Langrage Interpolating Formula Derivation
Hello readers, in today’s topic, we will discuss Langrage Interpolating Formula Derivation numerical analysis/techniques. This approach to determining the largest polynomial is crucial. Before these, we also discussed Secant Method in Numerical Analysis Regula Falsi Method In Numerical Method, if you don’t know Hurry up Now! Previously we also described 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:
The interpolation polynomial’s Lagrange form demonstrates both its singularity and the linear nature of polynomial interpolation. As a result, it is preferred in justifications and theoretical justifications. The invertibility of the Vandermonde matrix, which results from the Vandermonde determinant not vanishing, is another indication of uniqueness.
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
To find Langrage Interpolating Formula Derivation 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 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 the conclusion that
Now putting the value in equation 1
Hence,
This is equation 2
This equation is called Langrage`s Interpolating formula and it is the derivation of Langrage`s Interpolating formula.
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