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CA-S-MATH-804: Numerical Analysis
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Lagrange Interpolation

Lagrange Interpolation

As will be seen, interpolation of regularly space intervals can produce unsatisfactory results.

First, load the tools needed.

import numpy as np
import matplotlib.pyplot as plt
plt.style.use('seaborn-poster')

Now define the basis polynomial function

def lagrange(x_int, y_int, x_new):
    """
    This function takes pairs of points (x_int, y_int) and, from a set of points x_new
    computes the Lagrange polynomial to return the interpolated values y_new
    """
    y_new = np.zeros(x_new.shape, dtype=np.float)
    for xi, yi in zip(x_int, y_int):
        y_new += yi * np.prod( [(x_new - xj) / (xi - xj) for xj in x_int if xi != xj], axis=0)
    return y_new

Define the Runge function as \begin{equation} f\left(x\right) = \dfrac{1}{1+25 x^2}\end{equation}

runge = lambda x: 1.0 / (1.0 + 25.0 * x**2)

Let the range be $(-1,1)$.

x = np.linspace(-1, 1, 100)
x_int = np.linspace(-1, 1, 11)
y_int = runge(x_int)
x_new = np.linspace(-1, 1, 1000)
y_new = lagrange(x_int, runge(x_int), x_new)
x_new0 = np.linspace(-1, 1, 20)
y_new0 = lagrange(x_int, runge(x_int), x_new0)
plt.plot(x, runge(x), "k--", label="Runge function")

plt.plot(x_new, y_new, label="Uniform Interpolation n=1000")
plt.plot(x_new0, y_new0, label="Uniform Interpolation n=100")
plt.plot(x_int, y_int, "k*")
plt.legend()
plt.xlabel("x")
plt.ylabel("y")
plt.grid(True)
plt.show()

Runge phenomenon

    import numpy as np
    from scipy.interpolate import lagrange
    import matplotlib.pyplot as plt
    x = [0, 1, 2]
    y = [1, 3, 2]
    x_new = np.arange(-1.0, 3.1, 0.1)
    f = lagrange(x, y)
    fig = plt.figure()
    plt.plot(x_new, f(x_new), 'b', x, y, 'ro')
    plt.title(r'Lagrange Polynomial')
    plt.grid()
    plt.xlabel(r'$x$')
    plt.ylabel(r'$y$')
    plt.show()

Lagrange polynomial