Some advances in the theory of successive over-relaxation
For solving certain classes of linear equation Ax = b, the Young-Frankel method of successive over- and under-relaxation (the SOR method) is frequently used. Among the favourable cases of its application and analysis are those in which A belongs to one of the following classes. 1(a) Consistently ordered matrices (b) p-cyclic matrices; 2. Positive Definite Hermitian matrices; 3. Skew Hermitian matrices. The first two classes are considered in the thesis. For 1, the definition "π-consistently ordered of index p" is derived from the general definition of consistent ordering given by Verner and Bernal, so as to make the 'p' in the above coincide with the 'p' in "p-cyclic matrices". Definitions of strong and weak consistent orderings follow. It is shown that strong consistent orderings, in which the associated Jacobi matrix B has the "normal form" of a "weakly cyclic matrix of index p", are best suited for the application of the SOR method. A number of new results have been established under other and more general conditions on the eigenvalues of B than hitherto referred to in the literature. The main results relate to the conditions of convergence and the determination of the optimum relaxation factor which maximises the asymptotic rate of convergence of the SOR method. For 2, under certain conditions, a formula for the determination of a suitable value of the relaxation factor in terms of the largest positive eigenvalue of B is proposed. The discussed conditions are such that they extend the applicability of the SOR method. The thesis also deals with a more general method referred to as the "Generalised over-relaxation (GOR) method" in which two relaxation factors are used. Some circumstances in which the GOR method is superior to the SOR method are discussed. In particular, cases where the SOR method diverges whilst the GOR method converges are of interest.