The Theory and Applications of Iteration Methods PDF
Preview The Theory and Applications of Iteration Methods
systems engineering series series editor A. Terry Bahill, University of Arizona Linear Systems Theory Ferenc Szidarovszky, University of Arizona A. Terry Bahill, University of Arizona Engineering Modeling and Design William L. Chapman, Hughes Aircraft Company A. Terry Bahill, University of Arizona A. Wayne Wymore, Systems Analysis and Design Systems Model-Based Systems Engineering A. Wayne W5nnore, Systems Analysis and Design Systems The Theory and Applications of Iteration Methods loannis K. Argyros, Cameron University Ferenc Szidarovszky, University of Arizona 1 the theory and applications of iteration methods loannis K. Argyros Ferenc Szidarovszky CRC Press / Taylor 8i Francis Group Boca Raton London New York CRC Press is an imprint of the Taylor & Francis Croup, an informa business CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 1993 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works This book contains information obtained from authentic and highly regarded sources. Reason able efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. 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Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Weh site at http://www.crcpress.com Contents Preface vil 1 The convergence of algorithmic models 1 1.1 Algorithmic models.................................................................................1 1.2 Convergence criteria for algorithmic models.....................................4 1.3 Applications............................................................................................10 Exercises..................................................................................................41 2 The convergence of iteration sequences 45 2.1 The general convergence theorem......................................................45 2.2 Convergence of 1-step methods..........................................................47 2.3 Convergence of single-step methods.................................................51 2.4 Convergence of single-step methods with differentiable iteration functions..................................................................................55 2.5 Case studies............................................................................................60 Exercises................................................................................................119 3 Monotone convergence 123 3.1 General results.....................................................................................123 3.2 A general model in linear spaces.....................................................124 3.3 Case studies..........................................................................................129 Exercises................................................................................................153 4 Comparison theorems 157 4.1 General results.....................................................................................157 4.2 Comparison to fixed points...............................................................159 4.3 Case studies..........................................................................................160 Exercises................................................................................................178 5 The convergence of Newton methods and their variants 181 5.1 Convergence analysis and Ptâk error estimates.............................181 5.2 Convergence with a nondifferentiable term..................................190 5.3 Convergence of Newton-like methods............................................191 5.4 Case Studies..........................................................................................194 Exercises................................................................................................321 6 The monotone convergence of Newton methods and their variants 325 6.1 Monotone convergence of Newton-like methods.........................325 6.2 A lattice theoretical fixed point theorem........................................337 6.3 Case studies..........................................................................................339 Exercises................................................................................................343 References.........................................................................................................345 Index...................................................................................................................351 Preface This textbook was written for students in engineering, the physical sci ences, mathematics, and economics at an upper division undergraduate or graduate level. Prerequisites for using the text are calculus, linear al gebra, elements of functional analysis, and the fundamentals of differential equations. Students with some knowledge of the principles of numerical analysis and optimization will have an advantage, since the general schemes and concepts can be easily followed if particular methods, special cases, are already known. However, such knowledge is not essential in under standing the material of this book. A large number of problems in applied mathematics and also in en gineering are solved by finding the solutions of certain equations. For example, d)mamic systems are mathematically modeled by difference or differential equations, and their solutions usually represent the states of the systems. For the sake of simplicity, assume that a time-invariant system is driven by the equation x = f(x), where x is the state. Then the equilibrium states are determined by solving the equation f(x) = 0. Similar equations are used in the case of discrete systems. The urrknowns of engineering equations can be functions (difference, differential, and integral equations), vectors (systems of linear or nonlinear algebraic equations), or real or com plex numbers (single algebraic equations with single unknowns). Except in special cases, the most commonly used solution methods are iterative — when starting from one or several initial approximations a sequence is constructed that converges to a solution of the equation. Iteration methods are also applied for solving optimization problems. In such cases, the it eration sequences converge to an optimal solution of the problem at hand. Since all of these methods have the same recursive structure, they can be introduced and discussed in a general framework. In recent years, the study of general iteration schemes has included a substantial effort to identify properties of iteration schemes that will guar antee their convergence in some sense. A number of these results have used an abstract iteration scheme that consists of the recursive application of a point-to-set mapping. In this book, we are concerned with these types of results. Each chapter contains several new theoretical results and important applications in engineering, in dynamic economic systems, in input-output systems, in the solution of nonlhiear and linecu- differential equations, and in optimization problems. Chapter 1 gives an outline of general iteration schemes in which the convergence of such schemes is examined. We also show that oiu: condi tions are very general: most classical results can be obtained as special cases and, if the conditions are weakened slightly, then our results may not hold. In Chapter 2 the discrete time-scale Liapunov theory is extended to time dependent, higher order, nonlinear difference equations. In addition, the speed of convergence is estimated in most cases. The monotone conver gence to the solution is examined in Chapter 3 and comparison theorems are proven in Chapter 4. It is also shown that our results generalize well- known classical theorems such as the contraction mapping principle, the lemma of Kantorovich, the famous Gronwall lemma, and the well-known stability theorem of Uzawa. Chapter 5 examines conditions for the con vergence of special single-step methods such as Newton's method, mod ified Newton's method, and Newton-like methods generated by point-to- point mappings in a Banach space setting. The speed of convergence of such methods is examined using the theory of majorants and a method called "continuous induction", which builds on a special variant of Ban ach's closed graph theorem. Finally, Chapter 6 examines conditions for monotone convergence of special single-step methods such as Newton's method, Newton-like methods, and secant methods generated by point- to-point mappings in a partially ordered space setting. At the end of each chapter, case studies and numerical examples are presented from different fields of engineering and economy. Authors loannis K. Argyros is a Professor in the Department of Mathematics at Cameron University, Lawton, Oklahoma. He received his B.Sc. degree in 1979 from the University of Athens, Greece. In March 1982, he started his graduate studies and received his M.Sc. and Ph.D. in 1983 and 1984 from the University of Georgia, Athens. Professor Argyros is the author or co-author of over 180 journal articles. His primary interests are in numerical hmctional analysis, numerical analy sis, approximation theory, optimization theory, and applied mathematics. Ferenc Szidarovszky has been a Professor of Systems and Industrial En gineering at the University of Arizona in Tucson since 1988. He received his Ph.D. in mathematics from the Eotvos University of Science, Budapest, Hungary in 1970. He received a second Ph.D. in economics from the Bu dapest Economics University, Himgary in 1977. His research interests are in the field of d)mamic economic systems, game theory, numerical analysis, and their applications in natural resources management. He is the author of five textbooks and three monographs published in Hungarian, as well as Principles and Procedures of Numerical Analysis (with Sidney Yakowitz), Plenum, 1978, Introduction to Numerical Computations (with Sidney Yakowitz), Macmillan, 1986 (second edition, 1989), Techniques of Multiobjective Decision Making in System Managements (with Mark Gershon and Luden Duckstein), Elsevier, 1986, and The Thany of Oligopoly voith Multi Product Firms (with Koji Okuguchi), Springer, 1990, and Linear Systems Theory (with A. Terry Bahill), CRC Press, 1992.
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