0
$\begingroup$

The following matrix is symmetric and all entries are real,

In[147]:= mat == 0.5*(mat+ Transpose[mat])

In[153]:= Table[
 Select[mat[[i]], Not[Element[#1, Reals]] &], {i, 1, 30}]

Due to limitation of 30000 char, I don't show the results. The first is True, and 2nd all empty lists.

Here's the matrix.

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If you we do Eigenvalues[mat], then you'll get non-real eigenvalues. How can that be?

Out[148]= {(...), -1.58298*10^291 + 
  6.14057*10^290 I, -1.58298*10^291 - 6.14057*10^290 I, 
 (...), 8.28201*10^289 + 3.31445*10^290 I, 
 8.28201*10^289 - 3.31445*10^290 I}
$\endgroup$
  • $\begingroup$ Also, Mathematica doesn't show any error message when computing this... i.e. it won't say that it's ill-conditioned. $\endgroup$ – An old man in the sea. Apr 17 at 19:52
  • 3
    $\begingroup$ With large exponents you just have to standardize a bit: 10^306 Eigenvalues[m/10^306] works fine. $\endgroup$ – JimB Apr 17 at 20:15
  • $\begingroup$ @JimB So, it's a bug? By the way, thanks for the comment. How would I make that automatic. This matrix showed during a simulation. $\endgroup$ – An old man in the sea. Apr 17 at 20:31
  • $\begingroup$ @JimB I've tried dividing by the 'Max[mat]' , but it gives different values by something of the order of 10^-15 . $\endgroup$ – An old man in the sea. Apr 17 at 21:09
  • 1
    $\begingroup$ How about Eigenvalues[mat/Norm[mat]]*Norm[mat]? Does should automatically adapt to the range of number and does not produce imaginary numbers, at least in my computer. $\endgroup$ – Mauricio Fernández Apr 18 at 7:16
5
$\begingroup$

I would say it's not a bug.

Be aware that $MaxMachineNumber is around 10^308, in your situation, fed Mathematica with machine-precision-numbers, the underline hardware arithmetic will be called, and the intermediate results are very likely to be overflowed.

Luckily, you can renormalize your matrix, as suggested by JimB in the comment.

newMat = mat/10^305;

10^305 Eigenvalues@newMat

BTW, my experience tells me, even if the matrix itself is not ill-conditioned, the problem is practically ill-posted. Since one would expect the absolute error of each entity is around 10^305*$MachineEpsilon~10^289, which is enormous, you may probably go back and check whether the entities are trustable or check whether the large rescaling constant can be absorbed into a meaningful quantity.

$\endgroup$

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