Chapter 7 - GENERAL ITERATIVE METHODS

Abstract

This chapter discusses the processes that may be viewed as generalizations of methods for the solution of one equation in one unknown, such as the n-dimensional counterparts of Newton's method, the secant method, Steffensen's method, and their variations. The chapter discusses generalizations of iterative methods for linear systems of equations, with particular emphasis on successive over-relaxation methods. A number of other processes are also discussed in the chapter, which are attempts to provide a technique for overcoming the problem of finding a suitable initial approximation. In many applications, the system of equations to be solved arises in the attempt to find a minimizer or a critical point of a related nonlinear functional. The Newton and secant methods represent generalizations of one-dimensional iteration methods. The chapter discusses a class of processes that arise from iterative methods used for linear systems of equations. A characteristic of the generalized linear methods is that they all exhibit only a linear rate of convergence except under special circumstances.