LogLog Plot in MATLAB (Estimating PI, error vs N) - matlab

Question

TASK: Using the Monte_Carlo method, compute the approximation of PI for N=100,200,500,1000,2000,5000,10000,100000 and plot the Error of Approximation against N on a Log­Log Plot. Where Error=abs(Actual value­ - Approximate Value). In addition, compute PI using two other infinite series methods and compute pi for N=10,20,50,100,200,500,1000,2000,5000,10000. Plot out on the same graph the Error of Approximation on a Log­Log plot for all 2 formulae and the Monte­Carlo method.
Estimating PI using M_C Method.
clear all
clc
close all
%n = linspace(0, 100000, 100)
n = [100, 200, 500, 1000, 2000, 5000, 10000, 100000]
c = 0;
t = 0;
for q=1:length(n)
x = rand([1 n(q)]);
y = rand([1 n(q)]);
for i= 1:n(q)
t = t+1;
if x(i)^2 + y(i)^2 <= 1
c = c+1;
figure(2)
k(i) = x(i);
r(i) = y(i);
hold on;
else
figure(2)
p(i) = x(i);
j(i) = y(i);
end
end
figure(1)
hold on
if n(q) == 1000
plot(k, r, 'b.');
plot(p, j, 'r.');
end
ratio = c/t;
PI= 4*ratio
error = abs(pi - PI)
figure(2)
loglog(n(q), error, '-b');
end
loglog(n, error, 's-')
grid on
%% Calculating PI using the James Gregory
%n = 10000;
n = [10, 20, 50, 100, 200, 500, 1000, 2000, 5000, 10000]
d = 1;
c = 4;
for j = 1:n
d = d + 2;
c = c + ((-1)^(j))*(4)*(1/d);
end
PI_1 = c;
error = abs(n - PI_1);
loglog(n,error, '-s')
display(c);
%% Calculating PI using the Leibniz Formula for PI
%n = 10000;
n = [10, 20, 50, 100, 200, 500, 1000, 2000, 5000, 10000]
d = 1;
c = 1;
for k = 1:n
d = d + 2;
c = c + ((-1)^k)*(1/d);
end
PI_2 = c*4;
error = abs(n - PI_2);
figure(3)
loglog(n, error, '-s')
The problem I'm having is that the loglog plots don't display the expected information.

Answer 1

Plotting Discussion
For the Monte Carlo plotting, the line
loglog(n, error, 's-')
following the for-loop overwrites all of the plotting done by
loglog(n(q), error, '-b');
because a hold('on') was never issued for figure(2).
Also, in both cases, the plots will look odd due both to the style options and the fact that error is not a vector:
The call loglog(n, error, 's-') will generate a series of disconnected boxes since n is a vector and error is a scalar; Matlab interprets the elements of n as separated data sets each associated with the same scalar value error (the error from the last iteration of the for-loop; plot([1,2],0,'s-') is another example).
The called loglog(n(q), error, '-b'); has a similar problem. Since the style calls for a "solid blue line" but the data passed to loglog is a scalar-scalar pair each iteration, nothing will appear. Matlab cannot form a line for scalar-scalar input (consider the line plot plot(1,1,'-b') versus the circle plot plot(1,1,'ob') as another example).
These problems can be fixed by changing error to a vector of length(n):
error = zeros(1,length(n)); % before for-loop
...
error(q) = abs(pi - PI); % inside the q-for-loop
and performing the loglog plot after the for-loop only (this is also a performance increase since plotting calls are heavy relative to computation).
Performance Discussion
Speaking of performance (to speed up your Monte Carlo), the crowning virtue of Monte Carlo integration, besides not succumbing to the curse of dimensionality, is its ridiculously parallelizable (i.e., vectorizable) nature.
And this is a great thing since vanilla Monte Carlo requires a lot of samples to get accurate results.
Also, Matlab's logical indexing allows for a nice semantic way to pull values satisfying a number of criteria.
With that said, your i-for-loop for the Monte Carlo method can be vetorized with the following code:
% % ----- i-for-loop replacement
% Determine location of points
inCircle = (x.^2 + y.^2) <= 1;
% k = xIn, r = yIn
xIn = x(inCircle);
yIn = y(inCircle);
% p = xOut, j = yOut
xOut = x(~inCircle); % or x(not(inCircle));
yOut = y(~inCircle); % or y(not(inCircle));
% % ----- end of i-for-loop replacement
% Calculate MC pi and error
ratio = nnz(inCircle)/n(q);
PI = 4*ratio;
error(q) = abs(pi - PI);