Synchronous Multisplitting Iteration Methods Research Articles

Modulus-based synchronous multisplitting iteration methods without auxiliary variable for solving vertical linear complementarity problems

In this work, by applying the synchronous multisplitting technique to the non-auxiliary variable modulus equation of the vertical linear complementarity problems, a new parallel method is constructed, which can generalize the existing modulus-based matrix splitting iteration method. The convergence conditions of the proposed method are discussed in the cases of more than two system matrices, and the existing results are improved. Numerical examples under OpenACC framework show that the proposed method can solve the large sparse vertical linear complementarity problems efficiently with high parallel efficiency.

Modulus-based synchronous multisplitting iteration methods for large sparse vertical linear complementarity problems

Modulus-based synchronous multisplitting iteration methods for large sparse vertical linear complementarity problems

Relaxed Modulus-Based Synchronous Multisplitting Multi-Parameter Methods for Linear Complementarity Problems

In 2013, Bai and Zhang (Numer Linear Algebra Appl, 20:425–439 2013) constructed modulus-based synchronous multisplitting iteration methods by an equivalent reformulation of the linear complementarity problem into a system of fixed-point equations and studied the convergence of them. In this paper, we generalize Bai and Zhang’s method and study relaxed modulus-based synchronous multisplitting multi-parameter methods for linear complementarity problems. Furthermore, when the system matrix is an H+-matrix, we give convergence results for our new methods under weaker conditions than those in Bai and Zhang’s. Finally, numerical experiments are presented to illustrate the efficiency of the proposed methods.

A class of modified modulus-based synchronous multisplitting iteration methods for linear complementarity problems

In this paper, we present modified modulus-based synchronous multisplitting iteration methods based on multisplittings of the system matrix for solving the large sparse linear complementarity problems. The proposed methods extend the existing modulus-based synchronous multisplitting iteration methods to a more general case. We establish the convergence theory of these modified modulus-based synchronous multisplitting iteration methods when the system matrix is an H+-matrix. In particular, we investigate the optimal choice of the parameter matrices in theory. Numerical results confirm that the new iteration methods have higher parallel computational efficiency than the existing modulus-based synchronous multisplitting iteration methods. The proposed methods are applied in reconstruction of two-dimensional image data and show the efficiency.

Modulus-Based Synchronous Multisplitting Iteration Methods for a Restricted Class of Nonlinear Complementarity Problems

Modulus-Based Synchronous Multisplitting Iteration Methods for a Restricted Class of Nonlinear Complementarity Problems

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

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.

Modulus-based Synchronous Multisplitting Iteration Methods for an Implicit Complementarity Problem

AbstractWe construct modulus-based synchronous multisplitting iteration methods to solve a large implicit complementarity problem on parallel multiprocessor systems, and prove their convergence. Numerical results confirm our theoretical analysis and show that these new methods are efficient.

Parallel multisplitting iteration methods based on M-splitting for the PageRank problem

A class of iterative schemes based on the regular splitting is proposed for PageRank computation. By some specific splitting, these methods will reduce to some well-known iterative schemes, like Jacobi method and successive overrelaxation method. To suit computational requirements of the modern high-speed multiprocessor environments, we further study synchronous parallel counterparts for the new splitting iteration methods. By using the M-splitting of the coefficient matrix, we establish the convergence theory of these synchronous multisplitting iteration methods. Finally, some numerical experiments are presented to illustrate the feasibility, stability and effectiveness of the proposed algorithms.

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

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

New convergence proofs of modulus-based synchronous multisplitting iteration methods for linear complementarity problems

Modulus-based synchronous multisplitting iteration methods were recently proposed for solving linear complementarity problems. We give a much simpler approach to prove the convergence of these iteration methods when the system matrix is an H+-matrix. Moreover, this idea can also be applied to prove the convergence of two-step modulus-based synchronous multisplitting iteration methods, which avoids the construction of the irreducible system matrix and the introduction of the infinite norm.

A note on the convergence of the MSMAOR method for linear complementarity problems

SUMMARY Modulus-based splitting, as well as multisplitting iteration methods, for linear complementarity problems are developed by Zhong-Zhi Bai. In related papers (see Bai, Z.-Z., Zhang, L.-L.: Modulus-Based Synchronous Multisplitting Iteration Methods for Linear Complementarity Problems. Numerical Linear Algebra with Applications 20 (2013) 425–439, and the references cited therein), the problem of convergence for two-parameter relaxation methods (accelerated overrelaxation-type methods) is analyzed under the assumption that one parameter is greater than the other. Here, we will show how we can avoid this assumption and, consequently, improve the convergence area. Copyright © 2013 John Wiley & Sons, Ltd.

Modulus‐based synchronous multisplitting iteration methods for linear complementarity problems

SUMMARYBy an equivalent reformulation of the linear complementarity problem into a system of fixed‐point equations, we construct modulus‐based synchronous multisplitting iteration methods based on multiple splittings of the system matrix. These iteration methods are suitable to high‐speed parallel multiprocessor systems and include the multisplitting relaxation methods such as Jacobi, Gauss–Seidel, successive overrelaxation, and accelerated overrelaxation of the modulus type as special cases. We establish the convergence theory of these modulus‐based synchronous multisplitting iteration methods and their relaxed variants when the system matrix is an H + ‐matrix. Numerical results show that these new iteration methods can achieve high parallel computational efficiency in actual implementations. Copyright © 2012 John Wiley & Sons, Ltd.