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Abstract In this paper, by adopting the preconditioned technique for the accelerated generalized successive overrelaxation method (AGSOR) proposed by Edalatpour et al. (2015), we establish the preconditioned AGSOR (PAGSOR) iteration method for solving a class of complex symmetric linear systems. The convergence conditions, optimal iteration parameters and corresponding optimal convergence factor of the PAGSOR iteration method are determined. Besides, we prove that the spectral radius of the PAGSOR iteration method is smaller than that of the AGSOR one under proper restrictions, and its optimal convergence factor is smaller than that of the preconditioned symmetric block triangular splitting (PSBTS) one put forward by Zhang et al. (2018) recently. The spectral properties of the preconditioned PAGSOR matrix are also proposed. Numerical experiments illustrate the correctness of the theories and the effectiveness of the proposed iteration method and the preconditioner for the generalized minimal residual (GMRES) method. [ABSTRACT FROM AUTHOR]

Abstract By utilizing the successive-overrelaxation (SOR) acceleration technique for the generalized version of the double-step scale (DSS) iteration method, we construct the generalized DSS-SOR (GDSSOR) iteration method for solving a class of complex symmetric linear systems. The convergence theory of the GDSSOR iteration method is established and its optimal parameters are investigated. Meanwhile, a practical way to choose iteration parameters for the GDSSOR iteration method is developed. Inexact version of the GDSSOR iteration (IGDSSOR) method and its convergence properties are also presented. Numerical experiments illustrate that both GDSSOR and IGDSSOR iteration methods are feasible and effective for solving the complex symmetric linear systems, and perform better than some other commonly used iteration methods. [ABSTRACT FROM AUTHOR]

Abstract This paper is devoted to deriving some new upper and lower bounds for the minimum H -eigenvalue of irreducible nonsingular M -tensors. We demonstrate that the proposed bounds improve some existing ones and numerical examples are given to illustrate this fact. Additionally, we extend these bounds to general nonsingular M -tensors. The other main results in this paper is to provide sharper Ky Fan theorem and comparison theorem for nonsingular M -tensors which are better than the corresponding ones obtained by He and Huang (2014) [12]. [ABSTRACT FROM AUTHOR]

In this paper, a new two-step iterative method called the two-step parameterized (TSP) iteration method for a class of complex symmetric linear systems is developed. We investigate its convergence conditions and derive the quasi-optimal parameters which minimize the upper bound of the spectral radius of the iteration matrix of the TSP iteration method. Meanwhile, some more practical ways to choose iteration parameters for the TSP iteration method are proposed. Furthermore, comparisons of the TSP iteration method with some existing ones are given, which show that the upper bound of the spectral radius of the TSP iteration method is smaller than those of the modified Hermitian and skew-Hermitian splitting (MHSS), the preconditioned MHSS (PMHSS), the combination method of real part and imaginary part (CRI) and the parameterized variant of the fixed-point iteration adding the asymmetric error (PFPAE) iteration methods proposed recently. Inexact version of the TSP iteration (ITSP) method and its convergence properties are also presented. Numerical experiments demonstrate that both TSP and ITSP are effective and robust when they are used either as linear solvers or as matrix splitting preconditioners for the Krylov subspace iteration methods and they have comparable advantages over some known ones for the complex symmetric linear systems. [ABSTRACT FROM AUTHOR]

To solve nonsymmetric saddle point problems, the parameterized generalized shift-splitting (PGSS) preconditioner is presented and analyzed. The corresponding PGSS iteration method can be applied not only to the nonsingular saddle point problems but also to the singular ones. The convergence and semi-convergence of the PGSS iteration method are discussed carefully. Meanwhile, the spectral properties of the preconditioned matrix and the strategy of the choices of the parameters are given. Numerical experiments further demonstrate that the PGSS iteration method and the PGSS preconditioner are efficient and have better performance than some existing iteration methods and newly proposed preconditioners, respectively, for solving both the nonsingular and singular nonsymmetric saddle point problems. [ABSTRACT FROM AUTHOR]

For a nonsymmetric saddle point problem, the modified shift-splitting (MSS) preconditioner has been proposed by Zhou et al. By replacing the parameter α in (2,2)-block in the MSS preconditioner by another parameter β , a generalized MSS (GMSS) preconditioner is established in this paper, which results in a fixed point iteration called the GMSS iteration method. We provide the convergent and semi-convergent analysis of the GMSS iteration method, which show that this method is convergence and semi-convergence if the related parameters satisfy suitable restrictions. Meanwhile, the distribution of eigenvalues and the forms of the eigenvectors of the preconditioned matrix are analyzed in detail. Finally, numerical examples show that the GMSS method is more feasibility and robustness than the MSS, Uzawa-HSS and PU-STS methods as a solver, and the GMSS preconditioner outperforms the GSOR, Uzawa-HSS, MSS and LMSS preconditioners for the GMRES method for solving both the nonsingular and the singular saddle point problems with nonsymmetric positive definite and symmetric dominant (1,1) parts. [ABSTRACT FROM AUTHOR]

Based on the parameterized preconditioned Hermitian and skew-Hermitian splitting iteration method (PPHSS), proposed by Li et al. (2014) for saddle point problems, an improvement on the PPHSS method (IPPHSS) is presented in this paper. By adding a block lower triangular matrix to the coefficient matrix on two sides of the first equation of the PPHSS iterative scheme, both the number of iterations and the consume time are decreased. We provide the convergence and semi-convergence analysis of the IPPHSS method, which show that this method is convergence and semi-convergence if the related parameters satisfy suitable restrictions. Furthermore, we discuss the spectral properties of the corresponding preconditioned matrix of the IPPHSS method. Finally, numerical examples show that the IPPHSS method is better than the PPHSS, PHSS and AHSS methods both as a solver and as a preconditioner for the GMRES method. [ABSTRACT FROM AUTHOR]

In this paper, we present some new preconditioned GAOR methods for solving generalized least-squares problems and their comparison results. Comparison results show that the convergence rates of the new preconditioned GAOR methods are better than those of the preconditioned GAOR methods presented by Zhou et al. (2009) [19] and Wang et al. (2013) [18] whenever these methods are convergent. Lastly, numerical experiments are provided to confirm the theoretical results. [ABSTRACT FROM AUTHOR]

For the generalized saddle point problems from the Stokes equations, we propose an efficient preconditioned variant of the positive semidefinite and skew-Hermitian (PVPSS) preconditioner. The new preconditioner is established by adopting matrix preconditioning strategy and relaxation technique for the PSS one derived by Pan et al. (2006). Compared with the PSS one, the PVPSS preconditioner is much closer to the coefficient matrix and easier to be implemented if proper preconditioning matrices are adopted. We prove the convergence of the PVPSS iteration method under some restrictions and discuss the spectral properties of the PVPSS preconditioned matrix. Meanwhile, the implementation and a practical way to choose the parameter of the PVPSS preconditioner are discussed. Comparisons between the PVPSS preconditioner and some existing ones are also given. Numerical experiments are carried out to illustrate that the proposed preconditioner is effective for the generalized saddle point problems and outperforms several other commonly used ones. [ABSTRACT FROM AUTHOR]