Lagrange interpolation in Python
I'm almost a decade late to the party, but I found this searching for a simple implementation of Lagrange interpolation. @smichr's answer is great, but the Python is a little outdated, and I also wanted something that would work nicely with np.ndarrays
so I could do easy plotting. Maybe others will find this useful:
import numpy as np
import matplotlib.pyplot as plt
class LagrangePoly:
def __init__(self, X, Y):
self.n = len(X)
self.X = np.array(X)
self.Y = np.array(Y)
def basis(self, x, j):
b = [(x - self.X[m]) / (self.X[j] - self.X[m])
for m in range(self.n) if m != j]
return np.prod(b, axis=0) * self.Y[j]
def interpolate(self, x):
b = [self.basis(x, j) for j in range(self.n)]
return np.sum(b, axis=0)
X = [-9, -4, -1, 7]
Y = [5, 2, -2, 9]
plt.scatter(X, Y, c='k')
lp = LagrangePoly(X, Y)
xx = np.arange(-100, 100) / 10
plt.plot(xx, lp.basis(xx, 0))
plt.plot(xx, lp.basis(xx, 1))
plt.plot(xx, lp.basis(xx, 2))
plt.plot(xx, lp.basis(xx, 3))
plt.plot(xx, lp.interpolate(xx), linestyle=':')
plt.show()
Try
def interpolate(x, x_values, y_values):
def _basis(j):
p = [(x - x_values[m])/(x_values[j] - x_values[m]) for m in xrange(k) if m != j]
return reduce(operator.mul, p)
assert len(x_values) != 0 and (len(x_values) == len(y_values)), 'x and y cannot be empty and must have the same length'
k = len(x_values)
return sum(_basis(j)*y_values[j] for j in xrange(k))
You can confirm it as follows:
>>> interpolate(1,[1,2,4],[1,0,2])
1.0
>>> interpolate(2,[1,2,4],[1,0,2])
0.0
>>> interpolate(4,[1,2,4],[1,0,2])
2.0
>>> interpolate(3,[1,2,4],[1,0,2])
0.33333333333333331
So the result is the interpolated value based on the polynomial that goes through the points given. In this case, the 3 points define a parabola and the first 3 tests show that the stated y_value is returned for the given x_value.