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 f at five equally spaced points:

${\begin{aligned}&x_{1}\\&x_{2}=x_{1}+h\\&x_{3}=x_{1}+2h\\&x_{4}=x_{1}+3h\\&x_{5}=x_{1}+4h\end{aligned}}$

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

$\int _{x_{1}}^{x_{5}}f(x)\,dx={\frac {2h}{45}}{\bigl [}7f(x_{1})+32f(x_{2})+12f(x_{3})+32f(x_{4})+7f(x_{5}){\bigr ]}+{\text{error term}}$

where 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).

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 $[a,b]$, the composite Boole's rule might be applied. Given $N$ divisions, the integrated value amounts to

$$\int _{x_{0}}^{x_{N}}f(x)\,dx={\frac {2h}{45}}\left(7(f(x_{0})+f(x_{N}))+32\left(\sum _{i\in \{1,3,5,\ldots ,N-1\}}f(x_{i})\right)+12\left(\sum _{i\in \{2,6,10,\ldots ,N-2\}}f(x_{i})\right)+14\left(\sum _{i\in \{4,8,12,\ldots ,N-4\}}f(x_{i})\right)\right)$$

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

• Newton–Cotes formulas
• Simpson's rule
• Romberg's method