Solving Hermitian positive definite systems using indefinite incomplete factorizations

Haim Avron*, Anshul Gupta, Sivan Toledo

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


Incomplete LD* factorizations sometimes produce an indefinite preconditioner even when the input matrix is Hermitian positive definite. The two most popular iterative solvers for symmetric systems, CG and MINRES, cannot use such preconditioners; they require a positive definite preconditioner. One approach, that has been extensively studied to address this problem is to force positive definiteness by modifying the factorization process. We explore a different approach: use the incomplete factorization with a Krylov method that can accept an indefinite 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 indefinite. 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 indefinite preconditioner. Furthermore, we propose another optimal short recurrence method called IP-MINRES that can use an indefinite preconditioner, and a variant of PCG-ODIR, which we call IP-CG, that is more numerically stable and usually requires fewer iterations.

Original languageEnglish
Pages (from-to)126-138
Number of pages13
JournalJournal of Computational and Applied Mathematics
Issue number1
StatePublished - 2013


FundersFunder number
Israel Academy of Sciences and Humanities
Israel Science Foundation


    • Conjugate gradients methods
    • Incomplete factorizations
    • Krylov methods
    • Lanczos method
    • Preconditioning


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