Existence And Uniqueness Of Interpolating Polynomial
Existence And Uniqueness Of Interpolating Polynomial We now examine the existence and uniqueness of a polynomial that interpolates a function f(x) at a given set of distinct nodes x₀,x₁,x₂….xₙ . Note that the interpolation means if i≠j then xᵢ≠xⱼ . Suppose Pₙ(x) given by is the polynomial interpolating f at a set of n+1 distinct points x₀,x₁,x₂……xₙ . Then we have Pₙ(x) = fᵢ= f(xᵢ) ; i= 0,1,2….n a₀+a₁x₀+…..+aₙx₀ⁿ = f₀ ⇒ a₀ + a₁x₁ + ….+aₙx₁ⁿ = f₁ ……………………………….. }...(2.2.1) a₀ + a₁xₙ + …….+aₙxₙⁿ = fₙ This is a system of n+1 linear equations in n+1 unknowns : a₀,a₁,a₂…..aₙ; hence the system will have a unique solution if the determinant Δ = DET(f₀,f₁,......fₙ) ≠ 0 ..........(2.2.2) Indeed , the value of the determinant Δ is not zero since Δ = Π (xᵢ - xⱼ) 0≤j≤i≤n xᵢ ≠ xⱼ for i≠ j as the points x₀,x₁,....xₙ are distinct . Therefore an unique interpolating polynomial exists whose co_ effi