进行更高效的蒙特卡罗模拟

Question

So, I've written this code that should effectively estimate the area under the curve of the function defined as h(x). My problem is that i need to be able to estimate the area to within 6 decimal places, but the algorithm i've defined in estimateN seems to be using too heavy for my machine. Essentially the question is how can i make the following code more efficient? Is there a way i can get rid of that loop?

h = function(x) {
    return(1+(x^9)+(x^3))
}
estimateN = function(n) {
    count = 0
    k = 1
    xpoints = runif(n, 0, 1)
    ypoints = runif(n, 0, 3)
    while(k <= n){
    if(ypoints[k]<=h(xpoints[k]))
        count = count+1
    k = k+1
    }
    #because of the range that im using for y
    return(3*(count/n))
}
#uses the fact that err<=1/sqrt(n) to determine size of dataset
estimate_to = function(i) {
    n = (10^i)^2
    print(paste(n, " repetitions: ", estimateN(n)))
}

estimate_to(6)

Answer 1

Replace this code:

count = 0
k = 1
while(k <= n){
if(ypoints[k]<=h(xpoints[k]))
    count = count+1
k = k+1
}

With this line:

count <- sum(ypoints <= h(xpoints))

Answer 2

If it's truly efficiency you're striving for, integrate is several orders of magnitude faster (not to mention more memory efficient) for this problem.

integrate(h, 0, 1)

# 1.35 with absolute error < 1.5e-14

microbenchmark(integrate(h, 0, 1), estimate_to(3), times=10)

# Unit: microseconds
#                expr        min         lq     median         uq        max neval
#  integrate(h, 0, 1)     14.456     17.769     42.918     54.514     83.125    10
#      estimate_to(3) 151980.781 159830.956 162290.668 167197.742 174881.066    10