Modified Hermite interpolation

I asked similar questions here and here, but I tried to formulate this
one in a sharper way.

Is anyone aware of some literature on polynomial interpolation where we sample the function and its derivatives, but perhaps we sample the derivative at a point where we have not sampled the function?

For instance, given a function $f$, suppose we choose a polynomial $p$ of degree at most 3 where
$$p(a_1)=f(a_1)\\p(a_2)=f(a_2)\\p'(a_3)=f'(a_3)\\p'(a_4)=f'(a_4).$$

Finding such a polynomial $\alpha_3 x^3 + \alpha_2 x^2 + \alpha_1 x + \alpha_0$ is equivalent to solving
$$\left[
\begin{matrix}
1&a_1 & a_1^2 & a_1^3\\
1&a_2 & a_2^2 & a_2^3\\
0&1 & 2a_3 & 3a_3^2\\
0&1 & 2a_4 & 3a_4^2
\end{matrix}
\right]
\left[
\begin{matrix}
\alpha_0 \\ \alpha_1 \\ \alpha_2 \\ \alpha_3
\end{matrix}
\right] =
\left[
\begin{matrix}
f(a_1) \\ f(a_2) \\ f'(a_3) \\ f'(a_4)
\end{matrix}
\right].
$$

I claim that the determinant of that matrix is $\frac{\partial^2}{\partial x_3\partial x_4}\left( \prod_{i<j}(x_i-x_j) \right) \left. \right|_{x_i=a_i\,\,\forall i}$. Therefore, $\frac{\partial^2}{\partial x_3\partial x_4}\left( \prod_{i<j}(x_i-x_j) \right) \left. \right|_{x_i=a_i\,\,\forall i}\neq 0$ is a necessary and sufficient condition for such a polynomial to exist uniquely.

In particular, I am interested in estimating the error of such a polynomial interpolation, since I don’t see how to modify the normal Hermite estimation.

Solutions Collecting From Web of "Modified Hermite interpolation"

See my answer at your previous question.

If $V(x_1, \ldots, x_n)$ is the Vandermonde matrix whose determinant is
$$\det V(x_1, \ldots, x_n)$, the fact that $\det V(x_1,\ldots,x_n)$ is
linear as a function of the $i$’th row tells you that the partial derivative
of the determinant w.r.t. $x_i$ is the determinant of the matrix with row $i$ replaced by its derivative w.r.t. $x_i$. If you prefer, you could also get this
from the Laplace expansion (aka minor expansion) along the $i$’th row:

$$ \det V = \sum_k (-1)^{i+k} v_{ik} M_{ik} $$
where the $M_{ik}$ don’t depend on $x_i$, so
$$ \dfrac{\partial}{\partial x_i} \det V = \sum_{k} (-1)^{i+k} \dfrac{\partial v_{ik}}{\partial x_i} M_{ik} $$
which is the determinant of the matrix obtained from $V$ by replacing
each entry of row $i$ by its partial derivative wrt $x_i$.

Robert Israel has sagely pointed out that we should not expect a convenient error term, since sampling the derivative where you have not sampled the function does not improve the approximation:

Estimate the difference between $f$ and $p$ interpolating $f$

I have shown that a sufficient condition for the polynomial to exist uniquely is that we are sampling in Hermite fashion. I did it without resorting to divided differences, which is the usual way.

There is a unique polynomial interpolating $f$ and its derivatives