Newton Methods for Nonlinear Problems: Affine Invariance and Adaptive Algorithms

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Springer Science & Business Media, Sep 18, 2011 - Mathematics - 424 pages
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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.
 

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Contents

1 Introduction
7
Part I ALGEBRAIC EQUATIONS
42
Local Newton Methods
43
Global Newton Methods
109
GaussNewton Methods
173
Continuation Methods
232
Part II DIFFERENTIAL EQUATIONS
283
6 Stiff ODE Initial Value Problems
285
7 ODE Boundary Value Problems
315
8 PDE Boundary Value Problems
369
References
405
Software
416
Index
419
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About the author (2011)

Peter Deuflhard is founder and head of the internationally renowned Zuse Institute Berlin (ZIB) and full professor of Numerical Analysis and Scientific Computing at the Free University of Berlin. He is a regular invited speaker at international conferences and universities as well as industry places all over the world.