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

[maitri ksh]

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.

Maitri
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