A New Preconditioner For The Solution Of Large Sparse Linear Systems

To further study the Hermitian and skew-Hermitian splitting methods for a non-Hermitian and positive definite matrix, we introduce a so-called Generalized Hermitian and skew-Hermitian splitting with two different parametersαandβ, and then establish a class of Generalized Hermitian and skew-Hermitian splitting (GHSS) methods to solve the non-Hermitian and positive definite systems of linear equations.We theoretically prove that the GHSS converges to the unique solution of the linear system for a loose restriction on the parametersαandβ. Moreover, the contraction factor of the GHSS iteration is derived. The presented numerical examples illustrate the effectiveness of GHSS iteration.We also introduce the construction of the Hermitian and skew-Hermitian splitting preconditioner, and we derive bounds on the eigenvalues of the preconditioned matrix that arises in the solution of saddle point problems when the Hermitian and non-Hermitian splitting preconditioner is employed. A few numerical experiments are used to illustrate the effectiveness of the preconditioned GHSS method.