Author | Allgower, Eugene L. author |
---|---|
Title | Numerical Continuation Methods [electronic resource] : An Introduction / by Eugene L. Allgower, Kurt Georg |
Imprint | Berlin, Heidelberg : Springer Berlin Heidelberg, 1990 |
Connect to | http://dx.doi.org/10.1007/978-3-642-61257-2 |
Descript | XIV, 388 p. online resource |
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 Newtonโs Method as Corrector -- 3.1 Motivation -- 3.2 The Moore-Penrose Inverse in a Special Case -- 3.3 A Newtonโs Step for Underdetermined Nonlinear Systems -- 3.4 Convergence Properties of Newtonโs 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 Broydenโs โGoodโ 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 Merrillโs 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 Lemkeโs 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 Newtonโs 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 โ 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