Question

Lagrange Interpolation polynomial using python of degree 3

I understand the math for the question I cannot seem to get the correct python code please help with python to solve

For f (x) = x ln(x), (1) use appropriate Lagrange interpolating polynomial of degree three to approximate f(8.4). Use the following data: f(8.1) = 16.94410, f(8.3) = 17.56492, f(8.6) = 18.50515, f(8.7) = 18.82091. (2) use appropriate Newton interpolation polynomial again to redo the work.

Everything has to be done in python

Answer 1

1. first attempt:

# Python3 program for implementation

# of Lagrange's Interpolation

# To represent a data point corresponding to x and y = f(x)

class Data:

    def __init__(self, x, y):

        self.x = x

        self.y = y

# function to interpolate the given data points

# using Lagrange's formula

# xi -> corresponds to the new data point

# whose value is to be obtained

# n -> represents the number of known data points

def interpolate(f: list, xi: int, n: int) -> float:

    # Initialize result

    result = 0.0

    for i in range(n):

        # Compute individual terms of above formula

        term = f[i].y

        for j in range(n):

            if j != i:

                term = term * (xi - f[j].x) / (f[i].x - f[j].x)

        # Add current term to result

        result += term

    return result

# Driver Code

if __name__ == "__main__":

    # creating an array of 4 known data points

    f = [Data(0, 2), Data(1, 3), Data(2, 12), Data(5, 147)]

    # Using the interpolate function to obtain a data point

    # corresponding to x=3

    print("Value of f(3) is :", interpolate(f, 3, 4))

2. other code:

# Lagrange Interpolation

# Importing NumPy Library
import numpy as np

# Reading number of unknowns
n = int(input('Enter number of data points: '))

# Making numpy array of n & n x n size and initializing 
# to zero for storing x and y value along with differences of y
x = np.zeros((n))
y = np.zeros((n))


# Reading data points
print('Enter data for x and y: ')
for i in range(n):
    x[i] = float(input( 'x['+str(i)+']='))
    y[i] = float(input( 'y['+str(i)+']='))


# Reading interpolation point
xp = float(input('Enter interpolation point: '))

# Set interpolated value initially to zero
yp = 0

# Implementing Lagrange Interpolation
for i in range(n):
    
    p = 1
    
    for j in range(n):
        if i != j:
            p = p * (xp - x[j])/(x[i] - x[j])
    
    yp = yp + p * y[i]    

# Displaying output
print('Interpolated value at %.3f is %.3f.' % (xp, yp))