GuideInterpolationadvanced

Hermite interpolation

Interpolation that imposes value and derivative at each node: the polynomial $H_{2n+1}$ , its construction from the Lagrange bases, the error, the practical route via divided differences with repeated nodes and a Bessel-function example.

What Hermite adds

So far we forced the polynomial through the points. If we also know $f$ 's derivative at those points, we can impose it too: this controls the slope and often improves the approximation.

H_{i}(x)=\bigl[1-2(x-x_i)L_i'(x_i)\bigr]L_i^2(x),\qquad \hat H_{i}(x)=(x-x_i)L_i^2(x)

Base blocks built from the Lagrange functions and their derivatives.

Procedure and error

Five steps

Compute the Lagrange functions $L_i(x)$ .
Differentiate them to obtain $L_i'(x)$ .
Form the blocks $H_i(x)$ .
Form the blocks $\hat H_i(x)$ .
Sum everything to obtain $H_{2n+1}(x)$ .