Editing Simpson's rule (section)

== Simpson's 3/8 rule ==
Simpson's 3/8 rule, also called Simpson's second rule, is another method for numerical integration proposed by Thomas Simpson. It is based upon a cubic interpolation rather than a quadratic interpolation. Simpson's 3/8 rule is as follows:
<math display="block">
\begin{align}
  \int_a^b f(x)\, dx
  &\approx \frac{b - a}{8} \left[f(a) + 3f\left(\frac{2a + b}{3}\right) + 3f\left(\frac{a + 2b}{3}\right) + f(b)\right]\\
  &= \frac{3}{8} h\left[f(a) + 3f\left(a + h\right) + 3f\left(a + 2h\right) + f(b)\right],
\end{align}
</math>
where <math>h = (b - a)/3</math> is the step size.

The error of this method is
<math display="block">-\frac{3}{80} h^5f^{(4)}(\xi) = -\frac{(b - a)^5}{6480} f^{(4)}(\xi),</math>
where <math>\xi</math> is some number between <math>a</math> and <math>b</math>. Thus, the 3/8 rule is about twice as accurate as the standard method, but it uses one more function value. A composite 3/8 rule also exists, similarly as above.{{sfn|Matthews|2004}}

A further generalization of this concept for interpolation with arbitrary-degree polynomials are the [[Newton–Cotes formulas]].

=== Composite Simpson's 3/8 rule ===
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{3}{8} h\sum_{i = 1}^{n/3} \big[f(x_{3i - 3}) + 3f(x_{3i - 2}) + 3f(x_{3i - 1}) + f(x_{3i})\big]\\
  &= \frac{3}{8} h\big[f(x_0) + 3f(x_1) + 3f(x_2) + 2f(x_3) + 3f(x_4) + 3f(x_5) + 2f(x_6) + \dots \\&\qquad+ 2f(x_{n - 3}) + 3f(x_{n - 2}) + 3f(x_{n - 1}) + f(x_n)\big]\\
  &= \frac{3}{8} h\left[f(x_0) + 3 \sum_{i = 1,\ 3\nmid i}^{n-1} f(x_i) + 2 \sum_{i=1}^{n/3 - 1} f(x_{3i}) + f(x_n)\right].
\end{align}
</math>

While the remainder for the rule is shown as{{sfn|Matthews|2004}}
<math>-\frac{1}{80} h^4(b - a)f^{(4)}(\xi),</math>
we can only use this if <math>n</math> is a multiple of three. The 1/3 rule can be used for the remaining subintervals without changing the order of the error term (conversely, the 3/8 rule can be used with a composite 1/3 rule for odd-numbered subintervals).