%% Other Efficiency Issues

%%
% This is a script file to illustrate other "tricks" to improve execution
% such as preallocation and smart coding 
%
% Note: *Preallocate* large vectors or matrices
%
% As our example we create a Brownian path
randn('state',20)     % initialize the random number generator
n = 10000;            % set the number of steps

%%
% How long does a for-loop take?
tic;
dt = 1/n;
w(1) = 0;
for i=1:n
    w(i+1) = w(i) + sqrt(dt)*randn;
end
t1 = toc;
plot(0:dt:1,w)
fprintf('Time with for-loop: %f seconds\n\n', t1)

%%
% Now we preallocate the vector $w$ and try again
clear w
tic;
dt = 1/n;
w = zeros(n+1,1);
for i=1:n
    w(i+1) = w(i) + sqrt(dt)*randn;
end
t1 = toc;
fprintf('Time with preallocation and for-loop: %f seconds\n\n', t1)

%%
% Here we preallocate the vector $w$ and precompute $\sqrt{dt}$
clear w
tic;
dt = 1/n;
w = zeros(n+1,1);
sdt = sqrt(dt);
for i=1:n
    w(i+1) = w(i) + sdt*randn;
end
t1 = toc;
fprintf('Time with preallocation and smart for-loop: %f seconds\n\n', t1)

%%
% Finally, we vectorize the for-loop with |cumsum|
clear w
tic;
dt = 1/n;
w = sqrt(dt)*cumsum([0;randn(n,1)]);
t1 = toc;
fprintf('Time with vectorization: %f seconds\n\n', t1)