We study theoretical properties of two inexact Hermitian/skew-Hermitian splitting (IHSS) iteration methods for the large sparse non-Hermitian positive definite system of linear equations. In the inner iteration processes, we employ the conjugate gradient (CG) method to solve the linear systems associated with the Hermitian part, and the Lanczos or conjugate gradient for normal equations (CGNE) method to solve the linear systems associated with the skew-Hermitian part, respectively, resulting in IHSS(CG, Lanczos) and IHSS(CG, CGNE) iteration methods, correspondingly. Theoretical analyses show that both IHSS(CG, Lanczos) and IHSS(CG, CGNE) converge unconditionally to the exact solution of the non-Hermitian positive definite linear system. Moreover, their contraction factors and asymptotic convergence rates are dominantly dependent on the spectrum of the Hermitian part, but are less dependent on the spectrum of the skew-Hermitian part, and are independent of the eigenvectors of the matrices involved. Optimal choices of the inner iteration steps in the IHSS(CG, Lanczos) and IHSS(CG, CGNE) iterations are discussed in detail by * Corresponding author.

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For the positive semidefinite system of linear equations of a block two-by-two structure, by making use of the Hermitian/skew-Hermitian splitting iteration technique we establish a class of preconditioned Hermitian/skew-Hermitian splitting iteration methods. Theoretical analysis shows that the new method converges unconditionally to the unique solution of the linear system. Moreover, the optimal choice of the involved iteration parameter and the corresponding asymptotic convergence rate are computed exactly. Numerical examples further confirm the correctness of the theory and the effectiveness of the method.

For the augmented system of linear equations, Golub, Wu and Yuan recently studied an SOR-like method (BIT 41(2001)71-85). By further accelerating it with another parameter, in this paper we present a generalized SOR (GSOR) method for the augmented linear system. We prove its convergence under suitable restrictions on the iteration parameters, and determine its optimal iteration parameters and the corresponding optimal convergence factor. Theoretical analyses show that the GSOR method has faster asymptotic convergence rate than the SOR-like method. Also numerical results show that the GSOR method is more effective than the SOR-like method when they are applied to solve the augmented linear system. This GSOR method is further generalized to obtain a framework of the relaxed splitting iterative methods for solving both symmetric and nonsymmetric augmented linear systems by using the techniques of vector extrapolation, matrix relaxation and inexact iteration. Besides, we also demonstrate a complete version about the convergence theory of the SOR-like method.

For the large sparse linear complementarity problems, by reformulating them as implicit fixed-point equations based on splittings of the system matrices, we establish a class of modulus-based matrix splitting iteration methods and prove their convergence when the system matrices are positive-definite matrices and H + -matrices. These results naturally present convergence conditions for the symmetric positive-definite matrices and the M-matrices. Numerical results show that the modulus-based relaxation methods are superior to the projected relaxation methods as well as the modified modulus method in computing efficiency.Many problems arising in scientific computing and engineering applications may lead to solutions of LCPs of the form (1). For example, the Nash equilibrium point of a bimatrix game, the contact problem, and the free boundary problem for journal bearings; see [1,2].To compute a numerical solution of the LCP(q, A), we often utilize matrix splittings to construct feasible and efficient splitting iteration methods, especially when the system matrix A is large and sparse. For example, the projected successive overrelaxation (SOR) iteration [3], the general fixed-point iterations [4][5][6][7], and the matrix multisplitting iterations [8][9][10][11][12] are some standard sequential and parallel splitting methods for iteratively solving the LCP(q, A). In these works, most convergence results have been established for the case that the system matrix A is symmetric positive definite, symmetric positive semi-definite, or diagonally dominant. The case that A is nonsymmetric is much more difficult and, as we have known, only a few results are obtainable, especially when some zero entries appear on the diagonal of the matrix A.By reformulating the LCP(q, A) as an implicit fixed-point equation, Murty presented a modulus iteration method in [2]. This method avoids the projections of the iterates used in the projected relaxation iterations and the general fixed-point iterations. Recently, by generalizing this modulus iteration method with the introduction of an iteration parameter, Dong and Jiang proposed a modified modulus iteration method and studied its convergence in [13] when the system matrix A is symmetric positive definite. The modified modulus iteration method inherits all merits of the modulus iteration method. Moreover, the iteration parameter can be used to accelerate the convergence of the iteration sequence. A drawback of this class of methods is that a linear system with the coefficient matrix I + A or I + A, with >0 a given parameter, needs to be solved exactly at each iteration step, which may be more costly and complicated in actual implementations.In this paper, by reformulating the LCP(q, A) as an implicit fixed-point equation based on a splitting of the system matrix A, we establish a class of modulus-based matrix splitting iteration methods for solving large sparse LCPs. With suitable choices of the involved parameter matrices, the new method not only covers the known modulus iteration method ...

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By further generalizing the concept of Hermitian (or normal) and skew-Hermitian splitting for a non-Hermitian and positive-definite matrix, we introduce a new splitting, called positive-definite and skew-Hermitian (PS) splitting, and then establish a class of positivedefinite and skew-Hermitian splitting (PSS) methods similar to the Hermitian (or normal) and skew-Hermitian splitting (HSS or NSS) method for iteratively solving the positive definite systems of linear equations. Theoretical analysis shows that the PSS method converges