Iterative Solution of Nonlinear Equations in Several Variables

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.

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