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 $\displaystyle e^{-A}$.