one-dimensional integrals

December 25, 2010

(This article was first published on YGC » R, and kindly contributed to R-bloggers)

The foundamental idea of numerical integration is to estimate the area of the region in the xy-plane bounded by the graph of function f(x). The integral was esimated by divide x to small intervals, then add all the small approximations to give a total approximation.

Numerical integration can be done by trapezoidal rule, simpson’s rule and quadrature rules. R has a built-in function integrate, which performs adaptive quadrature.

Trapezoidal rule works by approximating the region under the graph f(x) as a trapezoid and calculating its area.

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trapezoid <- function(fun, a, b, n=100) {
	# numerical integral of fun from a to b
	# using the trapezoid rule with n subdivisions
	# assume a < b and n is a positive integer
	h <- (b-a)/n
	x <- seq(a, b, by=h)
	y <- fun(x)
	s <- h * (y[1]/2 + sum(y[2:n]) + y[n+1]/2)
	return(s)
}

Simpson’s rule subdivides the interval [a,b] into n subintervals, where n is even, then on each consecutive pairs of subintervals, it approximates the behaviour of f(x) by a parabola (polynomial of degree 2) rather than by the straight lines used in the trapezoidal rule.

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simpson <- function(fun, a, b, n=100) {
	# numerical integral using Simpson's rule
	# assume a < b and n is an even positive integer
	h <- (b-a)/n
	x <- seq(a, b, by=h)
	if (n == 2) {
		s <- fun(x[1]) + 4*fun(x[2]) +fun(x[3])
	} else {
		s <- fun(x[1]) + fun(x[n+1]) + 2*sum(fun(x[seq(2,n,by=2)])) + 4 *sum(fun(x[seq(3,n-1, by=2)]))
	}
	s <- s*h/3
	return(s)
}

To calculate an integrate over infinite interval, one way is to transform it into an integral over a finite interval as introduce in wiki.

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simpson_v2 <- function(fun, a, b, n=100) {
	# numerical integral using Simpson's rule
	# assume a < b and n is an even positive integer
	if (a == -Inf & b == Inf) {
		f <- function(t) (fun((1-t)/t) + fun((t-1)/t))/t^2
		s <- simpson_v2(f, 0, 1, n)
	} else if (a == -Inf & b != Inf) {
		f <- function(t) fun(b-(1-t)/t)/t^2
		s <- simpson_v2(f, 0, 1, n)
	} else if (a != -Inf & b == Inf) {
		f <- function(t) fun(a+(1-t)/t)/t^2
		s <- simpson_v2(f, 0, 1, n)
	} else {
		h <- (b-a)/n
		x <- seq(a, b, by=h)
		y <- fun(x)
		y[is.nan(y)]=0
		s <- y[1] + y[n+1] + 2*sum(y[seq(2,n,by=2)]) + 4 *sum(y[seq(3,n-1, by=2)])
		s <- s*h/3
	}
	return(s)
}

> phi <- function(x) exp(-x^2/2)/sqrt(2*pi)  ##normal distribution
> simpson_v2(phi, 0, Inf, n=100)
[1] 0.4986569
> simpson_v2(phi, -Inf,3, n=100)
[1] 0.998635
> pnorm(3)
[1] 0.9986501

Simpson’s rule is more accuracy than trapezoidal rule. To compare the accuracy between simpson’s rule and trapezoidal rule, I estimated $int_{0.01}^{1} 1/x dx$ = -log(0.01) for a sequence of increasing values of n.

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f <- function(x) 1/x
#integrate(f, 0.01, 1) == -log(0.01)
S.trapezoid <- function(n) trapezoid(f, 0.01, 1, n)
S.simpson <- function(n) simpson(f, 0.01, 1, n)
 
n <- seq(10, 1000, by = 10)
St <- sapply(n, S.trapezoid)
Ss <- sapply(n, S.simpson)
 
opar <- par(mfrow = c(2, 2))
plot(n,St + log(0.01), type="l", xlab="n", ylab="error", main="Trapezoidal rule")
plot(n,Ss + log(0.01), type="l", xlab="n", ylab="error",main="Simpson's rule")
plot(log(n), log(St+log(0.01)),type="l", xlab="log(n)", ylab="log(error)",main="Trapezoidal rule")
plot(log(n), log(Ss+log(0.01)),type="l", xlab="log(n)", ylab="log(error)",main="Simpson's rule")

The plot showed that log(error) against log(n) appears to have a slope of -1.90 and -3.28 for trapezoidal rule and simpson’s rule respectively.

Another way to compare their accuracy is to calculate how large of the partition size n for reaching a specific tolerance.

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yIntegrate <- function(fun, a, b, tol=1e-8, method= "simpson", verbose=TRUE) {
	# numerical integral of fun from a to b, assume a < b 
	# with tolerance tol
	n <- 4
	h <- (b-a)/4
	x <- seq(a, b, by=h)
	y <- fun(x)
	yIntegrate_internal <- function(y, h, n, method) {
		if (method == "simpson") {
			s <- y[1] + y[n+1] + 4*sum(y[seq(2,n,by=2)]) + 2 *sum(y[seq(3,n-1, by=2)])
			s <- s*h/3
		} else if (method == "trapezoidal") {
			s <- h * (y[1]/2 + sum(y[2:n]) + y[n+1]/2)
		} else {
		}		
		return(s)
	}
 
	s <- yIntegrate_internal(y, h, n, method)
	s.diff <- tol + 1 # ensures to loop at once.
	while (s.diff > tol ) {
		s.old <- s
		n <- 2*n
		h <- h/2
		y[seq(1, n+1, by=2)] <- y ##reuse old fun values
		y[seq(2,n, by=2)] <- sapply(seq(a+h, b-h, by=2*h), fun)
		s <- yIntegrate_internal(y, h, n, method)
		s.diff <- abs(s-s.old)
	}
	if (verbose) {
		cat("partition size", n, "n")
	}
	return(s)
}

> fun <- function(x) exp(x) - x^2
> yIntegrate(fun, 3,5,tol=1e-8, method="simpson")
partition size 512
[1] 95.66096
> yIntegrate(fun, 3,5,tol=1e-8, method="trapezoidal")
partition size 131072
[1] 95.66096

As show above, simpson’s rule converge much faster than trapezoidal rule.

Quadrature rule is more efficient than traditional algorithms. In adaptive quadrature, the subinterval width h is not constant over the interval [a,b], but instead adapts to the function.
Here, I presented the adaptive simpson’s method.

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quadrature <- function(fun, a, b, tol=1e-8) {
	# numerical integration using adaptive quadrature
 
	quadrature_internal <- function(S.old, fun, a, m, b, tol, level) {
		level.max <- 100
		if (level > level.max) {
			cat ("recursion limit reached: singularity likelyn")
			return (NULL)
		}
		S.left <- simpson(fun, a, m, n=2) 
		S.right <- simpson(fun, m, b, n=2)
		S.new <- S.left + S.right
		if (abs(S.new-S.old) > tol) {
			S.left <- quadrature_internal(S.left, fun, a, (a+m)/2, m, tol/2, level+1)
			S.right <- quadrature_internal(S.right, fun, m, (m+b)/2, b, tol/2, level+1)
			S.new <- S.left + S.right
		}
		return(S.new)
	}
 
	level = 1
	S.old <- (b-a) * (fun(a) + fun(b))/2
	S.new <- quadrature_internal(S.old, fun, a, (a+b)/2, b, tol, level+1)
	return(S.new)
}

Quadrature rule is effective when f(x) is steep.

> fun <- function(x) return(1.5 * sqrt(x))
> system.time(yIntegrate(fun, 0,1, tol=1e-9, method="simpson"))
partition size 524288
   user  system elapsed
   1.58    0.00    1.58
> system.time(quadrature(fun, 0,1, tol=1e-9))
   user  system elapsed
   0.25    0.00    0.25

Reference:
1. Robinson A., O. Jones, and R. Maillardet. 2009. Introduction to Scientific Programming and Simulation Using R. Chapman and Hall.
2. http://en.wikipedia.org/wiki/Numerical_integration
3. http://en.wikipedia.org/wiki/Trapezoidal_rule
4. http://en.wikipedia.org/wiki/Simpson%27s_rule
5. http://en.wikipedia.org/wiki/Adaptive_quadrature