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Math Help - Lagrange Interpolating Polynomial!

  1. #1
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    Lagrange Interpolating Polynomial!

    Can any one explain this formula "Lagrange Interpolating Polynomial" to me in plain English.
    Is there any program around to calc this formaula?

    Thanks.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by BanderHM
    Can any one explain this formula "Lagrange Interpolating Polynomial" to me in plain English.
    Is there any program around to calc this formaula?

    Thanks.
    The Lagrangeian interpolating polynomial for a set of data {(x_i,y_i),\ i=1,\ ..\ n}
    is a polynomial of degree n (the number of data points) which passes exactly
    through each of the data points. It can be written explicitly in terms of the
    data as shown below:

    Consider the polynomials:

    <br />
P_i(x)=\prod_{j=1,\ j \ne i}^n \frac{x-x_j}{x_i-x_j}<br />
,

    This equals 1 at x=x_i, and equals 0 when x=x_j\ ,j \ne i.

    Hence:

    <br />
P(x)=\sum_{i=1}^n\ y_i\ P_i(x)<br />

    is a polynomial such that P(x_i)=y_i, \ \mbox{i=1,\ ..\ n}.

    This is the Lagrange interpolating polynomial for the data (x_i,y_i),\ i=1,\ ..\ n.

    There is one more thing to say about Lagrangian interpolation, and that is
    it is of more importance to theory than practice. If you actualy want to
    interpolate on real data there are almost always better ways of doing so
    than Lagrangian interpolation.

    RonL
    Last edited by CaptainBlack; February 16th 2006 at 04:24 AM.
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  3. #3
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    How?

    Quote Originally Posted by CaptainBlack
    If you actualy want to
    interpolate on real data there are almost always better ways of doing so
    than Lagrangian interpolation.
    RonL
    How? What are they?


    Thanks.
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  4. #4
    Grand Panjandrum
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    Quote Originally Posted by BanderHM
    How? What are they?


    Thanks.
    Splines, othogonal polynomials, rational functions (Pade approximations)...

    the list is almost endless

    RonL
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