Chapter 8 - Iterative Methods

Abstract

This chapter discusses a few iterative methods based on relaxation. The iterative methods have more modest storage requirements than direct methods and are also faster, depending on the iterative method and the problem. They usually also have better vectorization and parallelization properties. The Jacobi method is sometimes known as the method of simultaneous displacements. For the Jacobi method, the order in which the equations are processed is immaterial, although storage considerations make some orderings preferable. However, for the Gauss–Seidel method, each different ordering of the equations actually corresponds to a different iterative process. As the Jacobi and Gauss–Seidel method give a new iterate by averaging current values at neighboring points, they were known historically as relaxation methods. Even when the Jacobi and Gauss–Seidel methods are convergent, the rate of convergence is so slow as to preclude their usefulness. For certain iterative methods such as Jacobi's method, a potentially attractive alternative is to allow the processors to proceed asynchronously, without any synchronization.