Monte-Carlo Method

This is a small, simple example of utilising the Monte-Carlo method to calculate the value of pi. It's easy because it is only a 2-dimensional problem and thus is easy to visualise, and is trivial in its code implementation.

Picture the graph shown in the parent article - that of a quarter circle plotted on a unit x-y graph:

http://imagestore.ugbox.net/image/montecarlo_7dd5a4bb001953bd326f.jpg

Below is a small example piece of Matlab code that does the above simulation:

Code:

 % Monte-carlo simulator for solving area of a quarter-circle of radius 1,
 % dictated by bounds of x^2+y^2 = 1 
 
 % Set up arrays and other constants
 x = 0; % x coordinate
 y = 0; % y coordinate
 total = 10000; % amount of points to generate
 count = 0; % number of points inside area
 
 % monte-carlo solver
 for i = 1:total
     % generate a random point in 0<x<1 and 0<y<1
     x = rand; % random number between 0 and 1
     y = rand;
     % monte-carlo check if inside bounds
     if (x^2+y^2 <= 1)
         count = count + 1;
     end
 end
   
 % calculate 'true' area and monte-carlo solved area
 circlearea = pi*1^2; % 'true' area
 montearea = count/total*4; % monte-carlo area

This, when run on my computer, produced the result of 0.783. The area of a quarter circle works out to be π/4 if the radius is 1. π/4 = 0.7854, so you can see that this is quite close. More iterations than 1000 would produce a closer answer.

Monte-Carlo Method

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