Dear PETSc,
I have been reading the sources about the GMRES (mostly the FGMRES)
solver, and there are a few things I do not understand.
- Aren't the arrays used to represent the Hessenberg matrix too
wide ? The hh_origin and hes_origin arrays are allocated using (max_k +
2) * (max_k + 1) and (max_k + 1) * (max_k + 1) scalars respectively. But
when I monitor those two matrices I see that the last columns (and
therefore the last line of hh_origin) are not used. Is there a reason
why the hh_origin array is not of size (max_k + 1) * (max_k) ?
- In KSPComputeExtremeSingularValues_GMRES we compute the singular
values of the hh_origin array, which is an upper diagonal (R) matrix
from a QR decomposition of the Hessenberg matrix. Therefore it has the
same singular values as the original Hessenberg matrix. In this
function, we don't forget to "zero the below diagonal garbage". But in
KSPComputeEigenvalues_GMRES we just use hh_origin right away, without
removing the sub-diagonal. I don't understand what's computed here : the
eigenvalues of the upper diagonal representation of the Hessenberg
matrix + the subdiagonal part of the Hessenberg matrix ? Why don't we
use the hes_origin matrix ? Finally, in KSPComputeRitz_GMRES we use
hes_ritz, the last full Hessenberg matrix if available (gmres->fullcycle
> 0), and hh_origin otherwise. But hes_ritz is a copy of the last full
hh_origin, so my question is the same as before :
*TLDR* : why do we use hh_origin to compute the eigenvalues, and not
hes_origin ? Why is hes_ritz a copy of hh_origin and not of hes_origin ?
- I have also noticed that the fullcycle attribute is not used for
FGMRES (it is for GMRES). Is that intentional ? This attribute is only
used in the KSPComputeRitz_GMRES function, does that mean that
KSPComputeRitz should not be called with a FGMRES type ? If so, should
there be a test to check if the KSP type is indeed GMRES and raise a
warning/error if not ?
I hope I'm not bothering too much, I've tried to answer theses questions
by myself but I can't seem to find out.
Thanks in advance,
Pierre Seize
