Iterative Solution of Nonlinear Equations in Several Variables

by Ortega, James M.

• ISBN13: 9780125285506
• ISBN10: 0125285507
• Format: Hardcover
• Copyright: 1970-06-01
• Publisher: Academic Pr

Author Biography

James M. Ortega is Professor Emeritus at the University of Virginia, and currently teaches in the Department of Mathematical Sciences at the Florida Institute of Technology. Werner C. Rheinboldt is Professor Emeritus at the University of Pittsburgh.

Table of Contents

Preface to Classics Edition	p. xv
Preface	p. xix
Acknowledgments	p. xxiii
Glossary of Symbols	p. xxv
Introduction	p. 1
Background Material
Sample Problems
Two-Point Boundary Value Problems	p. 9
Elliptic Boundary Value Problems	p. 14
Integral Equations	p. 18
Minimization Problems	p. 21
Two-Dimensional Variational Problems	p. 26
Linear Algebra
A Review of Basic Matrix Theory	p. 34
Norms	p. 38
Inverses	p. 45
Partial Ordering and Nonnegative Matrices	p. 51
Analysis
Derivatives and Other Basic Concepts	p. 59
Mean-Value Theorems	p. 68
Second Derivatives	p. 74
Convex Functionals	p. 82
Nonconstructive Existence Theorems
Gradient Mappings and Minimization
Minimizers, Critical Points, and Gradient Mappings	p. 93
Uniqueness Theorems	p. 98
Existence Theorems	p. 104
Applications	p. 110
Contractions and the Continuation Property
Contractions	p. 119
The Inverse and Implicit Function Theorems	p. 125
The Continuation Property	p. 132
Monotone Operators and Other Applications	p. 141
The Degree of a Mapping
Analytic Definition of the Degree	p. 147
Properties of the Degree	p. 156
Basic Existence Theorems	p. 161
Monotone and Coercive Mappings	p. 165
Additional Analytic Results	p. 169
Iterative Methods
General Iterative Methods
Newton's Method and Some of Its Variations	p. 181
Secant Methods	p. 189
Modification Methods	p. 206
Generalized Linear Methods	p. 214
Continuation Methods	p. 230
General Discussion of Iterative Methods	p. 236
Minimization Methods
Paraboloid Methods	p. 240
Descent Methods	p. 243
Steplength Algorithms	p. 249
Conjugate-Direction Methods	p. 260
The Gauss-Newton and Related Methods	p. 267
Convergence of the Conjugate Gradient and the Davidon--Fletcher--Powell Algorithms for Quadratic Functionals	p. 271
Search Methods for One-Dimensional Minimization	p. 275
Local Convergence
Rates of Convergence--General
The Quotient Convergence Factors	p. 281
The Root Convergence Factors	p. 287
Relations between the R and Q Convergence Factors	p. 295
One-Step Stationary Methods
Basic Results	p. 299
Newton's Method and Some of Its Modifications	p. 310
Generalized Linear Iterations	p. 320
Continuation Methods	p. 334
Comparison Theorems and Optimal [omega] for SOR Methods	p. 341
Multistep Methods and Additional One-Step Methods
Introduction and First Results	p. 347
Consistent Approximations	p. 355
The General Secant Method	p. 369
Semilocal and Global Convergence
Contractions and Nonlinear Majorants
Some Generalizations of the Contraction Theorem	p. 383
Approximate Contractions and Sequences of Contractions	p. 393
Iterated Contractions and Nonexpansions	p. 400
Nonlinear Majorants	p. 409
More General Majorants	p. 415
Newton's Method and Related Iterations	p. 421
Convergence under Partial Ordering
Contractions under Partial Ordering	p. 432
Monotone Convergence	p. 441
Convexity and Newton's Method	p. 447
Newton-SOR Interactions	p. 456
M-Functions and Nonlinear SOR Processes	p. 464
Convergence of Minimization Methods
Introduction and Convergence of Sequences	p. 473
Steplength Analysis	p. 479
Gradient and Gradient-Related Methods	p. 494
Newton-Type Methods	p. 501
Conjugate-Direction Methods	p. 509
Univariate Relaxation and Related Processes	p. 513
An Annotated List of Basic Reference Books	p. 521
Bibliography	p. 523
Author Index	p. 559
Subject Index	p. 566
Table of Contents provided by Syndetics. All Rights Reserved.