Derivation of Newton's Fundamental Interpolation Formula
Derivation of Newton's Fundamental Interpolation Formula: Let y = f (x) be a function with given values yᵢ = f(xᵢ) for (n+1) points x₀,x₁,x₂,.....,xₙ . Our aim is to construct a polynomial Φ(x) of degree not higher than n satisfying the following conditions Φ(xᵢ) =yᵢ=f(xᵢ) ........(1) for i = 0,1,2...,n Let us take the polynomial Φ(x) in the following form Φ(x) =a₀+a₁(x-x₀)+a₂(x-x₀)(x-x₁)+a₃(x-x₀)(x-x₁)(x-x₂) + ......+aₙ(x-x₀)(x-x₁)(x-x₂)....(x-xₙ) ........(2) where a₀,a₁,....aₙ i.e aᵢ's are constants to be determined . Putting i=0 in equation (1) ,we get Φ(x₀) = y₀ = f(x₀) i.e f(x₀) = Φ(x₀) Again , putting x= x₀ in equation (2) ,we get Φ(x₀) = a₀ ⇒a₀ = f(x₀) ⇒a₀ = f[x₀] .......(3) Putting i =1 in and x= x₁ in (2) ,we get f(x₁) = Φ(x₁) and Φ(x₁) = a₀ + a₁(x₁-x₀) or, f(x₁) = a₀ +a₁(x₁-x₀)=f(x₀) +a₁(