Inexact Arnoldi residual estimates and decay properties for functions of non-Hermitian matrices

Pozza, Stefano; Simoncini, Valeria

doi:10.1007/s10543-019-00763-6

Cited by 11 publications

(15 citation statements)

References 37 publications

(55 reference statements)

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“…These results have been extended to the non-normal case by the author and Boito in [7] and, more recently, by Pozza and Simoncini in [39]. Specifically, if A is a banded normal matrix and f is analytic in the interior of W( A) and bounded on the boundary ∂W(A), then an exponential off-diagonal decay bound can be established for the entries of f (A).…”

Section: The Field Of Values and Some Of Its Propertiesmentioning

confidence: 88%

“…Using this theorem, Pozza and Simoncini [39] obtained the following off-diagonal decay bound. We include the short and elegant proof for completeness.…”

Section: The Field Of Values and Some Of Its Propertiesmentioning

confidence: 96%

“…A more precise statement is possible to account for matrices with lower bandwidth β and upper bandwidth γ with β = γ , see [39]. Moreover, the result can be extended to more general sparse matrices.…”

Section: The Field Of Values and Some Of Its Propertiesmentioning

confidence: 99%

“…Theorem 3 [39] Let A ∈ C n×n be k-banded and such that W( A) ⊆ K, with K compact and convex. With φ and τ > 1 defined as before, we have…”

Section: The Field Of Values and Some Of Its Propertiesmentioning

confidence: 99%

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Some uses of the field of values in numerical analysis

Benzi

2020

Boll Unione Mat Ital

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Section: The Field Of Values and Some Of Its Propertiesmentioning

confidence: 88%

“…Using this theorem, Pozza and Simoncini [39] obtained the following off-diagonal decay bound. We include the short and elegant proof for completeness.…”

Section: The Field Of Values and Some Of Its Propertiesmentioning

confidence: 96%

Section: The Field Of Values and Some Of Its Propertiesmentioning

confidence: 99%

“…Theorem 3 [39] Let A ∈ C n×n be k-banded and such that W( A) ⊆ K, with K compact and convex. With φ and τ > 1 defined as before, we have…”

Section: The Field Of Values and Some Of Its Propertiesmentioning

confidence: 99%

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Some uses of the field of values in numerical analysis

Benzi

2020

Boll Unione Mat Ital

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“…In practical calculations, however, the subspaces spanned by the computed bases may not be Krylov subspaces 22,30 . For instance, the application of A may be inexact in practice, 31‐34 such that the constructed subspaces are not Krylov subspaces with respect to A . Moreover, it is well known that the bases built in the Arnoldi method, the Lanczos method, and the two‐sided Lanczos method may lose orthogonality or biorthogonality during iterations 4 .…”

Section: Introductionmentioning

confidence: 99%

New strategies for determining backward perturbation bound of approximate two‐sided Krylov subspaces

2020

Numerical Linear Algebra App

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Summary Given a nonsymmetric matrix A∈ℝn×n and two unit norm vectors, the two‐sided Krylov subspace methods construct a pair of bases for two Krylov subspaces with respect to A and AT, respectively. In practical calculations, however, the two subspaces spanned by the computed bases may not be Krylov subspaces. Given two subspaces 𝒦 and ℒ, in [G. Wu et al, Toward backward perturbation bounds for approximate dual Krylov subspaces, BIT, 53 (2013), pp. 225‐239], the authors considered how to determine a backward perturbation E whose norm is as small as possible, such that 𝒦 and ℒ are Krylov subspaces of A + E and (A + E)T, respectively. However, as the two bases used are biorthonormal, their results are nonoptimal in terms of unitarily invariant norms, and the perturbation bound can be greatly overestimated. In this work, we revisit this problem and use orthonormal bases instead of biorthonormal bases to derive new perturbation bounds. We propose two new strategies, the first one focuses on choosing optimal orthonormal basis matrices, and the second one resorts to solving small‐sized generalized Sylvester matrix equations. Numerical experiments show that our bounds improve the existing one substantially.

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Functions of rational Krylov space matrices and their decay properties

Pozza

Simoncini

2021

Numer. Math.

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Rational Krylov subspaces have become a fundamental ingredient in numerical linear algebra methods associated with reduction strategies. Nonetheless, many structural properties of the reduced matrices in these subspaces are not fully understood. We advance in this analysis by deriving bounds on the entries of rational Krylov reduced matrices and of their functions, that ensure an a-priori decay of their entries as we move away from the main diagonal. As opposed to other decay pattern results in the literature, these properties hold in spite of the lack of any banded structure in the considered matrices. Numerical experiments illustrate the quality of our results.

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Inexact Arnoldi residual estimates and decay properties for functions of non-Hermitian matrices

Cited by 11 publications

References 37 publications

Some uses of the field of values in numerical analysis

Some uses of the field of values in numerical analysis

New strategies for determining backward perturbation bound of approximate two‐sided Krylov subspaces

Functions of rational Krylov space matrices and their decay properties

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