On Fri, Jul 18, 2014 at 1:19 PM, Chetan Jhurani <[email protected]> wrote:
> > 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 > 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
