Interpolation of Hermite

The Hermite polynomial is the one that interpolates a set of points and the value of their derivatives in any points we want. That is, let's suppose that we have $(x_k,f_k)$ and $(x_k,f'_k)$.

Then we construct the same table as in Newton's method, placing $x_k$ in the first column and writing twice the same point if we know the value of the derivative at this point; in the second column the values of $f$ corresponding to the $x$ of the same line. Namely if we know the value of $f$ in $x_0$ and of its derivative we will write $x_0$ twice and next to them we will write $f_0$. For example,

From here on we proceed the same way, but with the difference that we have to define $f[x_i,x_i]=f'_i$, the value of the derivative in $x_i$.

Therefore, if we have $n +1$ values of the function and $n +1$ values of the derivatives, the Hermite polynomial will have a $2n +1$ degree .

Let's consider an example:

Let's suppose that we want to calculate $f\Big(\dfrac{1}{8}\Big)$ where $f(x)=\tan(\pi x)$ from Hermite interpolation in $0,\dfrac{1}{4}$.

To obtain the result, we draw a table as in Newton's interpolation but repeating every point which derivative we know. This is:

Proceeding as with Newton's interpolation, we get: $$ P_3(x)= \pi x +(16-4\pi)x^2+ (48\pi-128)x^2\Big( x-\dfrac{1}{4}\Big)$$

Now, $$\tan\Big(\dfrac{\pi}{8}\Big)\approx P_3\Big(\dfrac{1}{8}\Big)=0.4018\dots$$