A new class of parallel alternating-type iterative methods

This paper is concerned with parallel alternating-type iterative methods for solving large sparse linear systems of the form Au = b arising in the numerical solution of partial differential equations by finite difference methods. Examples of alternating-type methods include the alternating direction implicit (ADI) method and the unsymmetric SOR (USSOR) method. Each iteration of an alternating-type method involves the use of two parameters, say ϱ and ϱ′. We consider parallel alternating-type methods where, given an initial vector u (0) , the positive integer m, and two sets of m parameters “ϱ i” and “ϱ′ j” , one carries out m 2 single iterations in parallel, each involving one pair ( ϱ i , ϱ′ j ) of the parameters. It is shown that in some cases a linear combination v ∗ of the vectors thus obtained is the same as the vector v ∗∗ which would be obtained by a sequential process involving m iterations based on the successive use of the parameter pairs ( ϱ 1, ϱ′ 1), ( ϱ 2, ϱ′ 2),…, ( ϱ m , ϱ′ m ). Thus, the parallel procedure offers the potential of reducing the wall-clock time by a factor of m as compared with the sequential procedure. Preliminary numerical results based on the use of a virtual parallel system of sequential computers confirm the expected reductions in the number of iterations.