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]
```

__ADVANTAGES__

**1.** This polynomial can be used on functions that are tabulated at equal intervals as well as on the functions which are tabulated at unequal intervals.

**2.** It has an important theoretical role in the development of numerical differentiation and numerical integration of the function.

__DISADVANTAGES__

**1.** Computation of this method is difficult.

**2.** An nth degree Lagrange's interpolation at a point x is to be calculated by performing at least 2(n + 1) multiplications or divisions and (2n + 1) additions or subtractions.

__APPLICATIONS__

**1.** It is generally used for arguments that are spaced unequally.

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