Boole's rule

Template:Short description Boole's rule is a method of numerical quadrature arising in calculus, and named after George Boole. Boole's rule and the composite Boole rule approximate the area under a curve over a fixed interval using 5 equally spaced points, and is designed to exactly integrate 5th-order polynomials over that interval. It is a member of the Newton–Cotes family of rules.

Formula

Simple Boole's rule

It approximates an integral $${\int }_a^{b}f(x)dx$$ by using the values of Template:Mvar at five equally spaced points:Template:Sfn $$\begin{array}{cc}{x}_0& =a,\\ x_1& =x_0+h,\\ x_2& =x_0+2h,\\ x_3& =x_0+3h,\\ x_4& =x_0+4h=b.\end{array}$$

It is expressed thus in Abramowitz and Stegun's Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables:Template:Sfn $${\int }_{x_0}^{x_4}f(x)dx=\frac{2h}{45}[7f(x_0)+32f(x_1)+12f(x_2)+32f(x_3)+7f(x_4)]-\frac{8f^{(6)}(\xi )h^7}{945}$$ for some number Template:Tmath between Template:Tmath and Template:Tmath, and Template:Nowrap. It is sometimes erroneously referred to as Bode's rule, due to a typographical error that propagated from Abramowitz and Stegun.Template:Sfn

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 $[a,b]$, the composite Boole's rule might be applied. Given $N$ divisions, where Template:Nobr the integrated value amounts toTemplate:Sfn $${\int }_{x_0}^{x_N}f(x)dx=\frac{2h}{45}\left[7\right(f(x_0)+f(x_N))+32\sum _{i\in \{1,3,5,\dots ,N-1\}}f(x_i)+12\sum _{i\in \{2,6,10,\dots ,N-2\}}f(x_i)+14\sum _{i\in \{4,8,12,\dots ,N-4\}}f(x_i)]+\text{error term},$$ where the error term is similar to above. The following Common Lisp code implements the aforementioned formula: Template:Hidden

See also

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

References

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• Template:MathWorld