Two Minimal Residual NHSS Iteration Methods for Complex Symmetric Linear Systems

Abstract

For the large sparse complex symmetric linear systems, by applying the minimal residual technique to accelerate a preconditioned variant of new Hermitian and skew-Hermitian splitting (${\rm P}^∗{\rm NHSS}$) method and efficient parameterized ${\rm P}^∗{\rm NHSS}$ $({\rm PPNHSS})$ method, we construct the minimal residual ${\rm P}^∗{\rm NHSS}$ $({\rm MRP}^∗{\rm NHSS})$ method and the minimal residual ${\rm PPNHSS}$ $({\rm MRPPNHSS})$ method. The convergence properties of the two iteration methods are studied. Theoretical analyses imply that the ${\rm MRP}^∗{\rm NHSS}$ method and the ${\rm MRPPNHSS}$ method converge unconditionally to the unique solution. In addition, we also give the inexact versions of ${\rm MRP}^∗{\rm NHSS}$ method and ${\rm MRPPNHSS}$ method and their convergence proofs. Finally, numerical experiments show the high efficiency and robustness of our methods.