Chapter 20 - Basic Iterative Methods

Abstract

This chapter presents the basic iterative methods, Jacobi, Gauss-Seidel, and SOR that serve as models for more advanced methods. The Jacobi iteration uses the previous values of the iteration to advance, but Gauss-Seidel uses new component values as soon as they are computed. As such, it is generally more accurate. SOR (successive overrelaxation) computes a weighted average of the Gauss-Seidel components with the previous ones. The relaxation parameter, ω, must be in the range 0 < ω < 2. Convergence of these methods depends on the iteration matrix, a matrix such that xk = Bxk − 1 + c. If the norm of B for some subordinate norm is less than 1, the iteration converges. The iteration converges if and only if the spectral radius of B is less than 1. The spectral radius can be used as an approximation to how fast the iteration will converge; the smaller it is the faster the iteration converges. The Jacobi and Gauss-Seidel methods converge if A is strictly diagonally dominant, and the Gauss-Seidel iteration converges if B is positive definite. Convergence of the SOR iteration is guaranteed if 0 < ω < 2 and A is positive definite. If convergence is not guaranteed, it is possible for the one iteration to succeed and another fail. Choosing ω for SOR is difficult and a precise value is known for only a few matrices. Although costly, it is possible to estimate an optimal value by running a numerical experiment. The Poisson equation is very important in many fields of science and engineering. The chapter presents a five-point central difference approximation for the equation and uses the SOR iteration to develop an approximation to the solution of Poisson’s equation with boundary conditions of zero.