Bounds on the eigenvalue range and on the field of values of non-Hermitian and indefinite finite element matrices

Loghin, Daniel; Gijzen, Martin van; Jonkers, E.J.

doi:10.1016/j.cam.2005.04.040

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Theorem 32 (Classical Convergence Bound For Gmres1

Theorem 41 (Optimal Seed Frequency For Preconditioned Shifted Gmres) Let {S1

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Cited by 8 publications

(2 citation statements)

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“…Note that similar GMRES bounds exist that include the distance of the field of values of A to the origin. In particular in [22], the field of value for acoustic wave problems is studied and is shown to enclose the origin. The advantage of the bound in Theorem 3.2 is that we can analytically describe the area described by the circle (c, R), and that it describes the asymptotic convergence behavior of GMRES.…”

Section: Theorem 32 (Classical Convergence Bound For Gmresmentioning

confidence: 99%

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An efficient two-level preconditioner for multi-frequency wave propagation problems

Baumann

Gijzen

2019

Applied Numerical Mathematics

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Section: Theorem 32 (Classical Convergence Bound For Gmresmentioning

confidence: 99%

“…Moreover, we plot the angle along which we have optimized in (22) as a dashed line. In the right figure, we show the corresponding preconditioned spectrum with bounding circles for a surrogate problem.…”

Section: Theorem 41 (Optimal Seed Frequency For Preconditioned Shifted Gmres) Let {Smentioning

confidence: 99%

An efficient two-level preconditioner for multi-frequency wave propagation problems

Baumann

Gijzen

2019

Applied Numerical Mathematics

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Fields of Values of Linear Pencils and Spectral Inclusion Regions

Bebiano

Providência

Nata

et al. 2017

Springer Proceedings in Mathematics &Amp; Statistics

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We propose efficient methods for the numerical approximation of the field of values of the linear pencil A − λ B, when one of the matrix coefficients A or B is Hermitian and λ ∈ . Our approach builds on the fact that the field of values can be reduced under compressions to the bidimensional case, for which these sets can be exactly determined. The presented algorithms hold for matrices both of small and large size. Furthermore, we investigate spectral inclusion regions for the pencil based on certain fields of values. The results are illustrated by numerical examples. We point out that the given procedures complement the known ones in the literature.

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Efficient Adaptive Stochastic Collocation Strategies for Advection–Diffusion Problems with Uncertain Inputs

et al. 2023

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Bounds on the eigenvalue range and on the field of values of non-Hermitian and indefinite finite element matrices

Cited by 8 publications

References 8 publications

An efficient two-level preconditioner for multi-frequency wave propagation problems

An efficient two-level preconditioner for multi-frequency wave propagation problems

Fields of Values of Linear Pencils and Spectral Inclusion Regions

Efficient Adaptive Stochastic Collocation Strategies for Advection–Diffusion Problems with Uncertain Inputs

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