# Copyright (C) 1997-2005 The R Core Team ## Adaptive integration: Venables and Ripley pp. 105-110 ## This is the basic integrator. area <- function(f, a, b, ..., fa = f(a, ...), fb = f(b, ...), limit = 10, eps = 1.e-5) { h <- b - a d <- (a + b)/2 fd <- f(d, ...) a1 <- ((fa + fb) * h)/2 a2 <- ((fa + 4 * fd + fb) * h)/6 if(abs(a1 - a2) < eps) return(a2) if(limit == 0) { warning(paste("iteration limit reached near x = ", d)) return(a2) } area(f, a, d, ..., fa = fa, fb = fd, limit = limit - 1, eps = eps) + area(f, d, b, ..., fa = fd, fb = fb, limit = limit - 1, eps = eps) } ## The function to be integrated fbeta <- function(x, alpha, beta) { x^(alpha - 1) * (1 - x)^(beta - 1) } ## Compute the approximate integral, the exact integral and the error b0 <- area(fbeta, 0, 1, alpha=3.5, beta=1.5) b1 <- exp(lgamma(3.5) + lgamma(1.5) - lgamma(5)) c(b0, b1, b0-b1) ## Modify the function so that it records where it was evaluated fbeta.tmp <- function (x, alpha, beta) { val <<- c(val, x) x^(alpha - 1) * (1 - x)^(beta - 1) } ## Recompute and plot the evaluation points. val <- NULL b0 <- area(fbeta.tmp, 0, 1, alpha=3.5, beta=1.5) plot(val, fbeta(val, 3.5, 1.5), pch=0) ## Better programming style -- renaming the function will have no effect. ## The use of "Recall" as in V+R is VERY black magic. You can get the ## same effect transparently by supplying a wrapper function. ## This is the approved Abelson+Sussman method. area <- function(f, a, b, ..., limit=10, eps=1e-5) { area2 <- function(f, a, b, ..., fa = f(a, ...), fb = f(b, ...), limit = limit, eps = eps) { h <- b - a d <- (a + b)/2 fd <- f(d, ...) a1 <- ((fa + fb) * h)/2 a2 <- ((fa + 4 * fd + fb) * h)/6 if(abs(a1 - a2) < eps) return(a2) if(limit == 0) { warning(paste("iteration limit reached near x =", d)) return(a2) } area2(f, a, d, ..., fa = fa, fb = fd, limit = limit - 1, eps = eps) + area2(f, d, b, ..., fa = fd, fb = fb, limit = limit - 1, eps = eps) } area2(f, a, b, ..., limit=limit, eps=eps) }