basic monte carlo question: area of circle

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Question: Use monte carlo method to find area of circle radius 5 inside 7x7 square.
So this is the code I've put together so far to determine how many points land inside or on the circle (hits). My output for hits is 0 but I can't for the life of me figure out why, to me everything seems fine but clearly isn't.
clear;
N= 1000; % number of points generated
a = -7;
b = 7;
hits= 0;
for k = 1:N
x = a + (b-a).*rand(N,1);
y = a + (b-a).*rand(N,1);
% Count it if it is in/on the circle
radii = sqrt(x.^2+y.^2);
if radii <=5;
hits = hits + 1;
end
end
disp(hits)

Accepted Answer

pfb
pfb on 1 May 2015
Edited: pfb on 1 May 2015
You are mixing things up. Your code is part vectorized, part scalar.
You have a loop over N, but then at each iteration you generate N points (so you're generating N^2 points overall). The variables x, y, and radii are vectors.
Note that
[6 7 4 2 8 5] <=5
gives
[0 0 1 1 0 1]
Therefore condition radii<=5 has a very low probability to be met ( all of the N radii should be less than 5).
As a result hits is (almost) never accumulated.
Your code is structured for a scalar radius. It would work if you substitute the lines
x = a + (b-a).*rand(N,1);
y = a + (b-a).*rand(N,1);
with
x = a + (b-a).*rand;
y = a + (b-a).*rand;
In this case hits will be a number between 0 and N, and hits/N is going to be proportional to the ratio of the areas of the circle and square, pi*(5/14)^2 (note that you can use simple power operator ^ instead of elementwise .^)
You can avoid the loop altogether using the vectorized variables.
N= 1000; % number of points generated
a = -7;
b = 7;
x = a + (b-a).*rand(N,1);
y = a + (b-a).*rand(N,1);
radii = sqrt(x.^2+y.^2);
hits = sum(radii<=5);

More Answers (3)

Eric
Eric on 2 May 2015
Thanks both really helped a bunch.
How would I go about plotting the graph so that the hits are one colour and the rest are another?
I tried the following but I just get a linear line.
x = linspace(-7,7);
y = linspace(-7,7);
figure(1);
plot(x,y);
xlabel('x');
ylabel('y');
axis square
grid on;
title('Area of Circle');
  5 Comments
pfb
pfb on 2 May 2015
Edited: pfb on 2 May 2015
ok, then you can write the function sum with a loop. After creating the vector i (which you need for the plotting)
hits = 0;
for j = 1:N
hits=hits+i(j);
end
misses= N-hits;
If you're not allowed to use logical indexing in the plot you can pretend you do not know it and do this
% vectors for the hits
xh = zeros(1,hits);
yh = xh;
% vector for the misses
xm = zeros(1,misses);
ym = xm;
% counters
mc=1;
hc=1;
for j=1:N
if(i(j))
% put the point in the hits vector
xh(hc)=x(j);
yh(hc)=y(j);
% increment the hit counter
hc=hc+1;
else
% put the point in the misses vector
xm(mc)=x(j);
ym(mc)=y(j);
% increment the miss counter
mc=mc+1;
end
end
plot(xh,yh,'g.');
hold
plot(xm,ym,'r.');
etcetera

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Eric
Eric on 2 May 2015
Thanks a lot, appreciate it.
  1 Comment
pfb
pfb on 2 May 2015
you're welcome. Note that there was an error in my code above (in one instance I wrote hits instead of misses). Now it should be ok.

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munesh pal
munesh pal on 27 Feb 2018
Calculate the area of the circle using Monte Carlo Simulation
clc
clear all
close all
%%input circle radius and coordinate
r=2;
c_x=7;
c_y=7;
%%position of the outer rectangle
p_x=c_x-r;
p_y=c_y-r;
pos=[p_x,p_y,r^2,r^2]
%%random number generation within the sqaure
N=1000;
a=p_x;
b=p_x+(2*r);
x = a + (b-a).*rand(N,1);
y = a + (b-a).*rand(N,1);
radii = sqrt((x-c_x).^2+(y-c_y).^2);
i = radii <= r;
hits = sum(i);
misses = N-hits;
plot(x(i),y(i),'.g');
hold;
plot(x(~i),y(~i),'.r');
rectangle('Position',pos,'Curvature',[1 1])
rectangle('Position',pos,'EdgeColor','r')
actual_a=22/7*r^2;
ttl = sprintf('Estimate Actual Area of circle: %1.3f, Area of the circle: %1.3f',actual_a,hits/N*(2*r)^2);
title(ttl);
axis equal
  1 Comment
hosein bashi
hosein bashi on 30 Jul 2018
Edited: hosein bashi on 30 Jul 2018
can you explain please why we use random numbers? why we don't assume a grid and put our numbers in the middle of the cells? it would be more uniform in this way.

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