Boole's rule

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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 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; e.g. in Abramowitz and Stegun (1972, p. 886).[1]

See also[edit]

References[edit]

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