Re: LAPACK eigenvalues of singular generalized eigenproblem

stevenj_at_mit.edu
Date: 06/13/04 

Date: 13 Jun 2004 17:21:47 GMT

>stevenj@mit.edu wrote...
>> Hi, I want to used LAPACK to find the eigenvalues (and eigenvectors) of a
>> generalized eigenproblem Ax = \lambda Bx, e.g. using the ZGGEV or ZGGEVX
>> routines, for the case where A and B are nearly singular. As described
>> in:
>>
>> http://www.netlib.org/lapack/lug/node105.html
>>
>> this is generally an unstable situation. However, in my special case I
>> have reason to believe on physical grounds that the (nearly) null space of
>> A and B is the same, so I should be able to project out the non-singular
>> problem. (A and B are complex-symmetric.)
>
>the routines you plan to use apply the QZ algorithm. That avoids any
>inversion and is not affected by singularities as much. It may not
>give you n eigenvalues because they may not exist (not be finite, for
>example) but it will give you what is possible. I assume your matrices
>are not too large.

You misunderstand me. I agree that LAPACK's eigensolver is not affected
if *one* of the matrices is singular, but if *both* are, for the *same*
eigenvector, then the result is numerically unstable. See the above link
from LAPACK's own manual, if you don't believe me! I have also observed
this in practice.

However, in my special case, I know a-priori on physical grounds that the
null space of A and B should be the same, so I can remove this singularity
to some extent. I have also done this "by hand", as I described, and the
results definitely improved. However, the way I did this seemed kind of
clumsy to me and I thought there might be a better way, especially since
it seemed like this should be a well-studied problem.

I'd appreciate any pointers ore references on singular generalized
eigenproblems (in the sense defined above and by the LAPACK manual) with
"removable" singularities in the sense above.

Cordially,
Steven G. Johnson

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