%% How fast is Ax=b?
% Kathryn Gillow

%%
% Let's explore how the speed of solving $Ax=b$ scales with the size $n$ of
% the matrix $A$. We'll use a random $n \times n$ matrix with independent 
% normally distributed entries, and a random right-hand-side $b$.
n = 1000;
A = randn(n,n);
b = randn(n,1);
tic, x = A\b; toc

%% Vary n
% Now try for a range of matrix sizes
nn = 2.^(1:13);
% assign memory for tt vector
tt = zeros(size(nn)); % or tt = 0*nn;
for k = 1:length(nn)
    n = nn(k);
    A = randn(n,n);
    b = rand(n,1);
    tic, x = A\b; tt(k) = toc;
end
figure(1), clf
loglog(nn,tt,'x-','MarkerSize',12,'Linewidth',3)
xlabel('Matrix size n')
ylabel('Time in secs')
title('Time to solve Ax=b')

%% 
% We know that it takes O(n^3) time to solve $Ax=b$
% note that the second graph on the plot is in a separate section so the
% figure appears twice in the published version
hold on
loglog(nn, 1e-10*nn.^3,'r','Linewidth',3)
legend('time','predicted time','Location','SouthEast')