Hostname: page-component-76fb5796d-skm99 Total loading time: 0 Render date: 2024-04-25T12:44:29.156Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Partial differential equations, boundary value problems Nonlinear algebraic or transcendental equations Computer aspects of numerical algorithms

Published online by Cambridge University Press:  02 May 2017

Chen-Liang Li  and
Jun-Tao Hong
Chen-Liang Li*
Affiliation:
School of Mathematics and Computing Science, Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation, Guilin and University of Electronic Technology, Guilin, Guangxi, China541004
Jun-Tao Hong
Affiliation:
School of Mathematics and Computing Science, Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation, Guilin and University of Electronic Technology, Guilin, Guangxi, China541004
*
*Corresponding author. Email address:chenliang_li@hotmail.com (C.-L. Li)
Get access

Abstract

We 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.

Keywords

Implicit complementarity problemsynchronousiterative methodmodulus-based multisplitting

MSC classification

Secondary: 65Y05: Parallel computation 65H10: Systems of equations 65N55: Multigrid methods; domain decomposition
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 7 , Issue 2 , May 2017 , pp. 363 - 375
Copyright
Copyright © Global-Science Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Bai, Z-Z., The convergence of parallel iteration algorithms for linear complementarity problems, Computers and Mathematics with Applications 32, 1–17 (1996).Google Scholar
[2] Bai, Z-Z., On the convergence of the multisplitting methods for the linear complementarity problem, SIAM Journal of Matrix Analysis and Applications 21, 67–78 (1999).Google Scholar
[3] Bai, Z-Z., Modulus-based matrix splitting iteration methods for linear complementarity problems, Numerical Linear Algebra with Applications 17, 917–933 (2010).Google Scholar
[4] Bai, Z-Z. and Evans, D.J., Matrix multisplitting relaxation methods for linear complementarity problems, International Journal of Computer Mathematics 63, 309–326 (1997).Google Scholar
[5] Bai, Z-Z and Evans, D.J., Matrix multisplitting methods with applications to linear complementarity problems: Parallel synchronous and chaotic methods, Reseaux et Systemes Repartis: Calculateurs Paralleles 13, 125–154 (2001).Google Scholar
[6] Bai, Z-Z., Sun, J-C. and Wang, D-R., A unified framework for the construction of various matrix multisplitting iterative methods for large sparse system of linear equations, Computers and Mathematics with Applications 32, 51–76 (1996).Google Scholar
[7] Bai, Z-Z and Zhang, L-L., Modulus-based synchronous multisplitting iteration methods for linear complementarity problems, Numerical Linear Algebra with Applications 20, 425–439 (2013).Google Scholar
[8] Bai, Z-Z. and Zhang, L-L., Modulus-based synchronous two-stage multisplitting iteration methods for linear complementarity problems, Numerical Algorithms 62, 59–77 (2013).Google Scholar
[9] Cottle, R.W, Pang, J.-S. and Stone, R.E., The Linear Complementarity Problem, Academic Press (1992).Google Scholar
[10] Cvetkov, C.L. and V, Kostic, A note on the convergence of the MSMAOR method for linear complementarity problems, Numerical Linear Algebra with Applications 21, 534–539 (2014).Google Scholar
[11] Dong, J-L., Inexact multisplitting methods for linear complementarity problems, Journal of Computational and Applied Mathematics 223, 714–724 (2009).Google Scholar
[12] Dong, J-L. and Jiang, M-Q., A modified modulus method for symmetric positive-definite linear complementarity problems, Numerical Linear Algebra with Applications 16, 129–143 (2009).Google Scholar
[13] Ferris, M.C and Pang, J-S., Engineering and economic applications of complementarity problems, SIAM Review 39, 669–713 (1997).Google Scholar
[14] Frommer, A. and Szyld, D.B., H-splittings and two-stage iterative methods, Numerische Mathematik 63, 345–356 (1992).Google Scholar
[15] Hadjidimos, A., Lapidakis, M. and Tzoumas, M., On iterative solution for linear complementarity problems with an H+matrix, SIAM Journal of Matrix Analysis and Applications 33, 97–110 (2012).Google Scholar
[16] Hadjidimos, A. and Tzoumas, M., Nonstationary extrapolated modulus algorithms for the solution of the linear complementarity problem, Linear Algebra and Its Applications 431, 197–210 (2009).Google Scholar
[17] Hong, J-T. and Li, C-L., Modulus-based matrix splitting iteration methods for a class of implicit complementarity problems, Numerical Linear Algebra with Applications 23, 629–641 (2016).Google Scholar
[18] Kapple, N.W. and Waston, L.T., Iterative algorithms for the linear complementarity problems, International Journal of Computer Mathematics 19, 273–297 (1986).Google Scholar
[19] O’Leary, D.P. and White, R.E., Multisplitting of matrices and parallel solution of linear system, SIAM Journal on Algebraic Discrete Methods 6, 630–640 (1985).Google Scholar
[20] Li, C-L., Some New Methods for Variational Inequalities and Complementarity Problems, PhD Thesis, Hunan University, 44-50 (2004)Google Scholar
[21] Li, C-L. and Zeng, J-P., Multisplitting Iteration Schemes for solving a class of nonlinear complementarity problems, Acta Mathematicae Applicatae Sinca 1, 79–90 (2007).Google Scholar
[22] Li, W, A general modulus-based matrix splitting iteration methods for linear complementarity problems of H-matrices, Applied Mathematics Letters 26, 1159–1164 (2013).CrossRefGoogle Scholar
[23] Machida, N., Fukushima, N. and Ibaraki, T., A multisplitting method for symmetric linear complementarity problems, Journal of Computational and Applied Mathematics 62, 217–227 (1995).Google Scholar
[24] Murty, K., Linear Complementarity, Linear and Nonlinear Programming, Heldermann (1988).Google Scholar
[25] Naoki, M., Masao, F. and ToshideI, E., A multisplitting method for symmetric linear complementarity problems, Journal of Computational and Applied Mathematics 62, 217–227 (1995).Google Scholar
[26] Van Bokhoven, W.M.G., Piecewise-linear Modelling and Analysis, Proefschrift: Eindhoven. (1981).Google Scholar
[27] Zhang, L-L., Two-step modulus-based matrix splitting iteration methods for linear complementarity problems, Numerical Algorithms 57, 83–99 (2011).Google Scholar
[28] Zhang, L-L., Two-stage multisplitting iteration methods using modulus-based matrix splitting as inner iteration for linear complementarity problems, Journal of Optimization Theory and Applications 160, 189–203 (2014).Google Scholar
[29] Zhang, L-L., Two-step modulus-based synchronous multisplitting iteration methods for linear complementarity problems, Journal of Computational and Applied Mathematics 33, 100–112 (2015)Google Scholar
[30] Zhang, L-L. and Ren, Z-R., Improved convergence theorems of modulus-based matrix splitting iteration methods for linear complementarity problems, Applied Mathematics Letters 26, 638–642 (2013).Google Scholar
[31] Zheng, N and Yin, J-F., Accelerated modulus-based matrix splitting iteration methods for linear complementarity problem, Numerical Algorithms 64, 245–262 (2013).Google Scholar
[32] Zheng, N and Yin, J-F., Convergence of accelerated modulus-based matrix splitting iteration methods for linear complementarity problem with an H+-matrix, Journal of Computational and Applied Mathematics 260, 281–293 (2014).Google Scholar