Methods of Jacobi and Gauß-Seidel and SOR Iteration 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 §§5–6. The introductory Section 4.1 underlines the fact that the positive definite case is of practical interest: The Poisson-model matrix is positive definite. Chapter 4.2 is a recapitulation and generalisation of the algorithms already known from §1.4. Simple modifications of these iterations are discussed in §4.3 and §4.6. The convergence results of the qualitative and quantitative kind are given in §4.4. Section 4.8 introduces the symmetric iterations and, in particular, the SSOR method.