Structured Matrix Based Methods for Approximate Polynomial GCD

✍ Scribed by Paola Boito (auth.)

• Publisher: Edizioni della Normale
• Year: 2011
• Tongue: English
• Leaves: 207
• Series: Tesi/Theses 15
• Edition: 1
• Category: Library

✦ 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

Front Matter....Pages i-xvi
Approximate polynomial GCD....Pages 1-20
Structured and resultant matrices....Pages 21-43
The Euclidean algorithm....Pages 45-58
Matrix factorization and approximate GCDs....Pages 59-70
Optimization approach....Pages 71-82
New factorization-based methods....Pages 83-115
A fast GCD algorithm....Pages 117-131
Numerical tests....Pages 133-155
Generalizations and further work....Pages 157-165
Back Matter....Pages 167-199

✦ Subjects

Algebra