Proof of Lagrange Theorem
Let’s consider a nth-degree polynomial of the given form,
f(x) = A0(x – x1)(x – x2)(x – x3)…(x – xn) + A1(x – x1)(x – x2)(x – x3)…(x – xn) + … + A(n-1)(x – x1)(x – x2)(x – x3)…(x – xn)
Substitute observations xi to get Ai
Put x = x0 then we get A0
f(x0) = y0 = A0(x0 – x1)(x0 – x2)(x0 – x3)…(x0 – xn)
A0 = y0/(x0 – x1)(x0 – x2)(x0 – x3)…(x0 – xn)
By substituting x = x1 we get A1
f(x1) = y1 = A1(x1 – x0)(x1 – x2)(x1 – x3)…(x1 – xn)
A1 = y1/(x1 – x0)(x1 – x2)(x1 – x3)…(x1 – xn)
Similarly, by substituting x = xn we get An
f(xn) = yn = An(xn – x0)(xn – x1)(xn – x2)…(xn – xn-1)
An = yn/(xn – x0)(xn – x1)(xn – x2)…(xn – xn-1)
If we substitute all values of Ai in function f(x) where i = 1, 2, 3, …n then we get Lagrange Interpolation Formula as,
Lagrange Interpolation Formula
Lagrange Interpolation Formula finds a polynomial called Lagrange Polynomial that takes on certain values at an arbitrary point. It is an nth-degree polynomial expression of the function f(x). The interpolation method is used to find the new data points within the range of a discrete set of known data points.
In this article, we will learn about, Lagrange Interpolation, Lagrange Interpolation Formula, Proof for Lagrange Interpolation Formula, Examples based on Lagrange Interpolation Formula, and others in detail.