Richard S. Varga, Iterative methods for solving matrix equations, Amer. Math. Monthly 72(1965), 67-74. MR 30, 2677. Zbl. 151, p. 214. 

1. Introduction. Iterative methods are, in concept, well known to us all since it is likely that each of us has at one time or another used Newton's method to find square roots of numbers. It is probably not as well known that iterative methods, utilizing the great speeds of modern computing machines, are extensively used today in practical computations for solving matrix equations which arise from finite difference approximations to elliptic partial differential equations of reactor technology and petroleum technology.

The object of this paper is to illustrate the use of finite difference techniques and to illustrate the nature of iterative methods for solving the associated matrix equations. We shall do this by means of a very simple one-dimensional problem. It is hoped that this simple example will serve as an elementary introduction to the theory of such iterative methods which is covered in much more detail in [1,2,3,5,6].