TABLE OF CONTENTS The first iterative methods used for solving large linear systems were based on relaxation of the coordinates. Beginning with a given approximate solution, these methods modify the components of the approximation, one or a few at a time and in a certain order, until convergence is reached. Each of these modifications, called relaxation steps, is aimed at annihilating one or a few components of the residual vector. Now these techniques are rarely used separately. However, when combined with the more efficient methods described in later chapters, they can be quite successful. Moreover, there are a few application areas where variations of these methods are still quite popular.
This chapter begins by reviewing the basic iterative methods for solving linear systems. Given an n n real matrix A and a real n-vector b, the problem considered is as follows: Find x belonging to 
such that
Equation (4.1) is a linear system, A is the coefficient matrix, b is the right-hand side vector, and x is the vector of unknowns. Most of the methods covered in this chapter involve passing from one iterate to the next by modifying one or a few components of an approximate vector solution at a time. This is natural since there are simple criteria when modifying a component in order to improve an iterate. One example is to annihilate some component(s) of the residual vector b ?
