Consider solving large sparse symmetric positive semidefinite linear systems. Since the Conjugate Gradient (CG) method may not converge for inconsistent systems, we use the MINRES method. Moreover, we will use right preconditioning for MINRES, since it is easier to preserve the equivalence of the problem, especially for the inconsistent case.

First, we introduce the algorithm for the right preconditioned MINRES. We show that the right preconditioned MINRES converges to a weighted least squares solution without breakdown even if the linear systems is singular. Especially, when the linear system is consistent, we prove that the right preconditioned MINRES converges to a min-norm solution with respect to the metric defined by the preconditioner, if the initial vector is in the range space of the coefficient matrix. We then propose the right preconditioned MINRES using SSOR preconditioning with Eisenstat’s trick. Numerical experiments on semidefinite systems obtained from electromagnetic analysis using the Finite Element Method with edge elements, and some problems from the University of Florida Sparse Matrix Collection indicate that the right preconditoned MINRES using Eisenstat SSOR is more efficient and robust compared to right scaling preconditioned MINRES and MINRES without preconditioning for some problems. This is joint work with Mr. Kota Sugihara and Dr.Ning Zheng.