%
% m1 practice exam, problem 4
%

clear
clf
format compact

nn = (2000:100:3000)';
l = length(nn);
tt = zeros(l, 1);  % preallocate
nrep = 4;
rng(1); % seed the random number generator

for i = 1:l
    n = nn(i);
    A = randn(n, n);  
    b = rand(n, 1);

    tic()  % start a timer
    for j = 1:nrep   % repeat nrep times for more precise timing
        [L, U] = lu(A);  
        y = forwardsub(L, b);
        x = backsub(U, y);
    end
    tt(i) = toc()/nrep % record time
end

%%
% 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(nn, tt, 'ro-')
hold on
loglog(nn, tt(end)*(nn/nn(end)).^3, 'k--')
axis tight
xlabel('size of matrix')
ylabel('time (sec)')
title('Timing of LU factorization method of solution')
legend('lu','O(n^3)','location','southeast')
grid on
hold off