Extensions of the Ostrowski-Reich theorem for SOR iterations

Ortega, James M.; Plemmons, Robert J.

doi:10.1016/0024-3795(79)90131-9

Cited by 25 publications

(3 citation statements)

References 17 publications

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“…the convergence conditions of the iterative method (3). Yuan [26] investigated the convergence conditions of the iterative method (3) when M is nonsingular, and extended the Ortega-Plemmons theorem [31] to singular matrices. Now we modify the convergence conditions in [26] and further generalize some results with the singular matrix M.…”

Section: Remark 24mentioning

confidence: 99%

On the convergence of general stationary iterative methods for range‐Hermitian singular linear systems

Zhang¹,

Wei²

2009

Numerical Linear Algebra App

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General stationary iterative methods with a singular matrix M for solving range-Hermitian singular linear systems are presented, some convergence conditions and the representation of the solution are also given. It can be verified that the general Ortega-Plemmons theorem and Keller theorem for the singular matrix M still hold. Furthermore, the singular matrix M can act as a good preconditioner for solving range-Hermitian linear systems. Numerical results have demonstrated the effectiveness of the general stationary iterations and the singular preconditioner M.where A ∈ C n×n , x, b ∈ C n , and rank(A)

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Section: Remark 24mentioning

confidence: 99%

On the convergence of general stationary iterative methods for range‐Hermitian singular linear systems

Zhang¹,

Wei²

2009

Numerical Linear Algebra App

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“…It is a well known result [36,26] that if A is Hermitian positive definite and A = M − N is a P -regular splitting, then the splitting iterative method is convergent: ρ(M −1 N ) < 1. An extension of P -regular splitting was introduced by Ortega and Plemmons [25] and [6]. A splitting A = M − N with M nonsingular is called an extended P -regular splitting if the matrix M * (A −1 ) * A + N is positive definite.…”

mentioning

confidence: 99%

“…Theorem 2.3. (see [25] and [6]) Let A ∈ C n×n such that A = M − N is an extended P-regular splitting. Then ρ(T ) < 1, where T = M −1 N , if and only if A is positive definite.…”

mentioning

confidence: 99%

On parallel multisplitting methods for non-Hermitian positive definite linear systems

Zhang,

Luo,

Zhu

2014

Preprint

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To solve non-Hermitian linear system Ax = b on parallel and vector machines, some paralell multisplitting methods are considered. In this work, in particular: i) We establish the convergence results of the paralell multisplitting methods, together with its relaxed version, some of which can be regarded as generalizations of analogous results for the Hermitian positive definite case; ii) We extend the positive-definite and skew-Hermitian splitting (PSS) method methods in [SIAM J. Sci. Comput., 26:844-863, 2005] to the parallel PSS methods and propose the corresponding convergence results.

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Konvergenzkriterien für das Verallgemeinerte AOR‐Verfahren

Song

1992

Z Angew Math Mech

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Extensions of the Ostrowski-Reich theorem for SOR iterations

Cited by 25 publications

References 17 publications

On the convergence of general stationary iterative methods for range‐Hermitian singular linear systems

On the convergence of general stationary iterative methods for range‐Hermitian singular linear systems

On parallel multisplitting methods for non-Hermitian positive definite linear systems

Konvergenzkriterien für das Verallgemeinerte AOR‐Verfahren

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