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Math Help - Divided differences

  1. #1
    Member Mollier's Avatar
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    Divided differences

    Hi,

    my book starts to explain divided difference by saying that :

    Suppose that P_n(x) is the nth Lagrange polynomial that agrees with the function f at distinct numbers x_0,x_1,\cdots,x_n. The divided differences of f with respect to x_0,x_1,\cdots,x_n are used to express P_n(x) in the form

    P_n(x) = a_0+a_1(x-x_0)+a_2(x-x_0)(x-x_1)+\cdots+a_n(x-x_0)(x-x_1)\cdots(x-x_{n-1})

    Since I'm pretty slow, I do not see the logic behind this choice. I'm that I'm lacking some basic theory such that this appears like a mystical choice. Could someone point me in the right direction please?

    Note:

    I do understand why the interpolating polynomial between two points x_0 and x_1 can be written as,

    P_1(x) = f(x_0) + \frac{f(x_1)-f(x_0)}{x_1-x_0}(x-x_0),

    The literature often states that a convenient form of the 2-order polynomial is,

    P_2(x) =b_0 + b_1(x-x_0)+b_2(x-x_0)(x-x_1).

    How did man figure out that this is a convenient form? Trial and error? Thinking really hard?

    Thanks.
    Last edited by Mollier; August 30th 2010 at 03:46 AM.
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  2. #2
    Member Mollier's Avatar
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    Never mind guys, I found a great explanation at: 5.3 Newton Polynomials | Department of Electrical and Computer Engineering | University of Waterloo.
    I have been spoon-fed and am quite happy
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