Modulus-based circulant and skew-circulant splitting iteration method for the linear complementarity problem with a Toeplitz matrix (original) (raw)
Related papers
Numerical Mathematics: Theory, Methods and Applications
Linear complementarity problems have drawn considerable attention in recent years due to their wide applications. In this article, we introduce the two-step two-sweep modulus-based matrix splitting (TSTM) iteration method and two-sweep modulus-based matrix splitting type II (TM II) iteration method which are a combination of the two-step modulus-based method and the two-sweep modulus-based method, as two more effective ways to solve the linear complementarity problems. The convergence behavior of these methods is discussed when the system matrix is either a positive-definite or an H +-matrix. Finally, numerical experiments are given to show the efficiency of our proposed methods.
The Shifted Classical Circulant and Skew Circulant Splitting Iterative Methods for Toeplitz Matrices
Canadian Mathematical Bulletin, 2016
It is known that every Toeplitz matrix T enjoys a circulant and skew circulant splitting (denoted CSCS) i.e., T = C−S with C a circulantmatrix and S a skew circulantmatrix. Based on the variant of such a splitting (also referred to as CSCS), we first develop classical CSCS iterative methods and then introduce shifted CSCS iterative methods for solving hermitian positive deûnite Toeplitz systems in this paper. The convergence of each method is analyzed. Numerical experiments show that the classical CSCS iterative methods work slightly better than the Gauss–Seidel (GS) iterative methods if the CSCS is convergent, and that there is always a constant α such that the shifted CSCS iteration converges much faster than the Gauss–Seidel iteration, no matter whether the CSCS itself is convergent or not.
On the choice of parameters in MAOR type splitting methods for the linear complementarity problem
Numerical Algorithms, 2014
In the present work we consider the iterative solution of the Linear Complementarity Problem (LCP), with a nonsingular H + coefficient matrix A, by using all modulus-based matrix splitting iterative methods that have been around for the last couple of years. A deeper analysis shows that the iterative solution of the LCP by the modified Accelerated Overrelaxation (MAOR) iterative method is the "best", in a sense made precise in the text, among all those that have been proposed so far regarding the following three issues: i) The positive diagonal matrix-parameter ≥ diag(A) involved in the method is = diag(A), ii) The known convergence intervals for the two AOR parameters, α and β, are the widest possible, and iii) The "best" possible MAOR iterative method is the modified Gauss-Seidel one.
A new approach for the modulus-based matrix splitting algorithms
IEEE Access
We investigate the modulus-based matrix splitting iteration algorithms for solving the linear complementarity problems (LCPs) and propose a new model to solve it. The structure of the new model is straightforward and can be extended to other issues of type of complementarity. Convergence analysis of the new approach for the symmetric positive definite matrix is also discussed. Numerical results of the proposed approach are compared with the existing algorithms to show its efficiency.
Advances in Numerical Analysis, 2012
We study the CSCS method for large Hermitian positive definite Toeplitz linear systems, which first appears in Ng's paper published in (Ng, 2003), and CSCS stands for circulant and skew circulant splitting of the coefficient matrix . In this paper, we present a new iteration method for the numerical solution of Hermitian positive definite Toeplitz systems of linear equations. The method is a two-parameter generation of the CSCS method such that when the two parameters involved are equal, it coincides with the CSCS method. We discuss the convergence property and optimal parameters of this method. Finally, we extend our method to BTTB matrices. Numerical experiments are presented to show the effectiveness of our new method.
Verification of Iterative Methods for the Linear Complementarity Problem
In the present chapter, we give an overview of iterative methods for linear complementarity problems (abbreviated as LCPs). We also introduce these iterative methods for the problems based on fixed-point principle. Next, we present some new properties of preconditioned iterative methods for solving the LCPs. Convergence results of the sequence generated by these methods and also the comparison analysis between classic Gauss-Seidel method and preconditioned Gauss-Seidel (PGS) method for LCPs are established under certain conditions. Finally, the efficiency of these methods is demonstrated by numerical experiments. These results show that the mentioned models are effective in actual implementation and competitive with each other.
Modification of iterative methods for solving linear complementarity problems
Engineering Computations, 2013
Purpose – The purpose of this paper is to present the efficient iterative methods for solving linear complementarity problems (LCP), using a class of pre-conditioners. Design/methodology/approach – By using the concept of solving the fixed-point system of equations associated to the LCP, pre-conditioning techniques and Krylov subspace methods the authors design some projected methods to solve LCP. Furthermore, within the computational framework, some models of pre-conditioners candidates are investigated and evaluated. Findings – The proposed algorithms have a simple and graceful structure and can be applied to other complementarity problems. Asymptotic convergence of the sequence generated by the method to the unique solution of LCP is proved, along with a result regarding the convergence rate of the pre-conditioned methods. Finally, a computational comparison of the standard methods against pre-conditioned methods based on Example 1 is presented which illustrate the merits of simp...
A note on the convergence of the MSMAOR method for linear complementarity problems
Numerical Linear Algebra with Applications, 2013
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 twoparameter 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.
Numerical Algorithms, 2016
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.
On the Two SAOR Iterative Formats for Solving Linear Complementarity Problems
International Journal of Information …, 2011
Han et.al have applied two SAOR splitting formats for solving the linear complementarity problem. We improve them by introducing a class of preconditioners based on the SAOR methods. The convergences of the modified methods have been analyzed. We also show the applicability of the methods by numerical example.