Date: Sat, 6 Dec 2008 08:07:58 -0700
From: Pauli Virtanen
To: "lapack@cs.utk.edu" <lapack@cs.utk.edu>
Cc: "scipy-dev@scipy.org" <scipy-dev@scipy.org>
Subject: [Lapack] Invalid (?) output from DGGEV for some matrices
Dear Lapack authors,
DGGEV returns for some matrices an ALPHAI vector such that ALPHAI(i) = 0
but ALPHAI(i+1) < 0. The DGGEV documentation does not mention such
outputs. Does this imply that the i, i+1 entries describe a complex
eigenvalue pair, or does this indicate a bug in LAPACK or in the
compiler?
The problematic matrix pairs are of the form
[ 1 0 0 0 ] [ 0 0 1 0 ] c1 = omega^2 - 9
[ 0 1 0 0 ] [ 0 0 0 1 ] c2 = 2 omega
[ 0 0 c1 0 ] [ 1 0 0 -c2 ]
[ 0 0 0 c1 ] [ 0 1 c2 0 ]
A test program `2dof.f` illustrating this is attached. However, what it
outputs appears to depend on compiler, compiler version, and
optimization flags:
* LAPACK 3.2, compiled with gfortran 4.3.2-1ubuntu11, OPTS = -O0:
No problematic outputs.
BLAS provided by Atlas 3.6.0.
* LAPACK 3.2, compiled with gfortran 4.3.2-1ubuntu11, OPTS = -O1 (2/3):
Missing (?) complex pair for omega = 1.5656565656565657
Info: 0
I / ALPHAR(I) / ALPHAI(I) / BETA(I):
1 0.0000000000000000 1.7080358288156292 0.37410519259457370
2 0.0000000000000000 -1.7080358288156292 0.37410519259457370
3 0.0000000000000000 0.0000000000000000 0.0000000000000000
4 0.0000000000000000 -2.62037428091521910E-016 1.82688066063807497E-016
Missing (?) complex pair for omega = 1.9191919191919191
Info: 0
I / ALPHAR(I) / ALPHAI(I) / BETA(I):
1 0.0000000000000000 0.0000000000000000 0.0000000000000000
2 0.0000000000000000 -1.46001785734970358E-017 2.96800344717906839E-018
3 0.0000000000000000 3.6846073726768025 3.4091227092990981
4 0.0000000000000000 -3.6846073726768025 3.4091227092990981
BLAS provided by Atlas 3.6.0.
* LAPACK 3.2, compiled with gfortran 4.1.2 20061115, OPTS = -O0:
Missing (?) complex pair for omega = 1.86868686868687
BLAS provided by Atlas 3.6.0; same with Netlib BLAS.
* LAPACK 3.1.1, compiled with g77 3.4.6, OPTS = -O0/O1/O2/O3:
Missing (?) complex pair for omega = 1.86868687
BLAS provided by Atlas 3.6.0; same with Netlib BLAS.
What appears to matter for gfortran 4.3.2 is which optimization flags
were used to compile the file `dhgeqz.f`; recompiling this file can make
the problem appear (-O1) and disappear (-O0).
Output from LAPACK double eigenvalue tests, `dgd-O0/O1.out` is also
attached. It corresponds to gfortran 4.3.2-1ubuntu11, for -O0 and -O1.
There are some failures, but they are the same for both flags.
This issue is present at least in LAPACK libraries shipped by Ubuntu and
Debian. It may be that also other Linux distributions are affected.
***
We ran into this problem while wrapping DGGEV for use in the
higher-level language Python: the ambiguity in the return value caused
an error in repacking the complex eigenvectors in our code. [1] (Cc'd
to the developer list.)
If this is a LAPACK or compiler bug, suggestions on how the DGGEV return
values should be interpreted are appreciated: should we treat this as an
error condition, or should we attempt to treat the result as a complex
eigenvalue pair?
Best regards,
Pauli Virtanen
.. [1] http://scipy.org/scipy/scipy/ticket/709