[petsc-users] Krylov-Schur Tolerance

[Christopher Pierce]

Thanks,

That makes a lot of sense.  Is there any simple way to query the true
residual norm of the approximate eigenvector in the equivalent single
matrix problem?  IE ||B^{-1}AX-lambda*X||.  I think that would help
determine the reliability of the computed solution.

Chris





On 02/22/17 06:51, Jose E. Roman wrote:
>> El 20 feb 2017, a las 11:33, Jose E. Roman <jroman at dsic.upv.es> escribió:
>>
>>
>>> El 20 feb 2017, a las 11:01, Christopher Pierce <cmpierce at WPI.EDU> escribió:
>>>
>>> It seems to give the same results.  I exported the matrices to Matlab
>>> and checked the estimated condition number of the B matrix which came to
>>> ~15 and the 2-norm of the B matrix which was ~10^6.  I'm guessing that
>>> the large matrix norm is the problem.  I glanced over the source for the
>>> RQCG solver and it doesn't seem to use a linear solver which is likely
>>> why it showed better performance.  Do you have any suggestions for
>>> dealing with problems like this?
>>>
>>> Chris
>> Send the data files to my personal email and I will make some tests.
>> Jose
>>
> The large errors that you report can be explained by the fact that both A and B have large norm, in particular ~10^11 and ~10^5, respectively. Note that the norm of the residual ||A*x-lambda*B*X|| can be made arbitrarily large by increasing the matrix norms. So if you scale the matrices so that e.g. norm(B)=1, you will see a reduction of the reported error in several orders of magnitude.
>
> On the other hand, note that in case of solving the problem as a GHEP the returned eigenvector is not normalized to have ||x||_2=1, but to have ||x||_B=1, so when showing the relative residual error, SLEPc should divide by ||x||_2, which is quite small in your case. This can be considered a bug in SLEPc that we will fix for the next release. Doing this in your case would imply increasing a bit the reported errors (much less than the gain from the first comment).
>
> In conclusion, you can rely on the computed solution even though the reported residual norms are large in this case.
>
> Jose
>