1. ## Hermite interpolating polynomial?

Hello, I have just been given a question that I don't understand and after looking over my course notes I can't find anything resembling it. So could anyone could show me how to solve it, explain what exactly this question is about, or point me in the direction of a site that explains it clearly without too much reliance on other mathematics topics? I mainly just want to know the technique so I can do this thing. Any help whatsoever would be greatly appreciated!

Let $\displaystyle A= $\left( \begin{array}{ccccc} 1 & -3 & -4 & 4 & -2\\ 10 & 12 & 4 & 0 & 12\\ 2 & 1 & 1 & 2 & 2\\ -9 & -7 & 0 & -4 & -8 \end{array} \right)$$

Let $\displaystyle x_0$, $\displaystyle x_1$ be two distinct points at which the function $\displaystyle f(x)$ and its first derivative $\displaystyle f'(x)$ are defined and assume that the second derivative $\displaystyle f''(x)$ is also defined at the point $\displaystyle x_1$. A Hermite interpolating quartic polynomial of the form
$\displaystyle p(x)=\sum_{i=0}^{4} a_{i}x^{i}$
can be constructed for the function $\displaystyle f(x)$ from the given five data values by determining the unknown coefficients $\displaystyle a_{i}$ using the conditions:
$\displaystyle p(x_0)=f(x_0)$
$\displaystyle p'(x_0)=f'(x_0)$
$\displaystyle p(x_1)=f(x_1)$
$\displaystyle p'(x_1)=f'(x_1)$ and
$\displaystyle p''(x_1)=f''(x_1)$
Construct a Hermite quartic polynomial that interpolates the function $\displaystyle f(x)=e^{-x}$ at the eigenvalues of $\displaystyle A$ by solving an appropriately defined linear system whose solution provides the coefficients $\displaystyle a_i$, $\displaystyle i=1,...,5$. Use this polynomial to construct a matrix function approximation for $\displaystyle e^{-A}$.

2. Originally Posted by zealot
Hello, I have just been given a question that I don't understand and after looking over my course notes I can't find anything resembling it. So could anyone could show me how to solve it, explain what exactly this question is about, or point me in the direction of a site that explains it clearly without too much reliance on other mathematics topics? I mainly just want to know the technique so I can do the this thing. Any help whatsoever would be greatly appreciated!

Let $\displaystyle A= $\left( \begin{array}{ccccc} 1 & -3 & -4 & 4 & -2\\ 10 & 12 & 4 & 0 & 12\\ 2 & 1 & 1 & 2 & 2\\ -9 & -7 & 0 & -4 & -8 \end{array} \right)$$

Let $\displaystyle x_0$, $\displaystyle x_1$ be two distinct points at which the function $\displaystyle f(x)$ and its first derivative $\displaystyle f'(x)$ are defined and assume that the second derivative $\displaystyle f''(x)$ is also defined at the point $\displaystyle x_1$. A Hermite interpolating quartic polynomial of the form
$\displaystyle p(x)=\sum_{i=0}^{4} a_{i}x^{i}$
can be constructed for the function $\displaystyle f(x)$ from the given five data values by determining the unknown coefficients $\displaystyle a_{i}$ using the conditions:
$\displaystyle p(x_0)=f(x_0)$
$\displaystyle p'(x_0)=f'(x_0)$
$\displaystyle p(x_1)=f(x_1)$
$\displaystyle p'(x_1)=f'(x_1)$ and
$\displaystyle p''(x_1)=f''(x_1)$
Construct a Hermite quartic polynomial that interpolates the function $\displaystyle f(x)=e^{-x}$ at the eigenvalues of $\displaystyle A$ by solving an appropriately defined linear system whose solution provides the coefficients $\displaystyle a_i$, $\displaystyle i=1,...,5$. Use this polynomial to construct a matrix function approximation for $\displaystyle e^{-A}$.
Should not $\displaystyle A$ be square if you want eigenvalues or the matrix
exponential?

RonL

3. Originally Posted by CaptainBlack
Should not $\displaystyle A$ be square if you want eigenvalues or the matrix
exponential?

RonL
Oh yeah, I didn't think about that. Now this question makes even less sense to me . I'll probably have to ask the lecturer about it.

I see my post was edited for using the d-word. Sorry bout that, just got a little frustrated.

4. Hello.
I actually have this exact problem, only the matrix is 5x5. It is possible that zealot left out a line. The matrix i have is:

$\displaystyle A=$\left( \begin{array}{ccccc}1 & -3 & -4 & 4 & -2\\10 & 12 & 4 & 0 & 12\\2 & 1 & 1 & 2 & 2\\5 & 5 & 2 & 2 & 6\\-9 & -7 & 0 & -4 & -8\end{array} \right)$$

If anyone could point me in the right direction that would be a great help.
Thank you.