Initial Approximations and Root Finding Methods - Couverture souple

Kyurkchiev, Nikolay N.

Couverture souple

ISBN 10 : 3527401326 ISBN 13 : 9783527401321

Editeur : Wiley-VCH Verlag GmbH, 1998

Synopsis

This text provides an update on iterative methods of calculating simultaneously all roots of a polynomial. A survey on basic facts is followed by a discussion of methods and properties connected with the classical task of the approximative determination of roots. Finally, new ideas and research results for the computer determination are discussed, which facilitate the practical numerical treatment of polynomials.

A propos du livre

Quatrième de couverture

One of the crucial problems in solving polynomial equations is the construction of such initial conditions which provide a safe convergence of the considered numerical method. In general the problem of the choice of initial approximations is a very difficult one and the convergence of any iterative algorithm for finding all roots of a given polynomial is strongly connected with the distribution of its zeros. The author’s investigations concern the opposite problem, i. e. the divergent feature: Can we state such initial conditions under which the considered numerical method fails? Situations are considered in which the critical initial approximations fail, when the iterative methods of Weierstrass, Euler–Chebyshev, Alefeld–Herzberger, Nourein, Dvorcuk, Kyurkchiev–Andreev and Wang–Zheng are applied. Investigations of divergent starting point for every numerical method show that for any monic polynomial f of degree n there exists a set G f ??? C n such that these methods, starting from z o = z &Egr; G f, do not converge to the roots of f. This NS–set yields divergent starting points and is obtained as the set of solutions of a non–linear system of n equations. The study of the distribution and measure of NS–sets is very complicated and does not appear in the literature. Non–attractive starting points for iterative algorithms are presented which facilitate the choice of the initial approximations for the user. This gives an improvement of all results available until now.

Présentation de l'éditeur

Polynomials as mathematical objects have been studied extensively for a long time, and the knowledge collected about them is enormous. Polynomials appear in various fields of applied mathematics and engineering, from mathematics of finance up to signal theory or robust control. The calculation of the roots of a polynomial is a basic problems of numerical mathematics.
In this book, an update on iterative methods of calculating simultaneously all roots of a polynomial is given: a survey on basic facts, a lot of methods and properties of those methods connected with the classical task of the approximative determination of roots. For the computer determination the choice of the initial approximation is of special importance. Here the authors offers his new ideas and research results of the last decade which facilitate the practical numerical treatment of polynomials.

Détails de l'édition

• Éditeur: Wiley-VCH Verlag GmbH
• Date d'édition: 1998
• Langue: anglais
• ISBN 10: 3527401326
• ISBN 13: 9783527401321
• Reliure: Broché
• Nombre de pages: 200
• Coordonnées du fabricant: non disponible
• Personne responsable: non disponible