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In this paper, we further investigate the single proper nonnegative splittings and double proper nonnegative splittings of rectangular matrices. Two convergence theorems for the single proper nonnegative splitting of a semimonotone matrix are derived, and more comparison results for the spectral radii of matrices arising from the single proper nonnegative splittings and double proper nonnegative splittings of different rectangular matrices are presented. The obtained results generalize the previous ones, and it can be regarded as the useful supplement of the results in [Comput. Math. Appl., 67: 136–144, 2014] and [Results. Math., 71: 93–109, 2017].

Based on the relaxed alternating positive semi-definite splitting (RAPSS) preconditioner, in this paper, a new preconditioner, called variant of relaxed alternating positive semi-definite splitting (VRAPSS) preconditioner, is presented and discussed. Spectral properties of the VRAPSS preconditioned matrix are analyzed in detail. Numerical experiments are provided to verify the efficiency of the VRAPSS preconditioner.

The comparison results for K-double splittings of one K-monotone matrix are established in the literatures. As comparison theorems between the spectral radii of different matrices are a useful tool for judging the efficiency of preconditioners, we propose some comparison results for K-nonnegative double splittings of different K-monotone matrices in this note. The obtained results generalize the previous ones.

Picard-type methods are efficient methods for solving the absolute value equation $ Ax-|x| = b $. To further improve the performance of Picard iteration method, a new inexact Picard iteration method, named Picard-SHSS iteration method, is proposed to solve the absolute value equation. The sufficient condition for the convergence of the proposed method for solving the absolute value equation is given. A numerical example is given to demonstrate the effectiveness of the new method.

In this paper, new preconditioned GAOR methods are proposed for solving a class of $ 2 \times 2 $ block structure linear systems arising from the weighted linear least squares problems. Comparison theorems are derived. Comparison results show that the convergence rates of the new preconditioned GAOR methods are better than those of the preconditioned GAOR methods in the previous literatures whenever these methods are convergent. A numerical example is given to confirm our theoretical results.

In this paper, a new preconditioner for a class of $$2\times 2$$ block linear systems is proposed. The proposed new preconditioner is a better approximation to the original coefficient matrix than the previous ones. The eigenvalue distribution and an upper bound of the degree of the minimal polynomial of the preconditioned matrix are discussed. Finally, two numerical examples are provided to show the effectiveness of the proposed preconditioner.

In this paper, we investigate structured backward errors for two kinds of generalized saddle point systems. The explicit formulas of structured backward errors are derived, which are useful for tes...

Based on the single-step iteration (SSI) method for the non-Hermitian positive definite linear systems, we propose a Uzawa-SSI method for solving the saddle point problems with non-Hermitian positive definite (1, 1) block in this paper. The convergence and semi-convergence properties of the Uzawa-SSI method, respectively, for nonsingular and singular cases are analyzed. Numerical examples with experiments are given to show the robustness and the efficiency of Uzawa-SSI method.

For nonsymmetric saddle point problems, the fast shift-splitting method and its generalized form are studied recently. In this paper, a general fast shift-splitting iteration method is proposed for solving nonsymmetric saddle point problems. The convergence and the semi-convergence of the proposed iteration method, respectively, for nonsingular and singular nonsymmetric saddle point problems are studied, respectively. Meanwhile, the spectral properties of the corresponding preconditioned matrix are analyzed in details. Numerical examples with experiments are given to show the robustness and the efficiency of the proposed method.

In this paper, based on the simplified Hermitian and skew-Hermitian (SHSS) preconditioner, a new preconditioner, called the generalized SHSS (GSHSS) preconditioner, is considered to solve the generalized saddle point problem. We prove that the GSHSS iteration method is convergent if the iteration parameters satisfy appropriate conditions. In addition, it is proved that all eigenvalues of the GSHSS preconditioned matrix are real and non-unit eigenvalues are located in a positive interval. Numerical experiment is provided to show the effectiveness of the GSHSS preconditioner.

We study a preconditioned generalized shift-splitting iteration method for solving saddle point problems. The unconditional convergence theory of the preconditioned generalized shift-splitting iteration method is established. When the splitting matrix is used as a preconditioner, we analyze eigenvalue distribution of the preconditioned saddle point matrix. It is proved that complex eigenvalues having nonzero imaginary parts of the preconditioned matrix are located in an intersection of two circles and the real parts of all eigenvalues of the preconditioned matrix are located in a positive interval. Numerical experiments are used to verify our theoretical results and illustrate effectiveness of the proposed iteration method and the corresponding splitting preconditioner.

Recently, a class of generalized shift-splitting iteration methods were proposed for solving nonsingular nonsymmetric saddle point problems (Cao et al., 2015). In this paper, the generalized shift-splitting iteration method is extended to solve singular nonsymmetric saddle point problems. Semi-convergence of the generalized shift-splitting iteration method is carefully analyzed, which shows that the iterative sequence generated by the generalized shift-splitting iteration method converges to a solution of the singular saddle point problem unconditionally. Numerical results verify the robustness and efficiency of the generalized shift-splitting iteration method.