This is a pure Python package which provides routines for rational and polynomial approximation through the so-called barycentric representation. The advantage of this representation is (often significantly) improved stability over classical approaches.

See the API documentation for an overview of the available functions.

The package implements the novel BRASIL algorithm for best rational approximation; see the paper or the preprint to learn more.

The following example computes the best uniform rational approximation of degree 5 to a given function in the interval [0, pi]:

import numpy as np
import baryrat

def f(x): return np.sin(x) * np.exp(x)
r = baryrat.brasil(f, [0,np.pi], 5)

The rational function r can then be evaluated at arbitrary nodes, its poles computed, and more. See the documentation for details.

The package includes 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.

The package includes functions for polynomial interpolation, rational interpolation with either fixed poles or fixed interpolation nodes, Floater-Hormann interpolation, and more.

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

pip install baryrat

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

import numpy as np
from baryrat import aaa

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

r = aaa(Z, F, 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 baryrat.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