Fast Order Basis and Kernel Basis Computation and Related Problems
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Date
2013-01-28T16:03:52Z
Authors
Zhou, Wei
Journal Title
Journal ISSN
Volume Title
Publisher
University of Waterloo
Abstract
In this thesis, we present efficient deterministic algorithms
for polynomial matrix computation problems, including the computation
of order basis, minimal kernel basis, matrix inverse, column basis,
unimodular completion, determinant, Hermite normal form, rank and
rank profile for matrices of univariate polynomials over a field.
The algorithm for kernel basis computation also immediately provides
an efficient deterministic algorithm for solving linear systems. The
algorithm for column basis also gives efficient deterministic algorithms
for computing matrix GCDs, column reduced forms, and Popov normal
forms for matrices of any dimension and any rank.
We reduce all these problems to polynomial matrix multiplications.
The computational costs of our algorithms are then similar to the
costs of multiplying matrices, whose dimensions match the input matrix
dimensions in the original problems, and whose degrees equal the average
column degrees of the original input matrices in most cases. The use
of the average column degrees instead of the commonly used matrix
degrees, or equivalently the maximum column degrees, makes our computational
costs more precise and tighter. In addition, the shifted minimal bases
computed by our algorithms are more general than the standard minimal
bases.
Description
Keywords
polynomial matrix computation, algorithm, complexity, computer algebra, order basis, kernel basis, linear system solving, matrix inverse, determinant, unimodular completion, Popov form, Hermite form, column reduced form, GCD, matrix GCD, rank, rank profile, column basis