The Young–Eidson Algorithm: Applications and Extensions

Abstract

AhstractIn this paper it is assumed that the point (or block) Jacobi manix. B associated with the matrix A is weakly 2-cyclic consistently ordered with complex, in general, eigenvalue spectrum, a(B), lying in the interior of the infinite unit strip. It is then our objecl ive to apply and ex.tend the Young-Eidson algorithm in order to determine the real optimum relaxation factor in the following two cases: i) In the case of the Successive Overrelaxation (SOR) matrix associated with A when a(B) lies in a "bow-tie" region and ti) In the case of the Symmetric SOR (SSOR) matrix associated with A. It is noted that as a by-product of (ti) above both the relaxation factor for the SSOR matrix corresponding to a "bow-tie" spectrum a(8) and the optimum pairs of the relaxation factors for the Unsymmetric SOR (USSOR) matrix associated with A are also obtained.

A Young-Eidson's type algorithm for complex p-cyclic SOR spectra

“…Note: The reader may have noticed the complete analogy between the solution to the two•point problem just given and the solution to the two-point problem given by Young and Eidson for p = 2 (see [26], [25] or [6]).…”

“…Before we give in pseudocode the algorithm for the solution to the two-point problem (as in [6]) we give the algorithm for the solution to the one-point problem taken from [5].…”

“…We conclude this section by presenting the algorithm that solves the many-point problem. As one can observe this algorithm (Algorithm 3) is a systematic extension of Algorithm 2, that solves the two-point problem (k = 2), and is in complete analogy to the one developed by Young and Eidson [26] for the case p = 2 (see also [251, [6]). Table 2.…”