Iterative Solution of Nonlinear Equations in Several Variables

PrefaceAcknowledgmentsGlossary of SymbolsIntroductionPart 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 FunctionalsPart 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 ResultsPart 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 MinimizationPart 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 MethodPart 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 ProcessesAn Annotated List of Basic Reference BooksBibliographyAuthor IndexSubject Index