Английская Википедия:Boole's rule
Шаблон:Short description Шаблон:Hatnote
In mathematics, Boole's rule, named after George Boole, is a method of numerical integration.
Formula
Simple Boole's Rule
It approximates an integral: <math display="block"> \int_{a}^{b} f(x)\,dx </math> by using the values of Шаблон:Mvar at five equally spaced points:Шаблон:Sfn <math display="block">\begin{align} & x_0 = a\\ & x_1 = x_0 + h \\ & x_2 = x_0 + 2h \\ & x_3 = x_0 + 3h \\ & x_4 = x_0 + 4h = b \end{align}</math>
It is expressed thus in Abramowitz and Stegun:Шаблон:Sfn <math display="block"> \int_{x_0}^{x_4} f(x)\,dx = \frac{2 h}{45}\bigl[ 7f(x_0) + 32 f(x_1) + 12 f(x_2) + 32 f(x_3) + 7f(x_4) \bigr] + \text{error term}</math> where the error term is <math display="block"> -\,\frac{8f^{(6)}(\xi)h^7}{945} </math> for some number Шаблон:Tmath between Шаблон:Tmath and Шаблон:Tmath where Шаблон:Nowrap.
It is often known as Bode's rule, due to a typographical error that propagated from Abramowitz and Stegun.Шаблон:Sfn
The following constitutes a very simple implementation of the method in Common Lisp which ignores the error term:
(defun integrate-booles-rule (f x1 x5)
"Calculates the Boole's rule numerical integral of the function F in
the closed interval extending from inclusive X1 to inclusive X5
without error term inclusion."
(declare (type (function (real) real) f))
(declare (type real x1 x5))
(let ((h (/ (- x5 x1) 4)))
(declare (type real h))
(let* ((x2 (+ x1 h))
(x3 (+ x2 h))
(x4 (+ x3 h)))
(declare (type real x2 x3 x4))
(* (/ (* 2 h) 45)
(+ (* 7 (funcall f x1))
(* 32 (funcall f x2))
(* 12 (funcall f x3))
(* 32 (funcall f x4))
(* 7 (funcall f x5)))))))
Composite Boole's Rule
In cases where the integration is permitted to extend over equidistant sections of the interval <math>[a, b]</math>, the composite Boole's rule might be applied. Given <math>N</math> divisions, the integrated value amounts to:Шаблон:Sfn
<math display=block>
\int_{x_0}^{x_N} f(x)\,dx
= \frac{2 h}{45}
\left(
7(f(x_0) + f(x_N))
+ 32\left(\sum_{i \in \{1, 3, 5, \ldots, N-1\}} f(x_i)\right)
+ 12\left(\sum_{i \in \{2, 6, 10, \ldots, N-2\}} f(x_i)\right)
+ 14\left(\sum_{i \in \{4, 8, 12, \ldots, N-4\}} f(x_i)\right)
\right) + \text{error term}
</math>
where the error term is similar to above. The following Common Lisp code implements the aforementioned formula:
(defun integrate-composite-booles-rule (f a b n)
"Calculates the composite Boole's rule numerical integral of the
function F in the closed interval extending from inclusive A to
inclusive B across N subintervals."
(declare (type (function (real) real) f))
(declare (type real a b))
(declare (type (integer 1 *) n))
(let ((h (/ (- b a) n)))
(declare (type real h))
(flet ((f[i] (i)
(declare (type (integer 0 *) i))
(let ((xi (+ a (* i h))))
(declare (type real xi))
(the real (funcall f xi)))))
(* (/ (* 2 h) 45)
(+ (* 7 (+ (f[i] 0) (f[i] n)))
(* 32 (loop for i from 1 to (- n 1) by 2 sum (f[i] i)))
(* 12 (loop for i from 2 to (- n 2) by 4 sum (f[i] i)))
(* 14 (loop for i from 4 to (- n 4) by 4 sum (f[i] i))))))))
See also
Notes
References
