The AAA algorithm for rational approximation. We introduce a new algorithm for approximation by rational functions on a real or complex set of points, implementable in 40 lines of Matlab and requiring no user input parameters. Even on a disk or interval the algorithm may outperform existing methods, and on more complicated domains it is especially competitive. The core ideas are (1) representation of the rational approximant in barycentric form with interpolation at certain support points and (2) greedy selection of the support points to avoid exponential instabilities. The name AAA stands for ”adaptive Antoulas--Anderson” in honor of the authors who introduced a scheme based on (1). We present the core algorithm with a Matlab code and nine applications and describe variants targeted at problems of different kinds. Comparisons are made with vector fitting, RKFIT, and other existing methods for rational approximation.

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  1. Aumann, Quirin; Deckers, Elke; Jonckheere, Stijn; Desmet, Wim; Müller, Gerhard: Automatic model order reduction for systems with frequency-dependent material properties (2022)
  2. Benner, Peter; Goyal, Pawan; Heiland, Jan; Duff, Igor Pontes: Operator inference and physics-informed learning of low-dimensional models for incompressible flows (2022)
  3. Brubeck, Pablo D.; Trefethen, Lloyd N.: Lightning Stokes solver (2022)
  4. Causley, Matthew F.: The gamma function via interpolation (2022)
  5. De Marchi, Stefano; Elefante, Giacomo; Francomano, Elisa; Marchetti, Francesco: Polynomial mapped bases: theory and applications (2022)
  6. Deng, G.; Lustri, C. J.: Nanoptera in nonlinear woodpile chains with zero precompression (2022)
  7. Derevianko, Nadiia; Plonka, Gerlind: Exact reconstruction of extended exponential sums using rational approximation of their Fourier coefficients (2022)
  8. Díaz Millán, R.; Peiris, V.; Sukhorukova, N.; Ugon, J.: Multivariate approximation by polynomial and generalized rational functions (2022)
  9. Gosea, Ion Victor; Gugercin, Serkan: Data-driven modeling of linear dynamical systems with quadratic output in the AAA framework (2022)
  10. Kaltenbacher, Barbara; Khristenko, Ustim; Nikolić, Vanja; Rajendran, Mabel Lizzy; Wohlmuth, Barbara: Determining kernels in linear viscoelasticity (2022)
  11. Peiris, Vinesha; Sukhorukova, Nadezda: The extension of the linear inequality method for generalized rational Chebyshev approximation to approximation by general quasilinear functions (2022)
  12. Sukhorukova, Nadezda; Ugon, Julien: A generalisation of de la Vallée-Poussin procedure to multivariate approximations (2022)
  13. Wilber, Heather; Damle, Anil; Townsend, Alex: Data-driven algorithms for signal processing with trigonometric rational functions (2022)
  14. An, Dong; Lin, Lin; Xu, Ze: Split representation of adaptively compressed polarizability operator (2021)
  15. Deckers, Elke; Desmet, Wim; Meerbergen, Karl; Naets, Frank: Case studies of model order reduction for acoustics and vibrations (2021)
  16. Egger, H.; Schmidt, K.; Shashkov, V.: Multistep and Runge-Kutta convolution quadrature methods for coupled dynamical systems (2021)
  17. Farazandeh, Elham; Mirzaei, Davoud: A rational RBF interpolation with conditionally positive definite kernels (2021)
  18. Gosea, Ion Victor; Güttel, Stefan: Algorithms for the rational approximation of matrix-valued functions (2021)
  19. Jin, Bangti; Zhou, Zhi: Recovering the potential and order in one-dimensional time-fractional diffusion with unknown initial condition and source (2021)
  20. Keith, Brendan; Khristenko, Ustim; Wohlmuth, Barbara: A fractional PDE model for turbulent velocity fields near solid walls (2021)

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