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| | In mathematics, '''Boole's rule''', named after [[George Boole]], is a method of [[numerical integration]]. It approximates an integral |
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| | : <math> \int_{x_1}^{x_5} f(x)\,dx </math> |
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| | by using the values of ''ƒ'' at five equally spaced points |
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| | : <math> 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. \, </math> |
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| | It is expressed thus [[Abramowitz and Stegun]] (1972, p. 886): |
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| | : <math> \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}, </math> |
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| | and the error term is |
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| | : <math> -\,\frac{8}{945} h^7 f^{(6)}(c) </math> |
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| | for some number ''c'' between ''x''<sub>1</sub> and ''x''<sub>5</sub>. (945 = 1 × 3 × 5 × 7 × 9.) |
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| | It is often known as Bode's rule, due to a typographical error that propagated; e.g. in [[Abramowitz and Stegun]] (1972, p. 886).<ref>Weisstein, Eric W. "Boole's Rule." From MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/BoolesRule.html</ref> |
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| | ==See also== |
| | * [[Newton-Cotes formulas]] |
| | * [[Simpson's Rule]] |
| | * [[Romberg's method]] |
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| | ==References== |
| | {{reflist}} |
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| | {{DEFAULTSORT:Boole's Rule}} |
| | [[Category:Numerical integration (quadrature)]] |
In mathematics, Boole's rule, named after George Boole, is a method of numerical integration. It approximates an integral
by using the values of ƒ at five equally spaced points
It is expressed thus Abramowitz and Stegun (1972, p. 886):
and the error term is
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
References
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