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}
Improve this question
$\endgroup$
8
  • $\begingroup$ 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.
    Commented Apr 17, 2019 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
    Commented Apr 17, 2019 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.
    Commented Apr 17, 2019 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.
    Commented Apr 17, 2019 at 21:09
  • 2
    $\begingroup$ 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
    Commented Apr 18, 2019 at 7:16

1 Answer 1

Reset to default
7
$\begingroup$

I would say it's not a bug.

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

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

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-posed. Since one would expect that 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 trustworthy or check whether the large rescaling constant can be absorbed into a meaningful quantity.

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