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)

Time with for-loop: 0.106239 seconds

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)

Time with preallocation and for-loop: 0.122389 seconds

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)

Time with preallocation and smart for-loop: 0.121054 seconds

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)

Time with vectorization: 0.000239 seconds