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How to be sure of finding a root of a complex polynomial using Newton's method

How to be sure of finding a root of a complex polynomial using Newton's method The trouble with Newton's method for finding the roots of a complex polynomial is knowing where to start the iteration. In this paper we apply the theory of rational maps and some estimates based on distortion theorems for univalent functions to find lower bounds, depending only on the degreed, for the size of regions from which the iteration will certainly converge to a root. We can also bound the number of iterations required and we give a method that works for every polynomial and takes at most some constant timesd 2(logd)2 log(d 3/∈) iterations to find one root to within an accuracy of ∈. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the Brazilian Mathematical Society, New Series Springer Journals

How to be sure of finding a root of a complex polynomial using Newton's method

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References (22)

Publisher
Springer Journals
Copyright
Copyright © 1992 by Sociedade Brasileira de Matemática
Subject
Mathematics; Mathematics, general; Theoretical, Mathematical and Computational Physics
ISSN
1678-7544
eISSN
1678-7714
DOI
10.1007/BF01232940
Publisher site
See Article on Publisher Site

Abstract

The trouble with Newton's method for finding the roots of a complex polynomial is knowing where to start the iteration. In this paper we apply the theory of rational maps and some estimates based on distortion theorems for univalent functions to find lower bounds, depending only on the degreed, for the size of regions from which the iteration will certainly converge to a root. We can also bound the number of iterations required and we give a method that works for every polynomial and takes at most some constant timesd 2(logd)2 log(d 3/∈) iterations to find one root to within an accuracy of ∈.

Journal

Bulletin of the Brazilian Mathematical Society, New SeriesSpringer Journals

Published: Feb 11, 2005

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