Matrix Splitting Iteration Method Research Articles

A Relaxation Two-Step Newton-Based Matrix Splitting Iteration Method for Generalized Absolute Value Equations Associated with Circular Cones

A Relaxation Two-Step Newton-Based Matrix Splitting Iteration Method for Generalized Absolute Value Equations Associated with Circular Cones

Two-Step Simplified Modulus-Based Matrix Splitting Iteration Method for Linear Complementarity Problems

Two-Step Simplified Modulus-Based Matrix Splitting Iteration Method for Linear Complementarity Problems

Improved modulus-based matrix splitting iteration methods for a class of horizontal implicit complementarity problems

Recently, the modulus-based matrix splitting (MMS) iteration method has been successfully extended to solve the horizontal implicit complementarity problem (HICP). However, due to the existence of two system matrices and one nonlinear term, in addition to an equivalent fixed-point equation of the HICP, a system of linear equations has to be solved at each iteration to satisfy both the nonnegative constraint and the orthogonal constraint. This costs very expensive. In this paper, by introducing an additional judgement criterion after obtaining the approximate solution, a class of improved MMS (IMMS) iteration methods are proposed. By suitably choosing the judgement criterion, the IMMS iteration methods do not need to solve such system of linear equations at each iteration and thus greatly improves the computing efficiency. Convergence conditions of the IMMS iteration methods are studied when the system matrices are positive definite matrices and H+-matrices, respectively. Finally, two numerical examples are presented to show the effectiveness of the proposed IMMS iteration methods and their advantages over the existing MMS iteration methods when solving the HICP.

Two Variants of Robust Two-Step Modulus-Based Matrix Splitting Iteration Methods for Mixed-Cell-Height Circuit Legalization Problem

Two Variants of Robust Two-Step Modulus-Based Matrix Splitting Iteration Methods for Mixed-Cell-Height Circuit Legalization Problem

Modified two-step modulus-based matrix splitting iteration methods for implicit complementarity problems

Modified two-step modulus-based matrix splitting iteration methods for implicit complementarity problems

On the splitting iteration method for Pareto eigenvalue complementarity problems of H +-matrices

We present a class of inexact splitting-modulus iteration methods for solving the Pareto Eigenvalue Complementarity Problem (EiCP) when the system matrix A is an H + -matrix. Our method first employs the matrix splitting technique to transform the original eigenvalue complementarity problem (EiCP) into an easily solvable Linear Complementarity Problem (LCP), and the inner iterative process corresponds to the inexact split iterative process when the modulus iteration method is used, where we employed an effective and fast inner iterative stopping criterion. We then provide a convergence proof for inexact matrix splitting iteration methods. Numerical experiments demonstrate that our proposed algorithms exhibit significant superiority over the classical matrix splitting iteration method in terms of computational time, especially for solving large-scale symmetric and asymmetric eigenvalue complementarity problems of H + -matrices.

General double-relaxation two-sweep modulus-based matrix splitting iteration methods for horizontal linear complementarity problem

General double-relaxation two-sweep modulus-based matrix splitting iteration methods for horizontal linear complementarity problem

Two-parameter modified matrix splitting iteration method for Helmholtz equation

For effectively solving the nonsymmetric and indefinite linear system originating from the discretized Helmholtz equation with complex wavenumber, we propose a two-parameter modified matrix splitting iteration method. We establish the asymptotic convergence theory for this method and demonstrate its convergence under certain conditions. The proposed iteration method leads to a new preconditioner that can be efficiently inverted by using the discrete sine transform. Numerical experiments are carried out to show that the new preconditioner significantly improves the convergence rate of the GMRES method, which results in effective preconditioned GMRES method for solving the Helmholtz equation of high wavenumber.

The simplified modulus-based matrix splitting iteration method for the nonlinear complementarity problem

<abstract><p>In this paper, the simplified modulus-based matrix splitting iteration method was extended to solve the nonlinear complementarity problem, and the convergence conditions were presented from the spectral radius and the matrix norm. Then, for the special cases of this method, we provided some concrete convergence conditions as well as the quasi-optimal parameter matrix. Moreover, some numerical examples were illustrated to show the validity of the convergence results.</p></abstract>

A modified Newton-based matrix splitting iteration method for generalized absolute value equations

Many problems in scientific computing and engineering fields lead to the solution of generalized absolute value equations (GAVE). In this paper, by simultaneously splitting both coefficient matrices in the differential and non-differential parts of the GAVE, a modified Newton-based matrix splitting (MNMS) iteration method is proposed. The MNMS iteration method not only covers the well-known generalized Newton iteration method as well as the recent proposed modified Newton-based iteration method and the Newton-based matrix splitting iteration method for solving the GAVE, but also results in a series of relaxation versions that are very flexible in real applications. Convergence properties of the MNMS iteration method are studied in detail when the coefficient matrices are positive definite matrices and H+-matrices. Finally, three numerical examples are presented to show the feasibility and effectiveness of the proposed MNMS iteration method.

A two-step new modulus-based matrix splitting method for vertical linear complementarity problem

In this paper, for solving the vertical linear complementarity problem (VLCP) effectively, a two-step new modulus-based matrix splitting (TNMMS) iteration method is introduced. Its convergence properties are discussed in depth. The efficiency of the proposed method is confirmed by some numerical experiments from the 2-D boundary problem. Numerical results show that the proposed method is superior to some existing methods, such as the two modulus-based matrix splitting (TMMS) iteration method (Numer. Algor., 2011, 57: 83–99) and the new modulus-based matrix splitting (NMMS) iteration method (Optim., 2023, 72: 2499–2516).

Accelerated Relaxation Two-Sweep Modulus-Based Matrix Splitting Iteration Method for Linear Complementarity Problems

In this paper, by applying the accelerated technique and relaxation two-sweep strategy to the modulus-based matrix splitting iteration method, we establish the accelerated relaxation two-sweep modulus-based matrix splitting method. The proposed method contains the accelerated modulus-based matrix splitting and the generalized accelerated modulus-based matrix splitting methods presented recently as special cases. Compared with some existing methods, the proposed method can use more information during each iteration, which may lead to higher computational efficiency. We investigate the convergence properties of the new method, and prove that it is convergent under certain assumptions. Also, for the accelerated relaxation two-sweep modulus-based accelerated overrelaxation method, we analyze the convergence regions of the parameters in detail. Numerical examples are offered to show that the proposed method is efficient and performs better than the generalized accelerated modulus-based matrix splitting one in actual implementation.

The nonlinear lopsided PSS-like and HSS-like modulus-based matrix splitting iteration methods for horizontal linear complementarity problem

The nonlinear lopsided PSS-like and HSS-like modulus-based matrix splitting iteration methods for horizontal linear complementarity problem

A Two-Step Modulus-Based Matrix Splitting Iteration Method Without Auxiliary Variables for Solving Vertical Linear Complementarity Problems

A Two-Step Modulus-Based Matrix Splitting Iteration Method Without Auxiliary Variables for Solving Vertical Linear Complementarity Problems

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.

Relaxation modulus-based matrix splitting iteration method for vertical linear complementarity problem

For solving vertical linear complementarity problems (VLCPs), we propose a relaxation modulus-based matrix splitting iteration method and a two-step relaxation modulus-based matrix splitting iteration method. When the system matrices are H+-matrices, we analyze the convergence theory of the proposed methods. Numerical experiments show that relaxation modulus-based matrix splitting iteration method and the two-step relaxation modulus-based matrix splitting iteration method are promising and have great performance in higher dimension.

A two‐step matrix splitting iteration paradigm based on one single splitting for solving systems of linear equations

AbstractFor solving large sparse systems of linear equations, we construct a paradigm of two‐step matrix splitting iteration methods and analyze its convergence property for the nonsingular and the positive‐definite matrix class. This two‐step matrix splitting iteration paradigm adopts only one single splitting of the coefficient matrix, together with several arbitrary iteration parameters. Hence, it can be constructed easily in actual applications, and can also recover a number of representatives of the existing two‐step matrix splitting iteration methods. This result provides systematic treatment for the two‐step matrix splitting iteration methods, establishes rigorous theory for their asymptotic convergence, and enriches algorithmic family of the linear iteration solvers, for the iterative solutions of large sparse linear systems.

A block-diagonally preconditioned Uzawa splitting iteration method for solving a class of saddle-point problems

This paper develops a block diagonal preconditioned Uzawa splitting (BDP-US) method for solving saddle point problems by generalizing the Uzawa splitting iteration method proposed by Li and Ma ( Numer Math Theory Methods Appl 2018; 11: 235–246). A sufficient condition is then provided to ensure the convergence of the BDP-US method. Meanwhile, a preconditioner on the basis of the BDP-US method is proposed, the spectral properties of the preconditioned matrix is analyzed, and the choice of the parameters for this matrix splitting iteration method is discussed. Numerical results are provided to support the obtained results, and demonstrate the effectiveness of BDP-US method as well as the corresponding preconditioner.

Improved modulus-based matrix splitting iteration methods for quasi-complementarity problems

In this paper, the improved modulus-based matrix splitting iteration methods are proposed to solve the quasi-complementarity problems. It is proved that the proposed iteration methods are convergent under certain conditions by using some new arguments. Numerical experiments are given to illustrate the high efficiency of the new iteration methods. Numerical results show that the proposed methods have better performance than the existing methods.

An Improved Convergence Condition of the MMS Iteration Method for Horizontal LCP of H+-Matrices

In this paper, inspired by the previous work in (Appl. Math. Comput., 369 (2020) 124890), we focus on the convergence condition of the modulus-based matrix splitting (MMS) iteration method for solving the horizontal linear complementarity problem (HLCP) with H+-matrices. An improved convergence condition of the MMS iteration method is given to improve the range of its applications, in a way which is better than that in the above published article.