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Methods and Algorithms for Scientific Computing

Algorithms for the Rational Approximation of Matrix-Valued Functions

Abstract

A selection of algorithms for the rational approximation of matrix-valued functions are discussed, including variants of the interpolatory adaptive Antoulas--Anderson (AAA) method, the rational Krylov fitting (RKFIT) method based on approximate least squares fitting, vector fitting, and a method based on low-rank approximation of a block Loewner matrix. A new method, called the block-AAA algorithm, based on a generalized barycentric formula with matrix-valued weights, is proposed. All algorithms are compared in terms of obtained approximation accuracy and runtime on a set of problems from model order reduction and nonlinear eigenvalue problems, including examples with noisy data. It is found that interpolation-based methods are typically cheaper to run, but they may suffer in the presence of noise for which approximation-based methods perform better.

Keywords

  1. rational approximation
  2. block rational function
  3. Loewner matrix

MSC codes

  1. 41A20
  2. 65D15

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Information & Authors

Information

Published In

cover image SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Pages: A3033 - A3054
ISSN (online): 1095-7197

History

Submitted: 11 March 2020
Accepted: 18 April 2021
Published online: 9 September 2021

Keywords

  1. rational approximation
  2. block rational function
  3. Loewner matrix

MSC codes

  1. 41A20
  2. 65D15

Authors

Affiliations

Funding Information

Engineering and Physical Sciences Research Council https://doi.org/10.13039/501100000266 : EP/N510129/1

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