[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:

> 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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