Derivation of Newton's Forward Difference Interpolation Formula
Newton's Forward Difference Interpolation Formula: Let y = f(x) be a function of x and let us suppose that yᵢ = f(xᵢ) ...(1) for i = 1,2,3,.....,n satisfying the condition xᵢ = x₀+ih where 'h' is the interval of difference . Now our aim is to constuct a function Φ(x) of degree not higher than n such that Φ(xᵢ) = yᵢ ............(2) Since Φ(x) is a polynomial of degree n then we can write 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₋₁)...(3) Let us find the value of a₀,a₁,a₂.....aₙ satisfying the equations (2) and (3) From equation (2) , we get Φ(x₀) = y₀ From equation (3) , we get Φ(x₀) = a₀ so , a₀ = y₀ ...........(4) From equation (2) we get Φ(x₁) = y₁ From equation (3) we get Φ(x₁) = a₀+a₁(x₁-x₀) Therefore y₁ = a₀ + a₁(x₁-x₀) ⇒y₁ = y₀ + a₁h as a₀=y₀ and h = x₁-x₀ ⇒