Question

in Matlab, Use the Monte Carlo analysis to compute the area of a circle with radius 1. print out your code, at least one figure on which the circle and ‘dart hits’ are shown, and numerical results for N=10, 100,1000. For each N, repeat the calculation at least 5times.

Answer 1

%%Matlab code for Monte Carlo simulation for finding area of circle
clear all
close all
%Radius of the circle
r=1;
% number of points generated
N = 10;
a = -r;
b = r;
x = a + (b-a).*rand(N,1);
y = a + (b-a).*rand(N,1);
radii = sqrt(x.^2+y.^2);
hits = sum(radii<=1);
th=linspace(0,2*pi);
x_r=cos(th);
y_r=sin(th);

figure(1)
plot(x_r,y_r,'Linewidth',2)
%Plotting data
k=0;
hold on
for i=1:N
   
     radd=sqrt((x(i)).^2+(y(i)).^2);
     if radd<=1
         k=k+1;
         xx(k)=x(i);
         yy(k)=y(i);
         plot(x(i),y(i),'r*')
     else
         plot(x(i),y(i),'b*')
     end
   
end
xlim([-1 1])
ylim([-1 1])
title(sprintf('Monte Carlo plot for N=%d',N))
xlabel('x')
ylabel('y')
ar=4*hits/N;
fprintf('\n\tFor N=%d area of Circle using Monte Carlo method is %f.\n',N,ar)

            %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
          
clear all

%Radius of the circle
r=1;
% number of points generated
N = 100;
a = -r;
b = r;
x = a + (b-a).*rand(N,1);
y = a + (b-a).*rand(N,1);
radii = sqrt(x.^2+y.^2);
hits = sum(radii<=1);
th=linspace(0,2*pi);
x_r=cos(th);
y_r=sin(th);

figure(2)
plot(x_r,y_r,'Linewidth',2)
%Plotting data
k=0;
hold on
for i=1:N
   
     radd=sqrt((x(i)).^2+(y(i)).^2);
     if radd<=1
         k=k+1;
         xx(k)=x(i);
         yy(k)=y(i);
         plot(x(i),y(i),'r*')
     else
         plot(x(i),y(i),'b*')
     end
   
end
xlim([-1 1])
ylim([-1 1])
title(sprintf('Monte Carlo plot for N=%d',N))
xlabel('x')
ylabel('y')
ar=4*hits/N;
fprintf('\n\tFor N=%d area of Circle using Monte Carlo method is %f.\n',N,ar)

            %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
          
clear all
%Radius of the circle
r=1;
% number of points generated
N = 1000;
a = -r;
b = r;
x = a + (b-a).*rand(N,1);
y = a + (b-a).*rand(N,1);
radii = sqrt(x.^2+y.^2);
hits = sum(radii<=1);
th=linspace(0,2*pi);
x_r=cos(th);
y_r=sin(th);

figure(3)
plot(x_r,y_r,'Linewidth',2)
%Plotting data
k=0;
hold on
for i=1:N
   
     radd=sqrt((x(i)).^2+(y(i)).^2);
     if radd<=1
         k=k+1;
         xx(k)=x(i);
         yy(k)=y(i);
         plot(x(i),y(i),'r*')
     else
         plot(x(i),y(i),'b*')
     end
   
end
xlim([-1 1])
ylim([-1 1])
title(sprintf('Monte Carlo plot for N=%d',N))
xlabel('x')
ylabel('y')
ar=4*hits/N;
fprintf('\n\tFor N=%d area of Circle using Monte Carlo method is %f.\n',N,ar)

            %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%