Description

Book Synopsis
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.

Table of Contents
i. Introduction.- ii. Notation.- 1. Approximate polynomial GCD.- 2. Structured and resultant matrices.- 3. The Euclidean algorithm.- 4. Matrix factorization and approximate GCDs.- 5. Optimization approach.- 6. New factorization-based methods.- 7. A fast GCD algorithm.- 8. Numerical tests.- 9. Generalizations and further work.- 10. Appendix A: Distances and norms.- 11. Appendix B: Special matrices.- 12. Bibliography.- 13. Index.

Structured Matrix Based Methods for Approximate

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      Book details

      Publisher Birkhauser Verlag AG
      Published 12 July 2011
      ISBN-13 9788876423802
      978-8876423802
      ISBN-10 887642380X
      Also in:
      Algebra

      Description

      Book Synopsis
      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.

      Table of Contents
      i. Introduction.- ii. Notation.- 1. Approximate polynomial GCD.- 2. Structured and resultant matrices.- 3. The Euclidean algorithm.- 4. Matrix factorization and approximate GCDs.- 5. Optimization approach.- 6. New factorization-based methods.- 7. A fast GCD algorithm.- 8. Numerical tests.- 9. Generalizations and further work.- 10. Appendix A: Distances and norms.- 11. Appendix B: Special matrices.- 12. Bibliography.- 13. Index.

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