Numerical Continuation Methods

1. Front Matter

   Pages I-XIV

   PDF

2. Introduction

   • Eugene L. Allgower, Kurt Georg

   Pages 1-6

3. The Basic Principles of Continuation Methods

   • Eugene L. Allgower, Kurt Georg

   Pages 7-16

4. Newton’s Method as Corrector

   • Eugene L. Allgower, Kurt Georg

   Pages 17-27

5. Solving the Linear Systems

   • Eugene L. Allgower, Kurt Georg

   Pages 28-36

6. Convergence of Euler-Newton-Like Methods

   • Eugene L. Allower, Kurt Georg

   Pages 37-43

7. Steplength Adaptations for the Predictor

   • Eugene L. Allower, Kurt Georg

   Pages 44-60

8. Predictor-Corrector Methods Using Updating

   • Eugene L. Allgower, Kurt Georg

   Pages 61-74

9. Detection of Bifurcation Points Along a Curve

   • Eugene L. Allower, Kurt Georg

   Pages 75-90

10. Calculating Special Points of the Solution Curve

    • Eugene L. Allower, Kurt Georg

    Pages 91-95

11. Large Scale Problems

    • Eugene L. Allower, Kurt Georg

    Pages 96-111

12. Numerically Implementable Existence Proofs

    • Eugene L. Allower, Kurt Georg

    Pages 112-150

13. PL Continuation Methods

    • Eugene L. Allower, Kurt Georg

    Pages 151-172

14. PL Homotopy Algorithms

    • Eugene L. Allgower, Kurt Georg

    Pages 173-202

15. General PL Algorithms on PL Manifolds

    • Eugene L. Allgower, Kurt Georg

    Pages 203-232

16. Approximating Implicitly Defined Manifolds

    • Eugene L. Allower, Kurt Georg

    Pages 233-251

17. Update Methods and their Numerical Stability

    • Eugene L. Allower, Kurt Georg

    Pages 252-265

18. Back Matter

    Pages 346-388

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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.

• Algorithmus
• Approximation
• Numerical continuation
• Verzweigungspunkte
• bifurcation
• boundary element method
• complementary pivoting
• computation
• curve and surface approximation
• eigenvalue problem
• nichtlineare Differentialgleichung
• nonlinear systems
• nonlinear systems of equations
• numerische Kontinuitätsmethoden
• strategy

• Department of Mathematics, Colorado State University, Fort Collins, USA

  Eugene L. Allgower, Kurt Georg

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