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Structured Matrix Based Methods for Approximate Polynomial GCD

  • Book
  • © 2011

Overview

Authors:
  1. Paola Boito

    1. Université de Limoges / CNRS, Limoges Cedex, France

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  • Topics situated at the crossroads between two fields of increasing interest to the mathematical community: symbolic-numeric polynomial computation and structured numerical linear algebra
  • Survey of the main tools and techniques used in either domain
  • State-of-the-art methods that exploit matrix structure to improve the performance of polynomial computations

Part of the book series: Publications of the Scuola Normale Superiore (PSNS, volume 15)

Part of the book sub series: Theses (Scuola Normale Superiore) (TSNS)

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Table of contents (9 chapters)

  1. Front Matter

    Pages i-xvi
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  2. Approximate polynomial GCD

    • Paola Boito
    Pages 1-20

  3. Structured and resultant matrices

    • Paola Boito
    Pages 21-43

  4. The Euclidean algorithm

    • Paola Boito
    Pages 45-58

  5. Matrix factorization and approximate GCDs

    • Paola Boito
    Pages 59-70

  6. Optimization approach

    • Paola Boito
    Pages 71-82

  7. New factorization-based methods

    • Paola Boito
    Pages 83-115

  8. A fast GCD algorithm

    • Paola Boito
    Pages 117-131

  9. Numerical tests

    • Paola Boito
    Pages 133-155

  10. Generalizations and further work

    • Paola Boito
    Pages 157-165

  11. Back Matter

    Pages 167-199
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Keywords

About this book

Defining and computing a greatest common divisor of two polynomials with inexact coefficients is a classical problem in symbolic-numeric computation. The first part of this book reviews the main results that have been proposed so far in the literature. As usual with polynomial computations, the polynomial GCD problem can be expressed in matrix form: the second part of the book focuses on this point of view and analyses the structure of the relevant matrices, such as Toeplitz, Toepliz-block and displacement structures. New algorithms for the computation of approximate polynomial GCD are presented, along with extensive numerical tests. The use of matrix structure allows, in particular, to lower the asymptotic computational cost from cubic to quadratic order with respect to polynomial degree.

Authors and Affiliations

  • Université de Limoges / CNRS, Limoges Cedex, France

    Paola Boito

Bibliographic Information

  • Book Title: Structured Matrix Based Methods for Approximate Polynomial GCD

  • Authors: Paola Boito

  • Series Title: Publications of the Scuola Normale Superiore

  • DOI: https://doi.org/10.1007/978-88-7642-381-9

  • Publisher: Edizioni della Normale Pisa

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Scuola Normale Superiore Pisa 2011

  • Softcover ISBN: 978-88-7642-380-2Published: 12 July 2011

  • eBook ISBN: 978-88-7642-381-9Published: 13 March 2012

  • Series ISSN: 2239-1460

  • Series E-ISSN: 2532-1668

  • Edition Number: 1

  • Number of Pages: 250

  • Topics: Algebra

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