# The Theory and Applications of Iteration Methods

This text book and its first edition are written for students in engineering, the physical sciences, mathematics, and economics at an upper division undergraduate or graduate level. Prerequisites for using the text are calculus, linear algebra, elements of functional analysis, and the fundamentals of differential equations. Student 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 understanding the material of this book.

A plethora of problems in mathematics and also in engineering are solved by finding the solutions of certain equations. For example, dynamics systems are mathematical model by difference or differential equations, and their solutions usually represent the states of the systems. Assume for the sake of simplicity that a time invariant system is driven by 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 unknowns of engineering equations can be functions (difference, differential, and integral equations), vectors (systems
of linear or non-linear algebraic equations), or real or complex numbers (single al-gebraic equations with single unknowns). Except in the 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 iteration 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 identity properties of iteration schemes that will guarantee their convergence is 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 non-linear and linear differential equations, and in optimization problems.