Hermite_Interpolation

Scientific Computing

The Hermite interpolating polynomial interpolates function
as well as its certain order derivatives at given data points.

Conditions satisfied by Hermite polynomial :
H(X_i) = f(X_i) and
H^'(X_i) = f^'(X_i) where i = 0,1,2...n

Since there are 2n+2 conditions to be satisfied,
H(X) must be a polynomial of degree ≤ 2n+1.

The required polynomial is written as :
H(X) = ∑ⁿ_i=0 A_i(X)f(X_i) + ∑ⁿ_i=0 B_i(X)f^'(X_i)

Using Lagrange fundamental polynomials l_i(X), We have
A_i(X) = [1 - 2(X - X_i)l_i^'(X_i)]l_i²(X)
B_i(X) = (X - X_i)l_i²(X)

Error at a point "ʓ" is given by :
E(f ; ʓ) = (ʓ - X₀)*(ʓ - X₁)*...(ʓ - X_n)*f²ⁿ⁺²(ʓ)/(2n+2)!

Bound on error is given by :
|f(X) - H(X)| = max{(X - X₀)*(X - X₁)*...(X - X_n)}*M/(2n+2)!
Where M = max(f²ⁿ⁺²(ʓ)); X₀ ≤ ʓ ≤ X_n