On Different Splittings and the Associated Iteration Methods
Wilhelm Niethammer · SIAM Journal on Numerical Analysis · 1979
The well-known SOR-method can be derived from a splitting of the diagonal I of the matrix A of a given linear system. In different papers Sisler introduced a splitting of the lower triangular matrix L of A and a two-parametric method where the splitting of I and L is combined. It is proven that this two-parametric method is closely related to the SOR-method. Then some results are derived if A is noncyclic, and it is shown that many of the results of Sisler in the cyclic case can be more easily proven by using the relation mentioned above. For comparison there will always be considered symmetric and skew-symmetric matrices B, where $A = I - B$. It reveals that for cyclic matrices the two-parametric method usually is not superior to SOR whereas an example of a noncyclic matrix is given such that the two-parametric method with optimal parameters is twice as fast as SOR with optimal relaxation factor.