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Guassian Quadrature

  1. Nov 8, 2012 #1

    Pengwuino

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    Gold Member

    1. The problem statement, all variables and given/known data

    Why must the [itex]\mu_i[/itex] be the roots of [itex]P_N(\mu_i) = 0[/itex] to satisfy N even-moment conditions? Consider [itex]2\pi \int d\mu \mu^{N+n}[/itex] and write [itex]\mu^{N+n}[/itex] as [itex]P_N(\mu)q_N(\mu) + R_k(\mu)[/itex] which involves [itex]P_N[/itex] and the polynomials [itex]q_N[/itex] (quotient) and [itex]R_k[/itex] (remainder). If n<N, these are both of degree less than N.

    2. Relevant equations

    The [itex]P_N(\mu)[/itex] are the Legendre polynomials of order N.

    3. The attempt at a solution

    So my understanding is that the quadrature means that you can write some polynomial as [itex]\int_{-1}^1 H(\mu) = \sum_{i=0}^{N} w_i P_i(\mu_i)[/itex]. I really have no idea how to go about this. Is my understanding of quadrature even correct in the first place?
     
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