Boole's rule

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

$x_1, \quad x_2 = x_1 %2B h, \quad x_3 = x_1 %2B 2h, \quad x_4 = x_1 %2B 3h, \quad x_5 = x_1 %2B4h. \,$

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

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

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 in Abramowitz and Stegun (1972, p. 886) that propagated.^[1]

References

^ Weisstein, Eric W. "Boole's Rule." From MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/BoolesRule.html