Simpson's Rule

  1. Simpson's Estimate

    Yesterday, we saw that the Trapezoidal and Midpoint estimates provided better accuracy than the Left and Right endpoint estimates.  It turns out that a certain combination of the Trapezoid and Midpoint estimates is even better. 


     

                              Definition

    Let f(x) be a function defined on [a,b].  Then 

              S(n) = 1/3 T(n) + 2/3 M(n)

    where T(n) and M(n) are the Trapezoidal and Midpoint Estimates.

    Geometrically, if n is an even number then Simpson's Estimate gives the area under the parabolas defined by connecting three adjacent points.  

    Let n be even then using the even subscripted x values for the trapezoidal estimate and the midpoint estimate, gives

            S(n) = 1/3[(b - a)/2 (f(x0) + 2f( f(x2) + f(x4)+ ... + f(xn-2) + f(xn)] + 2/3[f(x1) + f(x3) + f(x5) + ... + f(xn-1)

            = (b - a)/6 ( f(x0) + 4f(x1) + 2f(x2) + 4f(x3) + 2f(x4) + 4f(x5) + ... + 2f(xn-2) + 4f(xn-1) + f(xn)

    Notice the 

            1 2 4 2 4 ... 2 4 2 4 1 

    pattern.


    Example

    Use Simpson's Estimate to approximate

           

    Using n = 6


    Solution

     We partition 

            0 < 1/3 < 2/3 < 1 < 4/3 < 5/3 < 2

    and calculate


           

    and put these into the formula for Simpson's Estimate

            (2 - 0)/6 [1 + 2 . 1.12 + 4 . 1.56 + 2 . 2.72 + 4 . 5.92 + 2 . 16.08 + 54.60]

            = 41.79

    Exercise

    Approximate 

           


  2. Error in Simpson's Estimate

    Without proof, we state

    Let 

            M = max |f''''(x)| 

    and let ES be the error in using Simpson's estimate then

            |ES|  < M(b - a)5/180n4 

 

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