[petsc-users] Krylov-Schur Tolerance

[Jose E. Roman]

> 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