Numerical Continuation Methods An Introduction
Over the past fifteen years two new techniques have yielded extremely important contributions toward the numerical solution of nonlinear systems of equations. This book provides an introduction to and an up-to-date survey of numerical continuation methods
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Editorial Board
R. L. Graham, Murray Hill J. Stoer, WOrzburg R. Varga, Kent (Ohio)
Eugene L. Allgower Kurt Georg
Numerical Continuation Methods An Introduction With 37 Figures
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong
Eugene L.Allgower Kurt Georg Department of Mathematics Colorado State University Fort Collins, CO 80523, USA
Mathematics Subject Classification (1980): 65H10, 65K05, 90C30 ISBN-13: 978-3-642-64764-2 e-ISBN-13: 978-3-642-61257-2 DOl: 10.1007/978-3-642-61257-2 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 microfilms or in other ways, and storage in data banks. Duplication of this publication or parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its current version, and a copyright fee must always be paid. Violations fall under the prosecution act of the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1990
Softcover reprint of the hardcover 18t edition 1990 2141/3140 - 5 4 3 210
Foreword
Over the past ten to fifteen years two new techniques have yielded extremely important contributions toward the numerical solution of nonlinear systems of equations. These two methods have been called by various names. One of the methods has been called the predictor-corrector or pseudo arc-length continuation method. This method has its historical roots in the imbedding and incremental loading methods which have been successfully used for several decades by engineers and scientists to improve convergence properties when an adequate starting value for an iterative method is not available. The second method is often referred to as the simplicial or piecewise linear method. This method has its historical roots in the Lemke-Howson algorithm for solving nonlinear complementarity problems. The idea of complementary pivoting has been adapted and applied to the calculation of fixed points of continuous maps and of semi-continuous set valued maps. In this book we endeavor to provide an easy access for scientific workers and students to the numerical aspects of both of these methods. As a by-product of our discussions we hope that it will become evident to the reader that these two seemingly very distinct methods are actually rather closely related in a number of ways. The two numerical methods have many common features and are based on similar general principles. This holds even for the numerical implementations. Hence we have elected to refer to both of these methods as continuation methods. The techniques based on predictor and corrector steps and exploiting differentiability are referred to as "predictor-corrector continuation methods". The techniques based on piecewise linear approximations are referred to as "piecewise linear continuation methods" . Chapters 3-10 treat the predictor-corrector methods primarily, and chapters 12-16
