THE BLOCK RATIONAL ARNOLDI METHOD

Steven Elsworth; Stefan Güttel

doi:10.1137/19M1245505

THE BLOCK RATIONAL ARNOLDI METHOD

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Abstract

The block version of the rational Arnoldi method is a widely used procedure for generating an orthonormal basis of a block rational Krylov space. We study block rational Arnoldi decompositions associated with this method and prove an implicit Q theorem. We show how to choose parameters to prevent a premature breakdown of the method and improve its numerical stability by relating the decompositions to nonlinear eigenvalue problems. We explain how rational matrixvalued functions are encoded in block rational Arnoldi decompositions and how they can be evaluated numerically. Two diﬀerent types of deﬂation strategies are discussed. Numerical illustrations using the MATLAB Rational Krylov Toolbox are included.

Original language	English
Journal	S I A M Journal on Matrix Analysis and Applications
DOIs	https://doi.org/10.1137/19M1245505
Publication status	Published - 9 Apr 2020

Keywords

block rational Krylov
rational matrix-valued function
vector autoregression

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10.1137/19M1245505Licence: CC BY

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AB - The block version of the rational Arnoldi method is a widely used procedure for generating an orthonormal basis of a block rational Krylov space. We study block rational Arnoldi decompositions associated with this method and prove an implicit Q theorem. We show how to choose parameters to prevent a premature breakdown of the method and improve its numerical stability by relating the decompositions to nonlinear eigenvalue problems. We explain how rational matrixvalued functions are encoded in block rational Arnoldi decompositions and how they can be evaluated numerically. Two diﬀerent types of deﬂation strategies are discussed. Numerical illustrations using the MATLAB Rational Krylov Toolbox are included.

KW - block rational Krylov

KW - rational matrix-valued function

KW - vector autoregression

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DO - 10.1137/19M1245505

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SN - 0895-4798

JO - S I A M Journal on Matrix Analysis and Applications

JF - S I A M Journal on Matrix Analysis and Applications

ER -

THE BLOCK RATIONAL ARNOLDI METHOD

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