# Run a large number of iterations without the use of a for loop

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Divye Kalra on 9 Jan 2022
Commented: Divye Kalra on 12 Jan 2022
Hello everyone, I was trying to implement the brownian motion of a very large number (1536256) of particles in MATLAB. The following code works fine as it is, but takes a very long time to execute. Is there a way to vectorize the following program so that it can work without the use of a for loop?
N=1000;
x_brown=cumsum(randn(1,N));
y_brown=cumsum(randn(1,N));
z_brown=cumsum(randn(1,N));
r0=[x_brown(1) y_brown(1) z_brown(1)];
rf=[x_brown(end) y_brown(end) z_brown(end)];
t = rf-r0;
for i = 2:1:1536256
disp(i);
x_brown=cumsum(randn(1,N));
y_brown=cumsum(randn(1,N));
z_brown=cumsum(randn(1,N));
% Storing r0 and rf values of the previous iteration in m and n
m = r0;
n = rf;
r0=[x_brown(1) y_brown(1) z_brown(1)];
rf=[x_brown(end) y_brown(end) z_brown(end)];
t = cat(1,t,rf-r0);
end
It would be great if someone could help me out here, thanks!

per isakson on 10 Jan 2022
Edited: per isakson on 12 Jan 2022
Preallocating the variable t will improve speed a lot.
Something like
t = nan( 1536256, 1 );
t(1) = rf-r0;
...
t(i) = rf-r0;
In response to comment
The word "bit" in your comment, "Yes, it did reduce the runtime a bit.", triggered me to make my own test on R2018b. My test shows that preallocation improves the speed 16 times for P=1e5 particles and more than twice as much for P=2e5. (I interupted the executions for P = 1536256. Too long to wait.) I guess you tested with a small value of P.
P = 100000. Improvement of speed: 16.0 times
P = 200000. Improvement of speed: 36.8 times
The results may vary with the hardware and the Matlab release.
% P = 1536256;
% P = 100000;
P = 200000;
tic, t2 = loop_pre( P ); e2 = toc;
tic, t1 = loop_cat( P ); e1 = toc;
fprintf( 1, 'P = %d. Improvement of speed: %4.1f times\n', P, e1/e2 )
%%
function t = loop_cat( P )
N=1000;
x_brown=cumsum(randn(1,N));
y_brown=cumsum(randn(1,N));
z_brown=cumsum(randn(1,N));
r0=[x_brown(1) y_brown(1) z_brown(1)];
rf=[x_brown(end) y_brown(end) z_brown(end)];
t = rf-r0;
for jj = 2:1:P
% disp(jj);
x_brown=cumsum(randn(1,N));
y_brown=cumsum(randn(1,N));
z_brown=cumsum(randn(1,N));
% Storing r0 and rf values of the previous iteration in m and n
m = r0;
n = rf;
r0=[x_brown(1) y_brown(1) z_brown(1)];
rf=[x_brown(end) y_brown(end) z_brown(end)];
t = cat(1,t,rf-r0);
end
end
function t = loop_pre( P )
N=1000;
x_brown=cumsum(randn(1,N));
y_brown=cumsum(randn(1,N));
z_brown=cumsum(randn(1,N));
r0=[x_brown(1) y_brown(1) z_brown(1)];
rf=[x_brown(end) y_brown(end) z_brown(end)];
t = nan( P, 3 );
t(1,:) = rf-r0;
for jj = 2:1:P
% disp(jj);
x_brown=cumsum(randn(1,N));
y_brown=cumsum(randn(1,N));
z_brown=cumsum(randn(1,N));
% Storing r0 and rf values of the previous iteration in m and n
m = r0;
n = rf;
r0=[x_brown(1) y_brown(1) z_brown(1)];
rf=[x_brown(end) y_brown(end) z_brown(end)];
t(jj,:) = rf-r0;
end
end
Divye Kalra on 12 Jan 2022
@per isakson Thank you for the very well explained answer. Sorry if my other comment triggered you, wasn't my intention. However, as I said before, I was wondering if the same thing could be done without the use of a for loop (one where the iterating variable is not used inside it ) and where the vectorization of code doesn't seem like a very obvious choice.

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