what


NUMERICAL QUADRATURE

We again start with a sum

what

And draw this function as follows:

what

where h is the lattice spacing. We now select two points a and b , separated by an even number of lattice spacings, N, where a is less than b : i.e. N=(b-a)/h. It is sufficient to derive a formula for the integral from -h to h since we can repeat this many times to get the integral from from a to b . (The interval a to b is made up of intervals from a to a +2h , a+2h to a +4h ,....., b-2h to b .) The idea is then to approximate f(x) over each interval by some function that can be integrated exactly.

We carry this scheme out by considering the intervals [-h,0] and [0,h], assume f(x) to be linear in both, and obtain, by the trapezoidal rule,

what

The error here is h2 f'' and we recall the definitions f(x=x1) =f1 , f(x=x-1) =f-1

A better approximation is to use

  1. the Taylor series as given in Equation 1 of the page on differentiation,
  2. the two point finite difference approximation for f' , and
  3. the expression for f'' from the bottom of the differentiation page to get

what

which is integrated to give

what

This is Simpson's Rule, and we can combine the intervals to give

what

Example: chap1b.f This routine evaluates the integral

what

for the value of N=1/h input with the combined Simpson's rule algorithm. Compile and run this program with different h selections. How do the errors behave as h is increased? Do they increase after some point? (Answer: no as there is not the subtraction as in the case of differentiation.)

How can we improve this? Smaller h is one way, but there are better ways to approximate the functions, e.g. with cubic or quartic polynomials leading to Simpson's 3/8 and Bode's rule, which have more terms for each interval. Table 1.3 of Koonin shows the comparative results.

We will return to quadrature in Lecture 6, after learning about special functions.


Back to Lecture 3 page.