Newton Interpolation Method in Python

Interpolation is the estimation of the value of two known values   in a range of values.

Newton's fractional difference interpolation formula is an interpolation technique used when the interval difference is not equal to all values.

The (n + 1) values   of the y = f (x) function correspond to the arguments x = x0, x1, x2 are f (x0), f (x1), f (x2) ……… f (xn)… Xn, where interval differences Are not identical.

Newton Interpolation Formula

Write a Program to find value of x using Newton Interpolation Formula

value at 52

Input: x = 45, 50, 55, 60

          y= f(x) = 0.7071, 0.7660, 0.8192, 0.8660

Python Code:
def u_cal(u,n):
    temp = u
    for i in range(1,n):
        temp = temp*(u-i)
    return temp
def fact(n):
    f = 1
    for i in range(2,n+1):
        f*=i
    return f
n = 4
x = [45,50,55,60]
y = [[0 for i in range(n)] for j in range(n)]
y[0][0]=0.7071
y[1][0]=0.7660
y[2][0]=0.8192
y[3][0]=0.8660
for i in range(1,n):
    for j in range(n-i):
        y[j][i]=y[j+1][i-1]-y[j][i-1]
for i in range (n):
    print(x[i],end="\t")
    for j in range(n-i):
        print(y[i][j],end="\t")
    print("")
value = 52
sum = y[0][0]
u =(value-x[0]/x[1]-x[0])
for i in range(1,n):
    sum = sum+(u_cal(u,i)*y[0][i]/fact(i))
print("\n value at",value,"is",round(sum,6))

Output:

45	0.7071	0.05890000000000006	-0.005700000000000038	-0.0007000000000000339	
50	0.766	0.053200000000000025	-0.006400000000000072	
55	0.8192	0.04679999999999995	
60	0.866	

value at 52 is 0.962846

Example 2:
Write a program to find value of x using Newton Interpolation Formula

value at 1925

Input: x = 1891,1901,1911,1921,1931

          y = f(x) = 46,66,81,93,101

Python Code
def u_cal(u,n): 
    temp = u
    for i in range(1,n):
        temp = temp*(u-i)
    return temp
def fact(n):
    f = 1
    for i in range(2,n+1):
        f*=i
    return f
n = 5
x = [1891,1901,1911,1921,1931]
y = [[0 for i in range(n)] for j in range(n)]
y[0][0]=46
y[1][0]=66
y[2][0]=81
y[3][0]=93
y[4][0]=101
for i in range(1,n):
    for j in range(n-i):
        y[j][i]=y[j+1][i-1]-y[j][i-1]
for i in range (n):
    print(x[i],end="\t")
    for j in range(n-i):
        print(y[i][j],end="\t")
    print("")
value=1925
sum=y[0][0]
u =(value-x[0]/x[1]-x[0])
for i in range(1,n):
    sum = sum+(u_cal(u,i)*y[0][i]/fact(i))
print("\n value at",value,"is",round(sum,6))
Output:
1891	46	20	-5	2	-3	
1901	66	15	-3	-1	
1911	81	12	-4	
1921	93	8	
1931	101	

 value at 1925 is -113859.489961