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.

References in zbMATH (referenced in 32 articles , 1 standard article )

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  1. An, Dong; Lin, Lin; Xu, Ze: Split representation of adaptively compressed polarizability operator (2021)
  2. Deckers, Elke; Desmet, Wim; Meerbergen, Karl; Naets, Frank: Case studies of model order reduction for acoustics and vibrations (2021)
  3. Egger, H.; Schmidt, K.; Shashkov, V.: Multistep and Runge-Kutta convolution quadrature methods for coupled dynamical systems (2021)
  4. Farazandeh, Elham; Mirzaei, Davoud: A rational RBF interpolation with conditionally positive definite kernels (2021)
  5. Gosea, Ion Victor; Güttel, Stefan: Algorithms for the rational approximation of matrix-valued functions (2021)
  6. Jin, Bangti; Zhou, Zhi: Recovering the potential and order in one-dimensional time-fractional diffusion with unknown initial condition and source (2021)
  7. Keith, Brendan; Khristenko, Ustim; Wohlmuth, Barbara: A fractional PDE model for turbulent velocity fields near solid walls (2021)
  8. Lushnikov, Pavel M.; Silantyev, Denis A.; Siegel, Michael: Collapse versus blow-up and global existence in the generalized Constantin-Lax-Majda equation (2021)
  9. McLean, William: Numerical evaluation of Mittag-Leffler functions (2021)
  10. Nakatsukasa, Yuji; Trefethen, Lloyd N.: Reciprocal-log approximation and planar PDE solvers (2021)
  11. Peiris, V.; Sharon, N.; Sukhorukova, N.; Ugon, J.: Generalised rational approximation and its application to improve deep learning classifiers (2021)
  12. Ramanantoanina, Andriamahenina; Hormann, Kai: New shape control tools for rational Bézier curve design (2021)
  13. Trefethen, Lloyd N.; Nakatsukasa, Yuji; Weideman, J. A. C.: Exponential node clustering at singularities for rational approximation, quadrature, and PDEs (2021)
  14. Alfaro Vigo, Daniel G.; Álvarez, Amaury C.; Chapiro, Grigori; García, Galina C.; Moreira, Carlos G.: Solving the inverse problem for an ordinary differential equation using conjugation (2020)
  15. Beattie, Christopher; Gugercin, Serkan; Tomljanović, Zoran: Sampling-free model reduction of systems with low-rank parameterization (2020)
  16. Berrut, J.-P.; De Marchi, S.; Elefante, G.; Marchetti, F.: Treating the Gibbs phenomenon in barycentric rational interpolation and approximation via the S-Gibbs algorithm (2020)
  17. Harizanov, Stanislav; Lazarov, Raytcho; Margenov, Svetozar: A survey on numerical methods for spectral space-fractional diffusion problems (2020)
  18. Hofreither, Clemens: A unified view of some numerical methods for fractional diffusion (2020)
  19. Hokanson, Jeffrey M.; Magruder, Caleb C.: ( \mathcalH_2)-optimal model reduction using projected nonlinear least squares (2020)
  20. Malachivskyy, P. S.; Pizyur, Ya. V.; Malachivsky, R. P.: Chebyshev approximation by a rational expression for functions of many variables (2020)

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