Iterative Solution of Nonlinear Equations in Several Variables

Iterative Solution of Nonlinear Equations in Several Variables

1st Edition - July 31, 1970

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  • Authors: J. M. Ortega, W. C. Rheinboldt
  • eBook ISBN: 9781483276724

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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


    Glossary of Symbols


    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


    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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