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

Jose E. Roman jroman at dsic.upv.es
Thu Dec 5 01:54:17 CST 2019

```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
>
>

```