Shi-Liang Wu

• Anyang Normal University

To our knowledge, the error and perturbation bounds of the general absolute value equations (AVE) are not discussed. In order to fill in this study gap, in this paper, by introducing a class of absolute value functions, we study the error and perturbation bounds of these two AVEs: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasys...

In this paper, an application of the generalized absolute value equation is exploited. Concretely, by using the generalized absolute value equation, we provide a new approach to efficiently obtain the hull of the solution set to interval linear equations with certain inverse positive coefficient interval matrix. Compared with some existed classical...

This paper focuses on analyzing the boundness of the solution set for the vertical tensor complementarity problem (VTCP). We introduce two novel quantities related to tensor sets, establish the interconnections between these introduced quantities, and present an equivalent condition regarding a \documentclass[12pt]{minimal} \usepackage{amsmath} \us...

In this paper, by using a general equivalent form of the minimum function, some new error bounds of the extended vertical linear complementarity problem under appropriate conditions are obtained, which cover some existing results. Not only that, these new error bounds skillfully avoid the inconvenience caused by the row rearrangement technique for...

In this paper, we introduce a modulus-based formulation for solving vertical tensor complementarity problems (VTCP) with an arbitrary number of tensors. This formulation allows us to design the modulus-based tensor splitting iterative method to fit different number of tensors. In this context, we especially analyze the modulus-based tensor splittin...

In this paper, we consider the numerical solution of a class of vertical tensor complementarity problems. By reformulating the involved vertical tensor complementarity problem (VTCP) as an equivalent projected fixed point equation, together with the relevant properties of the power Lipschitz tensor, we propose a projected fixed point method for the...

In this paper, we further study the projected-type method for the extended vertical linear complementarity problem. By making use of some basic absolute value inequalities, some new convergence properties of the projected-type method are obtained. Compared with the existing results in the literature, the convergence range of the projected-type meth...

In this paper, we introduce some constants with the tensors of special structures and present their some useful properties. Furthermore, some perturbation bounds of the tensor complementarity problem are obtained on the base of these constants.

A published counterexample motivates us to correct one of our theorems in the previously published work in Wu and Guo (J Optim Theory Appl 169:705–712, 2016).

In this paper, we introduce the extended vertical tensor complementarity problem and investigate the existence and uniqueness of its solution. We introduce several sets of special tensors and demonstrate their properties. We leverage the structure of tensors to obtain some properties of the solution of the extended vertical tensor complementarity p...

As a natural extension of the tensor complementarity problem, the vertical tensor complementarity problem VTCP\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( {\t...

In this paper, the vertical tensor complementarity problem (VTCP) is reformulated as the projected equation. After then, a projected splitting iterative method for the VTCP is proposed and corresponding monotone convergence analysis on the projected splitting method for the VTCP associated with Z-tensors is investigated under the condition that the...

In this paper, we go on studying the convergence property of the new modulus-based matrix splitting (NMMS) method for linear complementarity problem of H+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsid...

In this paper, we consider the global uniqueness and solvability (GUS) of tensor complementarity problems for special power Lipschitz tensors (SPL-tensors). It is shown that a SPL-tensor is a P-tensor, but not necessarily an H-tensor. And it is also proved that tensor complementarity problems of SPL-tensors have the GUS-property. In addition, we pr...

In this paper, by introducing a class of absolute value functions, we study the error bounds and perturbation bounds of two types of absolute value equations (AVEs): Ax -B|x|= b and Ax -|Bx|= b. Some useful error bounds and perturbation bounds for the above two types of absolute value equations are presented. By applying the absolute value equation...

In this paper, based on the shift splitting of the coefficient matrix, a generalized two-sweep shift splitting (GTSS) method is introduced to solve the non-Hermitian positive definite linear systems. Theoretical analysis shows that the GTSS method is convergent to the unique solution of the linear systems under a loose restriction on the iteration...

In this paper, by transforming the vertical linear complementarity problem (VLCP) as a certain absolute value equation, we design a class of modulus-based matrix splitting iteration methods for solving the VLCP. The convergence properties of the proposed methods are discussed in depth. By making use of some numerical experiments, we confirm the eff...

In this paper, combining the Newton method with matrix splitting technique, a class of Newton-based matrix splitting iteration methods is presented to solve the weakly nonlinear system with some special matrices. Theoretical analysis shows that this kind of iteration method for some special matrices is convergent under suitable conditions. Numerica...

In the paper, we are concerned with the existence and uniqueness of solution for tensor complementarity problem (TCP) and tensor absolute value equations (TAVEs) with special structure. First, we give a sufficient condition of P tensors by using the nonsingularity of the relevant tensors and the properties of eigenvalues. In addition, we give the c...

In this paper, to find the numerical solution of the linear systems with the non-Hermitian positive definite matrix, a preconditioned two-sweep shift splitting (PTSS) iteration method is introduced and its convergent conditions are discussed. When the PTSS method is used as a preconditioner, the eigenvalue distribution of the corresponding precondi...

In this paper, based on the previous published work by Wang et al. [Modified Newton-type iteration methods for generalized absolute value equations, Wang et al. (2019), by using the matrix splitting technique, Newton-based matrix splitting iterative method is established to solve the generalized absolute value equation. The proposed method not only...

In this paper, a modified LPMHSS (MLPMHSS) method is proposed to solve the problem of a class of complex symmetric linear systems with strong Hermitian parts. Theoretical analysis shows that the MLPMHSS method can converge to the unique solution of linear equations under appropriate conditions. Numerical experiments show that the method is effectiv...

In this paper, the absolute value equation (AVE) is equivalently reformulated as a nonlinear equation in the form of 2 times 2 blocks. A block diagonal inverse block diagonal iteration method based on block-diagonal and anti-block-diagonal splitting (BAS) is proposed. Theoretical analysis shows that BAS is convergent, and numerical experiments show...

In this paper, based on the shift splitting technique, a shift splitting (SS) iteration method is presented to solve the generalized absolute value equations. Convergence conditions of the SS method are discussed in detail when the involved matrices are some special matrices. Finally, numerical experiments show the effectiveness of the proposed met...

In this paper, a class of new Sylvester-like absolute value equation (AVE) $AXB-|CXD|=E$ with $A,C\in \mathbb{R}^{m\times n}$, $B,D\in \mathbb{R}^{p\times q}$ and $E\in \mathbb{R}^{m\times q}$ is considered, which is quite distinct from the published work by Hashemi [Applied Mathematics Letters, 112 (2021) 106818]. Some sufficient conditions for th...

In this paper, we execute the shift-splitting preconditioner for asymmetric saddle point problems with its (1,2) block’s transposition unequal to its (2,1) block under the removed minus of its (2,1) block. The proposed preconditioner is stemmed from the shift splitting (SS) iteration method for solving asymmetric saddle point problems, which is con...

In this paper, based on the work of Ke and Ma, a modified SOR-like method is presented to solve the absolute value equations (AVE), which is gained by equivalently expressing the implicit fixed-point equation form of the AVE as a two-by-two block nonlinear equation. Under certain conditions, the convergence conditions for the modified SOR-like meth...

In this paper, we propose a new preconditioner of the tensor splitting iterative method for solving multi-linear systems with \(\mathcal {M}\)-tensors. We theoretically show that the spectral radii of the preconditioner iterative tensor decrease as the parameters in the new preconditioners increase, if the preconditioned tensor is a strong \(\mathc...

In this paper, we present a generalized SOR-like iteration method to solve the non-Hermitian positive definite linear complementarity problem (LCP), which is obtained by reformulating equivalently the implicit fixed-point equation of the LCP as a two-by-two block nonlinear equation. The convergence properties of the generalized SOR-like iteration m...

In this note, we show that the singular value condition \(\sigma _{\max }(B) < \sigma _{\min }(A)\) leads to the unique solvability of the absolute value equation \(Ax + B|x| = b\) for any b. This result is superior to those appeared in previously published works by Rohn (Optim Lett 3:603–606, 2009).

In this paper, we further consider the SOR-like iteration method for solving absolute value equations. Some new convergence conditions are obtained from the involved iteration matrix of the SOR-like iteration method, which are different from the results by Ke and Ma (2017). Numerical experiments show that the SOR-like iteration method for solving a...

In this paper, based on the previous work by Bai [On the convergence of the multisplitting methods for the linear complementarity problem. SIAM J Matrix Anal Appl. 1999;21:67–78] and by Zhang et al. [Improved convergence theorems of multisplitting methods for the linear complementarity problem. Appl Math Comput. 2014;243:982–987], we further discus...

In this paper, we further analyze the generalized Newton method for the generalized absolute value equations. Some new convergent conditions for the generalized Newton method are obtained, which are superior to those appeared in previously published works. Numerical experiments are given to demonstrate the effectiveness of the generalized Newton me...

In this paper, based on the complex-symmetric and skew-Hermitian splitting (CSS) of the coefficient matrix, a modified complex-symmetric and skew-Hermitian-splitting (MCSS) iteration method is presented to solve a class of complex-symmetric indefinite linear systems from the classical state-space formulation of frequency analysis of the degree-of-f...

In this paper, we focus on the unique solution of the absolute value equations (AVE). Using an equivalence relation to the linear complementarity problem (LCP), two necessary and sufficient conditions for the unique solution of the AVE are presented. Based on the obtained results, some new sufficient conditions for the unique solution of the AVE ar...

In this note, the unique solution of the linear complementarity problem (LCP) is further discussed. Using the absolute value equations, some new results are obtained to guarantee the unique solution of the LCP for any real vector.

In this paper, for a class of the large sparse system of weakly nonlinear equations, combining separable property of the linear and nonlinear terms with the lopsided Hermitian and skew-Hermitian splitting (LHSS) of the coefficient matrix, the Picard-LPMHSS and nonlinear LPMHSS-like iteration methods are presented. Theoretical analysis shows that th...

In this paper, not requiring that the Hermitian part of the complex symmetric linear system must be Hermitian positive definite, a class of splitting methods is established by the modified positive/negative-stable splitting (PNS) of the coefficient matrix and is called the MPNS method. Theoretical analysis shows that the MPNS method is absolutely c...

In this note, based on the previous work by Pour and Goughery (New Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems Numer. Algor. 69 (2015) 207–225), we further discuss this new Hermitian and skew-Hermitian splitting (described as NHSS) methods for non-Hermitian positive definite linear systems. Some...

In this note, without one parameter being greater than the other, a new sufficient condition for convergence of the MSMAOR method for linear complementarity problems is obtained, which is superior to those that appeared in previously published works.

In this paper, based on the implicit fixed-point equation of the linear complementarity problem (LCP), a generalized Newton method is presented to solve the non-Hermitian positive definite linear complementarity problem. Some convergence properties of the proposed generalized Newton method are discussed. Numerical experiments are presented to illus...

In this paper, we will extend the two-sweep iteration methods to solve the linear complementarity problems and establish a class of two-sweep modulus-based matrix splitting iteration methods for the implicit fixed-point equation of the linear complementarity problems. Some convergence properties of two-sweep modulus-based matrix splitting iteration...

In this paper, we introduce a relaxed splitting preconditioner for saddle point problems. Spectral properties of the preconditioned matrix are analyzed and compared with the closely related preconditioner in recent paper [New preconditioners for saddle point problems, Appl. Math. Comput. 172 (2006), 762-771] by Pan et al. Numerical experiments are...

Based on the previous work by Zhang and Zheng (A parameterized splitting iteration method for complex symmetric linear systems, Japan J. Indust. Appl. Math., 31 (2014) 265–278), three block preconditioners for complex symmetric linear system with two-by-two block form are presented. Spectral properties of the preconditioned matrices are discussed i...

In this paper, two block triangular preconditioners for the asymmetric saddle point problems with singular (1,1) block are presented. The spectral characteristics of the preconditioned matrices are discussed in detail. Theoretical analysis shows that all the eigenvalues of the preconditioned matrices are strongly clustered. Numerical experiments ar...

In this paper, based on the Hermitian and skew-Hermitian splitting (HSS) iteration method, a single-step HSS (SHSS) iteration method is introduced to solve the non-Hermitian positive definite linear systems. Theoretical analysis shows that, under a loose restriction on the iteration parameter, the SHSS method is convergent to the unique solution of...

For the augmented system of linear equations, by considering a new splitting of the coefficient matrix, a modified SOR-like (MSOR-like) method is presented in this paper. The convergence of this method is discussed under suitable restrictions on iteration parameters, the optimal parameters and the corresponding optimal convergence factor are determ...

In this paper, we discuss the semi-convergence of the modified Hermitian and skew-Hermitian splitting (MHSS) iteration method for solving a broad class of complex singular linear systems. Some semi-convergence theories of the MHSS iteration method are established and are weaker than those appeared in previously published works.

In this paper, we derive some new bounds on the complex eigenvalues of the Hermitian and skew-Hermitian splitting (HSS) preconditioner for saddle point problems and improve those stated in previously published works.

This paper is concerned with a splitting iterative method for a class of complex symmetric linear systems from an nn-degree-of-freedom (nn-DOF) discrete system. This splitting iterative method is established by the complex-symmetric and skew-Hermitian splitting of the coefficient matrix and is called the CSS method, which is from the classical stat...

In this paper, we further investigate the double splitting iterative methods for solving linear systems. Building on the previous work by Song and Song [Convergence for nonnegative double splittings of matrices, Calcolo, (2011) 48: 245-260], some new comparison theorems for the spectral radius of double splittings of matrices under suitable conditi...

Based on the modified Hermitian and skew-Hermitian splitting (MHSS) and preconditioned MHSS (PMHSS) methods, a generalized preconditioned MHSS (GPMHSS) method for a class of complex symmetric linear systems is presented. Theoretical analysis gives an upper bound for the spectral radius of the iteration matrix. From a practical point of view, we hav...

Based on the preconditioned MHSS (PMHSS) and generalized PMHSS (GPMHSS) methods, a double-parameter GPMHSS (DGPMHSS) method for solving a class of complex symmetric linear systems from Helmholtz equation is presented. A parameter region of the convergence for DGPMHSS method is provided. From practical point of view, we have analyzed and implemented...

In this paper, an indefinite block triangular preconditioner for symmetric saddle point problems is discussed. The bounds for all the real eigenvalues of the preconditioned matrix are provided, and the bounds for the real and imaginary parts of all the complex eigenvalues of the preconditioned matrix are also provided. The corresponding theoretical...

We discuss spectral properties of the iteration matrix of the HSS method for saddle point problems and derive estimates for the region containing both the nonreal and real eigenvalues of the iteration matrix of the HSS method for saddle point problems.

In this note, some inaccuracies in the article (Numer. Linear Algebra Appl. 2012; 19:754–772) are pointed out and correct results are presented. Copyright © 2013 John Wiley & Sons, Ltd.

In this paper, based on the results presented by Jiang et al. [M.-Q. Jiang, Y. Cao, L.-Q. Yao, On parameterized block triangular preconditioners for generalized saddle point problems, Appl. Math. Comput. 216 (2010) 1777–1789], we consider an indefinite parameterized block triangular preconditioner for symmetric saddle point problems. The eigenvalue...

In this paper, building on the previous work by Greif and Schötzau [Preconditioners for the discretized time-harmonic Maxwell equations in mixed form, Numer. Linear Algebra Appl. 14 (2007) 281–297] and Benzi and Olshanskii [An augmented lagrangian-based approach to the Oseen problem, SIAM J. Sci. Comput. 28 (2006) 2095–2113], we present the improve...

Some comparison theorems for the spectral radius of double splittings of different matrices under suitable conditions are presented, which are superior to the corresponding results in the recent paper by Miao and Zheng (2009). Some comparison theorems between the spectral radius of single and double splittings of matrices are established and are ap...

Three kinds of preconditioners are proposed to accelerate the generalized AOR (GAOR) method for the linear system from the generalized least squares problem. The convergence and comparison results are obtained. The comparison results show that the convergence rate of the preconditioned generalized AOR (PGAOR) methods is better than that of the orig...

Technique of optimal vibration control with exponential decay rate and simulation for vehicle active suspension systems is developed. Mechanical model and dynamic system for a class of tracked vehicle suspension vibration control is established and the corresponding system of state space form is described. In order to prolong the working life of su...

In this paper, a new lower bound on a positive stable block triangular preconditioner for saddle point problems is derived; it is superior to the corresponding result obtained by Cao [Z.-H. Cao, Positive stable block triangular preconditioners for symmetric saddle point problems, Appl. Numer. Math. 57 (2007) 899–910]. A numerical example is reporte...

This paper is concerned with a modified version of the generalization of Hermitian and skew-Hermitian splitting iteration (MGHSS) to solve the non-Hermitian positive definite linear systems. The corresponding inexact MGHSS (IMGHSS) method is developed by employing some Krylov subspace methods as its inner process. Numerical examples are reported to...

We consider the SIMPLE preconditioning for block two-by-two generalized saddle point problems; this is the general nonsymmetric, nonsingular case where the (1,2) block needs not to equal the transposed (2,1) block, and the (2,2) block may not be zero. The eigenvalue analysis of the SIMPLE preconditioned matrix is presented. The relationship between...

New splitting iterative methods for Toeplitz systems are proposed by means of recently developed matrix splittings based on discrete sine and cosine transforms due to Kailath and Olshevsky [Displacement structure approach to discrete-trigonometric transform-based preconditioners of G. Strang type and of T. Chan type, SIAM J. Matrix Anal. Appl. 26 (...

The finite difference method discretization of Helmholtz equations usually leads to the large spare linear systems. Since the coefficient matrix is frequently indefinite, it is difficult to solve iteratively. In this paper, a modified symmetric successive overrelaxation (MSSOR) preconditioning strategy is constructed based on the coefficient matrix...

In this paper, we present some comparison theorems on preconditioned iterative method for solving L-matrices linear systems. Comparison results and numerical examples show that the rate of convergence of the preconditioned Gauss-Seidel iterative method is faster than the rate of convergence of the preconditioned SOR iterative method.

In this paper, we explore the block triangular preconditioning techniques applied to the iterative solution of the saddle point linear systems arising from the discretized Maxwell equations. Theoretical analysis shows that all the eigenvalues of the preconditioned matrix arestrongly clustered. Numerical experiments are given to demonstrate the effi...

In this paper, we consider the Hermitian and skew-Hermitian splitting (HSS) preconditioner for generalized saddle point problems with nonzero (2, 2) blocks. The spectral property of the preconditioned matrix is studied in detail. Under certain conditions, all eigenvalues of the preconditioned matrix with the original system being non-Hermitian will...

Golub, Wu and Yuan [G.H. Golub, X. Wu, J.Y. Yuan, SOR-like methods for augmented systems, BIT 41 (2001) 71–85] have presented the SOR-like algorithm to solve augmented systems. In this paper, we present the modified symmetric successive overrelaxation (MSSOR) method for solving augmented systems, which is based on Darvishi and Hessari’s work above....

Using the finite difference method to discretize Helmholtz equations usually leads to a large spare linear system of equations Ax=b. Since the coefficient matrix A is frequently indefinite, it is difficult to solve iteratively. The approach taken in this Letter is to precondition this linear system with positive stable preconditioners and then to s...

In this paper, spectral properties and computational performance of a generalized block triangular preconditioner for symmetric saddle point problems are discussed in detail. We will provide estimates for the region containing both the nonreal and the real eigenvalues and generalize the results of Simoncini (Appl Numer Math 49:63–80, 2004) and Cao...

Positive eigenvector of nonlinear perturbations of nonsymmetric M-matrix and its Newton iterative solution are studied. It is shown that any number greater than the smallest positive eigenvalue of the M-matrix is an eigenvalue of the nonlinear problem and that the corresponding positive eigenvector is unique and the Newton iteration of the positive...

In this paper, we improve the preconditioned AOR method of linear systems considered by Evans et al. [The AOR iterative method for new preconditioned linear systems, Comput. Appl. Math. 132 (2001) 461–466]. In Evans’ paper, the coefficient matrix of linear system have to be an L-matrix with ai,i+1ai+1,i>0, i=1,…,n-1 and 0a1nan11. When ai,i+1ai+1,i=...

To further study the Hermitian and non-Hermitian splitting methods for a non-Hermitian and positive-definite matrix, we introduce a so-called lopsided Hermitian and skew-Hermitian splitting and then establish a class of lopsided Hermitian/skew-Hermitian (LHSS) methods to solve the non-Hermitian and positive-definite systems of linear equations. The...

Milaszewicz [J.P. Milaszewicz, Improving Jacobi and Gauss–Seidel iterations, Linear Algebra Appl. 93 (1987) 161–170] and Gunawardena et al. [A.D. Gunawardena, S.K. Jain, L. Snyder, Modified iteration methods for consistent linear systems, Linear Algebra Appl. 154–156 (1991) 123–143] presented preconditioned methods for linear system in order to imp...