Lagrangian Interpolation Formula
Lagrangian Interpolation Formula : Let y = f(x) be a real valued function which is defined in an interval [a,b] . Let x₀ , x₁ ,x₂ ,............xₙ be n+1 distinct points in that interval at which the respective values y₀ ,y₁,y₂ ............yₙ are tabulated. Now our aim is to construct a polynomial Φ(x) of degree ≤ n , which interpolates f(x) such that Φ(xᵢ) = y(xᵢ) , i = 1,2,3,.......,n .........(1) Let us suppose that the polynomial Φ(x) n be given by Φ(x) = Σ lᵢ(x) y(xᵢ) .........(2) i= 0 where each lᵢ(x) is a polynomial of degree ≤n in xᵢ , called Lagrangian function. The function given in equation (1) if each lᵢ (x) satisfied lᵢ(xⱼ) = 0 where i ≠ j ,lᵢ(xⱼ)=1 when i=j .........(3) Now , the polynomial lᵢ(x) vanishes at the (n+1) points xₒ , x₁ , ........., xₙ therefore , we can write this polynomial in the following form lᵢ(x) = cᵢ (x-x₀)(x-x₁).......