Condition number of Hilbert matrix depending on MATLAB build??

Hi,

I am a teacher of Numerical Analysis and one of my student noticed that the condition number of the 20x20 Hilbert matrix computed by his MATLAB is different from the one obtained by mine. The MATLAB command given is

clear; format long e; cond(hilb(20))

My result is

2.106530558070438e+18

while the student's result is

5.194395744359904e+19

I asked other students to try, and about ten of them got my result (2.1e+18) but one of them got another different result:

1.531575599359190e+18

I know that the Hilbert matrix is terribly ill-conditioned, so very sensistive to even slighest changes in the algorithms, but I do not expect that different instances of MATLAB give different results. Any explanation? I have also collected the output of the ver command, but it is not very illuminating. Could you please try the experiment yourself?

Thanks

Alessandro

[Bjorn Gustavsson]

Sure thing:

2.121145569096015e+18

With matlab version 2020a in Ubuntu.

[Alessandro Russo]

So this is a new value! Meanwhile I collected another 1.53e+18.

Answer 1

[Bruno Luong]

You and the students should NOT expect COND returns identical result, especially for ill-conditionned matrix like Hilbert matrix. COND invokes SVD and the document here is what one can read officially

"Different machines and releases of MATLAB can produce different singular vectors that are still numerically accurate. Corresponding columns in U and V can flip their signs, since this does not affect the value of the expression A = U*S*V'."

[Alessandro Russo]

Thanks Bruno!