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

Abstract

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 twostep 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.

Two-Step Modulus-Based Synchronous Multisplitting Iteration Methods for Linear Complementarity Problems

“…From the numerical results in [26], we know that for some linear complementarity problems the two-step modulus-based matrix splitting iteration method is effective to decrease the number of iteration steps. Thus, for the MSM iteration method we take two sweeps at each iteration step to reduce the communication among the processors, which may improve the computing time for solving linear complementarity problems.…”

“…, ℓ) suitably such that the tasks distributed on the ℓ processors are well balanced. We remark that when ℓ = 1, the TMSM iteration method naturally reduces to the two-step modulus-based matrix splitting iteration method in [26].…”

“…In this section, we firstly improve the convergence theorems of the two-step modulus-based matrix splitting iteration method in [26]. Then, we prove the convergence of the TMSM and SMSMAOR iteration methods.…”

An accelerated Newton method of high-order convergence for solving a class of weakly nonlinear complementarity problems

“…On matrix multisplitting iteration aspects, also, there are many useful papers were introduced to solve (1.2), see [1,2,4,6,8] for more details. Many kinds of accelerated modulus-based matrix splitting iteration versions are also developed, see [30,31]. Moreover, the Modulus-based synchronous multisplitting iteration methods for (1.2) are considered in [9].…”

The modulus-based nonsmooth Newton’s method for solving linear complementarity problems

“…Recently, Bai [6] presented a modulus-based matrix splitting method which not only included the modified modulus method [7] and the nonstationary extrapolated modulus algorithms [8] as its special cases, but also yielded a series of iteration methods, such as modulusbased Jacobi, Gauss-Seidel, SOR and AOR iteration methods, which were extended to more general cases by Li [9]. In addition, Hadjidimos et al [10] and Zhang [11] proposed scaled extrapolated modulus algorithms and two-step modulus-based matrix splitting iteration methods, respectively. The global convergence conditions are discussed when the system matrix is either a positive definite matrix or an H + -matrix.…”