Jozsef Bakosi <[email protected]> writes: > On 07.17.2014 16:41, Jed Brown wrote: >> Jozsef Bakosi <[email protected]> writes: >> >> > Hi folks, >> > >> > I have a matrix as a result of a finite element discretization of the >> > Poisson >> > operator and associated right hand side. As it turns out, the matrix is >> > symmetric but not positive definite since it has at least two negative >> > small >> > eigenvalues. I have been solving this system without problem using the >> > conjugate >> > gradients (CG) algorithm with ML as a preconditioner, but I'm wondering >> > why it >> > works. >> >> Is the preconditioned matrix >> >> P^{-1/2} A P^{-T/2} >> >> positive definite? > > How would you extract P?
You don't, you apply P^{-1}. Have you checked whether your solution is
accurate? If you want to study the operator further, you can consider
the generalized eigenvalue problem
A x = \lambda P x
Note that eigensolvers can work on this system while only being able to
apply A and P^{-1}. You can find any negative eigenvalues of that
problem, get the eigenvectors, and see how it compares to the
unpreconditioned case.
pgptct0hH3YNo.pgp
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