[petsc-users] eigenvalue problem involving inverse of a matrix

[maitri ksh]

I used 'Matshell' and associated it with a custom-defined matrix-vector
multiplication and used it to solve the eigenvalue problem ((*((LU)^-H)*Q**
*(LU)^-1)***x = lmbda*x,* where LU is the factor of a *matrix **B*). I
compared the eigenvalue results with matlab, however in matlab, I computed
the matrix *A=(B^-H)*Q*B^-1* directly and used eig(A).  Here are the
results:

petsc('*eigVal.png'*): (method: krylovschur)
lmbd1 = 22.937184
lmbd2 = -6.306099
lmbd3 =  2.904980
lmbd4 = 0.026435

Matlab:
lmbd1 = 0.0021
lmbd2 = 0.0840
lmbd3 = 3.9060
lmbd4 = 22.7579

It appears that the iterative procedure that I have adopted (in petsc) is
accurate only for the largest eigenvalue. Is this correct? or is it due to
some error in my code?

Also, I tried using shift-invert-strategy ('*code_snippet_sinvert.png'*) to
see if I can get accurate non-largest eigenvalue, but it throws error ('
*error.png'*) related to '*MatSolverType mumps does not support matrix type
shell*', and it gives the same error message with petsc's native
MATSOLVERSUPERLU. How to resolve this?







On Mon, Aug 14, 2023 at 1:20 PM maitri ksh <maitri.ksh at gmail.com> wrote:

> got it, thanks Pierre & Jose.
>
> On Mon, Aug 14, 2023 at 12:50 PM Jose E. Roman <jroman at dsic.upv.es> wrote:
>
>> See for instance ex3.c and ex9.c
>> https://slepc.upv.es/documentation/current/src/eps/tutorials/index.html
>>
>> Jose
>>
>>
>> > El 14 ago 2023, a las 10:45, Pierre Jolivet <pierre.jolivet at lip6.fr>
>> escribió:
>> >
>> >
>> >
>> >> On 14 Aug 2023, at 10:39 AM, maitri ksh <maitri.ksh at gmail.com> wrote:
>> >>
>> >> 
>> >> Hi,
>> >> I need to solve an eigenvalue problem  Ax=lmbda*x, where
>> A=(B^-H)*Q*B^-1 is a hermitian matrix, 'B^-H' refers to the hermitian of
>> the inverse of the matrix B. Theoretically it would take around 1.8TB to
>> explicitly compute the matrix B^-1 . A feasible way to solve this
>> eigenvalue problem would be to use the LU factors of the B matrix instead.
>> So the problem looks something like this:
>> >>                      (((LU)^-H)*Q*(LU)^-1)*x = lmbda*x
>> >> For a guess value of the (normalised) eigen-vector 'x',
>> >> 1) one would require to solve two linear equations to get 'Ax',
>> >>         (LU)*y=x,             solve for 'y',
>> >>        ((LU)^H)*z=Q*y,   solve for 'z'
>> >>     then one can follow the conventional power-iteration procedure
>> >> 2) update eigenvector: x= z/||z||
>> >> 3) get eigenvalue using the Rayleigh quotient
>> >> 4) go to step-1 and loop through with a conditional break.
>> >>
>> >> Is there any example in petsc that does not require explicit
>> declaration of the matrix 'A' (Ax=lmbda*x) and instead takes a vector 'Ax'
>> as input for an iterative algorithm (like the one above). I looked into
>> some of the examples of eigenvalue problems ( it's highly possible that I
>> might have overlooked, I am new to petsc) but I couldn't find a way to
>> circumvent the explicit declaration of matrix A.
>> >
>> > You could use SLEPc with a MatShell, that’s the very purpose of this
>> MatType.
>> >
>> > Thanks,
>> > Pierre
>> >
>> >> Maitri
>>
>>
-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://lists.mcs.anl.gov/pipermail/petsc-users/attachments/20230815/a585feda/attachment-0001.html>
-------------- next part --------------
A non-text attachment was scrubbed...
Name: eigVal.png
Type: image/png
Size: 11110 bytes
Desc: not available
URL: <http://lists.mcs.anl.gov/pipermail/petsc-users/attachments/20230815/a585feda/attachment-0003.png>
-------------- next part --------------
A non-text attachment was scrubbed...
Name: code_snippet_sinvert.png
Type: image/png
Size: 29508 bytes
Desc: not available
URL: <http://lists.mcs.anl.gov/pipermail/petsc-users/attachments/20230815/a585feda/attachment-0004.png>
-------------- next part --------------
A non-text attachment was scrubbed...
Name: error.png
Type: image/png
Size: 72446 bytes
Desc: not available
URL: <http://lists.mcs.anl.gov/pipermail/petsc-users/attachments/20230815/a585feda/attachment-0005.png>