The Theory and Applications of Iteration Methods

by Argyros; Ioannis Konstantinos

• ISBN13: 9780849380143
• ISBN10: 0849380146
• Format: Hardcover
• Copyright: 1993-09-15
• Publisher: CRC Press

Summary

The Theory and Applications of Iteration Methods focuses on an abstract iteration scheme that consists of the recursive application of a point-to-set mapping. Each chapter presents new theoretical results and important applications in engineering, dynamic economic systems, and input-output systems. At the end of each chapter, case studies and numerical examples are presented from different fields of engineering and economics.Following an outline of general iteration schemes, the authors extend the discrete time-scale Liapunov theory to time-dependent, higher order, nonlinear difference equations. The monotone convergence to the solution is examined in and comparison theorems are proven . Results generalize well-known classical theorems, such as the contraction mapping principle, the lemma of Kantorovich, the famous Gronwall lemma, and the stability theorem of Uzawa. The book explores conditions for the convergence of special single- and two-step methods such as Newton's method, modified Newton's method, and Newton-like methods generated by point-to-point mappings in a Banach space setting. Conditions are examined for monotone convergence of Newton's methods and their variants. Students and professionals in engineering, the physical sciences, mathematics, and economics will benefit from the book's detailed examples, step-by-step explanations, and effective organization.

Table of Contents

Preface
The convergence of algorithmic models	p. 1
Algorithmic models	p. 1
Convergence criteria for algorithmic models	p. 4
The convergence of iteration sequences	p. 45
The general convergence theorem	p. 45
Convergence of 1-step methods	p. 47
Convergence of single-step methods	p. 51
Convergence of single-step methods with differentiable iteration functions	p. 55
Monotone convergence	p. 123
General results	p. 123
A general model in linear spaces	p. 124
Comparison theorems	p. 157
General results	p. 157
Comparison to fixed points	p. 159
The convergence of Newton methods and their variants	p. 181
Convergence analysis and Ptak error estimates	p. 181
Convergence with a nondifferentiable term	p. 190
Convergence of Newton-like methods	p. 191
The monotone convergence of Newton methods and their variants	p. 325
Monotone convergence of Newton-like methods	p. 325
A lattice theoretical fixed point theorem	p. 337
References	p. 345
Index	p. 351
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