Abstract
It has been shown that if A is a positive definite consistently ordered matrix, then a substantial improvement can be obtained using successive overrelaxation (SOR) method, with a suitable relaxation factor, as compared with Gauss–Seidel method. This chapter discusses the relaxation of the conditions on A. There is a need for some restrictions on A so that the SOR theory will hold. This chapter presents examples involving a consistently ordered matrix, a positive definite matrix, and an L-matrix where the SOR theory does not hold even approximately. The chapter also shows that if A is a Stieltjes matrix, then the SOR theory does hold to a large extent. On the other hand, for a given value of the spectral radius of the matrix B corresponding to the Jacobi method the best convergence is attained if A is consistently ordered.
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