error bound for polynomial interpolation with derivative matching

Question

We all know the following formula for the maximum error (evenly spaced) polynomial interpolation:

$|f(x) - p_n(x)| \leq \frac{h^{n+1}}{4(n+1)} \max_{x\in [a,b]} f^{(n+1)}(x)$

where $p_n(x)$ is the polynomial of degree $n$ coinciding at the points $x_i = a + i h$ ($i=0,\ldots,n-1, b = x_{n-1}$) with the given function $f(x)$ (i.e. $f(x_i) = p_n(x_i)$).

So far so good. But I would like to know whether there is a similar error bound if we do the matching only on $n/2$ points but also match the derivative (in case of even $n$).

The particular case I am interested in is for $n=3$, where we construct the interpolating polynomial $p_3(x)$ as

$p_3(a) = f(a)$
$p_3'(a) = f'(a)$
$p_3(b) = f(b)$
$p_3'(b) = f'(b)$

This of course uniquely defines $p_3$. The question is what can we say about

$|f(x) - p_n(x)| \leq ???$

perhaps in an analogous way as the formula above.

Answer 1

You asking about Hermit Interpolation which is a generalization of Newton Interpolation Polynomial the error in your case is limited by $$E(x)=\frac{f^{(4)}(c)}{4!} (x-a)^2(x-b)^2, $$ where $c\in [a,b]$

In the general case the error formula is given by $$E(x) =\frac{f^{(m)}(c)}{m!} \prod_{i=1}^{n}(x-x_i)^{m_i}, $$ where $c\in[x_0,x_n]$, $n$ is a number of points and $m$ is a total number of data points and $m_i$ in number of known values at $x_i$.