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Introduction to Numerical Continuation MethodsIntroduction to Numerical Continuation Methods

Eugene L. Allgower and Kurt Georg


Classics in Applied Mathematics 45

Numerical continuation methods have provided important contributions toward the numerical solution of nonlinear systems of equations for many years. The methods may be used not only to compute solutions, which might otherwise be hard to obtain, but also to gain insight into qualitative properties of the solutions. Introduction to Numerical Continuation Methods, originally published in 1979, was the first book to provide easy access to the numerical aspects of predictor corrector continuation and piecewise linear continuation methods. Not only do these seemingly distinct methods share many common features and general principles, they can be numerically implemented in similar ways. Introduction to Numerical Continuation Methods also features the piecewise linear approximation of implicitly defined surfaces, the algorithms of which are frequently used in computer graphics, mesh generation, and the evaluation of surface integrals.

To help potential users of numerical continuation methods create programs adapted to their particular needs, this book presents pseudo-codes and Fortran codes as illustrations. Since it first appeared, many specialized packages for treating such varied problems as bifurcation, polynomial systems, eigenvalues, economic equilibria, optimization, and the approximation of manifolds have been written. The original extensive bibliography has been updated in the SIAM Classics edition to include more recent references and several URLs so users can look for codes to suit their needs or write their own based on the models included in the book.

Audience

This book continues to be useful for researchers and graduate students in mathematics, sciences, engineering, economics, and business looking for an introduction to computational methods for solving a large variety of nonlinear systems of equations. A background in elementary analysis and linear algebra are adequate prerequisites for reading this book; some knowledge from a first course in numerical analysis may also be helpful.

Contents

Table of Pseudo Codes; Preface to the Classics Edition; Foreword; Chapter 1: Introduction; Chapter 2: The Basic Principles of Continuation Methods; Chapter 3: Newton's Method as Corrector; Chapter 4: Solving the Linear Systems; Chapter 5: Convergence of Euler-Newton-Like Methods; Chapter 6: Steplength Adaptations for the Predictor; Chapter 7: Predictor-Corrector Methods Using Updating; Chapter 8: Detection of Bifurcation Points Along a Curve; Chapter 9: Calculating Special Points of the Solution Curve; Chapter 10: Large Scale Problems; Chapter 11: Numerically Implementable Existence Proofs; Chapter 12: PL Continuation Methods; Chapter 13: PL Homotopy Algorithms; Chapter 14: General PL Algorithms on PL Manifolds; Chapter 15: Approximating Implicitly DeŻned Manifolds; Chapter 16: Update Methods and their Numerical Stability; Appendix 1: A Simple PC Continuation Method; Appendix 2: A PL Homotopy Method; Appendix 3: A Simple Euler Newton Update Method; Appendix 4: A Continuation Algorithm for Handling Bifurcation; Appendix 5: A PL Surface Generator; Appendix 6: SCOUT | Simplicial Continuation Utilities; Bibliography; Index and Notation.

2003 / xxvi + 388 pages / Softcover / ISBN-13: 978-0-898715-44-6 / ISBN-10: 0-89871-544-X /
List Price $74.50 / SIAM Member Price $52.15 / Order Code CL45
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