An accurate parallel block Gram-Schmidt algorithm without reorthogonalization

Vanderstraeten, Denis

doi:10.1002/1099-1506(200005)7:4<219::aid-nla196>3.0.co;2-l

Cited by 10 publications

(3 citation statements)

References 15 publications

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“…A similar block algorithm based upon CGS, justified only by numerical tests, is proposed by Stewart [12]. Other block Gram-Schmidt algorithms are presented by Jalby and Phillippe [10] and Vanderstaeten [13]. A goal of this paper is to establish the stability analysis results for BCGS2.…”

Section: Introductionmentioning

confidence: 99%

Reorthogonalized block classical Gram–Schmidt

Barlow

Smoktunowicz

2012

Numer. Math.

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A new reorthogonalized block classical Gram-Schmidt algorithm is proposed that factorizes a full column rank matrix A into A = QR where Q is left orthogonal (has orthonormal columns) and R is upper triangular and nonsingular.With appropriate assumptions on the diagonal blocks of R, the algorithm, when implemented in floating point arithmetic with machine unit ε M , produces Q and R such thatThe resulting bounds also improve a previous bound by Giraud et al. [Num. Math., 101(1):87-100, 2005] on the CGS2 algorithm originally developed by Abdelmalek [BIT, 11(4):354-367, 1971].

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Section: Introductionmentioning

confidence: 99%

Reorthogonalized block classical Gram–Schmidt

Barlow

Smoktunowicz

2012

Numer. Math.

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“…CGSI+ is preferred when one wants to selectively orthogonalize, which can reduce the total number of floating-point operations, usually at the expense of increased communication due to the norm computations used to determine which vectors to reorthogonalize. Many different selection criteria have been proposed; see, e.g., [11,30,36,51,56].…”

Section: Bcgs With Reorthogonalizationmentioning

confidence: 99%

“…Recent work by Barlow [4], as well as important results by Barlow and Smoktunowicz [5], Vanderstraeten [56], and Jalby and Philippe [31], appear to be the only rigorous stability treatments of BGS. Some of the most influential papers on block Krylov subspace methods (KSMs) devote little or no attention to the stability analysis of the underlying block Arnoldi, and implicitly, BGS, routine.…”

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confidence: 99%

An overview of block Gram-Schmidt methods and their stability properties

Carson¹,

Lund²,

Rozložńık³

et al. 2020

Preprint

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Block Gram-Schmidt algorithms comprise essential kernels in many scientific computing applications, but for many commonly used variants, a rigorous treatment of their stability properties remains open. This survey provides a comprehensive categorization of block Gram-Schmidt algorithms, especially those used in Krylov subspace methods to build orthonormal bases one block vector at a time. All known stability results are assembled, and new results are summarized or conjectured for important communication-reducing variants. A diverse array of numerical illustrations are presented, along with the Matlab code for reproducing the results in a publicly available at repository https://github.com/katlund/BlockStab. A number of open problems are discussed, and an appendix containing all algorithms type-set in a uniform fashion is provided.

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Gram‐Schmidt orthogonalization: 100 years and more

Leon

Björck

Gander

2012

Numerical Linear Algebra App

116

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In 1907, Erhard Schmidt published a paper in which he introduced an orthogonalization algorithm that has since become known as the classical Gram-Schmidt process. Schmidt claimed that his procedure was essentially the same as an earlier one published by J. P. Gram in 1883. The Schmidt version was the first to become popular and widely used. An algorithm related to a modified version of the process appeared in an 1820 treatise by P. S. Laplace. Although related algorithms have been around for almost 200 years, it is the Schmidt paper that led to the popularization of orthogonalization techniques. The year 2007 marked the 100th anniversary of that paper. In celebration of that anniversary, we present a comprehensive survey of the research on Gram-Schmidt orthogonalization and its related QR factorization. Its application for solving least squares problems and in Krylov subspace methods are also reviewed. Software and implementation aspects are also discussed.

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An accurate parallel block Gram-Schmidt algorithm without reorthogonalization

Cited by 10 publications

References 15 publications

Reorthogonalized block classical Gram–Schmidt

Reorthogonalized block classical Gram–Schmidt

An overview of block Gram-Schmidt methods and their stability properties

Gram‐Schmidt orthogonalization: 100 years and more

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