Lagrange interpolation method in Python

In this, you will learn about the Lagrange interpolation method. First you will learn what is interpolation method and principle of Lagrange interpolation method. Then you will learn how to implement Lagrange interpolation method in Python Programming.

What is Interpolation Method?

Interpolation is the method of finding new data points within an isolated set of known data points. In other words, for any intermediate value of an independent variable, the technique of estimating the value of a mathematical function is interpolation.

Here we can apply the Lagrange Interpolation principle to get our solution.

Lagrange interpolation principle:

If y = f (x) takes

 y0, y1,…, yn 

corresponding to x = x0, x1,…, xn,

This method is preferred over its counterparts, such as Newton's method, because it also applicable for unequal spacing of x.

Example 1: Program for Lagrange interpolation method

Python Code:

from math import *
def lagrange_interpolation(x,y,u):
    r = range(len(y))
    a = [y[i]/product(x[i]-x[j]for j in r if j!=i)for i in r]
    return sum(a[i]*product([u-x[j]for j in r if j!=i])for i in r)
def product(a):
    p = 1
    for i in a:p*=i
    return p
x = [0,1,2,5]
y =[2,3,12,147]
x0 = 2
esti = lagrange_interpolation(x,y,x0)
print("value at x0:",esti)

Output:

value at x0: 12.0

Example 2: Program for Lagrange interpolation method

Python code

from math import *
def lagrange_interpolation(x,y,u):
    r = range(len(y))
    a = [y[i]/product(x[i]-x[j]for j in r if j!=i)for i in r]
    return sum(a[i]*product([u-x[j]for j in r if j!=i])for i in r)
def product(a):
    p = 1
    for i in a:p*=i
    return p
x = [0,2.5,4,5.5,7]
y =[0.0,0.97,1.39,1.70,1.95]
x0 = 4
esti = lagrange_interpolation(x,y,x0)
print("value at x0:",esti)

Output:

value at x0: 1.39