November 4, 2005 --- Class 19 --- Numerical Integration in One Dimension

	Numerical Integration

	We talked about some of the standard methods for numerical
	integration in one dimension.  This is discussed in Section 11.1
	of CSM.  Additional material is available in Numerical Recipes
	or in Abramowitz and Stegun.  We compared the accuracy of
	various methods like the rectangle rule, trapezoid rule and
	Simpson's rule.  We indicated how Simpson's rule can simply
	be derived from removing the leading 1/N^2 error in the trapeziod
	rule with N and 2N points.  (This is detailed below.)
	This technique can be extended in the method
	known as Romberg integration.  The previous two points are
	expanded upon in Numerical Recipes.  Practical codes may be found
	there.

	Simpson's Rule Rederived

	Starting from the trapezoid rule, we used the formulae below
	for correcting the leading 1/N^2 errors.

	S_N  = A + E/N^2
	S_2N = A + E/(2N)^2

	so   A = 4/3 S_2N -1/3 S_N

	Applying this to a one-dimensional itegral, we will easily find that 
	the weighting of the sum of two expressions on the right hand side
	exactly coincides with the Simpson's rule formula.  We previously
	found that Simpson's Rule has a leading error of order 1/N^4.  Since
	the approach above removes the leading 1/N^2 error and since the
	error of the trapezoid rule only has even powers of 1/N (asserted, but
	not proved), this is not surprising.