Newton Backward Difference Interpolation Method | Python

				
					import math

# Define the function for Newton's Backward Difference Interpolation
def newton_backward_interpolation(x, y, x_new):
    # Get the number of data points
    n = len(y)

    # Get the common spacing between the x-values
    h = x[1] - x[0]

    # Compute the backward differences of the y-values using a nested loop
    deltas = [y] # initialize a list of lists with y as the first element
    for i in range(1, n):
        delta_i = []
        for j in range(n-i):
            delta_j = deltas[i-1][j+1] - deltas[i-1][j]
            delta_i.append(delta_j)
        deltas.append(delta_i)

    # Compute the coefficients of the backward difference interpolating polynomial using the backward differences
    b = [deltas[0][0] / math.factorial(i) for i in range(n)]
    for i in range(1, n):
        b[i] = deltas[i-1][0] / (math.factorial(i) * h**i)

    # Evaluate the interpolating polynomial at the desired point x_new using a nested loop
    Pn = b[-1] # start with the highest order coefficient
    for i in range(n-2, -1, -1):
        Pn = Pn*(x_new - x[i]) + b[i]
    
    # Return the estimated value of the function at x_new
    return Pn

# Example usage
# Define the data points
x = [0.0, 0.2, 0.4, 0.6, 0.8, 1.0]
y = [1.0000, 1.2214, 1.4918, 1.8221, 2.2255, 2.7183]

# Define the point where we want to estimate the function
x_new = 0.5

# Use the newton_backward_interpolation function to estimate the value of the function at x_new
Pn = newton_backward_interpolation(x, y, x_new)

# Print the estimated value of the function at x_new
print(f"Pn({x_new}) = {Pn}")

#Output - Pn(0.5) = 1.6604399999999998