SOLVING HERMITIAN POSITIVE DEFINITE SYSTEMS USING INDEFINITE INCOMPLETE FACTORIZATIONS

Abstract. Incomplete LDL ∗ factorizations sometimes produce an inde nite preconditioner even when the input matrix is Hermitian positive de nite. The two most popular iterative solvers for symmetric systems, CG and MINRES, cannot use such preconditioners; they require a positive de nite preconditioner. One approach, that has been extensively studied to address this problem is to force positive de niteness by modifying the factorization process. We explore a di erent approach: use the incomplete factorization with a Krylov method that can accept an inde nite preconditioner. The conventional wisdom has been that long recurrence methods (like GMRES), or alternatively non-optimal short recurrence methods (like symmetric QMR and BiCGStab) must be used if the preconditioner is inde nite. We explore the performance of these methods when used with an incomplete factorization, but also explore a less known Krylov method called PCG-ODIR that is both optimal and uses a short recurrence and can use an inde nite preconditioner. Furthermore, we propose another optimal short recurrence method called IP-MINRES that can use an inde nite preconditioner, and a variant of PCG-ODIR, which we call IP-CG, that is more numerically stable and usually requires fewer iterations. 1