First Semester in Numerical Analysis with Julia, 2020a

CHAPTER 4. NUMERICAL QUADRATURE AND DIFFERENTIATION 163
be 2 360 , which is about 10 100 , a number too large. In very large dimensions, the only general
method for numerical integration is the Monte Carlo method. In Monte Carlo, we generate
pseudorandom numbers from the domain, and evaluate the function average at those points.
In other words,
∫
f(x)dx ≈ 1
R n
n∑
f(x i )
where x i are pseudorandom vectors uniformly distributed in R. For the two-dimensional
integral we discussed before, the Monte Carlo estimate is
i=1
∫ b ∫ d
a c
f(x, y)dydx ≈
(b − a)(d − c)
n
n∑
f(a +(b − a)x i ,c+(d − c)y i ) (4.8)
i=1
where x i ,y i are pseudorandom numbers from (0, 1). In Julia, the function rand() generates
a pseudorandom number from the uniform distribution between 0 and 1. The following code
takes the endpoints of the intervals a, b, c, d, and the number of nodes n (which is called the
sample size in the Monte Carlo literature) as inputs, and returns the estimate for the integral
using Equation (4.8).
In [1]: function mc(f::Function,a,b,c,d,n)
sum=0.
for i in 1:n
sum=sum+f(a+(b-a)*rand(),c+(d-c)*rand())*(b-a)*(d-c)
end
return sum/n
end
Out[1]: mc (generic function with 1 method)
Now we use Monte Carlo to estimate the integral ∫ 1
0
In [2]: mc((x,y)->(pi^2/4)*sin(pi*x)*sin(pi*y),0,1,0,1,500)
Out[2]: 0.9441778334708931
∫ 1
( π sin πx)( π
sin πy) dydx:
0 2 2
With n = 500, we obtain a Monte Carlo estimate of 0.944178. An advantage of the Monte
Carlo method is its simplicity: the above code can be easily generalized to higher dimensions.
A disadvantage of Monte Carlo is its slow rate of convergence, which is O(1/ √ n). Figure
(4.3) displays 500 pseudorandom vectors from the unit square.