[Octave-bug-tracker] [bug #57196] eigs should detect symmetric/hermitian inputs and return real values only

[Rik]

URL:
  <https://savannah.gnu.org/bugs/?57196>

                 Summary: eigs should detect symmetric/hermitian inputs and
return real values only
                 Project: GNU Octave
            Submitted by: rik5
            Submitted on: Sat 09 Nov 2019 07:26:45 AM PST
                Category: Octave Function
                Severity: 3 - Normal
                Priority: 5 - Normal
              Item Group: Inaccurate Result
                  Status: Confirmed
             Assigned to: None
         Originator Name: 
        Originator Email: 
             Open/Closed: Open
         Discussion Lock: Any
                 Release: dev
        Operating System: Any

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Details:

A Hermitian matrix has only real eigenvalues.  However, due to various
well-known issues with numerical computation such as round-off error and
limited precision, Octave will return a complex result.

Sample code:


octave:50> x = ones (10,10);
octave:51> z = complex (x, x);
octave:52> A = [sparse(x), z; z', sparse(x)];
octave:53> eigs (A)
ans =

   2.4142e+01 - 7.8571e-16i
  -4.1421e+00 - 1.2042e-15i
   3.6792e-16 + 2.2611e-16i
  -5.6318e-31 + 6.3019e-31i
  -2.4149e-32 + 9.7647e-32i
   1.9359e-32 - 9.3915e-32i

octave:54> imag (ans)
ans =

  -7.8571e-16
  -1.2042e-15
   2.2611e-16
   6.3019e-31
   9.7647e-32
  -9.3915e-32


Matlab, however, returns real values.  Apparently there is a check for
symmetry/hermitian properties of the input and this decides whether real
values or complex values are returned.  As a test with the above matrix A,
calling eigs returns all real values, but a small change that disrupts the
symmettry causes complex values to be returned.


A(1,2) = eps;
eigs (A)


We could implement something similar in Octave, and it would also resolve bug
#57185.








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