A preconditioned general modulus-based matrix splitting iteration method for linear complementarity problems of <Emphasis Type="Italic">H</Emphasis>-matrices

In this paper, we propose a preconditioned general modulus-based matrix splitting iteration method for solving modulus equations arising from linear complementarity problems. Its convergence theory is proved when the system matrix is an H₊-matrix, from which some new convergence conditions can be derived for the (general) modulus-based matrix splitting iteration methods. Numerical results further show that the proposed methods are superior to the existing methods.     

Accelerated modulus-based matrix splitting iteration methods for linear complementarity problem

For the large sparse linear complementarity problem, a class of accelerated modulus-based matrix splitting iteration methods is established by reformulating it as a general implicit fixed-point equation, which covers the known modulus-based matrix splitting iteration methods. The convergence conditions are presented when the system matrix is either a positive definite matrix or an H ₊-matrix. Numerical experiments further show that the proposed methods are efficient and accelerate the convergence performance of the modulus-based matrix splitting iteration methods with less iteration steps and CPU time.     

Comparison results on preconditioned SOR-type iterative method for Z-matrices linear systems

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

New convergence results for alternating methods

The alternating methods for solving the large system of linear equations Ax=b are investigated. The convergence and the monotone convergence theories for the alternating method are formulated when the coefficient matrix is an H-matrix or a monotone matrix. Sufficient conditions are established for the induced splitting by the alternating method to be a regular splitting. Furthermore, new comparison theorems which improve previous comparison theorems are proved and several concrete applications are given.     

On convergence of two-stage splitting methods for linear complementarity problems

In this paper, we study the splitting method and two-stage splitting method for the linear complementarity problems. Convergence results for these two methods are presented when the system matrix is an H-matrix and the splittings used are H-splitting. Numerical experiments show that the two-stage splitting method has the same or even better numerical performance than the splitting method in some aspects under certain conditions.     

Convergence of two-stage multisplitting method using AOR or SSOR multisplittings

In this paper, we study the convergence of two-stage multisplitting method using AOR or SSOR multisplittings as inner splittings and an outer splitting for solving a linear system whose coefficient matrix is an H-matrix. We also introduce an application of the two-stage multisplitting method.     

Two-step modulus-based matrix splitting iteration method for linear complementarity problems

Bai has recently presented a modulus-based matrix splitting iteration method, which is a powerful alternative for solving the large sparse linear complementarity problems. In this paper, we further present a two-step modulus-based matrix splitting iteration method, which consists of a forward and a backward sweep. Its convergence theory is proved when the system matrix is an H ₊-matrix. Moreover, for the two-step modulus-based relaxation iteration methods, more exact convergence domains are obtained without restriction on the Jacobi matrix associated with the system matrix, which improve the existing convergence theory. Numerical results show that the two-step modulus-based relaxation iteration methods are superior to the modulus-based relaxation iteration methods for solving the large sparse linear complementarity problems.     

Modified modulus‐based matrix splitting iteration methods for linear complementarity problems

For solving the large sparse linear complementarity problems, we establish modified modulus‐based matrix splitting iteration methods and present the convergence analysis when the system matrices are H₊‐matrices. The optima of parameters involved under some scopes are also analyzed. Numerical results show that in computing efficiency, our new methods are superior to classical modulus‐based matrix splitting iteration methods under suitable conditions. Copyright © 2015 John Wiley & Sons, Ltd.     

Convergence of SSOR methods for linear complementarity problems

In this paper, by applying the SSOR splitting, we propose two new iterative methods for solving the linear complementarity problem LCP (M,q). Convergence results for these two methods are presented when M is an H-matrix (and also an M-matrix). Finally, two numerical examples are given to show the efficiency of the presented methods.     

Semiconvergence of extrapolated iterative methods for singular linear systems

In this paper, we discuss convergence of the extrapolated iterative methods for solving singular linear systems. A general principle of extrapolation is presented. The semiconvergence of an extrapolated method induced by a regular splitting and a nonnegative splitting is proved whenever the coefficient matrix A is a singular M-matrix with ‘property c’ and an irreducible singular M-matrix, respectively. Since the (generalized, block) JOR and AOR methods are respectively the extrapolated methods of the (generalized, block) Jacobi and SOR methods, so the semiconvergence of the (generalized, block) JOR and AOR methods for solving general singular systems are proved. Furthermore, the semiconvergence of the extrapolated power method, the (block) JOR, AOR and SOR methods for solving Markov chains are discussed.     

Iterative processes based on block <Emphasis Type="Italic">H</Emphasis>-splittings

Block H-splittings of block square matrices (which, in general, have complex entries) are examined. It is shown that block H-matrices are the only ones that admit this type of splittings. Iterative processes corresponding to these splittings are proved to be convergent. The concept of a simple splitting of a block matrix is introduced, and the convergence of iterative processes related to simple splittings of block H-matrices is investigated. Multisplitting and nonstationary iterative processes based on block H-splittings are considered. Sufficient conditions for their convergence are derived, and some estimates for the asymptotic convergence rate are given.     

Improved convergence theorems of the two-step modulus-based matrix splitting and synchronous multisplitting iteration methods for solving linear complementarity problems

Abstract

In this paper, the convergence conditions of the two-step modulus-based matrix splitting and synchronous multisplitting iteration methods for solving linear complementarity problems of H-matrices are weakened. The convergence domain given by the proposed theorems is larger than the existing ones.     

A splitting method for the nonlinear Schrödinger equation

We introduce a splitting method for the semilinear Schrödinger equation and prove its convergence for those nonlinearities which can be handled by the classical well-posedness L²(Rd)-theory. More precisely, we prove that the scheme is of first order in the L²(Rd)-norm for H²(Rd)-initial data.     

Modulus‐based matrix splitting iteration methods for linear complementarity problems

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. Copyright © 2009 John Wiley & Sons, Ltd.     

IGAOR and multisplitting IGAOR methods for linear complementarity problems

In this paper, we propose an interval version of the generalized accelerated overrelaxation methods, which we refer to as IGAOR, for solving the linear complementarity problems, LCP (M, q), and develop a class of multisplitting IGAOR methods which can be easily implemented in parallel. In addition, in regards to the H-matrix with positive diagonal elements, we prove the convergence of these algorithms and illustrate their efficiency through our numerical results.     

On the convergence of splittings for semidefinite linear systems

Recently, Lee et al. [Young-ju Lee, Jinbiao Wu, Jinchao Xu, Ludmil Zikatanov, On the convergence of iterative methods for semidefinite linear systems, SIAM J. Matrix Anal. Appl. 28 (2006) 634-641] introduce new criteria for the semi-convergence of general iterative methods for semidefinite linear systems based on matrix splitting. The new conditions generalize the classical notion of P-regularity introduced by Keller [H.B. Keller, On the solution of singular and semidefinite linear systems by iterations, SIAM J. Numer. Anal. 2 (1965) 281-290]. In view of their results, we consider here stipulations on a splitting A=M-N, which lead to fixed point systems such that, the iterative scheme converges to a weighted Moore-Penrose solution to the system Ax=b. Our results extend the result of Lee et al. to a more general case and we also show that it requires less restrictions on the splittings than Keller’s P-regularity condition to ensure the convergence of iterative scheme.     

Some extensions of the improved modified Gauss–Seidel iterative method for H‐matrices

In this paper, first we present a convergence theorem of the improved modified Gauss–Seidel iterative method, referred to as the IMGS method, for H‐matrices and compare the range of parameters α_i with that of the parameter ω of the SOR iterative method. Then with a more general splitting, the convergence analysis of this method for an H‐matrix and its comparison matrix is given. The spectral radii of them are also compared. Copyright © 2006 John Wiley & Sons, Ltd.     

The solution of the linear complementarity problem by the matrix analogue of the accelerated overrelaxation iterative method

The Linear Complementarity Problem (LCP), with an H₊?matrix coefficient, is solved by using the new “(Projected) Matrix Analogue of the AOR (MAAOR)” iterative method; this new method constitutes an extension of the “Generalized AOR (GAOR)” iterative method. In this work two sets of convergence intervals of the parameters involved are determined by the theories of “Perron-Frobenius” and of “Regular Splittings”. It is shown that the intervals in question are better than any similar convergence intervals found so far by similar iterative methods. A deeper analysis reveals that the “best” values of the parameters involved are those of the (projected) scalar Gauss-Seidel iterative method. A theoretical comparison of the “best” (projected) Gauss-Seidel and the “best” modulus-based splitting Gauss-Seidel method is in favor of the former method. A number of numerical examples support most of our theoretical findings.     

Two valued measure and summability of double sequences in asymmetric context

We extend the ideas of convergence and Cauchy condition of double sequences extended by a two valued measure (called ??-statistical convergence/Cauchy condition and convergence/Cauchy condition in ??-density, studied for real numbers in our recent paper [7]) to a very general structure like an asymmetric (quasi) metric space. In this context it should be noted that the above convergence ideas naturally extend the idea of statistical convergence of double sequences studied by Móricz [15] and Mursaleen and Edely [17]. We also apply the same methods to introduce, for the first time, certain ideas of divergence of double sequences in these abstract spaces. The asymmetry (or rather, absence of symmetry) of asymmetric metric spaces not only makes the whole treatment different from the real case [7] but at the same time, like [3], shows that symmetry is not essential for any result of [7] and in certain cases to get the results, we can replace symmetry by a genuinely asymmetric condition called (AMA).     

Convergence of descent methods for semi-algebraic and tame problems: proximal algorithms, forward–backward splitting, and regularized Gauss–Seidel methods

In view of the minimization of a nonsmooth nonconvex function f, we prove an abstract convergence result for descent methods satisfying a sufficient-decrease assumption, and allowing a relative error tolerance. Our result guarantees the convergence of bounded sequences, under the assumption that the function f satisfies the Kurdyka–?ojasiewicz inequality. This assumption allows to cover a wide range of problems, including nonsmooth semi-algebraic (or more generally tame) minimization. The specialization of our result to different kinds of structured problems provides several new convergence results for inexact versions of the gradient method, the proximal method, the forward–backward splitting algorithm, the gradient projection and some proximal regularization of the Gauss–Seidel method in a nonconvex setting. Our results are illustrated through feasibility problems, or iterative thresholding procedures for compressive sensing.