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

[Jed Brown]

Jozsef Bakosi <jbakosi at lanl.gov> writes:

> 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?

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.
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