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?
> 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.
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