Chapter 9 - Iterative Methods for Linear Systems

Abstract

In this chapter we present the idea of iterative solution methods for linear systems, in contrast with direct methods such as Gaussian elimination. The Jacobi, Gauss-Seidel, Successive Over-Relaxation, and Conjugate Gradient methods are presented. For each we discuss time to solution and the convergence of the method on an example problem and through a graphical demonstration. The concept of implementation improvement versus numerical improvement is introduced. In particular, the Jacobi method is shown to be faster than Gauss-Seidel even though it has more iterations to get the solution. Finally, we show that iterative methods tuned for the matrix structure can lead to even better performance.