## Convergence and Applications of Newton-type IterationsRecent 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. |

### What people are saying - Write a review

### Contents

1 | |

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 |

492 | |

Glossary of Symbols | 503 |

505 | |