Calling from Python

You can call the functions in this package from Python using the PythonCall/JuliaCall package.

Installation

It's recommended to create a new virtual environment to try this out. In Python, you need to install juliacall via

pip install juliacall

Then you start Python and run:

from juliacall import Main as jl

This will download and initialize a copy of Julia. Finally, you need to install this package in that Julia environment:

jl.seval('using Pkg; Pkg.add("RationalFunctionApproximation")')

That should be all you need in the Python environment.

Usage

In each new Python session, you need to load the packages:

from juliacall import Main as jl
jl.seval('using RationalFunctionApproximation, PythonCall')

All the functions and constants exposed to Julia by this package are available using the jl object. For example, to use the discrete AAA algorithm:

import numpy as np    # if installed in Python
x = np.linspace(-1, 1, 1000)
y = np.tanh(5 * (x - 0.2))
r = jl.aaa(x, y)
print(r)

Barycentric rational function of type (11,11)

This will return a wrapped Julia object that you can use in Python as if it were a Python object. For example, you can evaluate the approximation at a point:

r(0.5)

0.9051482536448658

If you want to apply the function at multiple points, you can use comprehensions or vectorize it in numpy:

rv = np.vectorize(r)
rv(np.array([0.5, 0.6, 0.7]))

array([0.90514825, 0.96402758, 0.9866143 ])

You can get information about the approximation using any documented function in the package, e.g.:

print(jl.poles(r))    # returns wrapped Julia type

ComplexF64[0.20000000000544785 - 0.31415926535542893im, 0.20000000000544788 + 0.31415926535542893im, 0.20000207991810143 - 0.942477292594254im, 0.20000207991810143 + 0.9424772925942541im, 0.20308324780986833 - 1.5724812056318853im, 0.20308324780986833 + 1.5724812056318853im, 0.29268586746842673 - 2.3408220889660796im, 0.29268586746842673 + 2.34082208896608im, 0.9695028397625358 + 4.390786420000105im, 0.969502839762536 - 4.390786420000105im, 21.59156666159181 + 0.0im]

print(np.array(jl.residues(r)))    # converts to numpy array

[  0.2       +2.72029915e-11j   0.2       -2.72031942e-11j
   0.19999893+6.55140711e-06j   0.19999893-6.55140637e-06j
   0.20352821+4.86975387e-03j   0.20352821-4.86975387e-03j
   0.33454619+6.91112099e-02j   0.33454619-6.91112099e-02j
   1.25164001-5.59634589e-01j   1.25164001+5.59634589e-01j
 -32.51419889+0.00000000e+00j]

Passing Python functions

To use continuous approximation, you can pass a Python function to the approximate function.

def f(x):
    return np.tanh(5 * (x - 0.2))

r = jl.approximate(f, jl.unit_interval)
r(.5)

0.9051482536448647