Intereting Posts

Matrices that commute with all matrices
Neighborhoods vs Open Neighborhoods?
Prove that the equation $x^4+(2k^2)x^2+2k^4$ is not a perfect square
Is there any trivial reason for $2$ is irreducible in $\mathbb{Z},\omega=e^{\frac{2\pi i}{23}}$?
Compact metrizable space has a countable basis (Munkres Topology)
Series in Real Analysis
Evaluate $\int _{ 0 }^{ 1 }{ \left( { x }^{ 5 }+{ x }^{ 4 }+{ x }^{ 2 } \right) \sqrt { 4{ x }^{ 3 }+5{ x }^{ 2 }+10 } \; dx } $
bound of the number of the primes on an interval of length n
Gradient descent with constraints
Continuous function with linear directional derivatives=>Total differentiability?
$2n^2-\lfloor m^b\rfloor=k$ has only finitely many integer solutions
Field bigger than $\mathbb{R}$
another form of the L'Hospital's rule
Define an infinite subset of primes such that the sum of reciprocals converges
What are Diophantine equations REALLY?

*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

$$p(a_1)=f(a_1)\\p(a_2)=f(a_2)\\p'(a_3)=f'(a_3)\\p'(a_4)=f'(a_4).$$

- Computing matrix-vector calculus derivatives
- Prove $\frac{1}{a^3+b^3+abc}+\frac{1}{a^3+c^3+abc}+\frac{1}{b^3+c^3+abc} \leq \frac{1}{abc}$
- Does $\sum\limits_{n=1}^\infty\frac{1}{\sqrt{n}+\sqrt{n+1}}$ converge?
- Integral of $\frac{1}{x^2+x+1}$ and$\frac{1}{x^2-x+1}$
- Is there a formula for $\sin(xy)$
- Show that $\lim_{x\to \infty}\left( 1-\frac{\lambda}{x} \right)^x = e^{-\lambda}$

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.

- If $\,\lim_{x\to 0} \Big(f\big({a\over x}+b\big) - {a\over x}\,f'\big({a\over x}+b\big)\Big)=c,\,$ find $\,\lim_{x\to\infty} f(x)$
- Evaluation of $ \int\frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}dx$
- Integrating Reciprocals of Polynomials
- Find expected number of successful trail in $N$ times
- Closed form of $\int_0^\infty \frac{dt}{\left(i t\sqrt{3}+\ln(2\sinh t)\right)^2}$
- Does the series $\sum_{n=1}^\infty (-1)^n \frac{\cos(\ln(n))}{n^{\epsilon}},\,\epsilon>0$ converge?
- Compute $\iint\limits_R\frac{y}{x+y^2}dA$ where $R=\times$
- Expressing the solutions of the equation $ \tan(x) = x $ in closed form.
- Find $\lim_{x \to - \infty} \left(\frac{4^{x+2}- 2\cdot3^{-x}}{4^{-x}+2\cdot3^{x+1}}\right)$
- Prove $f(x)=\int\frac{e^x}{x}\mathrm dx$ is not an elementary function

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

- Advice on finding counterexamples
- Does infinity and zero really exist?
- How many strings of $8$ English letters are there (repetition allowed)?
- Conjectures that have been disproved with extremely large counterexamples?
- What is the definition of “rotation” in a general metric space? (Or a Finsler manifold?)
- Maximum area of a isosceles triangle in a circle with a radius r
- Is $\mathbb{Q}=\{a+bα+cα^2 :a,b,c ∈ \mathbb{Q}\}$ with $α=\sqrt{2}$ a field?
- Pullback of maximal ideal in $k$ is not maximal in $k$.
- Prove that a linear operator $T:E \rightarrow E'$ such that $\langle Tx,y \rangle=\langle Ty,x\rangle$ is bounded
- A fractional part integral giving $\frac{F_{n-1}}{F_n}-\frac{(-1)^n}{F_n^2}\ln\left(\!\frac{F_{n+2}-F_n\gamma}{F_{n+1}-F_n\gamma}\right)$
- Is an Anti-Symmetric Relation also Reflexive?
- How to show countability of $\omega^\omega$ or $\epsilon_0$ in ZF?
- If $R$ is an infinite ring, then $R$ has either infinitely many zero divisors, or no zero divisors
- Proving that a doubly-periodic entire function $f$ is constant.
- Identity with Catalan numbers