Numerical stability of GMRES

Drkosová, Jitka; Greenbaum, Anne; Rozložńık, Miroslav; Strakoš, Zdeněk

doi:10.1007/bf01732607

Cited by 71 publications

(57 citation statements)

References 17 publications

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“…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…”

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confidence: 99%

“…HH GMRES was proved backward stable in [7]. That proof relied upon the fact that the Householder reflections keep the loss of orthogonality among the computed Arnoldi vectors close to the machine precision.…”

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confidence: 99%

“…Orthogonality among the Arnoldi vectors can be lost using MGS GMRES finite precision computations. Therefore the results from [7] could not be extended to MGS GMRES, and a different approach had to be used.…”

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confidence: 99%

“…The step from (2.4) to (2.5) requires orthogonality of the columns of V k+1 . However, even if orthogonality of the Arnoldi vectors computed using finite precision arithmetic is well preserved (as in HH GMRES), (2.7) will not hold for the computed quantities after the residual norm drops near the final accuracy level; see [7].…”

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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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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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confidence: 99%

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confidence: 99%

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confidence: 99%

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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 convergência de métodos iterativos é usualmente baseada em critérios de erro inverso em relação à norma, ver [11], [43] e [59]. Em [52], são propostos critérios para o controle da acurácia do produtos matriz-vetor e prova-se que eles garantem a convergência do GMRES tanto em relação à η(x k ), definido em (5.…”

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