%%
% We'll test the conclusion of $O(n^3)$ flops experimentally, using the
% built-in lu function instead of the purely instructive
% lufact.
n_ = (1000:200:4000)';
t_ = 0*n_;
for i = 1:length(n_)
    n = n_(i);
    A = randn(n,n);  
    tic()       % start a timer
    for j = 1:6
        [L,U] = lu(A);  
    end
    t = toc();  % read the timer
    t_(i) = t/6;  
end

%%
% We plot the performance on a log-log graph and compare it to O(n^3).
% The result could vary significantly from machine to machine. 
loglog(n_, t_, '.-')
hold on
loglog(n_, t_(end)*(n_/n_(end)).^3, '--')
xlabel('size of matrix')
ylabel('time (sec)')
title('Timing of LU factorization')
legend('lu','O(n^3)','location','southeast')
grid on