Open access
Methods and Algorithms for Scientific Computing

Algorithms for the Rational Approximation of Matrix-Valued Functions


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.


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

MSC codes

  1. 41A20
  2. 65D15

Formats available

You can view the full content in the following formats:


A. C. Antoulas, Approximation of Large-Scale Dynamical Systems, Adv. Des. Control 6, SIAM, Philadelphia, 2005,
A. C. Antoulas, C. A. Beattie, and S. Gugercin, Interpolatory model reduction of large-scale dynamical systems, in Efficient Modeling and Control of Large-Scale Systems, Springer, Boston, MA, 2010, pp. 3--58.
A. C. Antoulas, C. A. Beattie, and S. Gugercin, Interpolatory Methods for Model Reduction, Comput. Sci. Engrg. 21, SIAM, Philadelphia, 2020,
A. C. Antoulas, I. V. Gosea, and A. C. Ionita, Model reduction of bilinear systems in the Loewner framework, SIAM J. Sci. Comput., 38 (2016), pp. B889--B916,
A. C. Antoulas, S. Lefteriu, and A. C. Ionita, A tutorial introduction to the Loewner framework for model reduction, in Model Reduction and Approximation, Comput. Sci. Engrg. 15, SIAM, Philadelphia, 2017, pp. 335--376,
M. Berljafa, S. Elsworth, and S. Güttel, A Rational Krylov Toolbox for MATLAB, MIMS EPrint 2014.56, Manchester Institute for Mathematical Sciences, The University of Manchester, Manchester, UK, 2014,
M. Berljafa and S. Güttel, Generalized rational Krylov decompositions with an application to rational approximation, SIAM J. Matrix Anal. Appl., 36 (2015), pp. 894--916,
M. Berljafa and S. Güttel, The RKFIT algorithm for nonlinear rational approximation, SIAM J. Sci. Comput., 39 (2017), pp. A2049--A2071,
J.-P. Berrut and L. N. Trefethen, Barycentric Lagrange interpolation, SIAM Rev., 46 (2004), pp. 501--517,
Y. Chahlaoui and P. Van Dooren, Benchmark examples for model reduction of linear time-invariant dynamical systems, in Dimension Reduction of Large-Scale Systems, Lect. Notes Comput. Sci. Eng. 45, Springer, Berlin, 2005, pp. 379--392.
T. A. Driscoll, N. Hale, and L. N. Trefethen, Chebfun Guide,, 2014.
Z. Drmač, S. Gugercin, and C. Beattie, Vector fitting for matrix-valued rational approximation, SIAM J. Sci. Comput., 37 (2015), pp. A2346--A2379,
S. Elsworth and S. Güttel, Conversions between barycentric, RKFUN, and Newton representations of rational interpolants, Linear Algebra Appl., 576 (2019), pp. 246--257.
K. Gallivan, A. Vandendorpe, and P. Van Dooren, Model reduction of MIMO systems via tangential interpolation, SIAM J. Matrix Anal. Appl., 26 (2004), pp. 328--349,
B. Gustavsen, Improving the pole relocating properties of vector fitting, IEEE Trans. Power Delivery, 21 (2006), pp. 1587--1592.
B. Gustavsen and A. Semlyen, Rational approximation of frequency domain responses by vector fitting, IEEE Trans. Power Delivery, 14 (1999), pp. 1052--1061.
S. Güttel and F. Tisseur, The nonlinear eigenvalue problem, Acta Numer., 26 (2017), pp. 1--94.
N. J. Higham, G. M. Negri Porzio, and F. Tisseur, An updated set of nonlinear eigenvalue problems, Tech. Report MIMS EPrint 2019.5, Manchester, United Kingdom, 2019.
A. Hochman, FastAAA: A fast rational-function fitter, in Proceedings of the 26th Conference on Electrical Performance of Electronic Packaging and Systems (EPEPS) (San Jose, CA), IEEE, Washington, DC, 2017, pp. 1--3.
D. S. Karachalios, I. V. Gosea, and A. C. Antoulas, The Loewner framework for system identification and reduction, in Handbook on Model Reduction Volume 1: Methods and Algorithms, De Gruyter, Berlin, to appear.
P. Lietaert, J. Pérez, B. Vandereycken, and K. Meerbergen, Automatic Rational Approximation and Linearization of Nonlinear Eigenvalue Problems, preprint,, 2018.
A. J. Mayo and A. C. Antoulas, A framework for the solution of the generalized realization problem, Linear Algebra Appl., 425 (2007), pp. 634--662.
Y. Nakatsukasa, O. Sète, and L. N. Trefethen, The AAA algorithm for rational approximation, SIAM J. Sci. Comput., 40 (2018), pp. A1494--A1522,
A. Ruhe, Rational Krylov sequence methods for eigenvalue computation, Linear Algebra Appl., 58 (1984), pp. 391--405,
R. Van Beeumen, W. Michiels, and K. Meerbergen, Linearization of Lagrange and Hermite interpolating matrix polynomials, IMA J. Numer. Anal., 35 (2015), pp. 909--930.

Information & Authors


Published In

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


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


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

MSC codes

  1. 41A20
  2. 65D15



Funding Information

Engineering and Physical Sciences Research Council : EP/N510129/1

Metrics & Citations



If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Cited By

View Options

View options


View PDF







Copy the content Link

Share with email

Email a colleague

Share on social media

The SIAM Publications Library now uses SIAM Single Sign-On for individuals. If you do not have existing SIAM credentials, create your SIAM account