Numerical behaviour of the modified gram-schmidt GMRES implementation

Greenbaum, Anne; Rozložńık, Miroslav; Strakoš, Zdeněk

doi:10.1007/bf02510248

Cited by 56 publications

(55 citation statements)

References 9 publications

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“…We want to show that MGS GMRES succeeds despite the loss of orthogonality among the computed MGS Arnoldi vectors. A similar hypothesis was published in [11,24] with a justification based on the link between loss of orthogonality among the Arnoldi vectors and the size of the GMRES relative residual. Here is how we hope to prove a significantly stronger statement in [17] by using what is essentially the result (1.2) of this paper as a fundamental intermediate step.…”

mentioning

confidence: 64%

“…That is, significant loss of orthogonality in MGS GMRES apparently did not occur before convergence measured by r k /ρ 0 occurred. This fortuitous behavior was analyzed numerically in [11] and a partial explanation was offered there. A much stronger and more complete theoretical explanation of the observed behavior can be derived from the bounds (3.6)-(3.8).…”

Section: Delayed Convergence Of Gmresmentioning

confidence: 98%

“…It was observed that when MGS was used in (2.2), leading to the MGS GMRES method (2.1)-(2.6), loss of orthogonality in V k+1 was accompanied by a small relative residual norm r k /ρ 0 ; see [11]. That is, significant loss of orthogonality in MGS GMRES apparently did not occur before convergence measured by r k /ρ 0 occurred.…”

Section: Delayed Convergence Of Gmresmentioning

confidence: 99%

“…Thus, theoretical results on GMRES can, for example, provide lower bounds for the residuals of other methods using the same Krylov subspaces. GMRES is also interesting to study computationally, especially since a strong relationship has been noticed between convergence of GMRES and loss of orthogonality among the Arnoldi vectors computed via (finite precision) modified Gram-Schmidt (MGS) orthogonalization; see [11,24]. An understanding of this will be just as important for the practical use of the Arnoldi method as it will be for GMRES itself.…”

mentioning

confidence: 99%

“…Following the important work [5] of Björck, and that of Walker [32], the papers [7] and [11] showed a relationship between the finite precision loss of orthogonality in the MGS Arnoldi vectors and the condition number κ([v 1 ρ 0 , AV k ]). In particular, unless A is extremely ill-conditioned (close to numericaly singular), for computed quantities…”

mentioning

confidence: 99%

See 4 more Smart Citations

Residual and Backward Error Bounds in Minimum Residual Krylov Subspace Methods

Paige¹,

Strakoš²

2002

SIAM J. Sci. Comput.

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Paige and Z. Strakoš, Bounds for the least squares distance using scaled total least squares, Numer. Math., to appear] revealing upper and lower bounds on the residual norm of any linear least squares (LS) problem were derived in terms of the total least squares (TLS) correction of the corresponding scaled TLS problem. In this paper theoretical results of [C. Paige and Z. Strakoš, Bounds for the least squares distance using scaled total least squares, Numer. Math., to appear] are extended to the GMRES context. The bounds that are developed are important in theory, but they also have fundamental practical implications for the finite precision behavior of the modified Gram-Schmidt implementation of GMRES, and perhaps for other minimum norm methods.

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mentioning

confidence: 64%

Section: Delayed Convergence Of Gmresmentioning

confidence: 98%

Section: Delayed Convergence Of Gmresmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Residual and Backward Error Bounds in Minimum Residual Krylov Subspace Methods

Paige¹,

Strakoš²

2002

SIAM J. Sci. Comput.

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A new adaptive GMRES algorithm for achieving high accuracy

Sosonkina¹,

Watson²,

Kapania³

et al. 1998

Numer. Linear Algebra Appl.

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GMRES(k) is widely used for solving non‐symmetric linear systems. However, it is inadequate either when it converges only for k close to the problem size or when numerical error in the modified Gram–Schmidt process used in the GMRES orthogonalization phase dramatically affects the algorithm performance. An adaptive version of GMRES(k) which tunes the restart value k based on criteria estimating the GMRES convergence rate for the given problem is proposed here. This adaptive GMRES(k) procedure outperforms standard GMRES(k), several other GMRES‐like methods, and QMR on actual large scale sparse structural mechanics postbuckling and analog circuit simulation problems. There are some applications, such as homotopy methods for high Reynolds number viscous flows, solid mechanics postbuckling analysis, and analog circuit simulation, where very high accuracy in the linear system solutions is essential. In this context, the modified Gram–Schmidt process in GMRES, can fail causing the entire GMRES iteration to fail. It is shown that the adaptive GMRES(k) with the orthogonalization performed by Householder transformations succeeds whenever GMRES(k) with the orthogonalization performed by the modified Gram–Schmidt process fails, and the extra cost of computing Householder transformations is justified for these applications. © 1998 John Wiley & Sons, Ltd.

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An accurate parallel block Gram-Schmidt algorithm without reorthogonalization

Vanderstraeten

2000

Numer. Linear Algebra Appl.

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Numerical behaviour of the modified gram-schmidt GMRES implementation

Cited by 56 publications

References 9 publications

Residual and Backward Error Bounds in Minimum Residual Krylov Subspace Methods

Residual and Backward Error Bounds in Minimum Residual Krylov Subspace Methods

A new adaptive GMRES algorithm for achieving high accuracy

An accurate parallel block Gram-Schmidt algorithm without reorthogonalization

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