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):

$\int_{x_1}^{x_5} f(x)\,dx = \frac{2 h}{45}\left( 7f(x_1) + 32 f(x_2) + 12 f(x_3) + 32 f(x_4) + 7f(x_5) \right) + \text{error term},$

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).^[1][2]

See also

• Newton-Cotes formulas
• Simpson's Rule
• Romberg's method

References

1. Weisstein, Eric W., "Boole's Rule", MathWorld.
2. Zucker, Ruth (1983) [June 1964]. "Chapter 25.4.14: Numerical Interpolation, Differentiation, and Integration - Integration - Numerical Analysis". In Abramowitz, Milton; Stegun, Irene Ann. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C., USA; New York, USA: United States Department of Commerce, National Bureau of Standards; Dover Publications. p. 886. ISBN 0-486-61272-4. LCCN 64-60036. MR 0167642. ISBN 978-0-486-61272-0. LCCN 65-12253.