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http://dx.doi.org/10.4134/CKMS.c170423

PRECONDITIONED SSOR METHODS FOR THE LINEAR COMPLEMENTARITY PROBLEM WITH M-MATRIX  

Zhang, Dan (College of Mathematics and Statistics Northwest Normal University)
Publication Information
Communications of the Korean Mathematical Society / v.34, no.2, 2019 , pp. 657-670 More about this Journal
Abstract
In this paper, we consider the preconditioned iterative methods for solving linear complementarity problem associated with an M-matrix. Based on the generalized Gunawardena's preconditioner, two preconditioned SSOR methods for solving the linear complementarity problem are proposed. The convergence of the proposed methods are analyzed, and the comparison results are derived. The comparison results showed that preconditioned SSOR methods accelerate the convergent rate of the original SSOR method. Numerical examples are used to illustrate the theoretical results.
Keywords
linear complementarity problems; M-matrix; SSOR method; preconditioner; comparison theorem;
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1 B. H. Ahn, Solution of nonsymmetric linear complementarity problems by iterative methods, J. Optim. Theory Appl. 33 (1981), no. 2, 175-185.   DOI
2 O. Axelsson, Iterative Solution Methods, Cambridge University Press, Cambridge, 1994.
3 Z.-Z. Bai, Modulus-based matrix splitting iteration methods for linear complementarity problems, Numer. Linear Algebra Appl. 17 (2010), no. 6, 917-933.   DOI
4 Z.-Z. Bai and D. J. Evans, Matrix multisplitting relaxation methods for linear complementarity problems, Int. J. Comput. Math. 63 (1997), no. 3-4, 309-326.   DOI
5 A. Berman and R. J. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Academic Press, New York, 1979.
6 R. W. Cottle, J.-S. Pang, and R. E. Stone, The Linear Complementarity Problem, Computer Science and Scientific Computing, Academic Press, Inc., Boston, MA, 1992.
7 P.-F. Dai, J.-C. Lia, Y.-T. Lic, and J. Baia, A general preconditioner for linear complementarity problem with an M-matrix, J. Comput. Appl. Math. 317 (2017), 100-112.   DOI
8 J.-L. Dong and M.-Q. Jiang, A modified modulus method for symmetric positive-definite linear complementarity problems, Numer. Linear Algebra Appl. 16 (2009), no. 2, 129-143.   DOI
9 M. Dehghan and M. Hajarian, Convergence of SSOR methods for linear complementarity problems, Oper. Res. Lett. 37 (2009), no. 3, 219-223.   DOI
10 M. Dehghan and M. Hajarian, Improving preconditioned SOR-type iterative methods for L-matrices, Int. J. Numer. Methods Biomed. Eng. 27 (2011), no. 5, 774-784.   DOI
11 M. C. Ferris and Y. Zhang, Foreword: Special issue on mathematical programming in biology and medicine, Math. Program. 101 (2004), no. 2, Ser. B, 297-299.   DOI
12 A. D. Gunawardena, S. K. Jain, and L. Snyder, Modified iterative methods for consistent linear systems, Linear Algebra Appl. 154/156 (1991), 123-143.   DOI
13 A. Hadjidimos, D. Noutsos, and M. Tzoumas, More on modifications and improvements of classical iterative schemes for M-matrices, Linear Algebra Appl. 364 (2003), 253-279.   DOI
14 T. Kohno, H. Kotakemori, and H. Niki, Improving the modified Gauss-Seidel method for Z-matrices, Linear Algebra Appl. 267 (1997), 113-123.   DOI
15 A. Hadjidimos and M. Tzoumas, On the solution of the linear complementarity problem by the generalized accelerated overrelaxation iterative method, J. Optim. Theory Appl. 165 (2015), no. 2, 545-562.   DOI
16 A. Hadjidimos and M. Tzoumas, The solution of the linear complementarity problem by the matrix analogue of the accelerated overrelaxation iterative method, Numer. Algorithms 73 (2016), no. 3, 665-684.   DOI
17 N. W. Kappel and L. T. Watson, Iterative algorithms for the linear complementarity problems, I, Int. J.Comput. Math. 19 (1986), no. 3-4, 273-297.   DOI
18 D. Li, J. Zeng, and Z. Zhang, Gaussian pivoting method for solving linear complementarity problem, Appl. Math. J. Chinese Univ. Ser. B 12 (1997), no. 4, 419-426.   DOI
19 W. Li, The convergence of the modified Gauss-Seidel methods for consistent linear systems, J. Comput. Appl. Math. 154 (2003), no. 1, 97-105.   DOI
20 W. Li and W. Sun, Modified Gauss-Seidel type methods and Jacobi type methods for Z-matrices, Linear Algebra Appl. 317 (2000), no. 1-3, 227-240.   DOI
21 Y. Li and P. Dai, Generalized AOR methods for linear complementarity problem, Appl. Math. Comput. 188 (2007), no. 1, 7-18.   DOI
22 K. G. Murty, Linear Complementarity, Linear and Nonlinear Programming, Sigma Series in Applied Mathematics, 3, Heldermann Verlag, Berlin, 1988.
23 M. T. Yahyapour and S. A. Edalatpanah, Modified SSOR Modelling for Linear Complementarity Problems, Turkish Journal of Analysis and Number Theory. 2 (2014), no. 2, 47-52.   DOI
24 M. Neumann and R. J. Plemmons, Convergence of parallel multisplitting iterative methods for M-matrices, Linear Algebra Appl. 88/89 (1987), 559-573.   DOI
25 R. S. Varga, Matrix Iterative Analysis, Prentice-Hall, Inc., Englewood Cliffs, NJ, 1962.
26 L. Wang and Y. Song, Preconditioned AOR iterative methods for M-matrices, J. Comput. Appl. Math. 226 (2009), no. 1, 114-124.   DOI
27 L.-L. Zhang and Z.-R. Ren, Improved convergence theorems of modulus-based matrix splitting iteration methods for linear complementarity problems, Appl. Math. Lett. 26 (2013), no. 6, 638-642.   DOI
28 N. Zheng and J.-F. Yin, Accelerated modulus-based matrix splitting iteration methods for linear complementarity problem, Numer. Algorithms 64 (2013), no. 2, 245-262.   DOI