Newton Methods for Nonlinear Problems: Affine Invariance and Adaptive Algorithms

Peter Deuflhard (auth.)3540210997, 9783540210993, 3642059279, 9783642059278

This book deals with the efficient numerical solution of challenging nonlinear problems in science and engineering, both in finite dimension (algebraic systems) and in infinite dimension (ordinary and partial differential equations). Its focus is on local and global Newton methods for direct problems or Gauss-Newton methods for inverse problems. The term ‘affine invariance’ means that the presented algorithms and their convergence analysis are invariant under one out of four subclasses of affine transformations of the problem to be solved. Compared to traditional textbooks, the distinguishing affine invariance approach leads to shorter theorems and proofs and permits the construction of fully adaptive algorithms. Lots of numerical illustrations, comparison tables, and exercises make the text useful in computational mathematics classes. At the same time, the book opens many directions for possible future research.

Table of contents :
Front Matter….Pages i-xii
Introduction….Pages 7-41
Front Matter….Pages 43-43
Systems of Equations: Local Newton Methods….Pages 45-107
Systems of Equations: Global Newton Methods….Pages 109-172
Least Squares Problems: Gauss-Newton Methods….Pages 173-231
Parameter Dependent Systems: Continuation Methods….Pages 233-282
Front Matter….Pages 283-283
Stiff ODE Initial Value Problems….Pages 285-314
ODE Boundary Value Problems….Pages 315-368
PDE Boundary Value Problems….Pages 369-404
Back Matter….Pages 405-424