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Math Help - [SOLVED] Quadrature formula help

  1. #1
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    Nov 2009
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    Question [SOLVED] Quadrature formula help

    Hello,

    I have to determine a quadrature formula of this shape:



    with maximum degree of precision.

    I can't find any examples to start me off. I looked at some of the polynomials they use in examples to start off (Legendre, Laguerre etc.) but can't find one that matches, or even an example like this...

    Can anyone walk me through the solution, or at least start me off (what's the maximum precision, which polynomial do I start with)

    Thanks!
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  2. #2
    Newbie
    Joined
    Feb 2010
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    Berlin
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    Hey Tommy,

    I'm not on this site so often; I hope this thread isn't too cold that posting suggestions here is still worth while.

    The function W(x)=x^4 - 1 in the integral is a weight function. Notice that it is positive everywhere on the interval (1, 2).

    First you need to find the first few orthogonal polynomials for this weight, that is, polynomials p_0, p_1, \ldots satisfying

    \int_1^2W(x)p_m(x)p_n(x)~dx = 0, \quad m\neq n,

    and satisfying \deg p_n = n. You can do this by applying Gram-Schmidt to the polynomial basis 1, x, x^2,\ldots. You can also do this by hand, since it only involves definite integrals of polynomials.

    The quadrature rule you want has three "abscissae" x_1, x_2, x_3, which will be zeros of the polynomial p_3 that you find. This will give a quadrature rule of maximum precision.

    Note: x_1, x_2, x_3 \in (1,2). If not, you made a mistake somewhere!

    It does not matter if you normalize the p_n when calculating. Scaling a polynomial by a constant does not affect the location of its zeros. However the coefficients

    k_n = \int_1^2 W(x)p_n^2(x)~dx

    are relevant to the weights A_1, A_2, A_3 in the quadrature rule. There is a formula; also for R(f); if you want, I can look around and maybe find a reference online.

    Yours,
    Jerry
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