New Krylov-subspace Solvers for Hermitian Positive Definite Matrices with Indefinite Preconditioners

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 Hermitian systems, MINRES and CG, cannot use such preconditioners; they require a positive de nite preconditioner. We present two new Krylov-subspace solvers, a variant of MINRES and a variant of CG, both of which can be preconditioned using any non-singular Hermitian matrix as long as the original system is positive de nite. These algorithms allow the use of incomplete-factorization preconditioners for Hermitian positive de nite systems, even when the preconditioner is inde nite, without resorting to a more expensive non-symmetric iterative Krylov-subspace solver.