Numerical continuation methods - an introduction

1 Introduction. - 2 The Basic Principles of Continuation Methods. - 2.1 Implicitly Defined Curves. - 2.2 The Basic Concepts of PC Methods. - 2.3 The Basic Concepts of PL Methods. - 3 Newton's Method as Corrector. - 3.1 Motivation. - 3.2 The Moore-Penrose Inverse in a Special Case. - 3.3 A Newton's Step for Underdetermined Nonlinear Systems. - 3.4 Convergence Properties of Newton's Method. - 4 Solving the Linear Systems. - 4.1 Using a QR Decomposition. - 4.2 Givens Rotations for Obtaining a QR Decomposition. - 4.3 Error Analysis. - 4.4 Scaling of the Dependent Variables. - 4.5 Using LU Decompositions. - 5 Convergence of Euler-Newton-Like Methods. - 5.1 An Approximate Euler-Newton Method. - 5.2 A Convergence Theorem for PC Methods. - 6 Steplength Adaptations for the Predictor. - 6.1 Steplength Adaptation by Asymptotic Expansion. - 6.2 The Steplength Adaptation of Den Heijer & Rheinboldt. - 6.3 Steplength Strategies Involving Variable Order Predictors. - 7 Predictor-Corrector Methods Using Updating. - 7.1 Broyden's "Good" Update Formula. - 7.2 Broyden Updates Along a Curve. - 8 Detection of Bifurcation Points Along a Curve. - 8.1 Simple Bifurcation Points. - 8.2 Switching Branches Via Perturbation. - 8.3 Branching Off Via the Bifurcation Equation. - 9 Calculating Special Points of the Solution Curve. - 9.1 Introduction. - 9.2 Calculating Zero Points f(c(s)) = 0. - 9.3 Calculating Extremal Points minsf((c(s)). - 10 Large Scale Problems. - 10.1 Introduction. - 10.2 General Large Scale Solvers. - 10.3 Nonlinear Conjugate Gradient Methods as Correctors. - 11 Numerically Implementable Existence Proofs. - 11.1 Preliminary Remarks. - 11.2 An Example of an Implementable Existence Theorem. - 11.3 Several Implementations for Obtaining Brouwer Fixed Points. - 11.4 Global Newton and Global Homotopy Methods. - 11.5 Multiple Solutions. - 11.6 Polynomial Systems. - 11.7 Nonlinear Complementarity. - 11.8 Critical Points and Continuation Methods. - 12 PL Continuation Methods. - 12.1 Introduction. - 12.2 PL Approximations. - 12.3 A PL Algorithm for Tracing H(u) = 0. - 12.4 Numerical Implementation of a PL Continuation Algorithm. - 12.5 Integer Labeling. - 12.6 Truncation Errors. - 13 PL Homotopy Algorithms. - 13.1 Set-Valued Maps. - 13.2 Merrill's Restart Algorithm. - 13.3 Some Triangulations and their Implementations. - 13.4 The Homotopy Algorithm of Eaves & Saigal. - 13.5 Mixing PL and Newton Steps. - 13.6 Automatic Pivots for the Eaves-Saigal Algorithm. - 14 General PL Algorithms on PL Manifolds. - 14.1 PL Manifolds. - 14.2 Orientation and Index. - 14.3 Lemke's Algorithm for the Linear Complementarity Problem. - 14.4 Variable Dimension Algorithms. - 14.5 Exploiting Special Structure. - 15 Approximating Implicitly Defined Manifolds. - 15.1 Introduction. - 15.2 Newton's Method and Orthogonal Decompositions Revisited. - 15.3 The Moving Frame Algorithm. - 15.4 Approximating Manifolds by PL Methods. - 15.5 Approximation Estimates. - 16 Update Methods and their Numerical Stability. - 16.1 Introduction. - 16.2 Updates Using the Sherman-Morrison Formula. - 16.3 QR Factorization. - 16.4 LU Factorization. - P1 A Simple PC Continuation Method. - P2 A PL Homotopy Method. - P3 A Simple Euler-Newton Update Method. - P4 A Continuation Algorithm for Handling Bifurcation. - P5 A PL Surface Generator. - P6 SCOUT - Simplicial Continuation Utilities. - P6.1 Introduction. - P6.2 Computational Algorithms. - P6.3 Interactive Techniques. - P6.4 Commands. - P6.5 Example: Periodic Solutions to a Differential Delay Equation. - Index and Notation.