Results 1 to 2 of 2

Math Help - Numerical Methods - Quadrature formula

  1. #1
    Newbie
    Joined
    May 2010
    Posts
    7

    Numerical Methods - Quadrature formula

    Numerical methods - Quadrature formula?

    Can somebody show me how to do this...thx

    Use the method of undetermined coefficients to determine the values w0, w1, x0 and x1 which ensure that the quadrature formula Q[f] = w0f(x0) + w1f(x1) has degree of precision equal to three for integrals of the form

    1
    ∫ f(x)dx
    -1
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor chiph588@'s Avatar
    Joined
    Sep 2008
    From
    Champaign, Illinois
    Posts
    1,163
    Quote Originally Posted by pico24 View Post
    Numerical methods - Quadrature formula?

    Can somebody show me how to do this...thx

    Use the method of undetermined coefficients to determine the values  \omega_0,\; \omega_1,\; x_0,\; x_1 which ensure that the quadrature formula  Q(f) = \omega_0f(x_0) + \omega_1f(x_1) has degree of precision equal to three for integrals of the form

    \displaystyle \int_{-1}^1f(x)dx
    Let's try to make  Q(f) give the correct answer up to cubic polynomials.

    So we have  \displaystyle \int_{-1}^1 f(x)dx = \omega_0(c_0+c_1x_0+c_2x_0^2+c_3x_0^3)+\omega_1(c_  0+c_1x_1+c_2x_1^2+c_3x_1^3) .

    Rearranging we get  \displaystyle c_0\left(\omega_0+\omega_1-\int_{-1}^1dx\right)+c_1\left(\omega_0x_0+\omega_1x_1-\int_{-1}^1xdx\right)+c_2\left(\omega_0x_0^2+\omega_1x_1^  2-\int_{-1}^1x^2dx\right)+c_3\left(\omega_0x_0^3+\omega_1x_  1^3-\int_{-1}^1x^3dx\right)=0 .

    As  c_i are arbitrary we see  \begin{cases} \displaystyle\omega_0+\omega_1=2\\\displaystyle\om  ega_0x_0+\omega_1x_1=0\\\displaystyle\omega_0x_0^2  +\omega_1x_1^2=\tfrac23\\\displaystyle\omega_0x_0^  3+\omega_1x_1^3=0\end{cases}

    Applying algebra leads to  \begin{cases}\displaystyle\omega_0=\omega_1=1\\\di  splaystyle x_0=-\frac1{\sqrt3}\\\displaystyle x_1=\frac1{\sqrt3}\end{cases}
    Last edited by chiph588@; August 31st 2010 at 09:46 PM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Quadrature formula
    Posted in the Calculus Forum
    Replies: 1
    Last Post: November 20th 2011, 10:32 PM
  2. Numerical Analysis - Using Gaussian quadrature
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: November 30th 2010, 04:59 AM
  3. Numerical methods help
    Posted in the Calculus Forum
    Replies: 1
    Last Post: March 21st 2010, 11:39 PM
  4. Numerical Methods: Numerical Differentiation
    Posted in the Calculus Forum
    Replies: 0
    Last Post: February 25th 2009, 02:20 AM
  5. Numerical Methods
    Posted in the Algebra Forum
    Replies: 5
    Last Post: September 10th 2007, 08:36 AM

Search Tags


/mathhelpforum @mathhelpforum