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Structured Matrix Based Methods for Approximate Polynomial GCD / by Paola Boito
Resource type: Ressourcentyp: Buch (Online)Book (Online)Language: English Series: Tesi/Theses ; 15 | SpringerLink BücherPublisher: Pisa : Edizioni della Normale, 2011Description: Online-Ressource (250p, digital)ISBN:- 9788876423819
- 512.9422
- 512
- QA150-272
Contents:
Summary: 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.Summary: 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. .PPN: PPN: 1651395764Package identifier: Produktsigel: ZDB-2-SEB | ZDB-2-SXMS | ZDB-2-SMA
Title Page; Copyright Page; Table of Contents; Introduction; Acknowledgements; Notation; Chapter 1 Approximate polynomial GCD; 1.1. Coefficient-based definitions; 1.1.1. Quasi-GCD; 1.1.2. ε−GCD; 1.1.3. AGCD; 1.2. A geometrical interpretation; 1.3. Pseudozeros and root neighborhoods; 1.4. A root-based definition; 1.5. Graph-theoretical techniques for bounding the degree of the approximate GCDs; 1.6. Formulations of the approximate GCD problem; Chapter 2 Structured and resultant matrices; 2.1. Toeplitz and Hankel matrices; 2.2. Displacement structure; 2.2.1. Toeplitz-like matrices
2.2.2. Cauchy-like matrices2.3. Computation of displacement generators; 2.3.1. Sum of displacement structured matrices; 2.3.2. Product of displacement structured matrices; 2.3.3. Inverse of a displacement structured matrix; 2.4. Fast GEPP for structured matrices; 2.4.1. The GKO algorithm; 2.4.2. The Toeplitz-like case; 2.5. The Sylvester matrix; 2.6. The Bézout matrix; 2.7. More results on resultant matrices; Chapter 3 The Euclidean algorithm; 3.1. The classical algorithm; 3.1.1. Matrix representation; 3.1.2. Generalizations; 3.2. Stabilized numerical versions; 3.3. Relative primality
3.4. A look-ahead method3.5. Padè approximation; 3.6. Euclidean algorithm and factorization of the Bezoutian; Chapter 4 Matrix factorization and approximate GCDs; 4.1. Approximate rank and the SVD; 4.1.1. The SVD of Sylvester and Bézout matrices; 4.1.2. The SVD of subresultants; 4.1.3. Statistical approaches; 4.2. An SVD-based algorithm; 4.3. QR factorization of the Sylvester matrix; Chapter 5 Optimization approach; 5.1. The case of degree one; 5.2. Structured low-rank approximation; 5.2.1. Lift-and-project algorithm; 5.2.2. Structured total least norm; 5.3. The divisor - quotient iteration
5.4. More optimization strategies5.4.1. Another STLN approach; 5.4.2. Sums of squares relaxation; 5.4.3. Gradient-projection methods; Chapter 6 New factorization-based methods; 6.1. Real vs. complex coefficients; 6.2. Sylvester vs. Bèzout matrices; 6.3. QR Factorization; 6.3.1. Stability of the QR factorization; 6.3.2. Degree of an ε-GCD; 6.3.3. Coefficients of an ε-GCD; 6.3.4. Iterative refinement; 6.3.5. Rank-deficient Jacobian; 6.3.6. An algorithm based on QR decomposition; 6.4. Conditioning; 6.4.1. Degree; 6.4.2. Coefficients; 6.5. Tridiagonalization; 6.5.1. The exact case
6.5.2. The approximate case: rank determination6.5.3. The algorithm; 6.6. More factorization-based algorithms; 6.6.1. SVD of the Bèzout matrix; 6.6.2. QR decomposition of the Bèzout matrix; 6.6.3. QR decomposition of the companion matrix resultant; Chapter 7 A fast GCD algorithm; 7.1. Stability issues; 7.2. Fast and stable factorization of rectangular matrices; 7.3. Computing a tentative GCD; 7.4. Fast iterative refinement; 7.4.1. Iterative refinement with line search; 7.5. Choice of a tentative degree; 7.5.1. Is LU factorization rank-revealing?; 7.6. The Fastgcd algorithm
Chapter 8 Numerical tests
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