Playing with Barycentric Interpolation

Copyright (C) 2020 Andreas Kloeckner

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In [29]:
import numpy as np

import matplotlib.pyplot as plt
In [49]:
x = np.linspace(-1, 1, 5)

xx = np.linspace(-1, 1, 100)



def f(x):

    return np.sin(5/(1.6-x))
In [50]:
plt.plot(xx, f(xx))

plt.plot(x, f(x), "o")
Out[50]:
[<matplotlib.lines.Line2D at 0x7f071da8f790>]
In [80]:
dmat = x.reshape(-1, 1) - x

np.fill_diagonal(dmat, 1)

dmat[0, 2], x[0]-x[2]
Out[80]:
(-1.0, -1.0)
In [128]:
w = 1/dmat.prod(axis=1)

w
Out[128]:
array([ 0.66666667, -2.66666667,  4.        , -2.66666667,  0.66666667])
In [119]:
def bary_first_form(z, f):

    z = z[:, np.newaxis]

    ell = (z-x).prod(axis=-1)

    return ell * (w / (z - x) * f(x)).sum(axis=-1)
In [129]:
plt.plot(xx, bary_first_form(xx, f))

plt.plot(xx, f(xx))

/home/andreas/src/env-3.7/lib/python3.7/site-packages/ipykernel_launcher.py:4: RuntimeWarning: divide by zero encountered in true_divide
after removing the cwd from sys.path.
/home/andreas/src/env-3.7/lib/python3.7/site-packages/ipykernel_launcher.py:4: RuntimeWarning: invalid value encountered in multiply
after removing the cwd from sys.path.
Out[129]:
[<matplotlib.lines.Line2D at 0x7f071be80650>]
In [134]:
def bary_second_form(z, f, weights=None):

    if weights is None:

        weights = w

    z = z[:, np.newaxis]

    num = (weights / (z - x) * f(x)).sum(axis=-1)

    denom = (weights / (z - x)).sum(axis=-1)

    return num/denom
In [140]:
wr = w.copy()

wr[:] = np.random.randn(len(x))



plt.plot(xx, bary_second_form(xx, f))

plt.plot(xx, bary_second_form(xx, f, weights=wr))

plt.plot(xx, f(xx))

plt.plot(x, f(x), "o")

plt.ylim([-2, 2])

/home/andreas/src/env-3.7/lib/python3.7/site-packages/ipykernel_launcher.py:5: RuntimeWarning: divide by zero encountered in true_divide
"""
/home/andreas/src/env-3.7/lib/python3.7/site-packages/ipykernel_launcher.py:6: RuntimeWarning: divide by zero encountered in true_divide

/home/andreas/src/env-3.7/lib/python3.7/site-packages/ipykernel_launcher.py:7: RuntimeWarning: invalid value encountered in true_divide
import sys
Out[140]:
(-2, 2)
In [ ]: