Ortega / Rheinboldt Iterative Solution of Nonlinear Equations in Several Variables

1. Auflage 2014
ISBN: 978-1-4832-7672-4
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 592 Seiten, Web PDF

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.

Autoren/Hrsg.

Ortega, J. M.

Rheinboldt, W. C.

Rheinboldt, Werner

Inhaltsverzeichnis

1;Front Cover;1
2;Iterative Solution of Nonlinear Equations in Several Variables;4
3;Copyright Page;5
4;Table of Contents;8
5;PREFACE;14
6;ACKNOWLEDGMENTS;18
7;GLOSSARY OF SYMBOLS;20
8;INTRODUCTION;22
9;PART I: BACKGROUND MATERIAL;28
9.1;CHAPTER 1. SAMPLE PROBLEMS;30
9.1.1;1.1. TWO-POINT BOUNDARY VALUE PROBLEMS;30
9.1.2;1.2. ELLIPTIC BOUNDARY VALUE PROBLEMS;35
9.1.3;1.3. INTEGRAL EQUATIONS;39
9.1.4;1.4. MINIMIZATION PROBLEMS;42
9.1.5;1.5. TWO-DIMENSIONAL VARIATIONAL PROBLEMS;47
9.2;CHAPTER 2. LINEAR ALGEBRA;55
9.2.1;2.1. A REVIEW OF BASIC MATRIX THEORY;55
9.2.2;2.2. NORMS;59
9.2.3;2.3. INVERSES;66
9.2.4;2.4. PARTIAL ORDERING AND
NONNEGATIVE MATRICES;72
9.3;CHAPTER 3. ANALYSIS;80
9.3.1;3.1. DERIVATIVES AND
OTHER BASIC CONCEPTS;80
9.3.2;3.2. MEAN-VALUE THEOREMS;89
9.3.3;3.3. SECOND DERIVATIVES;95
9.3.4;3.4. CONVEX FUNCTIONALS;103
10;PART II: NONCONSTRUCTIVE EXISTENCE THEOREMS;112
10.1;CHAPTER 4. GRADIENT MAPPINGS AND MINIMIZATION;114
10.1.1;4.1. MINIMIZERS, CRITICAL POINTS, AND GRADIENT MAPPINGS;114
10.1.2;4.2. UNIQUENESS THEOREMS;119
10.1.3;4.3. EXISTENCE THEOREMS;125
10.1.4;4.4. APPLICATIONS;131
10.2;CHAPTER 5. CONTRACTIONS AND THE CONTINUATION PROPERTY;140
10.2.1;5.1. CONTRACTIONS;140
10.2.2;5.2. THE INVERSE AND IMPLICIT
FUNCTION THEOREMS;146
10.2.3;5.3. THE CONTINUATION PROPERTY;153
10.2.4;5.4. MONOTONE OPERATORS AND OTHER APPLICATIONS;162
10.3;CHAPTER 6. THE DEGREE OF A MAPPING;168
10.3.1;6.1. ANALYTIC DEFINITION OF THE DEGREE;168
10.3.2;6.2. PROPERTIES OF THE DEGREE;177
10.3.3;6.3. BASIC EXISTENCE THEOREMS;182
10.3.4;6.4. MONOTONE AND COERCIVE MAPPINGS;186
10.3.5;6.5. APPENDIX. ADDITIONAL ANALYTIC RESULTS;190
11;PART III: ITERATIVE METHODS;200
11.1;CHAPTER 7. GENERAL ITERATIVE METHODS;202
11.1.1;7.1. NEWTON'S METHOD AND SOME OF ITS VARIATIONS;202
11.1.2;7.2. SECANT METHODS;210
11.1.3;7.3. MODIFICATION METHODS;227
11.1.4;7.4. GENERALIZED LINEAR METHODS;235
11.1.5;7.5. CONTINUATION METHODS;251
11.1.6;7.6. GENERAL DISCUSSION OF ITERATIVE METHODS;257
11.2;CHAPTER 8. MINIMIZATION METHODS;261
11.2.1;8.1. PARABOLOID METHODS;261
11.2.2;8.2. DESCENT METHODS;264
11.2.3;8 3. STEPLENGTH ALGORITHMS;270
11.2.4;8.4. CONJUGATE-DIRECTION METHODS;281
11.2.5;8.5. THE GAUSS-NEWTON AND RELATED METHODS;288
11.2.6;8.6. APPENDIX 1. CONVERGENCE OF THE CONJUGATE GRADIENT AND THE DAVIDON-FLETCHER-POWELL ALGORITHMS FOR QUADRATIC FUNCTIONALS;292
11.2.7;8.7. APPENDIX 2. SEARCH METHODS FOR ONE-DIMENSIONAL MINIMIZATION;296
12;PART IV: LOCAL CONVERGENCE;300
12.1;CHAPTER 9. RATES OF CONVERGENCE—GENERAL;302
12.1.1;9.1. THE QUOTIENT CONVERGENCE FACTORS;302
12.1.2;9.2. THE ROOT-CONVERGENCE FACTORS;308
12.1.3;9.3. RELATIONS BETWEEN THE R AND Q CONVERGENCE FACTORS;316
12.2;CHAPTER 10. ONE-STEP STATIONARY METHODS;320
12.2.1;10.1. BASIC RESULTS;320
12.2.2;10.2. NEWTON'S METHOD AND SOME OF ITS MODIFICATIONS;331
12.2.3;10.3. GENERALIZED LINEAR ITERATIONS;341
12.2.4;10.4. CONTINUATION METHODS;355
12.2.5;10.5. APPENDIX. COMPARISON THEOREMS AND OPTIMAL w FOR SOR METHODS;362
12.3;CHAPTER 11. MULTISTEP METHODS AND ADDITIONAL ONE-STEP METHODS;368
12.3.1;11.1. INTRODUCTION AND FIRST RESULTS;368
12.3.2;11.2. CONSISTENT APPROXIMATIONS;376
12.3.3;11.3. THE GENERAL SECANT METHOD;390
13;PART V: SEMILOCAL AND GLOBAL CONVERGENCE;402
13.1;CHAPTER 12. CONTRACTIONS AND NONLINEAR MAJORANTS;404
13.1.1;12.1. SOME GENERALIZATIONS OF THE CONTRACTION THEOREM;404
13.1.2;12.2. APPROXIMATE CONTRACTIONS AND SEQUENCES
OF CONTRACTIONS;414
13.1.3;12.3. ITERATED CONTRACTIONS AND NONEXPANSIONS;421
13.1.4;12.4. NONLINEAR MAJORANTS;430
13.1.5;12.5. MORE GENERAL MAJORANTS;436
13.1.6;12.6. NEWTON'S METHOD AND RELATED ITERATIONS;442
13.2;CHAPTER 13. CONVERGENCE UNDER PARTIAL ORDERING;453
13.2.1;13.1. CONTRACTIONS UNDER PARTIAL ORDERING;453
13.2.2;13.2. MONOTONE CONVERGENCE;462
13.2.3;13.3. CONVEXITY AND NEWTON'S METHOD;468
13.2.4;13.4. NEWTON-SOR ITERATIONS;477
13.2.5;13.5. M-FUNCTIONS AND NONLINEAR SOR PROCESSES;485
13.3;CHAPTER 14. CONVERGENCE OF MINIMIZATION METHODS;494
13.3.1;14.1. INTRODUCTION AND CONVERGENCE OF SEQUENCES;494
13.3.2;14.2. STEPLENGTH ANALYSIS;500
13.3.3;14.3. GRADIENT AND
GRADIENT-RELATED METHODS;515
13.3.4;14.4. NEWTON-TYPE METHODS;522
13.3.5;14.5. CONJUGATE-DIRECTION METHODS;530
13.3.6;14.6. UNIVARIATE RELAXATION AND RELATED PROCESSES;534
14;AN ANNOTATED LIST OF BASIC REFERENCE BOOKS;542
15;BIBLIOGRAPHY;544
16;AUTHOR INDEX;580
17;SUBJECT INDEX;587