Editing Simpson's rule (section)

=== Composite Simpson's 1/3 rule ===
If the interval of integration <math>[a, b]</math> is in some sense "small", then Simpson's rule with <math>n = 2</math> subintervals will provide an adequate approximation to the exact integral. By "small" we mean that the function being integrated is relatively smooth over the interval <math>[a, b]</math>. For such a function, a smooth quadratic interpolant like the one used in Simpson's rule will give good results.

However, it is often the case that the function we are trying to integrate is not smooth over the interval. Typically, this means that either the function is highly oscillatory or lacks derivatives at certain points. In these cases, Simpson's rule may give very poor results. One common way of handling this problem is by breaking up the interval <math>[a, b]</math> into <math>n > 2</math> small subintervals. Simpson's rule is then applied to each subinterval, with the results being summed to produce an approximation for the integral over the entire interval. This sort of approach is termed the ''composite Simpson's 1/3 rule'', or just ''composite Simpson's rule''.

Suppose that the interval <math>[a, b]</math> is split up into <math>n</math> subintervals, with <math>n</math> an even number. Then, the composite Simpson's rule is given by

Dividing the interval <math>[a, b]</math> into <math>n</math> subintervals of length <math>h = (b - a)/n</math> and introducing the points <math>x_i = a + ih</math> for <math>0 \leq i \leq n</math> (in particular, <math>x_0 = a</math> and <math>x_n = b</math>), we have
<math display="block">
\begin{align}
  \int_a^b f(x)\, dx
  &\approx \frac{1}{3} h\sum_{i = 1}^{n/2}\big[f(x_{2i - 2}) + 4f(x_{2i - 1}) + f(x_{2i})\big]\\
  &= \frac{1}{3} h\big[f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + 2f(x_4) + \dots + 2f(x_{n - 2}) + 4f(x_{n - 1}) + f(x_n)\big]\\
  &= \frac{1}{3} h\left[f(x_0) + 4\sum_{i = 1}^{n/2} f(x_{2i - 1}) + 2\sum_{i = 1}^{n/2 - 1} f(x_{2i}) + f(x_n)\right].
\end{align}
</math>
This composite rule with <math>n = 2</math> corresponds with the regular Simpson's rule of the preceding section.

The error committed by the composite Simpson's rule is
<math display="block">-\frac{1}{180} h^4(b - a)f^{(4)}(\xi),</math>
where <math>\xi</math> is some number between <math>a</math> and <math>b</math>, and <math>h = (b - a)/n</math> is the "step length".{{sfn|Atkinson|1989|pp=257{{ndash}}258}}{{sfn|Süli|Mayers|2003|loc=§7.5}} The error is bounded (in absolute value) by
<math display="block">\frac{1}{180} h^4(b - a) \max_{\xi \in [a, b]} \left|f^{(4)}(\xi)\right|.</math>

This formulation splits the interval <math>[a, b]</math> in subintervals of equal length. In practice, it is often advantageous to use subintervals of different lengths and concentrate the efforts on the places where the integrand is less well-behaved. This leads to the [[adaptive Simpson's method]].