Iterative Solution of Nonlinear Equations in Several Variables
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Computer Science and Applied Mathematics: Iterative Solution of Nonlinear Equations in Several Variables presents a survey of the basic theoretical results about nonlinear equations in n dimensions and analysis of the major iterative methods for their numerical solution. This book discusses the gradient mappings and minimization, contractions and the continuation property, and degree of a mapping. The general iterative and minimization methods, rates of convergence, and one-step stationary and multistep methods are also elaborated. This text likewise covers the contractions and nonlinear majorants, convergence under partial ordering, and convergence of minimization methods. This publication is a good reference for specialists and readers with an extensive functional analysis background.
Table of Contents
Preface
Acknowledgments
Glossary of Symbols
Introduction
Part I Background Material
1. Sample Problems
1.1. Two-Point Boundary Value Problems
1.2. Elliptic Boundary Value Problems
1.3. Integral Equations
1.4. Minimization Problems
1.5. Two-Dimensional Variational Problems
2. Linear Algebra
2.1. A Review of Basic Matrix Theory
2.2. Norms
2.3. Inverses
2.4. Partial Ordering and Nonnegative Matrices
3. Analysis
3.1. Derivatives and Other Basic Concepts
3.2. Mean-Value Theorems
3.3. Second Derivatives
3.4. Convex Functionals
Part II Nonconstructive Existence Theorems
4. Gradient Mappings and Minimization
4.1. Minimizers, Critical Points, and Gradient Mappings
4.2. Uniqueness Theorems
4.3. Existence Theorems
4.4. Applications
5. Contractions and the Continuation Property
5.1. Contractions
5.2. The Inverse and Implicit Function Theorems
5.3. The Continuation Property
5.4. Monotone Operators and Other Applications
6. The Degree of a Mapping
6.1. Analytic Definition of the Degree
6.2. Properties of the Degree
6.3. Basic Existence Theorems
6.4. Monotone and Coercive Mappings
6.5. Appendix. Additional Analytic Results
Part III Iterative Methods
7. General Iterative Methods
7.1. Newton's Method and Some of Its Variations
7.2. Secant Methods
7.3. Modification Methods
7.4. Generalized Linear Methods
7.5. Continuation Methods
7.6. General Discussion of Iterative Methods
8. Minimization Methods
8.1. Paraboloid Methods
8.2. Descent Methods
8.3. Steplength Algorithms
8.4. Conjugate-Direction Methods
8.5. The Gauss-Newton and Related Methods
8.6. Appendix 1. Convergence of the Conjugate Gradient and the Davidon- Fletcher-Powell Algorithms for Quadratic Functionals
8.7. Apppendix 2. Search Methods for One-Dimensional Minimization
Part IV Local Convergence
9. Rates of Convergence-General
9.1. The Quotient Convergence Factors
9.2. The Root Convergence Factors
9.3. Relations between the R and Q Convergence Factors
10. One-Step Stationary Methods
10.1. Basic Results
10.2. Newton's Method and Some of Its Modifications
10.3. Generalized Linear Iterations
10.4. Continuation Methods
10.5. Appendix. Comparison Theorems and Optimal ω for SOR Methods
11. Multistep Methods and Additional One-Step Methods
11.1. Introduction and First Results
11.2. Consistent Approximations
11.3. The General Secant Method
Part V Semilocal and Global Convergence
12. Contractions and Nonlinear Majorants
12.1. Some Generalizations of the Contraction Theorem
12.2. Approximate Contractions and Sequences of Contractions
12.3. Iterated Contractions and Nonexpansions
12.4. Nonlinear Majorants
12.5. More General Majorants
12.6. Newton's Method and Related Iterations
13. Convergence under Partial Ordering
13.1. Contractions under Partial Ordering
13.2. Monotone Convergence
13.3. Convexity and Newton's Method
13.4. Newton-SOR Interactions
13.5. M-Functions and Nonlinear SOR Processes
14. Convergence of Minimization Methods
14.1. Introduction and Convergence of Sequences
14.2. Steplength Analysis
14.3. Gradient and Gradient-Related Methods
14.4. Newton-Type Methods
14.5. Conjugate-Direction Methods
14.6. Univariate Relaxation and Related Processes
An Annotated List of Basic Reference Books
Bibliography
Author Index
Subject Index
Product details
- No. of pages: 592
- Language: English
- Copyright: © Academic Press 1970
- Published: July 31, 1970
- Imprint: Academic Press
- eBook ISBN: 9781483276724
About the Authors
J. M. Ortega
W. C. Rheinboldt
About the Editor
Werner Rheinboldt
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