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Boole's rule

{{Short description|Method of numerical integration}}

{{hatnote|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.

Formula

=Simple Boole's Rule=

It approximates an integral:

\int_{a}^{b} f(x)\,dx

by using the values of {{mvar|f}} at five equally spaced points:{{sfn|Boole|1880|p=47|loc=Eq(21)}}

\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}

It is expressed thus in Abramowitz and Stegun:{{sfn|Davis|Polonsky|1983}}

\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}

where the error term is

-\,\frac{8f^{(6)}(\xi)h^7}{945}

for some number {{tmath|\xi}} between {{tmath|x_0}} and {{tmath|x_4}} where {{nowrap|1=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.{{sfn|Weisstein}}

The following constitutes a very simple implementation of the method in Common Lisp which ignores the error term:

role="presentation" class="wikitable mw-collapsible mw-collapsed"

|Example implementation in Common Lisp

(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, where N mod 4 = 0, the integrated value amounts to:{{sfn|Sablonnière|Sbibih|Tahrichi|2010|p=852}}

\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}

where the error term is similar to above. The following Common Lisp code implements the aforementioned formula:

role="presentation" class="wikitable mw-collapsible mw-collapsed"

|Example implementation in Common Lisp

(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))))))))

role="presentation" class="wikitable mw-collapsible mw-collapsed"

|Example implementation in R

booleQuad <- function(fx, dx) {

# Calculates the composite Boole's rule numerical

# integral for a function with a vector of precomputed

# values fx evaluated at the points in vector dx.

n <- length(dx)

h <- diff(dx)

stopifnot(exprs = {

length(fx) == n

n > 8L

h[1L] >= 0

n >= 2L

n %% 4L == 1L

isTRUE(all.equal(h, rep(h[1L], length(h))))

})

nm2 <- n - 2L

cf <- double(nm2)

cf[seq.int(1, nm2, 2L)] <- 32

cf[seq.int(2, nm2, 4L)] <- 12

cf[seq.int(4, nm2, 4L)] <- 14

cf <- c(7, cf, 7)

sum(cf * fx) * 2 * h[1L] / 45

}

See also

Notes

{{reflist}}

References

{{sfn whitelist|CITEREFDavisPolonsky1983|CITEREFWeisstein}}

  • {{cite book

|last= Boole

|first= George

|author-link= George Boole

|year=1880

|orig-year=1860

|edition = 3rd

|title=A Treatise on the Calculus of Finite Differences

|url=https://archive.org/details/cu31924031240934/page/n61/mode/2up?q=7

|publisher=Macmillan and Company

}}

  • {{AS ref|25, eqn 25.4.14|886|first1=Philip J. |last1=Davis |link1=Philip J. Davis |first2=Ivan |last2=Polonsky}}
  • {{cite journal

| last1 = Sablonnière

| first1 = P.

| last2 = Sbibih

| first2 = D.

| last3 = Tahrichi

| first3 = M.

| date = 2010

| title = Error estimate and extrapolation of a quadrature formula derived from a quartic spline quasi-interpolant

| journal = BIT Numerical Mathematics

| volume = 50

| issue = 4

| pages = 843{{ndash}}862

| doi = 10.1007/s10543-010-0278-0

}}

  • {{MathWorld |title=Boole's Rule|id=BoolesRule}}

{{Numerical integration}}

{{DEFAULTSORT:Boole's Rule}}

Category:Integral calculus

Category:Numerical analysis

Category:Numerical integration

Category:Articles with example Lisp (programming language) code