The modulus-based matrix splitting iteration methods for second-order cone linear complementarity problems

This paper presents a new computational technique for solving fractional pantograph differential equations. The fractional derivative is described in the Caputo sense. The main idea is to use Müntz-Legendre wavelet and its operational matrix of fractional-order integration. First, the Müntz-Legendre wavelet is presented. Then a family of piecewise functions is proposed, based on which the fractional order integration of the Müntz-Legendre wavelets are easy to calculate. The proposed approach is used this operational matrix with the collocation points to reduce the under study problem to a system of algebraic equations. An estimation of the error is given in the sense of Sobolev norms. The efficiency and accuracy of the proposed method are illustrated by several numerical examples.     

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.     

Accelerated modulus-based matrix splitting iteration methods for a restricted class of nonlinear complementarity problems

To solve a class of nonlinear complementarity problems, accelerated modulus-based matrix splitting iteration methods are presented and analyzed. Convergence analysis and the choice of the parameters are given when the system matrix is either positive definite or an H ₊-matrix. Numerical experiments further demonstrate that the proposed methods are efficient and have better performance than the existing modulus-based iteration method in aspects of the number of iteration steps and CPU time.     

Improved convergence theorems of modulus-based matrix splitting iteration methods for linear complementarity problems

We weaken the convergence conditions of modulus-based matrix splitting and matrix two-stage splitting iteration methods for linear complementarity problems. Thus their applied scopes are further extended.     

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

Numerical Algorithms - In this paper, a class of two-step modulus-based matrix splitting (TMMS) iteration methods are proposed to solve the implicit complementarity problems. It is proved that the...     

New modulus-based matrix splitting methods for implicit complementarity problem

Numerical Algorithms - In this paper, based on the published works by Bai and Tong (J. Univ. Electr. Sci. Tech. China 22:420–424, 1993) and Bai and Huang (J. Univ. Electr. Sci. Tech. China...     

Two-step modulus-based matrix splitting iteration methods for retinex problem

Numerical Algorithms - Retinex theory was proposed by Land in the 1970s. Its theoretical basis is that the color of an object is determined by the reflection ability of the object to the light of...     

A relaxation modulus-based matrix splitting iteration method for solving linear complementarity problems

In this paper, a relaxation modulus-based matrix splitting iteration method is established, which covers the known general modulus-based matrix splitting iteration methods. The convergence analysis and the strategy of the choice of the parameters are given. Numerical examples show that the proposed methods are efficient and accelerate the convergence performance with less iteration steps and CPU times.     

A two-step modulus-based matrix splitting iteration method for horizontal linear complementarity problems

Numerical Algorithms - In this paper, for solving horizontal linear complementarity problems, a two-step modulus-based matrix splitting iteration method is established. The convergence analysis of...     

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 modified modulus-based matrix splitting iteration method for solving implicit complementarity problems

In this paper, a modified modulus-based matrix splitting iteration method is established for solving a class of implicit complementarity problems. The global convergence conditions are given when the system matrix is a positive definite matrix or an H₊-matrix, respectively. In addition, some numerical examples show that the proposed method is efficient.

     

A generalized variant of two-sweep modulus-based matrix splitting iteration method for solving horizontal linear complementarity problems

Numerical Algorithms - Based on the two-sweep modulus-based matrix splitting iteration (TMMS) method for linear complementarity problems developed by Wu and Li (Comput. Appl. Math. 302:...     

A preconditioned general two-step modulus-based matrix splitting iteration method for linear complementarity problems of H+-matrices

Numerical Algorithms - In this paper, we present a preconditioned general two-step modulus-based iteration method to solve a class of linear complementarity problems. Its convergence theory is...     

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.     

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 modulus-based matrix splitting iteration method for a class of nonlinear complementarity problems

Numerical Algorithms - In this paper, we discuss the modulus-based matrix splitting iteration method for solving a class of nonlinear complementarity problems under a weakened condition, and...     

Modulus-based matrix splitting methods for horizontal linear complementarity problems

Numerical Algorithms - In this paper, we extend modulus-based matrix splitting iteration methods to horizontal linear complementarity problems. We consider both standard and accelerated methods and...     

A two-step modulus-based multisplitting iteration method for the nonlinear complementarity problem

In this paper, we construct a two-step modulus-based multisplitting iteration method based on multiple splittings of the system matrix for the nonlinear complementarity problem. And we prove its convergence when the system matrix is an $H$-matrix with positive diagonal elements. Numerical experiments show that the proposed method is efficient.     

Convergence analysis on matrix splitting iteration algorithm for semidefinite linear complementarity problems

Numerical Algorithms - In this paper, we present some novel observations for the semidefinite linear complementarity problems, abbreviated as SDLCPs. Based on these new results, we establish the...