[Trilinos-Users] PyTrilinos + Anasazi - Generalized eigenproblem

[Chris Baker]

Hello Marko,

I am currently looking into the reason that Anasazi+PyTrilinos becomes
unresponsive when using Auxilliary vectors.

Am I right to assume that one use of AuxVecs would be to used for removing
> "known" eigenvectors from the process? In my case, I will often have
> row-stochastic matrices. Most of the times, they will have the largest
> eigenvalue at 1 with eigenspace being the vector filled with ones (the
> diagonal of the vector space). Would passing a vector filled with ones ( via
> say PutScalar(1.0) ) to AuxVecs (and choosing "LM" as the sorting method)
> make the code return the second largest evalue/evector as the result
> (assuming NEV = 1)?


Yes.


> In any case, I have tried working a bit more with the AuxVecs to see why
> the code was not responsive before. It seems that AuxVecs vectors have to be
> of length 1


Also correct. In fact, for multiple auxiliary vectors, they must form an
orthonormal basis.

In this case however, this is the last thing the code prints to the screen
> (even with all verbosity options turned on) but it still continues to run at
> 100% CPU. I waited for some time but it never exited or printed other
> messages so I ended up killing the process.


Yeah, I'm currently investigating this. We don't see this problem with
Anasazi in general, so I am leaning toward some Python issue.


> When I pass the unit length vector to AuxVecs, the code runs but fails (and
> exits to command prompt) with
>
> Anasazi::SVQBOrthoManager::findBasis(): Orthogonalization constraints not
> feasible


This is because the auxiliary vectors add another constraint on the
orthogonalization routine. Your 5x5 example just doesn't have enough room to
move around: 1 vector for the auxiliary vector plus 3 vectors minimum for
BKS plus 1 additional vector because the problem is non-Hermitian (this
gives us extra room to keep from splitting a conjugate pair).

Having said that, I don't believe this is the cause for the problem with
setAuxVecs(). I ran on a larger matrix and still saw the freezing.

To summarize, my main question is the one stated at the beginning, about the
> use ot AuxVecs, the rest of the e-mail was more of a followup to your
> answer. Of course, I am still interested in an answer to my previous
> question about why the generalized eigenproblem acts as the regular
> eigenproblem in PyTrilinos (explained in my previous e-mails).


Solving generalized eigenvalue problems with BlockKrylovSchur requires a
little more work. I neglected to tell you this in my previous email, because
my brain doesn't work properly on the weekend. In short, the generalized
eigenvalue problem must be transformed into a standard eigenvalue problem to
work with BKS. (*This is not the case with the other Anasazi eigensolvers,
but they don't solve non-Hermitian problems, so they are not relevant to
you.*)

In your particular case (searching for the largest magnitude eigenvalues of
(W,Dsum)), construct a linear operator
Op = inv(Dsum) W
This transforms the eigenvalue problem to
inv(Dsum) W v = v lambda
The eigenvectors and eigenvalues are unchanged from the generalized
eigenvalue problem.

Pass Op to the BasicEigenproblem class, either by setOp() or setA(). If a
matrix is passed via setM(), it is used for the inner product defining
orthogonality. The benefit of doing that is to restore symmetry in Op and to
handle certain special cases. In your case, W was not symmetric, so that
symmetry in inv(Dsum) W cannot be easily restored. I would not pass anything
to setM().

However, in your case, because Dsum is a diagonal matrix, applying the
operator inv(Dsum) W serves to implicitly scale the entries of W on every
application of the operator. In this case, it is probably better to simply
scale them once in situ, as you were doing before, and consider simply the
standard eigenvalue problem.

I hope this helps. Even more, I hope this is correct; after all, it is only
Monday.

I'm still looking at the setAuxVecs() problem with Anasazi+PyTrilinos. I
will respond to you and the list when it is diagnosed.

Chris
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