Point Estimation of Root Finding Methods
This book sets out to state computationally verifiable initial conditions for predicting the immediate appearance of the guaranteed and fast convergence of iterative root finding methods. Attention is paid to iterative methods for simultaneous determinati
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1933
Miodrag Petkovi´c
Point Estimation of Root Finding Methods
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Miodrag Petkovi´c Department of Mathematics Faculty of Electronic Engineering University of Niš A. Medvedeva 14 P.O. Box 73 18000 Niš Serbia [email protected] [email protected] www.miodragpetkovic.com
ISBN: 978-3-540-77850-9 e-ISBN: 978-3-540-77851-6 DOI: 10.1007/978-3-540-77851-6 Lecture Notes in Mathematics ISSN print edition: 0075-8434 ISSN electronic edition: 1617-9692 Library of Congress Control Number: 2008921391 Mathematics Subject Classification (2000): 65H05 c 2008 Springer-Verlag Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: WMXDesign GmbH Printed on acid-free paper 987654321 springer.com
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Preface
The problem of solving nonlinear equations and systems of equations ranks among the most significant in the theory and practice, not only of applied mathematics but also of many branches of engineering sciences, physics, computer science, astronomy, finance, and so on. A glance at the bibliography and the list of great mathematicians who have worked on this topic points to a high level of contemporary interest. Although the rapid development of digital computers led to the effective implementation of many numerical methods, in practical realization, it is necessary to solve various problems such as computational efficiency based on the total central processor unit time, the construction of iterative methods which possess a fast convergence in the presence of multiplicity (or clusters) of a desired solution, the control of rounding errors, information about error bounds of obtained approximate solution, stating computationally verifiable initial conditions that ensure a safe convergence, etc. It is the solution of these challenging problems that was the principal motivation for the present study. In this book, we are mainly concerned with the statement and study of initial conditions that provide the guaranteed convergence of an iterative method for solving equations of the form f (z) = 0. The traditional approach to this problem is mainly based on asymptotic convergence analysis using some strong hypotheses on differentiability and derivative bounds in a rather wide domain. This kind of conditions often involves some
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