Numerical Linear Algebra with Applications

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 ...