Boole's Rule

In mathematics, Boole's rule, named after George Boole, is a method of numerical integration.

Formula

Simple Boole's Rule

It approximates an integral:
$\int_^ f(x)\,dx$
by using the values of at five equally spaced points:
$\begin & x_0 = a\\ & x_1 = x_0 + h \\ & x_2 = x_0 + 2h \\ & x_3 = x_0 + 3h \\ & x_4 = x_0 + 4h = b \end$

It is expressed thus in Abramowitz and Stegun:
\int_^ f(x)\,dx = \frac\bigl 7f(x_0) + 32 f(x_1) + 12 f(x_2) + 32 f(x_3) + 7f(x_4) \bigr+ \text
where the error term is
$-\,\frac$
for some number between and where .

It is often known as Bode's rule, due to a typographical error that propagated from Abramowitz and Stegun.

The following constitutes a very simple implementation of the method in Common Lisp which ignores the error term:

Composite Boole's Rule

In cases where the integration is permitted to extend over equidistant sections of the interval $$, b/math>, the composite Boole's rule might be applied. Given $N$ divisions, where $N$ mod $4 = 0$, the integrated value amounts to:

$\int_^ f(x)\,dx = \frac \left( 7(f(x_0) + f(x_N)) + 32\left(\sum_ f(x_i)\right) + 12\left(\sum_ f(x_i)\right) + 14\left(\sum_ f(x_i)\right) \right) + \text$

where the error term is similar to above. The following Common Lisp code implements the aforementioned formula:

See also

* Newton–Cotes formulas
* Simpson's rule
* Romberg's method

Notes

References

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{{DEFAULTSORT:Boole's Rule
Integral calculus
Numerical analysis
Numerical integration
Articles with example Lisp (programming language) code