p-cyclic matrices and the symmetric successive overrelaxation method

Varga, Richard S.; Niethammer, Wilhelm; Cai, Dong-ling

doi:10.1016/0024-3795(84)90223-4

Cited by 39 publications

(10 citation statements)

References 9 publications

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“…More specifically, in Section 3 we derive the matrix analogue of (1.9) and prove the equivalence of the MAOR method to a cerntin 2-step one. In Section 4 the precise convergence domains of the MAOR method when cr(T) is real or pure imaginary are determined extending in this way previous results by Hadjidimos [5] In order to derive (1.9) we follow an approach due to Varga, Niethammer and Cai [18], which was already used successfully in [7] and [8]. Also, to simplify the analysis we work out the case (p,q) = (5,2).…”

Section: Introductionmentioning

confidence: 99%

On the convergence of the modified accelerated overrelaxation (MAOR) method

Hadjidimos¹,

Psimarni²,

Yeyios³

1992

Applied Numerical Mathematics

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In this paper we introduce the Modified Accelerated Overrelaxation (MAOR) method, a generalization of the AOR one, for the iterative solution of the nonsingular linear system Ax = b. We assume that A is in a p x p partitioned fonn and belongs to a subclass of the p-cyclic consistently ordered matrices. It is pointed out that for specific choices of the "acceleration" and "relaxation" matrices the MAOR method reduces to an extrapolation of the Jacobi or the Modified (M)SOR method with different parameters corresponding to the row blocks of A. First, an eigenvalue relationship connecting the spectra of the block Jacobi and MAOR matrices associated with A is derived from which many wellRknown eigenvalue relationships can be recovered. Then, by considering the particular case p = 2 it is shown that a matrix analogue of the aforementioned eigenvalue relationship holds and the MAOR method is equivalent to a cenain 2-step one. Finally, the precise domains of convergence of the MAOR method are derived, when the spectrum of the Jacobi matrix is real or pure imaginary and a brief discussion follows which concludes the present study.

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Section: Introductionmentioning

confidence: 99%

On the convergence of the modified accelerated overrelaxation (MAOR) method

Hadjidimos¹,

Psimarni²,

Yeyios³

1992

Applied Numerical Mathematics

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“…As is known [15], due to D'SyIva and Miles [6] and Lynn [11] (see also [13]). However, as was proved The above "algorithm" can be directly appiled to the cases of i) alB) real with p(B) < I and ill-alB) pure imaginary to yield well known results.…”

Section: Development Of the Basic Theorymentioning

confidence: 79%

The Young–Eidson Algorithm: Applications and Extensions

Hadjidimos¹,

Noutsos²

1990

SIAM J. Matrix Anal. & Appl.

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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.

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“…Then, since 0<a<2 and co> 1, 2co-a+0. Hence, condition (c) of Theorem 2.1 reduces to r q o As < 1, it is evident that, for each q (or, equivalently, p) sufficiently large, ~-2& = in order for (3.22) The result in Corollary 3.2 is recently derived in Varga et al [21]. That there is given for H-matrices is not essential since the intersection of the set of H-matrices and the set of p-cyclic matrices is non-empty.…”

Section: To(~)=~m ~ [~P__o~q-k Flr-k(o~fl_~_ ~)(~)K ~;]mentioning

confidence: 86%

“…In [13], as a generalization of recent eigenvalue results (cf. [21,17,9]) for the USSOR and SSOR methods, it is shown: Theorem 2.1. Given the matrix A of (2.2), let B of (2.4) be its associated, weakly cyclic of index p, block Jacobi iteration matrix.…”

Section: Eigenvalue Properties Of the Ussor Methodsmentioning

confidence: 98%