ABSTRACT

In Chapter 3, we have discussed direct methods for solving a system of linear equations. Using the direct method, the solution vector can be obtained after a finite number of operations. In this chapter, we will discuss iterative solution methods that iteratively approach the solution of a system of linear equations:

A x b = (4.1)

Given an arbitrary initial solution vector x0, we use the iterative methods to generate a series of solution vectors according to certain rules such that the series of solution vectors converges to the exact solution x:

x x x0 1, , , ,k  (4.2)

lim k

= x x (4.3)

The iterative approach for the solution of linear systems is well suited for systems with very large sparse coefficient matrices or system stiffness matrices. With sparse matrices, it is not economical to carry out the standard triangular decomposition procedure used in the direct solution methods. It should also be noted that for most systems of nonlinear equations, the solution vector is obtained through iterative methods. However, the solution methods in solving system of nonlinear equations require special treatment and they will be discussed in Chapter 6. The iterative methods discussed in this chapter are iterative solution methods for system of linear equations.