Convergence and Applications of Newton-type Iterations:
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| Format: | Book |
| Language: | English |
| Published: |
New York, NY
Springer
2008
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| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Physical Description: | XV, 506 S. 235 mm x 155 mm |
| ISBN: | 9780387727417 9780387727431 0387727418 |
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| 245 | 1 | 0 | |a Convergence and Applications of Newton-type Iterations |c Ioannis K. Argyros |
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Contents
Introduction
. xi
1 Operators
and Equations
. 1
1.1
Operators on linear spaces
. 1
1.2
Divided differences of operators
. 9
1.3
Fixed points of operators
. 25
1.4
Exercises
. 29
2
The Newton-Kantorovich (NK) Method
. 41
2.1
Linearization of equations
. 41
2.2
Semilocal convergence of the NK method
. 42
2.3
New sufficient conditions for the secant method
. 54
2.4
Concerning the "terra incognita" between convergence regions of
two Newton methods
. 62
2.5
Enlarging the convergence domain of the NK method under regular
smoothness conditions
. 75
2.6
Convergence of NK method and operators with values in a cone
. 80
2.7
Convergence theorems involving center-Lipschitz conditions
. 84
2.8
The radius of convergence for the NK method
. 90
2.9
On a weak NK method
. 102
2.10
Bounds on manifolds
. 103
2.11
The radius of convergence and one-parameter operator embedding
. 106
2.12
NK method and Riemannian manifolds
. 110
2.13
Computation of shadowing orbits
. 113
2.14
Computation of continuation curves
. 116
2.15
Gauss-Newton method
. 121
2.16
Exercises
. 125
3
Applications of the Weaker Version of the NK Theorem
. 133
3.1
Comparison of Kantorovich and Moore theorems
. 133
3.2
Comparison of Kantorovich and Miranda theorems
. 137
viii Contents
3.3
The secant method and nonsmooth equations
. 142
3.4
Improvements on curve tracing of the homotopy method
. 153
3.5
Nonlinear finite element analysis
. 157
3.6
Convergence of the structured PSB update in Hubert space
. 162
3.7
On the shadowing lemma for operators with chaotic behavior
. 166
3.8
The mesh independence principle and optimal shape design
problems
. 170
3.9
The conditioning of
semidefinite
programs
. 180
3.10
Exercises
. 186
4
Special Methods
. 193
4.1
Broyden's method
. 193
4.2
Stirling's method
. 202
4.3
Steffensen's method
. 207
4.4
Computing zeros of operator satisfying autonomous differential
equations
. 215
4.5
The method of tangent hyperbolas
. 219
4.6
A modified secant method and function optimization
. 230
4.7
Local convergence of a King-Werner-type method
. 233
4.8
Secant-type methods
. 235
4.9
Exercises
. 239
5
Newton-like Methods
. 261
5.1
Newton-like methods of "bounded deterioration"
. 261
5.2
Weak conditions for the convergence of a certain class of iterative
methods
. 269
5.3
Unifying convergence analysis for two-point Newton methods
. 275
5.4
On a two-point method of convergent order two
. 290
5.5
Exercises
. 304
6
Analytic Computational Complexity: We Are Concerned with the
Choice of Initial Approximations
. 325
6.1
The general problem
. 325
6.2
Obtaining good starting points for Newton's method
. 328
6.3
Exercises
. 336
7
Variational Inequalities
. 339
7.1
Variational inequalities and partially relaxed monotone mapping
. 339
7.2
Monotonicity and solvability of nonlinear variational inequalities
. 345
7.3
Generalized variational inequalities
. 352
7.4
Semilocal convergence
. 354
7.5
Results on generalized equations
. 358
7.6
Semilocal convergence for quasivariational inequalities
. 362
7.7
Generalized equations in Hubert space
. 365
7.8
Exercises
. 371
Contents ix
8
Convergence
Involving Operators with Outer or Generalized Inverses
379
8.1
Convergence with no Lipschitz conditions
. 379
8.2
Exercises
. 388
9
Convergence on Generalized Banach Spaces: Improving Error
Bounds and Weakening of Convergence Conditions
. 395
9.1
íT-normed
spaces
. 395
9.2
Generalized Banach spaces
. 408
9.3
Inexact Newton-like methods on Banach spaces with a convergence
structure
. 417
9.4
Exercises
. 436
10
Point-to-Set-Mappings
. 445
10.1
Algorithmic models
. 445
10.2
A general convergence theorem
. 449
10.3
Convergence of
ł-step
methods
. 451
10.4
Convergence of single-step methods
. 454
10.5
Convergence of single-step methods with differentiable iteration
functions
. 458
10.6
Monotone convergence
. 468
10.7
Exercises
. 471
11
The Newton-Kantorovich Theorem and Mathematical Programming
475
11.1
Case
1:
Interior point methods
. 475
11.2
Case
2:
LP methods
. 482
11.3
Exercises
. 489
References
. 493
Glossary of Symbols
. 503
Index
. 505 |
| adam_txt |
Contents
Introduction
. xi
1 Operators
and Equations
. 1
1.1
Operators on linear spaces
. 1
1.2
Divided differences of operators
. 9
1.3
Fixed points of operators
. 25
1.4
Exercises
. 29
2
The Newton-Kantorovich (NK) Method
. 41
2.1
Linearization of equations
. 41
2.2
Semilocal convergence of the NK method
. 42
2.3
New sufficient conditions for the secant method
. 54
2.4
Concerning the "terra incognita" between convergence regions of
two Newton methods
. 62
2.5
Enlarging the convergence domain of the NK method under regular
smoothness conditions
. 75
2.6
Convergence of NK method and operators with values in a cone
. 80
2.7
Convergence theorems involving center-Lipschitz conditions
. 84
2.8
The radius of convergence for the NK method
. 90
2.9
On a weak NK method
. 102
2.10
Bounds on manifolds
. 103
2.11
The radius of convergence and one-parameter operator embedding
. 106
2.12
NK method and Riemannian manifolds
. 110
2.13
Computation of shadowing orbits
. 113
2.14
Computation of continuation curves
. 116
2.15
Gauss-Newton method
. 121
2.16
Exercises
. 125
3
Applications of the Weaker Version of the NK Theorem
. 133
3.1
Comparison of Kantorovich and Moore theorems
. 133
3.2
Comparison of Kantorovich and Miranda theorems
. 137
viii Contents
3.3
The secant method and nonsmooth equations
. 142
3.4
Improvements on curve tracing of the homotopy method
. 153
3.5
Nonlinear finite element analysis
. 157
3.6
Convergence of the structured PSB update in Hubert space
. 162
3.7
On the shadowing lemma for operators with chaotic behavior
. 166
3.8
The mesh independence principle and optimal shape design
problems
. 170
3.9
The conditioning of
semidefinite
programs
. 180
3.10
Exercises
. 186
4
Special Methods
. 193
4.1
Broyden's method
. 193
4.2
Stirling's method
. 202
4.3
Steffensen's method
. 207
4.4
Computing zeros of operator satisfying autonomous differential
equations
. 215
4.5
The method of tangent hyperbolas
. 219
4.6
A modified secant method and function optimization
. 230
4.7
Local convergence of a King-Werner-type method
. 233
4.8
Secant-type methods
. 235
4.9
Exercises
. 239
5
Newton-like Methods
. 261
5.1
Newton-like methods of "bounded deterioration"
. 261
5.2
Weak conditions for the convergence of a certain class of iterative
methods
. 269
5.3
Unifying convergence analysis for two-point Newton methods
. 275
5.4
On a two-point method of convergent order two
. 290
5.5
Exercises
. 304
6
Analytic Computational Complexity: We Are Concerned with the
Choice of Initial Approximations
. 325
6.1
The general problem
. 325
6.2
Obtaining good starting points for Newton's method
. 328
6.3
Exercises
. 336
7
Variational Inequalities
. 339
7.1
Variational inequalities and partially relaxed monotone mapping
. 339
7.2
Monotonicity and solvability of nonlinear variational inequalities
. 345
7.3
Generalized variational inequalities
. 352
7.4
Semilocal convergence
. 354
7.5
Results on generalized equations
. 358
7.6
Semilocal convergence for quasivariational inequalities
. 362
7.7
Generalized equations in Hubert space
. 365
7.8
Exercises
. 371
Contents ix
8
Convergence
Involving Operators with Outer or Generalized Inverses
379
8.1
Convergence with no Lipschitz conditions
. 379
8.2
Exercises
. 388
9
Convergence on Generalized Banach Spaces: Improving Error
Bounds and Weakening of Convergence Conditions
. 395
9.1
íT-normed
spaces
. 395
9.2
Generalized Banach spaces
. 408
9.3
Inexact Newton-like methods on Banach spaces with a convergence
structure
. 417
9.4
Exercises
. 436
10
Point-to-Set-Mappings
. 445
10.1
Algorithmic models
. 445
10.2
A general convergence theorem
. 449
10.3
Convergence of
ł-step
methods
. 451
10.4
Convergence of single-step methods
. 454
10.5
Convergence of single-step methods with differentiable iteration
functions
. 458
10.6
Monotone convergence
. 468
10.7
Exercises
. 471
11
The Newton-Kantorovich Theorem and Mathematical Programming
475
11.1
Case
1:
Interior point methods
. 475
11.2
Case
2:
LP methods
. 482
11.3
Exercises
. 489
References
. 493
Glossary of Symbols
. 503
Index
. 505 |
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| illustrated | Not Illustrated |
| index_date | 2024-07-02T21:50:05Z |
| indexdate | 2024-07-20T09:48:47Z |
| institution | BVB |
| isbn | 9780387727417 9780387727431 0387727418 |
| language | English |
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| physical | XV, 506 S. 235 mm x 155 mm |
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| spelling | Argyros, Ioannis K. Verfasser (DE-588)14160820X aut Convergence and Applications of Newton-type Iterations Ioannis K. Argyros New York, NY Springer 2008 XV, 506 S. 235 mm x 155 mm txt rdacontent n rdamedia nc rdacarrier Convergencia Método de Newton-Raphson Métodos iterativos (Matemáticas) Convergence Iterative methods (Mathematics) Newton-Raphson method Iteration (DE-588)4123457-1 gnd rswk-swf Konvergenz (DE-588)4032326-2 gnd rswk-swf Newton-Verfahren (DE-588)4171693-0 gnd rswk-swf Newton-Verfahren (DE-588)4171693-0 s Konvergenz (DE-588)4032326-2 s Iteration (DE-588)4123457-1 s DE-604 text/html http://deposit.dnb.de/cgi-bin/dokserv?id=2944732&prov=M&dok_var=1&dok_ext=htm Inhaltstext Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016700708&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Argyros, Ioannis K. Convergence and Applications of Newton-type Iterations Convergencia Método de Newton-Raphson Métodos iterativos (Matemáticas) Convergence Iterative methods (Mathematics) Newton-Raphson method Iteration (DE-588)4123457-1 gnd Konvergenz (DE-588)4032326-2 gnd Newton-Verfahren (DE-588)4171693-0 gnd |
| subject_GND | (DE-588)4123457-1 (DE-588)4032326-2 (DE-588)4171693-0 |
| title | Convergence and Applications of Newton-type Iterations |
| title_auth | Convergence and Applications of Newton-type Iterations |
| title_exact_search | Convergence and Applications of Newton-type Iterations |
| title_exact_search_txtP | Convergence and Applications of Newton-type Iterations |
| title_full | Convergence and Applications of Newton-type Iterations Ioannis K. Argyros |
| title_fullStr | Convergence and Applications of Newton-type Iterations Ioannis K. Argyros |
| title_full_unstemmed | Convergence and Applications of Newton-type Iterations Ioannis K. Argyros |
| title_short | Convergence and Applications of Newton-type Iterations |
| title_sort | convergence and applications of newton type iterations |
| topic | Convergencia Método de Newton-Raphson Métodos iterativos (Matemáticas) Convergence Iterative methods (Mathematics) Newton-Raphson method Iteration (DE-588)4123457-1 gnd Konvergenz (DE-588)4032326-2 gnd Newton-Verfahren (DE-588)4171693-0 gnd |
| topic_facet | Convergencia Método de Newton-Raphson Métodos iterativos (Matemáticas) Convergence Iterative methods (Mathematics) Newton-Raphson method Iteration Konvergenz Newton-Verfahren |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=2944732&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016700708&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT argyrosioannisk convergenceandapplicationsofnewtontypeiterations |