Monte Carlo Pi (Fortran)

Numerical Methods

This Fortran program estimates π using the Monte Carlo method by randomly sampling points in a unit square and counting how many fall inside a quarter circle.

Monte Carlo Method

The area of a quarter circle of radius 1 is $π /4$ . The area of the unit square is 1. By randomly sampling points in the square, the ratio of points inside the circle to total points approximates $π /4$ .

π \approx 4 \times \frac{points inside circle}{total points}

Implementation

program monte_carlo_pi
    implicit none
    integer, parameter :: n = 10000  ! Number of random points
    integer :: i, count
    real :: x, y, pi_estimate
    real :: rand1, rand2

    ! Initialize random number generator
    call random_seed()

    ! Open output file
    open(unit=10, file='monte_carlo_points.csv', status='replace', action='write')
    write(10, *) 'x, y, inside_circle'

    count = 0

    ! Generate random points and count points inside the quarter circle
    do i = 1, n
        call random_number(rand1)
        call random_number(rand2)
        x = rand1
        y = rand2
        if (x**2 + y**2 <= 1.0) then
            count = count + 1
            write(10, '(F6.4, 1X, F6.4, 1X, I1)') x, y, 1
        else
            write(10, '(F6.4, 1X, F6.4, 1X, I1)') x, y, 0
        end if
    end do

    ! Estimate pi
    pi_estimate = 4.0 * real(count) / real(n)

    ! Print the estimate
    print *, 'Estimated Pi:', pi_estimate

    ! Close the output file
    close(10)

end program monte_carlo_pi

Key Features

Uses random_seed() to initialize the random number generator
Uses random_number() to generate uniform random numbers in [0, 1)
Checks if points satisfy $x^{2} + y^{2} \leq 1$ (inside unit circle)
Outputs all points to CSV file for visualization
Converts integer to real using real() function

Accuracy

The accuracy improves with the square root of the number of samples. For $n = 10, 000$ , the estimate typically has an error of about 0.01.