> From: Jed Brown <[email protected]>
>
> 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?
This does not answer the original question regarding CG/ML, but
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.
