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Math Help - Hermite interpolating polynomial?

  1. #1
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    Unhappy 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=<br />
\[ \left( \begin{array}{ccccc}<br />
1 & -3 & -4 & 4 & -2\\<br />
10 & 12 & 4 & 0 & 12\\<br />
2 & 1 & 1 & 2 & 2\\<br />
-9 & -7 & 0 & -4 & -8<br />
\end{array} \right)\]<br />

    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}.
    Last edited by zealot; April 22nd 2006 at 04:16 AM.
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  2. #2
    Grand Panjandrum
    Joined
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    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=<br />
\[ \left( \begin{array}{ccccc}<br />
1 & -3 & -4 & 4 & -2\\<br />
10 & 12 & 4 & 0 & 12\\<br />
2 & 1 & 1 & 2 & 2\\<br />
-9 & -7 & 0 & -4 & -8<br />
\end{array} \right)\]<br />

    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
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  3. #3
    Newbie
    Joined
    Nov 2005
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    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 . 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.
    Last edited by zealot; April 22nd 2006 at 04:19 AM.
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  4. #4
    julian2501
    Guest
    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:

    <br />
A=\[ \left( \begin{array}{ccccc}1 & -3 & -4 & 4 & -2\\10 & 12 & 4 & 0 & 12\\2 & 1 & 1 & 2 & 2\\5 & 5 & 2 & 2 & 6\\-9 & -7 & 0 & -4 & -8\end{array} \right)\]<br />

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