[petsc-users] Symmetric non-positive definite system solved with CG+ML - why does it not fail?

[Jozsef Bakosi]

On 07.17.2014 16:41, Jed Brown wrote:
> Jozsef Bakosi <jbakosi at lanl.gov> 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?