The AAA algorithm for rational approximation

This is a Python implementation of the AAA algorithm for rational approximation described in the paper "The AAA Algorithm for Rational Approximation" by Yuji Nakatsukasa, Olivier Sète, and Lloyd N. Trefethen, SIAM Journal on Scientific Computing 2018 40:3, A1494-A1522. (doi)

A MATLAB implementation of this algorithm is contained in Chebfun. The present Python version is a more or less direct port of the MATLAB version.

The "cleanup" feature for spurious poles and zeros is not currently implemented.

Installation

The implementation is in pure Python and requires only numpy and scipy as dependencies. Install it using pip:

pip install aaa-approx

Usage

Here's an example of how to approximate a function in the interval [0,1]:

import numpy as np
from aaa import aaa

Z = np.linspace(0.0, 1.0, 1000)
F = np.exp(Z) * np.sin(2*np.pi*Z)

r = aaa(F, Z, mmax=10)

Instead of the maximum number of terms mmax, it's also possible to specify the error tolerance tol. Both arguments work exactly as in the MATLAB version.

The returned object r is an instance of the class aaa.BarycentricRational and can be called like a function. For instance, you can compute the error on Z like this:

err = F - r(Z)
print(np.linalg.norm(err, np.inf))

If you are interested in the poles and residues of the computed rational function, you can query them like

pol,res = r.polres()

and the zeroes using

zer = r.zeros()

Finally, the nodes, values and weights used for interpolation (called zj, fj and wj in the original implementation) can be accessed as properties:

r.nodes
r.values
r.weights