Hermite Interpolation

Hypotheses

Let \( f \in C^1([a, b]) \) be a differentiable function. Suppose we know the function values and first derivatives at \( n+1 \) distinct points \( x_0, x_1, \dots, x_n \). We seek a polynomial \( p(x) \) such that:

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

Since we have \( 2n+2 \) conditions, we choose a polynomial of degree at most \( 2n+1 \).

Hermite Interpolation Polynomial

The Hermite interpolation polynomial \( p \in \mathbb{P}_{2n+1} \) satisfies:

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

Construction (Modified Lagrange Form)

Let \( \ell_i(x) \) be the Lagrange polynomials:

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

Define \( q_i(x) = \ell_i(x)^2 \). Then the Hermite polynomial can be written as:

\[ p(x) = \sum_{i=0}^n \left[ H_i(x) f(x_i) + K_i(x) f'(x_i) \right] \] with \[ H_i(x) = \left(1 - 2(x - x_i)\ell_i'(x_i) \right) \ell_i(x)^2, \quad K_i(x) = (x - x_i)\ell_i(x)^2 \]

Theorems

Existence and Uniqueness: The polynomial \( p \) defined above is unique and satisfies the Hermite interpolation conditions.

Hermite Interpolation Error: If \( f \in C^{2n+2}([a,b]) \), then for any \( x \in [a,b] \), 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

• Mixed Lagrange-Hermite interpolations are possible if only some derivatives are known.
• Higher-order derivatives can be included if available, for higher-degree Hermite polynomials.

Example

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

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

Here \( x_0 = 0, x_1 = 1 \). The Lagrange polynomials are:

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

Compute \( H_i(x) \) and \( K_i(x) \):

\[ \begin{aligned} H_0(x) &= (1 - 2x)(1-x)^2 = 2x^3 - 3x^2 + 1 \\ K_0(x) &= x(1-x)^2 = x^3 - 2x^2 + x \\ H_1(x) &= (1 - 2(x-1))x^2 = 2x^3 - 3x^2 \\ K_1(x) &= (x-1)x^2 = -x^3 + x^2 \end{aligned} \]

Therefore, the Hermite polynomial is:

\[ \begin{aligned} p(x) &= H_0(x) \cdot 1 + K_0(x) \cdot 2 + H_1(x) \cdot 0 + K_1(x) \cdot 1 \\ &= (2x^3 - 3x^2 + 1) + 2(x^3 - 2x^2 + x) + (-x^3 + x^2) \\ &= 5x^3 - 8x^2 + 2x + 1 \end{aligned} \]

This polynomial exactly matches both the function values and derivatives at the specified points.