Machinery (ACM)
Publisher Date	2003-08-03
Publisher Place	New York
Access Restriction	Subscribed
Subject Keyword	Minimal basis Matrix gcd Column reduced form Polynomial matrix multiplication Determinant Matrix polynomial
Content Type	Text
Resource Type	Article

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On the complexity of polynomial matrix computations