A Method of Normalized Block Iteration

Cuthill, Elizabeth; Varga, Richard S.

doi:10.1145/320964.320981

Cited by 43 publications

(5 citation statements)

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“…The basic successive overrelaxation iterative method of YOUNG [17] and FRANKEL [4] and its newer variant, the cyclic Chebyshev semi-iterative method [6], have both been theoretically and practically coupled successfully with the use of block or multi-line techniques, considered in [1,3,8,11,13,16]. The first purpose of this paper is to theoretically couple the use of these block or multi-line techniques to the iterative solution of the cyclically reduced matrix equation (1.4), and to determine conditions which insure that these block techniques, applied to the cyclically reduced matrix problem of (t.4), are asymptotically iteratively faster than the corresponding block techniques applied to the original matrix problem of (t.1).…”

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confidence: 99%

Block iterative methods for cyclically reduced matrix equations

Hageman¹,

Varga²

1964

Numer. Math.

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confidence: 99%

Block iterative methods for cyclically reduced matrix equations

Hageman¹,

Varga²

1964

Numer. Math.

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“…Reference [20]. The Normalized Explicit Preconditioning Conjugate Gradient (NEPCG) method for solving linear systems can be stated as follows.…”

Section: Normalized Explicit Preconditioned Conjugate Gradient Methodsmentioning

confidence: 99%

Normalized explicit approximate inverse preconditioning for solving 3D boundary value problems on uniprocessor and distributed systems

Gravvanis

Giannoutakis

2005

Int. J. Numer. Meth. Engng

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SUMMARYNormalized explicit approximate inverse matrix techniques, based on normalized approximate factorization procedures, for solving sparse linear systems resulting from the finite difference discretization of partial differential equations in three space variables are introduced. Normalized explicit preconditioned conjugate gradient schemes in conjunction with normalized approximate inverse matrix techniques are presented for solving sparse linear systems. The convergence analysis with theoretical estimates on the rate of convergence and computational complexity of the normalized explicit preconditioned conjugate gradient method are also derived. A Parallel Normalized Explicit Preconditioned Conjugate Gradient method for distributed memory systems, using message passing interface (MPI) communication library, is also given along with theoretical estimates on speedups, efficiency and computational complexity. Application of the proposed method on a three-dimensional boundary value problem is discussed and numerical results are given for uniprocessor and multicomputer systems.

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“…The SOR theory applies for the block SOR method, and faster convergence is obtained than for the point SOR method. Cuthill and Varga (1959) showed how the extra work per iteration for line SOR can be essentially eliminated.…”

Section: Some Exaiilplesmentioning

confidence: 99%

On the iterative methods for solving linear systems of equations

Dutto

1993

Revue Européenne des Éléments Finis

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The solution of large systems of linear equations is a problem frequently met in numerical calculations, for example, from finite difference or finite element approximations to partial differential equations. The use of direct methods for solving linear systems of equations is limited by both storage requirement and computing time, and in practice, the only alternative is to use iterative algorithms. This paper presents some basic iterative methods, and also a short survey of recent research on this subject, with emphasis on linear systems whose coefficient matrix is non-Hermitian. We briefly discuss the notion of preconditioning, and we give some practical choices for the preconditioning matrix. REsUME. La resolution de systemes d'equations lineaires de grande taille est un probleme frequemment rencontre dans le calcul numerique, par exemple, lors de /'approximation d'equations aux derivees partielles par des methodes d'elements finis ou de differences finies. Les couts en temps CPU et en place memoire des methodes directes de resolution limitent sensiblement leur utilisation en pratique et imposent l'emploi des methodes iteratives comme seule alternative. Le but de 'cet article est de presenter quelques methodes iteratives bien connues, ainsi que les progres recents accomplis dans ce domaine . .En particulier, nous concentrons notre attention sur la resolution de systemes lineaires dont Ia matrice associee est non hermitienne. Nous esquissons Ia notion de preconditionnement et donnons quelques exemples. KEY WORDS : large and sparse linear systems, non-Hermitian matrices, iterative methods, Petrov-Galerkin methods, Krylov subspace methods, conjugate gradient, like methods, preconditioning. MOTS-CLEs : systemes lineaires creux et de grande taille, matrices non hermitiennes, methodes iteratives, methodes de Petrov-Galerkin, methodes des sous-espaces de Krylov, methodes de type gradient conjugue, preconditionnement. Revue europeenne des elements finis. Volume 2 -n° 4/1993, pages 423 a 448 Downloaded by [New York University] at 01:08 09 February 2015 424 Revue europeenne des elements finis. Vol. 2-n° 4/1993

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A Method of Normalized Block Iteration

Cited by 43 publications

References 10 publications

Block iterative methods for cyclically reduced matrix equations

Block iterative methods for cyclically reduced matrix equations

Normalized explicit approximate inverse preconditioning for solving 3D boundary value problems on uniprocessor and distributed systems

On the iterative methods for solving linear systems of equations

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