On solving sparse symmetric linear systems whose definiteness is unknown

Solving a large, sparse, symmetric linear system Ax=b iteratively must use appropriate methods. The conjugate gradient (CG) method can break down if A is indefinite while algorithms such as SYMMLQ and MINRES, though stable for indefinite systems, are computationally more expensive than CG when applied to positive definite matrices. In this paper, we present an iterative method for the case where the definiteness of A is not known a priori. We demonstrate that this method reduces to the CG method when applied to positive definite systems and is numerically stable when applied to indefinite systems.