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In numerical this method is used for polynomial interpolation. This method is used to find a polynomial by taking certain values at arbitrary points. The set of points are given such that no two points are of the same values, and this function works in such a way that the functions coincide at each point.

DERIVATION

Let y = f(x) be a real valued function defined in an interval [a, b] and let ${\mathrm{x}}_0, {\mathrm{x}}_1, {\mathrm{x}}_2, ..... , {\mathrm{x}}_{\mathrm{n}}$ be (n + 1) distinct points in the intervals at which the respective values ${\mathrm{y}}_0, {\mathrm{y}}_1, {\mathrm{y}}_2, ..... , {\mathrm{y}}_{\mathrm{n}}$ are given. 

Assume the polynomial 

This is called the Lagrangian function.

Since

Since

It should be in the form:

i.e.

Thus we have 

Using the product notation, we have, 

Again, we can write 

Differentiating with respect to x we get

Now, 

Therefore, 

Thus, the interpolating formula can be written as:

GRAPHICAL REPRESENTATION

INPUT: 

The values of f(x) and x with the value for which the f(x) is to be calculated

OUTPUT: 

The value of f(x) for the required x

PROCESS:

Step 1: [Taking the inputs from the user]
	Read n [The number of points]
	for i = 1 to n repeat
		Read x[i] [The values of x]
	[End of ‘for’ loop]
	for i = 1 to n repeat
		Read y[i] [the values of f(x)]
	[End of ‘for’ loop]
	Read x1 [The point of interpolation]

Step 2: [Lagrange’s Interpolation]
	Set u ← 1.0
	Set s ← 0.0
	for i = 1 to n repeat
		Set d[i] ← 1.0
		for j = 1 to n repeat
			If j = i then
				Set a[i][j] ← x1 - x[j]
			else
				Set a[i][j] ← x[i] - x[j]
			[End of ‘if’]	
Set d[i] ← d[i] × a[i][j]
		[End of ‘for’ loop]
		Set s ← s + y[i]/d[i]
		Set u ← u × a[i][i]
	[End of ‘for’ loop]
	for i = 1 to n repeat
		Print x[i], y[i]
	[End of ‘for’ loop]
	Print u × s
[End of Lagrange’s Interpolation]