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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
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.
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.