[petsc-users] Symmetric non-positive definite system

[Matthew Knepley]

On Fri, Jul 18, 2014 at 1:19 PM, Chetan Jhurani <chetan.jhurani at gmail.com>
wrote:

> > From: Jed Brown <jed at jedbrown.org>
> >
> > 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?
>
> This does not answer the original question regarding CG/ML, but
>

It is possible for CG to convergence with an indefinite matrix, but it is
also
possible for it to fail.

  Matt


> will reduce some debugging work.  P^{-1/2} A P^{-T/2} cannot be
> positive definite since A is not positive definite.  This is due
> to Sylvester's inertia theorem.
>
> http://books.google.com/books?id=P3bPAgAAQBAJ&pg=PA202
> Applied Numerical Linear Algebra By James W. Demmel, p 202
>
> Chetan
>
> > > Shouldn't CG fail for a non-positive-definite matrix? Does PETSc do
> some
> > > additional magic if it detects that the dot-product in the CG
> algorithm is
> > > negative? Does it solve the system using the normal equations, A'A,
> instead?
> >
> > CG will report divergence in case a direction of negative curvature is
> > found.
>
>


-- 
What most experimenters take for granted before they begin their
experiments is infinitely more interesting than any results to which their
experiments lead.
-- Norbert Wiener
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