% m1_howfast.m - how fast can we solve Ax=b with Matlab "\" command?


A = rand(5)                                              

b = ones(5,1)                                            

x = A\b                                                  

A*x                                                      

A*x-b                                                    

A = randn(500);                                          

b = ones(500,1);                                         

x = A\b;                                                 

whos                                                     

A*x-b                                                    

ans(end)                                                 

N = 1000; A = randn(N); b = ones(N,1); tic, A\b; toc     
N = 2000; A = randn(N); b = ones(N,1); tic, A\b; toc     
N = 4000; A = randn(N); b = ones(N,1); tic, A\b; toc     
N = 8000; A = randn(N); b = ones(N,1); tic, A\b; toc     

format bank

for i = 1:12, N = 2^i; A = randn(N);                        
   tic, A\ones(N,1); t=toc; disp([N t]), end                
 
for i = 1:12, N = 2^i; A = randn(N);
   tic, A\ones(N,1); t = toc; NN(i) = N; tt(i) = t; end

plot(NN,tt)

plot(NN,tt,'.-')

loglog(NN,tt,'.-')

grid on, hold on

loglog(NN,1e-11*NN.^3,'--r')

ylim([1e-5 1e2])

xlabel('dimension')

ylabel('time (secs)')

title('speed of Matlab backslash')

gtext('N^3','fontsize',24,'color','r')