[petsc-dev] Symmetry acceleration of the Jacobi-Davidson method (in SLEPc)

Krzysztof Gawarecki dkg2140 at gmail.com
Sat Feb 14 07:51:12 CST 2015

```Thank you for a fast response and very interesting idea. The matrix A is
Hermitian.
However I forgot, that T contains not only the matrix but also the operator
of complex conjugation:

T = M K, where K is a operator of complex conjugation and M has a form

( 0   0   0   0  -1   0   0   0 )
( 0   0   0   0   0    .   0   0 )
( 0   0   0   0   0   0    .   0 )
( 0   0   0   0   0   0   0  -1 )
M =  ( 1   0   0   0   0   0   0   0 )
( 0    .   0   0   0   0   0   0 )
( 0   0    .   0   0   0   0   0 )
( 0   0   0   1   0   0   0   0 )

So T b1 = M (b1)* = a1.

Is there still possibility to separate eigenvectors by S^{-1} A S    like
transformation?

2015-02-13 17:29 GMT+01:00 Tobin Isaac <tisaac at ices.utexas.edu>:

>
> On Fri, Feb 13, 2015 at 03:06:38PM +0100, Krzysztof Gawarecki wrote:
> > Dear All,
> >
> > I'm calculating eigenvalues and eigenvectors of the matrix which has
> > specific kind of symmetry.
> > Due to this symmetry I obtain the eigenvalues which are doubly
> degenerated.
> > So eg. eigeinvalue 'e1' has eigenvectors 'a1' and 'b1'. These
> eigenvectors
> > are related to each other by the relation a1 = T b1, where T is a matrix
> > (given for my problem).
> > So it is enough to calculate only one eigenvector for each eigenvalue
> (and
> > the second one can be calculated by matvec operation). This situation has
> > been described in http://dl.acm.org/citation.cfm?id=2494747.
> >
> > How could I take advantage on this in EPSSolve in Jacobi-Davidson method?
> > Could I add two vectors to the subspace (the second one would be
> calculated
> > by multiplying the first one by matrix T) in every iteration? Should I
> > modify function "dvd_updateV_update_gen" in dvd_updatev.c ?
> >
> > I would be very grateful for any suggestion.
>
> In that paper you're looking at a Hermitian operator, right?  In this
> case, can't you use the symmetry to make the problem smaller? If you
> run Jacobi-Davidson for the operator SAS, where S=0.5(I+T), you'll get
> zero eigenvalues for the anti-symmetric vectors, but the other
> eigenvalues should have multiplicity 1 and can be used to reconstruct
> what you want.
>
>   Toby
>
> >
> > Krzysztof
>
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