Mixed forward–backward stability of the two-level orthogonal Arnoldi method for quadratic problems

Karl Meerbergen; Javier Pérez

doi:10.1016/j.laa.2018.04.020

Mixed forward–backward stability of the two-level orthogonal Arnoldi method for quadratic problems

Karl Meerbergen
, Javier Pérez

Mathematical Sciences

KU Leuven

Research output: Contribution to journal › Article › peer-review

1 Scopus citations

Abstract

We revisit the numerical stability of the two-level orthogonal Arnoldi (TOAR) method for computing an orthonormal basis of a second-order Krylov subspace associated with two given matrices. We show that the computed basis is close to a basis for a second-order Krylov subspace associated with nearby matrices, provided that the norms of the given matrices are not too large or too small. Thus, we provide conditions that guarantee the numerical stability of the TOAR method in computing orthonormal bases of second-order Krylov subspaces. We also study scaling the quadratic problem for improving the numerical stability of the TOAR procedure when the norms of the matrices are too large or too small.

Original language	English
Pages (from-to)	1-15
Number of pages	15
Journal	Linear Algebra and Its Applications
Volume	553
DOIs	https://doi.org/10.1016/j.laa.2018.04.020
State	Published - Sep 15 2018

Funding

This paper presents research results of the Belgian Network IAP VII/19 (DYSCO) (Dynamical Systems, Control, and Optimization), funded by the Interuniversity Attraction Poles Programme , initiated by the Belgian State Science Policy Office. The research is also partially funded by the KU-Leuven Research Council grant OT/14/074 . The scientific responsibility rests with its authors.

Funder number
OT/14/074
IAP VII/19

Keywords

Arnoldi algorithm
Krylov subspace
Numerical stability
Second-order Arnoldi algorithm
Second-order Krylov subspace
Two-level orthogonal Arnoldi algorithm

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title = "Mixed forward–backward stability of the two-level orthogonal Arnoldi method for quadratic problems",

abstract = "We revisit the numerical stability of the two-level orthogonal Arnoldi (TOAR) method for computing an orthonormal basis of a second-order Krylov subspace associated with two given matrices. We show that the computed basis is close to a basis for a second-order Krylov subspace associated with nearby matrices, provided that the norms of the given matrices are not too large or too small. Thus, we provide conditions that guarantee the numerical stability of the TOAR method in computing orthonormal bases of second-order Krylov subspaces. We also study scaling the quadratic problem for improving the numerical stability of the TOAR procedure when the norms of the matrices are too large or too small.",

keywords = "Arnoldi algorithm, Krylov subspace, Numerical stability, Second-order Arnoldi algorithm, Second-order Krylov subspace, Two-level orthogonal Arnoldi algorithm",

author = "Karl Meerbergen and Javier P{\'e}rez",

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AU - Meerbergen, Karl

AU - Pérez, Javier

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N2 - We revisit the numerical stability of the two-level orthogonal Arnoldi (TOAR) method for computing an orthonormal basis of a second-order Krylov subspace associated with two given matrices. We show that the computed basis is close to a basis for a second-order Krylov subspace associated with nearby matrices, provided that the norms of the given matrices are not too large or too small. Thus, we provide conditions that guarantee the numerical stability of the TOAR method in computing orthonormal bases of second-order Krylov subspaces. We also study scaling the quadratic problem for improving the numerical stability of the TOAR procedure when the norms of the matrices are too large or too small.

AB - We revisit the numerical stability of the two-level orthogonal Arnoldi (TOAR) method for computing an orthonormal basis of a second-order Krylov subspace associated with two given matrices. We show that the computed basis is close to a basis for a second-order Krylov subspace associated with nearby matrices, provided that the norms of the given matrices are not too large or too small. Thus, we provide conditions that guarantee the numerical stability of the TOAR method in computing orthonormal bases of second-order Krylov subspaces. We also study scaling the quadratic problem for improving the numerical stability of the TOAR procedure when the norms of the matrices are too large or too small.

KW - Arnoldi algorithm

KW - Krylov subspace

KW - Numerical stability

KW - Second-order Arnoldi algorithm

KW - Second-order Krylov subspace

KW - Two-level orthogonal Arnoldi algorithm

UR - https://www.scopus.com/pages/publications/85046748269

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DO - 10.1016/j.laa.2018.04.020

M3 - Article

AN - SCOPUS:85046748269

SN - 0024-3795

VL - 553

SP - 1

EP - 15

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

ER -

Mixed forward–backward stability of the two-level orthogonal Arnoldi method for quadratic problems

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