Iterative solution of nonlinear equations in several variables:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Academic Press
1970 [erschienen] 1972
|
| Edition: | 2. print. |
| Series: | Computer science and applied mathematics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XX, 572 S. |
| ISBN: | 0125285507 |
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Record in the Search Index
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| adam_text | Titel: Iterative solution of nonlinear equations in several variables
Autor: Ortega, James M
Jahr: 1972
CONTENTS Preface xiii Acknowledgments xvii Glossary of Symbols xix Introduction 1 Part I BACKGROUND MATERIAL 1 ? Sample Problems 1.1. Two-Point Boundary Value Problems 9 Notes and Remarks 12 Exercises 13 1.2. Elliptic Boundary Value Problems 14 Notes and Remarks 16 1.3. Integral Equations 18 Notes and Remarks 20 Exercises 20 1.4. Minimization Problems 21 Notes and Remarks 25 Exercises 26 1.5. Two-Dimensional Variational Problems 26 Notes and Remarks 30 Exercises 32 vn
Vlil CONTENTS 2. Linear Algebra 2.1. A Review of Basic Matrix Theory 34 Notes and Remarks 37 Exercises 37 2.2. Norms 38 Notes and Remarks 44 Exercises 44 2.3. Inverses 45 Notes and Remarks 50 Exercises 51 2.4. Partial Ordering and Nonnegative Matrices 51 Notes and Remarks 57 Exercises 57 3. Analysis 3.1. Derivatives and Other Basic Concepts 59 Notes and Remarks _ 65 Exercises 66 3.2. Mean-Value Theorems 68 Notes and Remarks 73 Exercises 74 3.3. Second Derivatives 74 Notes and Remarks 81 Exercises 82 3.4. Convex Functionals 82 Notes and Remarks 88 Exercises 89 Part II NONCONSTRUCTIVE EXISTENCE THEOREMS 4. Gradient Mappings and Minimization 4.1. Minimizers, Critical Points, and Gradient Mappings 93 Notes and Remarks 97 Exercises 97 4.2. Uniqueness Theorems 98 Notes and Remarks 101 Exercises 102 4.3. Existence Theorems 104 Notes and Remarks 108 Exercises 109 4.4. Applications 110 Notes and Remarks 115 Exercises 116
CONTENTS IX 5. Contractions and the Continuation Property 5.1. Contractions 119 Notes and Remarks 124 Exercises 125 5.2. The Inverse and Implicit Function Theorems 125 Notes and Remarks 131 Exercises 132 5.3. The Continuation Property 132 Notes and Remarks 139 Exercises 140 5.4. Monotone Operators and Other Applications 141 Notes and Remarks 145 Exercises 145 6. The Degree of a Mapping 6.1. Analytic Definition of the Degree 147 Notes and Remarks 154 Exercises 155 6.2. Properties of the Degree 156 Notes and Remarks 160 Exercises 160 6.3. Basic Existence Theorems 161 Notes and Remarks 163 Exercises 164 6.4. Monotone and Coercive Mappings 165 Notes and Remarks 167 Exercises 168 6.5. Appendix. Additional Analytic Results 169 Part III ITERATIVE METHODS 7. General Iterative Methods 7.1. Newton’s Method and Some of Its Variations 181 Notes and Remarks 187 Exercises 188 7.2. Secant Methods 189 Notes and Remarks 200 Exercises 205 7.3. Modification Methods 206 Notes and Remarks 212 Exercises 214 7.4. Generalized Linear Methods 214 Notes and Remarks 222 Exercises 229
X CONTENTS 7.5. Continuation Methods 230 Notes and Remarks 234 Exercises 235 7.6. General Discussion of Iterative Methods 236 Notes and Remarks 239 8. Minimization Methods 8.1. Paraboloid Methods 240 Notes and Remarks 242 Exercises 243 8.2. Descent Methods 243 Notes and Remarks 247 Exercises 248 8.3. Steplength Algorithms 249 Notes and Remarks — 258. Exercises 259 8.4. Conjugate-Direction Methods 260 Notes and Remarks 263 Exercises ~ 266 8.5. The Gauss-Newton and Related Methods 267 Notes and Remarks 269 Exercises 271 8.6. Appendix 1. Convergence of the Conjugate Gradient and the Davidon- Fletcher-Powell Algorithms for Quadratic Functionals 271 8.7. Appendix 2. Search Methods for One-Dimensional Minimization 275 Part IV LOCAL CONVERGENCE 9. Rates of Convergence—General 9.1. The Quotient Convergence Factors 281 Notes and Remarks 286 Exercises 286 9.2. The Root Convergence Factors 287 Notes and Remarks 293 Exercises 294 9.3. Relations between the R and Q Convergence Factors 295 Exercises 298 10. One-Step Stationary Methods 10.1. Basic Results 299 Notes and Remarks 305 Exercises 308
CONTENTS XI 10.2. Newton’s Method and Some of Its Modifications 310 Notes and Remarks 316 Exercises 318 10.3. Generalized Linear Iterations 320 Notes and Remarks 331 Exercises 332 10.4. Continuation Methods 334 Notes and Remarks 340 10.5. Appendix. Comparison Theorems and Optimal ui for SOR Methods 341 11. Multistep Methods and Additional One-Step Methods 11.1. Introduction and First Results 347 Notes and Remarks 352 Exercises 354 11.2. Consistent Approximations 355 Notes and Remarks 363 Exercises 366 11.3. The General Secant Method 369 Notes and Remarks 378 Exercises 379 Part V SEMILOCAL AND GLOBAL CONVERGENCE 12. Contractions and Nonlinear Majorants 12.1. Some Generalizations of the Contraction Theorem 383 Notes and Remarks 390 Exercises 391 12.2. Approximate Contractions and Sequences of Contractions 393 Notes and Remarks 398 Exercises 399 12.3. Iterated Contractions and Nonexpansions 400 Notes and Remarks 406 Exercises 407 12.4. Nonlinear Majorants 409 Notes and Remarks 414 Exercises 414 12.5. More General Majorants 415 Notes and Remarks 419 Exercises 420 12.6. Newton’s Method and Related Iterations 421 Notes and Remarks 428 Exercises 430
CONTENTS xii 13. Convergence under Partial Ordering 13.1. Contractions under Partial Ordering 432 Notes and Remarks 439 Exercises 440 13.2. Monotone Convergence 441 Notes and Remarks 446 Exercises 446 13.3. Convexity and Newton’s Method 447 Notes and Remarks 454 Exercises 454 13.4. Newton-SOR Interactions 456 Notes and Remarks 463 Exercises 464 13.5. Ai-Functions and Nonlinear SOR Processes 464 Notes and Remarks 470 Exercises —- 471.. 14. Convergence of Minimization Methods 14.1. Introduction and Convergence of Sequences 473 Notes and Remarks 478 Exercises 478 14.2. Steplength Analysis 479 Notes and Remarks 493 Exercises 493 14.3. Gradient and Gradient-Related Methods 494 Notes and Remarks 499 Exercises 500 14.4. Newton-Type Methods 501 Notes and Remarks 506 Exercises 507 14.5. Conjugate-Direction Methods 509 Notes and Remarks 512 Exercises 513 14.6. Univariate Relaxation and Related Processes 513 Notes and Remarks 518 Exercises 519 An Annotated List of Basic Reference Books 521 Bibliography 523 Author Index Subject Index 559 566
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| any_adam_object | 1 |
| author | Ortega, James M. 1932- Rheinboldt, Werner C. |
| author_GND | (DE-588)114190852 |
| author_facet | Ortega, James M. 1932- Rheinboldt, Werner C. |
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| author_sort | Ortega, James M. 1932- |
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| bvnumber | BV019381915 |
| classification_rvk | SK 910 |
| ctrlnum | (OCoLC)837898209 (DE-599)BVBBV019381915 |
| discipline | Mathematik |
| edition | 2. print. |
| format | Book |
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| id | DE-604.BV019381915 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T19:58:59Z |
| institution | BVB |
| isbn | 0125285507 |
| language | English |
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| physical | XX, 572 S. |
| publishDate | 1970 |
| publishDateSearch | 1972 |
| publishDateSort | 1972 |
| publisher | Academic Press |
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| series2 | Computer science and applied mathematics |
| spelling | Ortega, James M. 1932- Verfasser (DE-588)114190852 aut Iterative solution of nonlinear equations in several variables J. M. Ortega and W. C. Rheinboldt 2. print. New York [u.a.] Academic Press 1970 [erschienen] 1972 XX, 572 S. txt rdacontent n rdamedia nc rdacarrier Computer science and applied mathematics Mehrere Variable (DE-588)4277015-4 gnd rswk-swf Nichtlineare Gleichung (DE-588)4455337-7 gnd rswk-swf Numerisches Verfahren (DE-588)4128130-5 gnd rswk-swf Nichtlineare Gleichung (DE-588)4455337-7 s Numerisches Verfahren (DE-588)4128130-5 s DE-604 Mehrere Variable (DE-588)4277015-4 s Rheinboldt, Werner C. Verfasser aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=012844947&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Ortega, James M. 1932- Rheinboldt, Werner C. Iterative solution of nonlinear equations in several variables Mehrere Variable (DE-588)4277015-4 gnd Nichtlineare Gleichung (DE-588)4455337-7 gnd Numerisches Verfahren (DE-588)4128130-5 gnd |
| subject_GND | (DE-588)4277015-4 (DE-588)4455337-7 (DE-588)4128130-5 |
| title | Iterative solution of nonlinear equations in several variables |
| title_auth | Iterative solution of nonlinear equations in several variables |
| title_exact_search | Iterative solution of nonlinear equations in several variables |
| title_full | Iterative solution of nonlinear equations in several variables J. M. Ortega and W. C. Rheinboldt |
| title_fullStr | Iterative solution of nonlinear equations in several variables J. M. Ortega and W. C. Rheinboldt |
| title_full_unstemmed | Iterative solution of nonlinear equations in several variables J. M. Ortega and W. C. Rheinboldt |
| title_short | Iterative solution of nonlinear equations in several variables |
| title_sort | iterative solution of nonlinear equations in several variables |
| topic | Mehrere Variable (DE-588)4277015-4 gnd Nichtlineare Gleichung (DE-588)4455337-7 gnd Numerisches Verfahren (DE-588)4128130-5 gnd |
| topic_facet | Mehrere Variable Nichtlineare Gleichung Numerisches Verfahren |
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