On Improved PMHSS Iteration Methods For Continuous Sylvester Equations

The preconditioned MHSS(PMHSS),the modified PMHSS(MPMHSS)and the double modified PMHSS(DMPMHSS)iteration method for the complex symmetric linear systems were introduced respectively.In this paper,on the basis of the PMHSS iteration method,we present the MPMHSS and the DMPMHSS iteration method for solving large sparse continuous Sylvester equations with non-Hermitian and positive definite/semi-definite matrices.The spectral radius of the iteration matrix of two methods is derived.By choosing an appropriate parameter,the MPMHSS iteration method converges faster than the PMHSS.Moreover,to reduce the computing cost,we establish an inexact variant of the MPMHSS iteration method and analyze its convergence property in detail.The DMPMHSS iteration method is four-step alternating iterative that is developed upon the two-step alternating iteration method MPMHSS.We discuss the choice of the parameters and prove the convergence of the DMPMHSS iteration method.In particular,we show that DMPMHSS converges faster than PMHSS at the parameter free situation in most cases.Numerical results show that the MPMHSS,the inexact MPMHSS and the DMPMHSS are efficient and robust solvers for this class of continuous Sylvester equations.