Results 1 to 3 of 3

Math Help - Quadrature Using Legendre Polynomials

  1. #1
    Senior Member
    Joined
    Mar 2009
    Posts
    378

    Quadrature Using Legendre Polynomials

    I am working on studying for my last Qualifying Exam and ran across a problem that has stumped me. I have seen someone else's work for this, but I am pretty it is wrong. Anyway...

    Let f(x) be a smooth function and consider the integral
    I=\int_{-1}^1\! f(x) \, dx
    and consider p_{n-1} that interpolates f at the roots of the Legendre polynomial of degree n.

    Use p_{n-1} to derive a quadrature method for the integral I
    The solution I have seen is as follows
    Let \left\{x_i\right\}_{i=1}^n be the n distinct roots of \widetilde{p}_n(x) (the Legendre Polynomial of degree n)

    Define
    w_i = \int_{-1}^1\! \prod_{\stackrel{j=1}{j\ne i}}^n \frac{x-x_j}{x_i-x_j}\, dx \Rightarrow Q(f) = \sum_{i=1}^n w_i f(x_i)
    So my issue is that he didn't really use Legendre polynomials. Instead he bipassed the Legendre polynomials in favor of a Lagrange polynomial that interpolates at the Legendre roots. Maybe I am missing something or just confused. Could someone help me either understand the work above or help me with the problem? Thanks in advanced.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by lvleph View Post

    So my issue is that he didn't really use Legendre polynomials. Instead he bipassed the Legendre polynomials in favor of a Lagrange polynomial that interpolates at the Legendre roots. Maybe I am missing something or just confused. Could someone help me either understand the work above or help me with the problem? Thanks in advanced.
    A polynomial with the same roots as a Legendre polynomial differs from the Legendre polynomial only by a multiplicative constant (or it is a Legendre polynomial with a different normalisation).

    CB
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member
    Joined
    Mar 2009
    Posts
    378
    I figured that was the case. Thank you.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Least Squares with Legendre Polynomials
    Posted in the Advanced Applied Math Forum
    Replies: 1
    Last Post: December 9th 2011, 12:36 AM
  2. Zeros of Legendre polynomials
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: November 7th 2011, 03:55 AM
  3. bessel function-legendre polynomials
    Posted in the Differential Geometry Forum
    Replies: 9
    Last Post: June 19th 2011, 01:10 AM
  4. Legendre Polynomials
    Posted in the Calculus Forum
    Replies: 2
    Last Post: April 23rd 2008, 08:18 AM
  5. legendre polynomials
    Posted in the Calculus Forum
    Replies: 0
    Last Post: February 8th 2007, 06:23 AM

Search Tags


/mathhelpforum @mathhelpforum