parametric-AAA

This project contains MATLAB implementations for the parametric adaptive Antoulas-Anderson (p-AAA) algorithm. The p-AAA algorithm computes a multivariate rational function that approximates a given data set. Several variants of the algorithm and efficient implementations for barycentric forms of rational functions are implemented.

Basic example for approximating the function $f(x,y) = \frac{x + y}{4 + cos(x) + cos(y)}$:

% Generate test data
x = linspace(-5,5,100);
y = linspace(-5,5,100);
samples = (x.' + y) ./ (4 + cos(x.') + cos(y));

% Approximate data via p-AAA with error tolerance 1e-5
bf = paaa(samples,{x,y},1e-5)

% Evaluate rational approximation
bf.eval([0,0])

Dependencies

• MATLAB R2022a or later
• For low-rank p-AAA implementation: Tensor Toolbox for MATLAB, version 3.4 or later

Contents

Barycentric forms:

• BarycentricForm Implemention of the barycentric form of a multivariate rational function.
• BlockBarycentricForm Implemention of the multivariate barycentric form for vector/matrix valued functions.
• LowRankBarycentricForm Implementation of the multivariate barycentric form with barycentric coefficients represented by low-rank tensors.

Algorithms:

• paaa Standard p-AAA algorithm.
• sv_paaa Set-valued p-AAA for vector/matrix valued functions.
• lr_paaa Low-rank p-AAA that represents barycentric coefficients in terms of low-rank tensors.

Reference

For the original algorithm:

A. Carracedo Rodriguez, L. Balicki, and S. Gugercin,
The p-AAA Algorithm for Data-Driven Modeling of Parametric Dynamical Systems,
SIAM J. Sci. Comput., vol. 45, no. 3, pp. A1332–A1358, 2023.
https://doi.org/10.1137/20M1322698

For the low-rank version:

L. Balicki and S. Gugercin,
Multivariate Rational Approximation via Low-Rank Tensors and the p-AAA Algorithm,
arXiv:2502.03204 [math.NA], 2025.
https://arxiv.org/abs/2502.03204

License

The code is available under the MIT license (see LICENSE.md for details).