Robust rational approximations of nonlinear eigenvalue problems

Güttel, Stefan and Negri Porzio, Gian Maria and Tisseur, Francoise (2020) Robust rational approximations of nonlinear eigenvalue problems. [MIMS Preprint]

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Abstract

We develop algorithms that construct robust (i.e., reliable for a given tolerance and scaling independent) rational approximations of matrix-valued functions on a given subset of the complex plane. We consider matrix-valued functions provided in both split form (i.e., as a sum of scalar functions times constant coefficient matrices) and in black box form. We use an error analysis to construct stopping criteria, one for each form. The criterion for split forms adds weights chosen relative to the importance of each scalar function, which we use in our weighted AAA algorithm, a variant of the set-valued AAA algorithm that guarantees to return a rational approximant with a user chosen accuracy. We propose two-phase approaches for black box matrix-valued functions that construct a surrogate AAA approximation in phase one and refine it in phase two, leading to the surrogate AAA algorithm with exact search and the surrogate AAA algorithm with cyclic Leja--Bagby refinement. The stopping criterion for black box matrix-valued functions is updated at each step of phase two to include information from the previous step. When convergence occurs, our two-phase approaches return rational approximants with a user chosen accuracy. We select problems from the NLEVP collection that represent a variety of matrix-valued functions of different sizes and properties and use them to benchmark our algorithms.

Item Type: MIMS Preprint
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis
Depositing User: Dr Françoise Tisseur
Date Deposited: 13 Nov 2020 15:21
Last Modified: 13 Nov 2020 15:21
URI: https://eprints.maths.manchester.ac.uk/id/eprint/2792

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