Hermite interpolating polynomial?

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April 18th 2006, 02:07 AM
zealot

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 $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 $x_0$ , $x_1$ be two distinct points at which the function $f(x)$ and its first derivative $f'(x)$ are defined and assume that the second derivative $f''(x)$ is also defined at the point $x_1$ . A Hermite interpolating quartic polynomial of the form
$p(x)=\sum_{i=0}^{4} a_{i}x^{i}$
can be constructed for the function $f(x)$ from the given five data values by determining the unknown coefficients $a_{i}$ using the conditions:
$p(x_0)=f(x_0)$
$p'(x_0)=f'(x_0)$
$p(x_1)=f(x_1)$
$p'(x_1)=f'(x_1)$ and
$p''(x_1)=f''(x_1)$
Construct a Hermite quartic polynomial that interpolates the function $f(x)=e^{-x}$ at the eigenvalues of $A$ by solving an appropriately defined linear system whose solution provides the coefficients $a_i$ , $i=1,...,5$ . Use this polynomial to construct a matrix function approximation for $e^{-A}$ .
April 21st 2006, 04:15 AM
CaptainBlack

Quote:

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

Should not $A$ be square if you want eigenvalues or the matrix
exponential?

RonL
April 22nd 2006, 04:15 AM
zealot

Quote:

Originally Posted by CaptainBlack

Should not $A$ 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 :confused:. 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. :)
April 27th 2006, 08:53 PM
julian2501

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:

$ 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.