Güttel, Stefan and Negri Porzio, Gian Maria and Tisseur, Francoise (2020) Robust rational approximations of nonlinear eigenvalue problems. [MIMS Preprint] (Submitted)
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Abstract
We develop algorithms that construct robust (i.e., reliable for a given tolerance, and scaling independent) rational approximants of matrixvalued functions on a given subset of the complex plane. We consider matrixvalued functions provided in both split form (i.e., as a sum of scalar functions times constant coefficient matrices) and in black box form. We develop a new error analysis and use it for the construction of stopping criteria, one for each form. Our criterion for split forms adds weights chosen relative to the importance of each scalar function, leading to the weighted AAA algorithm, a variant of the setvalued AAA algorithm that can guarantee to return a rational approximant with a userchosen accuracy. We propose twophase approaches for black box matrixvalued 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 LejaBagby refinement. The stopping criterion for black box matrixvalued functions is updated at each step of phase two to include information from the previous step. When convergence occurs, our twophase approaches return rational approximants with a userchosen accuracy. We select problems from the NLEVP collection that represent a variety of matrixvalued 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:  15 Nov 2020 17:02 
Last Modified:  15 Nov 2020 17:02 
URI:  https://eprints.maths.manchester.ac.uk/id/eprint/2796 
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Robust rational approximations of nonlinear eigenvalue problems. (deposited 13 Nov 2020 15:21)
 Robust rational approximations of nonlinear eigenvalue problems. (deposited 15 Nov 2020 17:02) [Currently Displayed]
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