Boole's rule

The widely propagated typographical error Bode's rule redirects here. For Bode's Law, see Titius–Bode law.

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

$\int _{x_{1}}^{x_{5}}f(x)\,dx$

by using the values of ƒ at five equally spaced points

$x_{1},\quad x_{2}=x_{1}+h,\quad x_{3}=x_{1}+2h,\quad x_{4}=x_{1}+3h,\quad x_{5}=x_{1}+4h.\,$

It is expressed thus in Abramowitz and Stegun (1972, p. 886):

$\int _{x_{1}}^{x_{5}}f(x)\,dx={\frac {2h}{45}}\left(7f(x_{1})+32f(x_{2})+12f(x_{3})+32f(x_{4})+7f(x_{5})\right)+{\text{error term}},$

and the error term is

$-\,{\frac {8}{945}}h^{7}f^{(6)}(c)$

for some number c between x₁ and x₅. (945 = 1 × 3 × 5 × 7 × 9.)

It is often known as Bode's rule, due to a typographical error that propagated from Abramowitz and Stegun (1972, p. 886).

See also

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