Volume 51, pp. 451-468, 2019.

Biorthogonal rational Krylov subspace methods

Niel Van Buggenhout, Marc Van Barel, and Raf Vandebril


A general framework for oblique projections of non-Hermitian matrices onto rational Krylov subspaces is developed. To obtain this framework we revisit the classical rational Krylov subspace algorithm and prove that the projected matrix can be written efficiently as a structured pencil, where the structure can take several forms such as Hessenberg or inverse Hessenberg. One specific instance of the structures appearing in this framework for oblique projections is a tridiagonal pencil. This is a direct generalization of the classical biorthogonal Krylov subspace method, where the projection becomes a single non-Hermitian tridiagonal matrix and of the Hessenberg pencil representation for rational Krylov subspaces. Based on the compact storage of this tridiagonal pencil in the biorthogonal setting, we can develop short recurrences. Numerical experiments confirm the validity of the approach.

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Key words

rational Krylov, biorthogonal, short recurrence, oblique projection, matrix pencil

AMS subject classifications

15A22, 47A75, 65F99, 65Q30

Links to the cited ETNA articles

[21]Vol. 43 (2014-2015), pp. 100-124 Thomas Mach, Miroslav S. Pranić, and Raf Vandebril: Computing approximate (block) rational Krylov subspaces without explicit inversion with extensions to symmetric matrices

ETNA articles which cite this article

Vol. 60 (2024), pp. 327-350 Stefan Kindermann and Werner Zellinger: A short-term rational Krylov method for linear inverse problems

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