Let's repeat the experiment of the previous figure for more, and larger, values of n.

n_ = (400:200:6000)';

t_ = 0*n_;

for i = 1:length(n_)

n = n_(i);

A = randn(n,n); x = randn(n,1);

tic % start a timer

for j = 1:10 % repeat ten times

A*x;

end

t = toc; % read the timer

t_(i) = t/10;

end

Plotting the time as a function of n on log-log scales is equivalent to plotting the logs of the variables, but is formatted more neatly.

loglog(n_,t_,'.-')

xlabel('size of matrix'), ylabel('time (sec)')

title('Timing of matrix-vector multiplications')

You can see that the graph is trending to a straight line of positive slope. For comparison, we can plot a line that represents growth exactly. (All such lines have slope equal to 2.)

hold on, loglog(n_,t_(1)*(n_/n_(1)).^2,'--')

axis tight

legend('data','O(n^2)','location','southeast')

The full story of the execution times is complicated, but the asymptotic approach to is unmistakable.