## 2016.32: Parallelization of the rational Arnoldi algorithm

2016.32:
Mario Berljafa and Stefan Güttel
(2016)
*Parallelization of the rational Arnoldi algorithm.*

*This is the latest version of this eprint.*

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## Abstract

Rational Krylov methods are applicable to a wide range of scientific computing problems, and the rational Arnoldi algorithm is a commonly used procedure for computing an orthonormal basis of a rational Krylov space. Typically, the computationally most expensive component of this algorithm is the solution of a large linear system of equations at each iteration. We explore the option of solving several linear systems simultaneously, thus constructing the rational Krylov basis in parallel. If this is not done carefully, the basis being orthogonalized may become badly conditioned, leading to numerical instabilities in the orthogonalization process. We introduce the new concept of continuation pairs which gives rise to a near-optimal parallelization strategy that allows to control the growth of the condition number of this nonorthogonal basis. As a consequence we obtain a significantly more accurate and reliable parallel rational Arnoldi algorithm. The computational benefits are illustrated using several numerical examples from different application areas.

Item Type: | MIMS Preprint |
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Uncontrolled Keywords: | rational Krylov, orthogonalization, parallelization |

Subjects: | MSC 2000 > 65 Numerical analysis MSC 2000 > 68 Computer science |

MIMS number: | 2016.32 |

Deposited By: | Stefan Güttel |

Deposited On: | 21 September 2016 |

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