[petsc-users] Correct Eigenvalue but Large Error

[Habib]

Hi Everyone,I hope this is the right medium to post my question as it is related to slepc4py.I have just installed slepc4py with complex scaler and I intend to use it to solve large sparse generalised eigenvalue problems that result from the stability of flows that can be described by linearised Navier-Stokes equations. For my problem, Ax = kBx, matrices A and B are non-Hermitian, B, in particular, is singular and I am interested in eigenvalue with maximum real part which could be positive if the flow is unstable or negative for a stable flow. Following the steps described in exp1.py, I wrote a code for Non-Hermitian Generalised Eigenvalue problem. I test the code on a problem that is stable and with known solution using Shift and Invert transformation with complex shift value and the solution seems okay but I have the following concerns that I would be grateful if you could clarify on.
(1) If I used a shift value that is close to the solution (which I already know for this test problem), the code converge at the correct eigenvalue but the computed relative error is very large. I was wondering why the error is large and the implication on my eigenvectors. How can this be corrected? Below is a typical command line option I entered and the results I got from solution:
python ComputeEigenValueUsingSlepc4Python.py -eps_nev 1 -eps_tol 1e-10 -st_type sinvert -eps_target  -105.01+252.02i -eps_converged_reason -eps_conv_abs -st_pc_factor_shift_type NONZERO -st_pc_factor_shift_amount 1e-10.
***  SLEPc Solution Results  ***
 
Number of iterations of the method: 1
Solution method: krylovschur
Number of requested eigenvalues: 1
Stopping condition: tol=1e-10, maxit=23923
Number of converged eigenpairs 1
 
        k          ||Ax-kx||/||kx|| 
----------------- ------------------
 -106.296653+251.756509 j  3.64763e+11
(2) Since my goal is to use the code to test the stability of flows in which I have no idea of what the maximum eigenvalue would be, I tried solving the problem I stated in 1 with different target values that are not close to the eigenvalue, but it was not successful. The code kept running for a very long time that I had to cancel it. With a target value of zero, I had convergence in finite time but to a spurious-like eigenvalue. Any suggestion on how I could go about getting the true eigenvalue for my main problem for which I have no idea of what a close target value to the maximum value would be and considering that matrix B is singular?
Looking forward to your suggestions.
Regards,
Habib
Department of Chemical Engineering,
Ahmadu Bello University, Zaria
Samaru Campus, 
Kaduna State, Nigeria
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