Numerical Continuation Methods: An Introduction

✍ Scribed by Eugene L. Allgower, Kurt Georg (auth.)

• Publisher: Springer-Verlag Berlin Heidelberg
• Year: 1990
• Tongue: English
• Leaves: 401
• Series: Springer Series in Computational Mathematics 13
• Edition: 1
• Category: Library

✦ Synopsis

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 (tracing of implicitly defined curves) of both predictor-corrector and piecewise-linear types. It presents and analyzes implementations aimed at applications to the computation of zero points, fixed points, nonlinear eigenvalue problems, bifurcation and turning points, and economic equilibria. Many algorithms are presented in a pseudo code format. An appendix supplies five sample FORTRAN programs with numerical examples, which readers can adapt to fit their purposes, and a description of the program package SCOUT for analyzing nonlinear problems via piecewise-linear methods. An extensive up-to-date bibliography spanning 46 pages is included. The material in this book has been presented to students of mathematics, engineering and sciences with great success, and will also serve as a valuable tool for researchers in the field.

✦ Table of Contents

Front Matter....Pages I-XIV
Introduction....Pages 1-6
The Basic Principles of Continuation Methods....Pages 7-16
Newton’s Method as Corrector....Pages 17-27
Solving the Linear Systems....Pages 28-36
Convergence of Euler-Newton-Like Methods....Pages 37-43
Steplength Adaptations for the Predictor....Pages 44-60
Predictor-Corrector Methods Using Updating....Pages 61-74
Detection of Bifurcation Points Along a Curve....Pages 75-90
Calculating Special Points of the Solution Curve....Pages 91-95
Large Scale Problems....Pages 96-111
Numerically Implementable Existence Proofs....Pages 112-150
PL Continuation Methods....Pages 151-172
PL Homotopy Algorithms....Pages 173-202
General PL Algorithms on PL Manifolds....Pages 203-232
Approximating Implicitly Defined Manifolds....Pages 233-251
Update Methods and their Numerical Stability....Pages 252-265
Back Matter....Pages 346-388

✦ Subjects

Numerical Analysis; Economic Theory