Hermite Interpolation

Hermite interpolation is an extension of Lagrange interpolation. It consists, for a differentiable function \( f \) and a given set of points \( x_i \), \( i = 0, \dots, n \), in constructing a polynomial \( p \) that matches not only the values of \( f \) at the points \( x_i \), but also the values of its derivative \( f' \).

This method helps avoid the Runge phenomenon in polynomial interpolation, while constructing polynomials with properties close to those of the interpolated function.

Assumption

Let \( f \in C^1([a, b]) \). Suppose we know \( f(x_i) \) and \( f'(x_i) \) at \( n + 1 \) distinct points \( x_0, x_1, \dots, x_n \). We seek a polynomial \( p \) such that:

\[ \begin{cases} p(x_i) = f(x_i) \\ p'(x_i) = f'(x_i) \end{cases} \quad \text{for } i = 0, \dots, n \]

We therefore need to determine \( 2n + 2 \) coefficients, which is why we choose a polynomial of degree at most \( 2n + 1 \).

Hermite Interpolation Polynomial

The Hermite interpolation polynomial of \( f \) at the distinct points \( x_0, \dots, x_n \) is the polynomial \( p \in \mathbb{P}_{2n+1} \) such that:

\[ p(x_i) = f(x_i), \quad p'(x_i) = f'(x_i), \quad \text{for } i = 0, \dots, n \]

Construction (Modified Lagrange Form)

We use the Lagrange polynomials \( \ell_i(x) \), defined by:

\[ \ell_i(x) = \prod_{\substack{j=0 \\ j \ne i}}^n \frac{x - x_j}{x_i - x_j} \]

Then define \( q_i(x) = \ell_i(x)^2 \), and build the polynomial \( p \) as follows:

\[ p(x) = \sum_{i=0}^n \left[ H_i(x) f(x_i) + K_i(x) f'(x_i) \right] \]

where:

\[ H_i(x) = \left(1 - 2(x - x_i)\ell_i'(x_i) \right)\ell_i(x)^2, \quad K_i(x) = (x - x_i)\ell_i(x)^2 \]

Theorem (Existence and Uniqueness)

The polynomial \( p \in \mathbb{P}_{2n+1} \) defined above is unique and is the Hermite interpolation polynomial of \( f \) at the points \( x_0, \dots, x_n \).

Theorem (Hermite Interpolation Error)

If \( f \in C^{2n+2}([a, b]) \), then for all \( x \in [a, b] \), there exists \( \xi \in (a, b) \) such that:

\[ f(x) - p(x) = \frac{f^{(2n+2)}(\xi)}{(2n+2)!} \prod_{i=0}^n (x - x_i)^2 \]

Remarks

• It is possible to define mixed Lagrange-Hermite interpolations where only some derivatives are known.
• Higher-order derivatives can also be used if available.

Example

Construct a polynomial \( p \) of degree 3 such that:

\[ p(0) = 1, \quad p'(0) = 2, \quad p(1) = 0, \quad p'(1) = 1 \]

We have \( x_0 = 0, x_1 = 1 \), so:

\[ \ell_0(x) = 1 - x, \quad \ell_1(x) = x, \quad \ell_0'(0) = -1, \quad \ell_1'(1) = 1 \]

\[ H_0(x) = (1 - 2x)(1 - x)^2 = 2x^3 - 3x^2 + 1 \]

\[ K_0(x) = x(1 - x)^2 = x^3 - 2x^2 + x \]

\[ H_1(x) = (1 - 2(x - 1))x^2 = 2x^3 - 3x^2 \]

\[ K_1(x) = (x - 1)x^2 = -x^3 + x^2 \]

Hence:

\[ p(x) = H_0(x) \cdot 1 + K_0(x) \cdot 2 + H_1(x) \cdot 0 + K_1(x) \cdot 1 \]

\[ p(x) = (2x^3 - 3x^2 + 1) + 2(x^3 - 2x^2 + x) + (-x^3 + x^2) \]

\[ p(x) = 5x^3 - 8x^2 + 2x + 1 \]