Some recent results on the modified SOR theory

Hadjidimos, A.; Yeyios, A.

doi:10.1016/0024-3795(91)90370-c

Cited by 11 publications

(10 citation statements)

References 26 publications

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“…ii) For r = 0) '" 0)' (MSOR method) for aCT) real with I! = 0 they can be found in [21], while in all other cases in [22] or they can be directly obtained from the analysis in [8]. iii) For r = 0, co '¢ co' (two-parameter Extrapolated Jacobi method) the convergence domains obtained in the previous section, at least in some cases, must be new.…”

Section: S Concluding Remarks and Discussionmentioning

confidence: 99%

“…It can be checked that (3.6) and (3.7) coincide with the schemes (4.4) and (4.5) in [8] respectively when in the former ones r2 = 002 (MSOR case) and in the latter ones p = 2. So, it has just been proved that:…”

Section: Ro Ro mentioning

confidence: 99%

“…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%

“…Despite the fact that in the cases of real and pure imaginary spectra aCT) we are concerned with in the sequel the study is made by using the MAOR (the EMSOR, to be precise,) method, for more general spectra it would be more convenient if we used the 2-step method. For such an equivalence to be established a matrix analogue of the eigenvalue relationship (1.9) will provide us with the key point needed (see, e.g., [2] - [4] and [8]). For this the following theorem is stated and proved: Theorem 2: Let p = 2; then under the assumptions of Theorem 1, and the additional assumption that R n-1 = sI, with s (#: 0) being a scalar, the following matrix identities hold…”

Section: Il E A(t)dmentioning

confidence: 99%

“…and the constant vector term in (3.6) can be simplified (see e.g., [8] for a similar simplification). So, one finally has…”

Section: Ro Ro mentioning

confidence: 99%

See 4 more Smart Citations

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: S Concluding Remarks and Discussionmentioning

confidence: 99%

Section: Ro Ro mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Il E A(t)dmentioning

confidence: 99%

“…and the constant vector term in (3.6) can be simplified (see e.g., [8] for a similar simplification). So, one finally has…”

Section: Ro Ro mentioning

confidence: 99%

See 3 more Smart Citations

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

Hadjidimos¹,

Psimarni²,

Yeyios³

1992

Applied Numerical Mathematics

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On a matrix identity connecting iteration operators associated with a p-cyclic matrix

Hadjidimos

Noutsos

1993

Linear Algebra and its Applications

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Modified successive overrelaxation (MSOR) and equivalent 2-step iterative methods for collocation matrices

Hadjidimos

Saridakis

1992

Journal of Computational and Applied Mathematics

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We consider a class of consistently ordered matrices which arise from the discretization of Boundary Value Problems (BVPs) when the finite element collocation method, with Hermite elements, is used. Through a recently derived equivalence relationship for the asymptotic rates of convergence of the Modified Successive Overrelaxation (MSOR) and a certain 2-step iterative method, we determine the optimum values for the parameters of the MSOR method, as it pertains to collocation matrices. A geometrical algorithm, which utilizes 'capturing ellipse' arguments, has been successfully used. The fast convergence properties of the optimum MSOR method are revealed after its comparison to several well-known iterative schemes. Numerical examples, which include the solution of Poisson's equation, are used to verify our results.'"

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Some recent results on the modified SOR theory

Cited by 11 publications

References 26 publications

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

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

On a matrix identity connecting iteration operators associated with a p-cyclic matrix

Modified successive overrelaxation (MSOR) and equivalent 2-step iterative methods for collocation matrices

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