Hermite Interpolation

Hermite interpolation is an extension of Lagrange interpolation. While Lagrange interpolation constructs a polynomial that passes through given points \((x_i, f(x_i))\), Hermite interpolation goes one step further: it also incorporates information about the derivatives of the function at those points.

This approach allows us to build polynomials that not only match the function values but also its slopes, which helps in reducing oscillations (Runge's phenomenon) and better approximates the local behavior of the function. Hermite interpolation is especially useful when derivative information is available and when higher accuracy is desired.

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 \( 2n+1 \).