# [petsc-users] Eigensolution with matrices from mixed formulation problems

Jose E. Roman jroman at dsic.upv.es
Thu Dec 5 08:31:03 CST 2019

> El 5 dic 2019, a las 15:12, Manav Bhatia <bhatiamanav at gmail.com> escribió:
>
> Thanks!
>
> Does the purify option only changes the eigenvector without influencing the eigenvalue?

Yes.

>
> -Manav
>
>> On Dec 5, 2019, at 1:54 AM, Jose E. Roman <jroman at dsic.upv.es> wrote:
>>
>> In symmetric problems (GHEP) you might have large residuals due to B being singular. The explanation is the following. By default, in SLEPc GHEPs are solved via a B-Lanczos recurrence, that is, using the inner product induced by B. If B is singular this is not a true inner product and numerical problems may arise, in particular, the computed eigenvectors may be corrupted with components in the null space of B. This is easily solved by a small computation called 'eigenvector purification' which is on by default in SLEPc, see EPSSetPurify(). This is described with a bit more detail in section 3.4.4 of the manual.
>>
>> In non-symmetric problems (GNHEP) you should not see this problem. The only precaution is not to solve systems with matrix B, e.g., using shift-and-invert.
>>
>> Jose
>>
>>
>>> El 5 dic 2019, a las 5:04, Manav Bhatia <bhatiamanav at gmail.com> escribió:
>>>
>>> Hi,
>>>
>>> I am working on mixed form finite element discretization which leads to eigenvalues of the form
>>>
>>> A x = lambda B x
>>>
>>> With the matrices defined in a block structure as
>>>
>>> A =    [ K  D^T ]
>>>         [ D  0     ]
>>>
>>> B =   [ M  0 ]
>>>        [ 0   0 ]
>>>
>>> The second row of equations come from Lagrange multipliers in our discretization scheme. A system with m Lagrange multiplier is expected to have m Inf eigenvalues. We are testing the standard eigensolvers in Matlab and as the system size increases the eigensolves are stopping with larger residuals, || r_i ||, of the eigensystem:
>>>
>>> r_i = A x_i - lambda_i B x_i
>>>
>>> I am working towards setting this up in SLEPc. In the meantime I am curious about the following:
>>>
>>> 1.  Is the eigensolution of such systems known to be problematic?
>>> 2.  Are there standard tricks in SLEPc or elsewhere that are geared towards more robust solutions of such systems?
>>>
>>>  I would appreciate guidance on this.
>>>
>>> Regards,
>>> Manav
>>>
>>>
>>
>