Block Gram-Schmidt algorithms and their stability properties

Carson, Erin; Lund, Kathryn; Rozložńık, Miroslav; Thomas, Stephen J.

doi:10.1016/j.laa.2021.12.017

Cited by 20 publications

(48 citation statements)

References 55 publications

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“…Like CholQR or BCGS, the orthogonlization error is of order O(κ 2 ε). For a detailed stability analysis see [5]. The pseudocode for this algorithm is shown in Algorithm 3.1.…”

Section: Bcgs-pipmentioning

confidence: 99%

“…Especially the classical Gram-Schmidt procedure may produce a significant orthogonalization error. The stability properties of various block Gram-Schmidt procedures are analyzed in the paper of Carson et al [4] and an excellent overview of existing block Gram-Schmidt methods is given in [5], taking performance and stability into account.…”

mentioning

confidence: 99%

“…Yamazaki et al [28] already presented a method that combines the projection and normalization step in a single reduction in the context of the classical Gram-Schmidt method. This was later extended to block Gram-Schmidt [5] under the name BCGS-PIP. The same algorithm can be derived in our setting as we will see in section 3.…”

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confidence: 99%

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A Hardware-aware and Stable Orthogonalization Framework

Dreier¹,

Engwer²

2022

Preprint

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The orthogonalization process is an essential building block in Krylov space methods, which takes up a large portion of the computational time. Commonly used methods, like the Gram-Schmidt method, consider the projection and normalization separately and store the orthogonal base explicitly. We consider the problem of orthogonalization and normalization as a QR decomposition problem on which we apply known algorithms, namely CholeskyQR and TSQR. This leads to methods that solve the orthogonlization problem with reduced communication costs, while maintaining stability and stores the orthogonal base in a locally orthogonal representation. Furthermore, we discuss the novel method as a framework which allows us to combine different orthogonalization algorithms and use the best algorithm for each part of the hardware. After the formulation of the methods, we show their advantageous performance properties based on a performance model that takes data transfers within compute nodes as well as message passing between compute nodes into account. The theoretic results are validated by numerical experiments.

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“…Like CholQR or BCGS, the orthogonlization error is of order O(κ 2 ε). For a detailed stability analysis see [5]. The pseudocode for this algorithm is shown in Algorithm 3.1.…”

Section: Bcgs-pipmentioning

confidence: 99%

mentioning

confidence: 99%

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A Hardware-aware and Stable Orthogonalization Framework

Dreier¹,

Engwer²

2022

Preprint

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“…Here, BGS amounts to a block version of CGS. For more details on BGS, we refer the reader to [52,53] and references therein. BGS and TSQR are both regarded as kernels (see [187,141]).…”

Section: S-step Gmresmentioning

confidence: 99%

“…Numerical results reported promising performance of the low-sync Gram-Schmidt and GMRES algorithms. A good survey of Gram-Schmidt up to 2010 was given by Leon et al [162] in 2013; we also refer the interested reader to [23,22,52,53] and their references for the discussion of block Gram-Schmidt algorithms. On the other hand, a low-sync GMRES variant based on Householder QR was derived by Walker [277] in 1988.…”

Section: Low-sync Gmresmentioning

confidence: 99%

GMRES algorithms over 35 years

Zou

2021

Preprint

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A flexible block classical Gram–Schmidt skeleton with reorthogonalization

Zou

2023

Numerical Linear Algebra App

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We investigate a variant of the reorthogonalized block classical Gram–Schmidt method for computing the QR factorization of a full column rank matrix. Our aim is to bound the loss of orthogonality even when the first local QR algorithm is only conditionally stable. In particular, this allows the use of modified Gram–Schmidt instead of Householder transformations as the first local QR algorithm. Numerical experiments confirm the stable behavior of the new variant. We also examine the use of non‐QR local factorization and show by example that the resulting variants, although less stable, may also be applied to ill‐conditioned problems.

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Block Gram-Schmidt algorithms and their stability properties

Cited by 20 publications

References 55 publications

A Hardware-aware and Stable Orthogonalization Framework

A Hardware-aware and Stable Orthogonalization Framework

GMRES algorithms over 35 years

A flexible block classical Gram–Schmidt skeleton with reorthogonalization

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