# Convergence and Applications of Newton-type Iterations

Springer Science & Business Media, Jun 12, 2008 - Mathematics - 56 pages

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 Copyright