I am supposed to plot the size k of a Hilbert matrix and condition number . I am supposed to plot k = 5, 10, 15, 20, 25, 30 vs condition # wrt to 1-norm, cond inf-norm, cond 2-norm

I have the code below which outputs the respective matrices and condition numbers but I am confused on how I am supposed to plot the actual matrix vs the norms. code below;

k = [ 5 10 15 20 25 30];
for k = 1:5
  hilbert = hilb(k)
  cond(hilbert,2)
end
for n = 1:5
      hilbert = hilb(k)
      cond(hilbert,1)
 end
 for n = 1:5
      hilbert = hilb(k)
        cond(hilbert, inf)
 end

%k = (5, 10, 15, 20, 25, 30);
%condNum = (1, 2, inf)
%plot(

Is there a better way to write the loop instead of repeating loops? Additionally, how would I plot the size versus the cond numbers ? Would it be better to do each separately for the three norms then record the values and then plot them? I am stuck on how to proceed and wonder if I am even writing this code correctly according to the instructions. Any pointers in the right direction would be greatly appreciated

Answer 1

[Dohyun Kim]

k = [5,10,15, 20, 25, 30];
cond1 = zeros(1,length(k));
cond2 = zeros(1,length(k));

    condinf = zeros(1,length(k));

for i = 1 : length(k)
  H = hilb(k(i));
  cond1(i) = cond(H,1);
  cond2(i) = cond(H,2);
  condinf(i) = cond(H,inf);
end

plot(k,cond1,k,cond2,k,condinf)

I suggest you to use

semilogy(k,cond1,k,cond2,k,condinf)

instead of just plot. Since condition number grows fast, it is better to use log scale for y-axis

Also, you may get warning while get condition number for large k.

In my case, from k = 15, I get warning when we get condition number in 1-norm and inf-norm.

Warning: Matrix is close to singular or badly scaled. Results may be inaccurate. RCOND =  2.765788e-19.

I think we cannot trust the result for k>=15.

In wikipedia, we can see that the condition number should grow as we e increase k.

However, result shows that they are bounded by 10^19 which appears to similar in the warning message.

FYI, RCOND is reciprocal of condition number (= 1/cond(A))