Abstract
In this paper, a general accelerated modulus-based matrix splitting iteration method is established, which covers the known general modulus-based matrix splitting iteration methods and the accelerated modulus-based matrix splitting iteration methods. The convergence analysis is given when the system matrix is an \(H_+\)-matrix. Numerical examples show that the proposed methods are efficient and accelerate the convergence performance with less iteration steps and CPU times.
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References
Ahn, B.H.: Iterative methods for linear complementarity problems with upper-bounds on primary variables. Math. Program. 26, 295–315 (1983)
Berman, A., Plemmons, R.J.: Nonnegative Matrix in the Mathematical Sciences. SIAM Publisher, Philadelphia (1994)
Bai, Z.-Z.: Modulus-based matrix splitting iteration methods for linear complementarity problems. Numer. Linear Algebra Appl. 17, 917–933 (2010)
Bai, Z.-Z.: On the convergence of the multisplitting methods for the linear complementarity problem. SIAM J. Matrix Anal. Appl. 21, 67–78 (1999)
Bai, Z.-Z., Zhang, L.-L.: Modulus-based synchronous multisplitting iteration methods for linear complementarity problems. Numer. Linear Algebra Appl. 20, 425–439 (2013)
Bai, Z.-Z., Zhang, L.-L.: Modulus-based synchronous two-stage multisplitting iteration methods for linear complementarity problems. Numer. Algorithms 62, 59–77 (2013)
Cottle, R.W., Pang, J.-S., Stone, R.E.: The Linear Complementarity Problem. Academic, San Diego (1992)
Dong, J.-L., Jiang, M.-Q.: A modified modulus method for symmetric positive-definite linear complementarity problems. Numer. Linear Algebra Appl. 16, 129–143 (2009)
Frommer, A., Mayer, G.: Convergence of relaxed parallel multisplitting methods. Linear Algebra Appl. 119, 141–152 (1989)
Hadjidimos, A., Lapidakis, M., Tzoumas, M.: On iterative solution for linear complementarity problem with an H-matrix. SIAM J. Matrix Anal. Appl. 33, 97–110 (2011)
Hadjidimos, A., Tzoumas, M.: Nonstationary extrapolated modulus algorithms for the solution of the linear complementarity problem. Linear Algebra Appl. 431, 197–210 (2009)
Hu, J.-G.: Estimates of \(\Vert {B^{-1}C} {B^{-1}C}\Vert _\infty \) and their applications. Math. Numer. Sinica 4, 272–282 (1982)
Li, W.: A general modulus-based matrix splitting method for linear complementarity problems of H- matrices. Appl. Math. Lett. 26, 1159–1164 (2013)
Murty, K.G.: Linear Complementarity, Linear and Nonlinear Programming. Heldermann Verlag, Berlin (1988)
van Bokhoven, W.M.G.: Piecewise-Linear Modelling and Analysis. Proefschrift, Eindhoven (1981)
Zhang, L.-L.: Two-step modulus based matrix splitting iteration for linear complementarity problems. Numer. Algorithms 57, 83–99 (2011)
Zheng, N., Yin, J.-F.: accelerated modulus-based matrix splitting iteration methods for linear complementarity problem. Numer. Algorithms 64, 245–262 (2013)
Acknowledgments
The authors would like to thank the referees for their helpful comments. This work is supported in part by National Natural Science Foundations of China (No. 11271144), Guangdong Provincial Natural Science Foundations (No. s2012010009985) and Project of Department of Education of Guangdong Province (No. 2013KJCX0053).
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Liu, S., Zheng, H. & Li, W. A general accelerated modulus-based matrix splitting iteration method for solving linear complementarity problems. Calcolo 53, 189–199 (2016). https://doi.org/10.1007/s10092-015-0143-2
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DOI: https://doi.org/10.1007/s10092-015-0143-2