Langrage`s Interpolating Formula Derivation

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 (x0,y0), (x1,y2), (x3,y3),……. (xn,yn) are n+1 calculated point. Let their points are exist

Xi= x0+ih where i =1, 2, 3….n

Let

 y = f(x) =     a0(x-x1)(x-x2)(x-x3)……(x-xn)

                  + a1(x-x0)(x-x2)(x-x3)……(x-xn)

                   +a2(x-x0)(x-x1)(x-x3)……(x-xn)

                    ……………………………….

                   +an(x-x0)(x-x1)(x-x2)……(xn-xn-1)

This is the equation 1

Now putting x=x0 in equation 1 we get

y0  = f(x0)=    a0(x0-x1)(x0-x2)(x0-x3)……(x0-xn)

                   + a1(x0-x0)(x0-x2)(x0-x3)……(x0-xn)

                   +a2(x0-x0)(x0-x1)(x0-x3)……(x0-xn)

                    ……………………………….

                   +an(x0-x0)(x0-x1)(x0-x2)……(x0-xn-1)

Finally we got 

y0 = a0(x0-x1)(x0-x2)(x0-x3)…….(x0-xn)

Again putting x=x1 in equation 1 we get

y1  = f(x1)=    a0(x1-x1)(x1-x2)(x1-x3)……(x1-xn)

                   + a1(x1-x0)(x1-x2)(x1-x3)……(x1-xn)

                   +a2(x1-x0)(x1-x1)(x1-x3)……(x1-xn)

                    ……………………………….

                   +an(x1-x0)(x1-x1)(x1-x2)……(x1-xn-1)

Finally we got 

y1 = a1(x1-x0)(x1-x2)(x1-x3)……(x1-xn)

Again putting x=x2 in equation 1 we get

Y2  = f(x2)=    a0(x2-x1)(x2-x2)(x2-x3)……(x2-xn)

                   + a1(x2-x0)(x2-x2)(x2-x3)……(x2-xn)

                   +a2(x2-x0)(x2-x1)(x2-x3)……(x2-xn)

                    ……………………………….

                   +an(x2-x0)(x2-x1)(x2-x2)……(x2-xn-1)

Finally we got 

y2 = a2(x2-x0)(x2-x1)(x2-x3)……(x2-xn)

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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