Convergence and Applications of Newton-type Iterations

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Springer Science & Business Media, Jun 12, 2008 - Mathematics - 56 pages
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Recent results in local convergence and semi-local convergence analysis constitute a natural framework for the theoretical study of iterative methods. This monograph provides a comprehensive study of both basic theory and new results in the area. Each chapter contains new theoretical results and important applications in engineering, dynamic economic systems, input-output systems, optimization problems, and nonlinear and linear differential equations. Several classes of operators are considered, including operators without Lipschitz continuous derivatives, operators with high order derivatives, and analytic operators. Each section is self-contained. Examples are used to illustrate the theory and exercises are included at the end of each chapter.

The book assumes a basic background in linear algebra and numerical functional analysis. Graduate students and researchers will find this book useful. It may be used as a self-study reference or as a supplementary text for an advanced course in numerical functional analysis.

 

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Contents

Operators and Equations
1
12 Divided differences of operators
9
13 Fixed points of operators
25
14 Exercises
29
The NewtonKantorovich NK Method
41
22 Semilocal convergence of the NK method
42
23 New sufficient conditions for the secant method
54
24 Concerning the terra incognita between convergence regions of two Newton methods
62
49 Exercises
239
Newtonlike Methods
261
52 Weak conditions for the convergence of a certain class of iterative methods
269
53 Unifying convergence analysis for twopoint Newton methods
275
54 On a twopoint method of convergent order two
290
55 Exercises
304
Analytic Computational Complexity We Are Concerned with the Choice of Initial Approximations
325
62 Obtaining good starting points for Newtons method
328

25 Enlarging the convergence domain of the NK method under regular smoothness conditions
75
26 Convergence of NK method and operators with values in a cone
80
27 Convergence theorems involving centerLipschitz conditions
84
28 The radius of convergence for the NK method
90
29 On a weak NK method
102
210 Bounds on manifolds
103
211 The radius of convergence and oneparameter operator embedding
106
212 NK method and Riemannian manifolds
110
213 Computation of shadowing orbits
113
214 Computation of continuation curves
116
215 GaussNewton method
121
216 Exercises
125
Applications of the Weaker Version of the NK Theorem
133
32 Comparison of Kantorovich and Miranda theorems
137
33 The secant method and nonsmooth equations
142
34 Improvements on curve tracing of the homotopy method
153
35 Nonlinear finite element analysis
157
36 Convergence of the structured PSB update in Hilbert space
162
37 On the shadowing lemma for operators with chaotic behavior
166
38 The mesh independence principle and optimal shape design problems
170
39 The conditioning of semidefinite programs
180
310 Exercises
186
Special Methods
193
42 Stirlings method
202
43 Steffensens method
207
44 Computing zeros of operator satisfying autonomous differential equations
215
45 The method of tangent hyperbolas
219
46 A modified secant method and function optimization
230
47 Local convergence of a KingWernertype method
233
48 Secanttype methods
235
63 Exercises
336
Variational Inequalities
338
72 Monotonicity and solvability of nonlinear variational inequalities
345
73 Generalized variational inequalities
352
74 Semilocal convergence
354
75 Results on generalized equations
358
76 Semilocal convergence for quasivariational inequalities
362
77 Generalized equations in Hilbert space
365
78 Exercises
371
Convergence Involving Operators with Outer or Generalized Inverses
379
82 Exercises
388
Convergence on Generalized Banach Spaces Improving Error Bounds and Weakening of Convergence Conditions
395
92 Generalized Banach spaces
408
93 Inexact Newtonlike methods on Banach spaces with a convergence structure
417
94 Exercises
436
PointtoSetMappings
444
102 A general convergence theorem
449
103 Convergence of Kstep methods
451
104 Convergence of singlestep methods
454
105 Convergence of singlestep methods with differentiable iteration functions
458
106 Monotone convergence
468
107 Exercises
471
The NewtonKantorovich Theorem and Mathematical Programming
475
LP methods
482
113 Exercises
489
References
492
Glossary of Symbols
503
Index
505
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Argyros is affiliated with the Department of Mathematics.

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