Error-Free Polynomial Matrix Computations
Error-Free Polynomial Matrix Computations

Author: E.V. Krishnamurthy

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 170

ISBN-13: 1461251184

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This book is written as an introduction to polynomial matrix computa tions. It is a companion volume to an earlier book on Methods and Applications of Error-Free Computation by R. T. Gregory and myself, published by Springer-Verlag, New York, 1984. This book is intended for seniors and graduate students in computer and system sciences, and mathematics, and for researchers in the fields of computer science, numerical analysis, systems theory, and computer algebra. Chapter I introduces the basic concepts of abstract algebra, including power series and polynomials. This chapter is essentially meant for bridging the gap between the abstract algebra and polynomial matrix computations. Chapter II is concerned with the evaluation and interpolation of polynomials. The use of these techniques for exact inversion of poly nomial matrices is explained in the light of currently available error-free computation methods. In Chapter III, the principles and practice of Fourier evaluation and interpolation are described. In particular, the application of error-free discrete Fourier transforms for polynomial matrix computations is consi dered.

Fast Error-free Algorithms for Polynomial Matrix Computations
Fast Error-free Algorithms for Polynomial Matrix Computations

Author: John S. Baras

Publisher:

Published: 1990

Total Pages: 29

ISBN-13:

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In this paper we develop highly efficient, error-free algorithms, for most of the important computations needed in linear systems over fields or rings. We show that the structure of the underlying rings and modules is critical in designing such algorithms. We also discuss the importance of such algorithms for controller synthesis."

Polynomial Algorithms in Computer Algebra
Polynomial Algorithms in Computer Algebra

Author: Franz Winkler

Publisher: Springer Science & Business Media

Published: 2012-12-06

Total Pages: 284

ISBN-13: 3709165717

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For several years now I have been teaching courses in computer algebra at the Universitat Linz, the University of Delaware, and the Universidad de Alcala de Henares. In the summers of 1990 and 1992 I have organized and taught summer schools in computer algebra at the Universitat Linz. Gradually a set of course notes has emerged from these activities. People have asked me for copies of the course notes, and different versions of them have been circulating for a few years. Finally I decided that I should really take the time to write the material up in a coherent way and make a book out of it. Here, now, is the result of this work. Over the years many students have been helpful in improving the quality of the notes, and also several colleagues at Linz and elsewhere have contributed to it. I want to thank them all for their effort, in particular I want to thank B. Buchberger, who taught me the theory of Grabner bases nearly two decades ago, B. F. Caviness and B. D. Saunders, who first stimulated my interest in various problems in computer algebra, G. E. Collins, who showed me how to compute in algebraic domains, and J. R. Sendra, with whom I started to apply computer algebra methods to problems in algebraic geometry. Several colleagues have suggested improvements in earlier versions of this book. However, I want to make it clear that I am responsible for all remaining mistakes.

Effective Polynomial Computation
Effective Polynomial Computation

Author: Richard Zippel

Publisher: Springer Science & Business Media

Published: 1993-07-31

Total Pages: 386

ISBN-13: 9780792393757

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Effective Polynomial Computation is an introduction to the algorithms of computer algebra. It discusses the basic algorithms for manipulating polynomials including factoring polynomials. These algorithms are discussed from both a theoretical and practical perspective. Those cases where theoretically optimal algorithms are inappropriate are discussed and the practical alternatives are explained. Effective Polynomial Computation provides much of the mathematical motivation of the algorithms discussed to help the reader appreciate the mathematical mechanisms underlying the algorithms, and so that the algorithms will not appear to be constructed out of whole cloth. Preparatory to the discussion of algorithms for polynomials, the first third of this book discusses related issues in elementary number theory. These results are either used in later algorithms (e.g. the discussion of lattices and Diophantine approximation), or analogs of the number theoretic algorithms are used for polynomial problems (e.g. Euclidean algorithm and p-adic numbers). Among the unique features of Effective Polynomial Computation is the detailed material on greatest common divisor and factoring algorithms for sparse multivariate polynomials. In addition, both deterministic and probabilistic algorithms for irreducibility testing of polynomials are discussed.

Computing Matrix Canonical Forms of Ore Polynomials
Computing Matrix Canonical Forms of Ore Polynomials

Author: Mohamed Khochtali

Publisher:

Published: 2017

Total Pages: 72

ISBN-13:

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We present algorithms to compute canonical forms of matrices of Ore polynomials while controlling intermediate expression swell. Given a square non-singular input matrix of Ore polynomials, we give an extension of the algorithm by Labhalla et al. 1992, to compute the Hermite form. We also give a new fraction-free algorithm to compute the Popov form, accompanied by an implementation and experimental results that compare it to the best known algorithms in the literature. Our algorithm is output-sensitive, with a cost that depends on the orthogonality defect of the input matrix: the sum of the row degrees in the input matrix minus the sum of the row degrees in the Popov form. We also use the recent advances in polynomial matrix computations, including fast inversion and rank profile computation, to describe an algorithm that computes the transformation matrix corresponding to the Popov form.

A Fast Las Vegas Algorithm for Computing the Smith Normal Form of a Polynomial Matrix
A Fast Las Vegas Algorithm for Computing the Smith Normal Form of a Polynomial Matrix

Author: Arne Storjohann

Publisher:

Published: 1994

Total Pages: 17

ISBN-13:

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Abstract: "A Las Vegas probabilistic algorithm is presented that finds the Smith normal form S [element of] Q[x][superscript nxn] of a nonsingular input matrix A [element of] Z[x][superscript nxn]. The algorithm requires an expected number of O(̃n3d(d+n2log [parallel]A)) bit operations (where [parallel]A bounds the magnitude of all integer coefficients appearing in A and d bounds the degrees of entries of A). In practice, the main cost of the computation is obtaining a non-unimodular triangularization of a polynomial matrix of same dimension and with similar size entries as the imput matrix. We show how to accomplish this in O(̃n5d(d+log [parallel]A) log [parallel]A) bit operations using standard integer, polynomial and matrix arithmetic. These complexity results improve significantly on previous algorithms in both a theoretical and practical sense."

A New Approach to Fast Polynomial Interpolation and Multipoint Evaluation
A New Approach to Fast Polynomial Interpolation and Multipoint Evaluation

Author: International Computer Science Institute

Publisher:

Published: 1992

Total Pages: 6

ISBN-13:

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Abstract: "The fastest known algorithms for the problems of polynomial evaluation and multipoint interpolation are devastatingly unstable numerically because of their recursive use of polynomial divisions. We apply a completely distinct approach to compute approximate solutions to both problems equally fast but with improved numerical stability. Our approach relies on new techniques, so far not used in this area: we reduce the problems to Vandermonde matrix computations and then exploit some recent methods for improving computations with structured matrices."