[Octave-bug-tracker] [bug #60357] Failing test for `qz` with LAPACK 3.9.1

[Rik]

Update of bug #60357 (project octave):

              Item Group:                    None => Incorrect Result       
                  Status:                    None => Confirmed              

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Follow-up Comment #16:

Some interesting data points on this bug report.

First, Matlab no longer documents that there is a final output from qz() which
is the eigenvalues for Octave.  See
https://www.mathworks.com/help/matlab/ref/qz.html.  Maybe that means
something?  Maybe the eigenvalue output is unstable?  Should Octave drop
support for the eigenvalue output?

Second, for the test case in this bug report, all outputs (aa, bb, q, z, v, w)
agree with Matlab.  This part of the calculation seems correct.

Third, although undocumented, you can ask for a seventh output from Matlab's
qz() function and it appears to be the eigenvalues.  The output is always 3x1,
as expected, because there should be 3 eigenvalues for 3x3 input matrices A
and B.  The fact that Octave returns only 2 eigenvalues is incorrect.  It
should return 3 eigenvalues, as it does with the LAPACK 3.9.1.

Fourth, there seems to be some convergence/roundoff issues in Matlab.  The
first 6 outputs are always the same, but the eigenvalues are sometimes all 0
([0;0;0]) and sometimes something very small (10^-306 x [1; 1; 1]) which is
effectively all zeros.  The two Octave eigenvalues are of normal size ([-4.2,
0.2]) which seems wrong.

Fifth, obviously this is a very hard test case for the QZ algorithm.  The rank
of the matrix B is 2 which means the last eigenvalue is not really determined
and could be anything.




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