Repetition

Firstly we will solve some examples with Lagrange interpolating polynomial

from IPython.lib.display import YouTubeVideo
vid = YouTubeVideo("7LhmLrJNVJg")
display(vid)

Newton interpolating polynomial

vid = YouTubeVideo("ziQTh7I6NfE")
display(vid)

Let us now look at how to implement an interpolation polynomial in Newton's form. The following code contains a function that evaluates a polynomial at given points, and calculates the coefficients of the interpolation polynomial in Newton's basis.

import numpy as np
from numpy import array,arange
from math import pi,cos
import matplotlib.pyplot as plt


def evalPoly(c,x,xu):
    n = len(x) - 1 # Degree of polynomial
    p = c[n]
    for k in range(1,n+1):
        p = c[n-k] + (xu -x[n-k])*p
    return p

def polycoef(x,y):
    n = len(x) # Number of data points
    c = y.copy()
    for j in range(1,n):
        c[j:n] = (c[j:n] - c[j-1])/(x[j:n] - x[j-1])
    return c

In the next part, we solve an example from the beginning of a lecture on interpolation

x=array([2 ,4 , 6, 8, 10 , 12, 14, 16, 18, 20, 22, 24])
y=array([0.0 ,0.35 , 0.46, 0.48, 0.50, 0.39,0.27,0.18,0.13,0.09, 0.07, 0.05])
c = polycoef(x,y)

xp = np.linspace(2, 24, 1000)


yu = evalPoly(c,x,xp)
    
fig, ax = plt.subplots(figsize=(6.5, 4))   
plt.plot(x, y, 'bo', label="Data")
plt.plot(xp, yu,label="yInterp")
ax.legend(loc='upper right')
plt.savefig("interolacija.eps") 

Problems related to Newton's interpolating polynomial

We will now solve a couple of problems related to Newton's interpolating polynomial and divided differences.

vid = YouTubeVideo("tvHZovLh3c4")
display(vid)