On the conditioning analysis of block approximate factorization methods

Monga-Made, Magolu; Notay, Yvan

doi:10.1016/0024-3795(91)90395-d

Cited by 17 publications

(7 citation statements)

References 11 publications

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“…[21]), the above corollary leads, when all "goes right", to O(h-1) upper bound on the spectral condition number of B-IA provided that the condition (4.3) is satisfied with k (smaller than) O(h-1). The upper bound 1/(1-%) of Theorem 4.1 has been found in [21] and [22] to have the broadest field of application, compared to alternate approaches.…”

Section: Theorem 41 We Assume (H 1) and (H3) Let P Be A N-diagonalmentioning

confidence: 85%

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Modified block-approximate factorization strategies

Monga-Made

1992

Numer. Math.

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Section: Theorem 41 We Assume (H 1) and (H3) Let P Be A N-diagonalmentioning

confidence: 85%

“…[5,10,13,[20][21][22]25]). Unfortunately, performances of modified methods are sometimes spoiled by round off errors in the presence of strongly isolated largest eigenvalues [27,28].…”

Section: Introductionmentioning

confidence: 98%

Modified block-approximate factorization strategies

Monga-Made

1992

Numer. Math.

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“…The code is an extension of the augmented maximumspanning-tree preconditioning code described in References [15,16].…”

Section: A Numerical Examplementioning

confidence: 99%

“…• Algebraic eigenvalue-bounds for block-incomplete-factorization preconditioners that depend on the number of blocks in the matrix partitioning, such as References [13,15]. These, too, are not directly applicable to MWB preconditioners, in which there is no natural partitioning into blocks.…”

Section: Introductionmentioning

confidence: 97%

“…Early results, such as References [16][17][18], depended on these perturbation, but later the unperturbed variants were shown to be equally e ective [19][20][21]. For further details regarding spectral analysis of preconditioners, see the monograph [11], the survey [13], the papers cited above, and the papers [13,15,18,[22][23][24][25][26][27][28].…”

Section: Introductionmentioning

confidence: 98%

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Maximum‐weight‐basis preconditioners

Boman

Chen

Hendrickson

et al. 2004

Numerical Linear Algebra App

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SUMMARYThis paper analyses a novel method for constructing preconditioners for diagonally dominant symmetric positive-deÿnite matrices. The method discussed here is based on a simple idea: we construct M by simply dropping o diagonal non-zeros from A and modifying the diagonal elements to maintain a certain row-sum property. The preconditioners are extensions of Vaidya's augmented maximum-spanning-tree preconditioners. The preconditioners presented here were also mentioned by Vaidya in an unpublished manuscript, but without a complete analysis.The preconditioners that we present have only O(n + t 2 ) nonzeros, where n is the dimension of the matrix and 16t6n is a parameter that one can choose. Their construction is e cient and guarantees that the condition number of the preconditioned system is O(n 2 =t 2 ) if the number of nonzeros per row in the matrix is bounded by a constant.We have developed an e cient algorithm to construct these preconditioners and we have implemented it. We used our implementation to solve a simple model problem; we show the combinatorial structure of the preconditioners and we present encouraging convergence results.

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Algebraic multigrid and algebraic multilevel methods: a theoretical comparison

Notay

2005

Numerical Linear Algebra App

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Dedicated to Owe Axelsson on the occasion of his 70th birthday SUMMARYWe consider algebraic methods of the two-level type for the iterative solution of large sparse linear systems. We assume that a ÿne/coarse partitioning and an algebraic interpolation have been deÿned in one way or another, and review di erent schemes that may be built with these ingredients. This includes algebraic multigrid (AMG) schemes, two-level approximate block factorizations, and several methods that exploit generalized hierarchical bases. We develop their theoretical analysis in a uniÿed way, gathering some known results, rewriting some other and stating some new. This includes lower bounds, that is, we do not only investigate su cient conditions of convergence, but also look at necessary conditions.

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On the conditioning analysis of block approximate factorization methods

Cited by 17 publications

References 11 publications

Modified block-approximate factorization strategies

Modified block-approximate factorization strategies

Maximum‐weight‐basis preconditioners

Algebraic multigrid and algebraic multilevel methods: a theoretical comparison

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