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Legendre polynomial

#!/usr/bin/env python

import numpy as np
import matplotlib.pyplot as plt

data = np.array(((-9, 5), (-4, 2), (-1, -2), (7, 9)), dtype=float)
xcoords = np.linspace(-10, 10, 201)
pts = np.arange(0, len(data))

l_x = []
for x in xcoords:
    l_j = []
    for j in pts:
        lval = 1.0
        for m in np.r_[pts[:j], pts[j + 1:]]:
            x_m, x_j = data[[m, j], 0]
            lval *= (x - x_m) / (x_j - x_m)
        l_j.append(lval)
    l_x.append(np.dot(data[:, 1], l_j))

plt.plot(xcoords, l_x)
plt.plot(data[:, 0], data[:, 1], 'o')
plt.show()

Finite difference coefficient - Wikipedia

https://doi.org/10.1090/S0025-5718-1988-0935077-0

Radial grid

#!/usr/bin/env python

import numpy as np
from matplotlib.patches import Circle
import matplotlib.pyplot as plt

# ln(r_n+1) = ln(r_n) + h
# h = 1, r_0 = 1
ln_r = np.arange(11)
radii = np.exp(ln_r)
fig, ax = plt.subplots()
for r in radii:
    ax.add_patch(Circle((0, 0), radius=r, fill=False))
plt.axis('scaled')
ax.set_aspect('equal')
plt.show()

https://doi.org/10.1016/0010-4655(70)90008-1