Hello readers, in today’s topic, we will discuss the Lagrange Interpolation Formula derivation in numerical analysis/techniques. This is a very important method to find out the largest polynomial. Before these, we also discussed about *Regula Falsi Method In Numerical Method*, and Secant Method in Numerical Analysis/Techniques if you don’t know Hurry up Now! Previously we also describe briefly various topics such as *BiSection Method In Numerical Techniques, Newton’s Raphson Method,* and many more which are really helpful for your better understanding. So without wasting time let’s dive into our topic:

## Langrage`s Interpolating Formula Derivation

Basically in this post, we are discourse about the Lagrange interpolation formula derivation. By using some polynomial.

Here we are interpolating a polynomial y = f(x). Where (x_{0},y_{0}), (x_{1},y_{2}), (x_{3},y_{3}),……. (x_{n},y_{n}) are n+1 calculated point. Let their points are exist

X_{i}= x_{0}+ih where i =1, 2, 3….n

Let

y = f(x) = a_{0}(x-x_{1})(x-x_{2})(x-x_{3})……(x-x_{n})

+ a_{1}(x-x_{0})(x-x_{2})(x-x_{3})……(x-x_{n})

+a_{2}(x-x_{0})(x-x_{1})(x-x_{3})……(x-x_{n})

……………………………….

+a_{n}(x-x_{0})(x-x_{1})(x-x_{2})……(x_{n}-x_{n-1})

This is the equation 1

Now putting x=x_{0 }in equation 1 we get

y_{0 } = f(x_{0})= a_{0}(x_{0}-x_{1})(x_{0}-x_{2})(x_{0}-x_{3})……(x_{0}-x_{n})

+ a_{1}(x_{0}-x_{0})(x_{0}-x_{2})(x_{0}-x_{3})……(x_{0}-x_{n})

+a_{2}(x_{0}-x_{0})(x_{0}-x_{1})(x_{0}-x_{3})……(x_{0}-x_{n})

……………………………….

+a_{n}(x_{0}-x_{0})(x_{0}-x_{1})(x_{0}-x_{2})……(x_{0}-x_{n-1})

Finally we got

y_{0 }= a_{0}(x_{0}-x_{1})(x_{0}-x_{2})(x_{0}-x_{3})…….(x_{0}-x_{n})

Again putting x=x_{1 }in equation 1 we get

y_{1 } = f(x_{1})= a_{0}(x_{1}-x_{1})(x_{1}-x_{2})(x_{1}-x_{3})……(x_{1}-x_{n})

+ a_{1}(x_{1}-x_{0})(x_{1}-x_{2})(x_{1}-x_{3})……(x_{1}-x_{n})

+a_{2}(x_{1}-x_{0})(x_{1}-x_{1})(x_{1}-x_{3})……(x_{1}-x_{n})

……………………………….

+a_{n}(x_{1}-x_{0})(x_{1}-x_{1})(x_{1}-x_{2})……(x_{1}-x_{n-1})

Finally we got

y_{1 }= a_{1}(x_{1}-x_{0})(x_{1}-x_{2})(x_{1}-x_{3})……(x_{1}-x_{n})

Again putting x=x_{2 }in equation 1 we get

Y_{2 } = f(x_{2})= a_{0}(x_{2}-x_{1})(x_{2}-x_{2})(x_{2}-x_{3})……(x_{2}-x_{n})

+ a_{1}(x_{2}-x_{0})(x_{2}-x_{2})(x_{2}-x_{3})……(x_{2}-x_{n})

+a_{2}(x_{2}-x_{0})(x_{2}-x_{1})(x_{2}-x_{3})……(x_{2}-x_{n})

……………………………….

+a_{n}(x_{2}-x_{0})(x_{2}-x_{1})(x_{2}-x_{2})……(x_{2}-x_{n-1})

Finally we got

y_{2 }= a_{2}(x_{2}-x_{0})(x_{2}-x_{1})(x_{2}-x_{3})……(x_{2}-x_{n})

Hence from the above calculation we come to conclusion that

Now putting the value in equation 1

Hence,

This is the equation 2

This equation is called as **Langrage`s Interpolating formula** and it is the derivation of **Langrage`s Interpolating formula**

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