FOR FREE CONTENT

This formula is used for (2n + 1) odd equispaced arguments and the value is to be tabulated near the middle point of those arguments. This interpolation is obtained by considering the arithmetic mean of Gauss’s forward and backward interpolation for odd (2n+1) arguments.

DERIVATION

The formula of Stirling’s interpolation is:

GRAPHICAL REPRESENTATION

INPUT: 

The values of x and f(x)

OUTPUT: 

The f(x) after calculating the value of specified x

PROCESS:

Step 1: [Taking the values of x and f(x)]
	Read n [number of given terms]
	for i = 0 to n - 1 repeat
		Read x[i], y[i]
	[End of for loop]
	Read x1 [value for which f(x) is to be calculated]

Step 2: [Stirling interpolation]
		Set h ← x[1] - x[0] 
		Set s ← n / 2 
		Set a ← x[s] 
		Set u ← (x1 - a) / h 
		for i = 0 to  n – 2 repeat
			Set del[i][0] ←  y[i + 1] - y[i] 
		[End of ‘for’ loop]
		for i = 1 to n – 2 repeat
			for j = 0 to n - i –2 repeat
				Set del[j][i] ← del[j + 1][i - 1] - del[j][i - 1]
			[End of ‘for’ loop]
		[End of ‘for’ loop]
Set y1 ← y[s] 
		for i = 1 to n – 1 repeat
			if i mod 2 ≠ 0) then
				if k ≠ 2 then
					Set t1 ← t1×(u^k-(k - 1)^2)
				else
					Set t1 ← t1 × (u^2 - (k - 1)^2)
				[End of ‘if’]
				Set k ← k + 1
				Set d ← d × i
				Set s ← (n - i) / 2 
				Set y1 ← y1 + ((t1 / (2 × d)) × (del[s][i - 1] + del[s - 1][i - 1]))
			else  
				Set t2 ← t2 × (u^2- (l - 1)^2)
				Set l ← l + 1
				Set d ← d × i
				Set s ← (n - i) / 2 
				Set y1 ← y1 + ((t2 / (d)) × (del[s][i - 1])) 
			[End of ‘if’]
		[End of ‘for’ loop]
	Print y1 
[End of stirling interpolation]