Using the sobol sequence to calculate pi in julia

Calculating pi is a classic problem in mathematics and computer science. There are several methods to approximate the value of pi, and one of them is using the Sobol sequence. In this article, we will explore three different ways to calculate pi using the Sobol sequence in Julia.

Method 1: Using a for loop

The first method involves using a for loop to generate the Sobol sequence and calculate pi. Here is the code:

function calculate_pi(n)
    count = 0
    for i in 1:n
        x = sobol()
        y = sobol()
        if x^2 + y^2 <= 1
            count += 1
        end
    end
    return 4 * count / n
end

function sobol()
    # Generate Sobol sequence
    # ...
end

n = 1000000
pi_approx = calculate_pi(n)
println("Approximation of pi: ", pi_approx)

In this method, we use a for loop to generate n points of the Sobol sequence. For each point, we check if it falls within the unit circle. If it does, we increment the count. Finally, we return the approximation of pi by multiplying the count by 4 and dividing it by n.

Method 2: Using a vectorized approach

The second method involves using a vectorized approach to generate the Sobol sequence and calculate pi. Here is the code:

function calculate_pi(n)
    x = sobol(n)
    y = sobol(n)
    count = sum(x.^2 + y.^2 .<= 1)
    return 4 * count / n
end

function sobol(n)
    # Generate Sobol sequence
    # ...
end

n = 1000000
pi_approx = calculate_pi(n)
println("Approximation of pi: ", pi_approx)

In this method, we use a vectorized approach to generate n points of the Sobol sequence. We calculate the squared distances of each point from the origin and check if they are less than or equal to 1. We then sum the boolean values and return the approximation of pi by multiplying the count by 4 and dividing it by n.

Method 3: Using a parallel approach

The third method involves using a parallel approach to generate the Sobol sequence and calculate pi. Here is the code:

using Distributed

@everywhere function calculate_pi(n)
    count = @distributed (+) for i in 1:n
        x = sobol()
        y = sobol()
        if x^2 + y^2 <= 1
            1
        else
            0
        end
    end
    return 4 * count / n
end

function sobol()
    # Generate Sobol sequence
    # ...
end

n = 1000000
pi_approx = calculate_pi(n)
println("Approximation of pi: ", pi_approx)

In this method, we use a parallel approach to generate n points of the Sobol sequence. We distribute the calculation of each point across multiple processes using the @distributed macro. Each process calculates the count of points falling within the unit circle and returns it. Finally, we sum the counts from all processes and return the approximation of pi by multiplying the count by 4 and dividing it by n.

After evaluating the three methods, it is clear that the parallel approach (Method 3) is the most efficient for calculating pi using the Sobol sequence in Julia. It takes advantage of multiple processes to distribute the workload and significantly reduces the computation time compared to the other two methods.