Classical Linear Iterations in the Positive Definite Case

Abstract

The methods of Jacobi and Gauss–Seidel and the SOR method are closely connected and therefore they will be analysed simultaneously. The analysis, however, is essentially different for the case of positive definite matrices A discussed below and other cases studied in Chapter 4. The introductory Section 3.1 underlines the fact that the positive definite case is of practical interest. The Poisson model matrix is an example of a positive definite matrix. Section 3.2 describes the traditional iterations of Richardson, Jacobi, Gauss–Seidel, and the SOR iteration. Block versions of these iterations are discussed in Section 3.3. The computational work required by the mentioned iterations is described in Section 3.4. Convergence results of qualitative and quantitative kind are given in Section 3.5. The convergence analysis of the Richardson iteration leads to convergence criteria for general positive definite iterations. The Gauss–Seidel and SOR iteration is analysed. In particular the improvement of the order of convergence by SOR is investigated. The convergence statements are illustrated in Section 3.6 for the Poisson model case and numerical examples are given.