Abstract
In this paper, based on the shift splitting technique, a shift splitting (SS) iteration method is presented to solve the generalized absolute value equations. Convergence conditions of the SS method are discussed in detail when the involved matrices are some special matrices. Finally, numerical experiments show the effectiveness of the proposed method.
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 11961082
Funding statement: This research was supported by National Natural Science Foundation of China (No. 11961082).
Acknowledgements
The author would like to thank two anonymous referees for providing helpful suggestions, which greatly improved the paper.
References
[1] O. Axelsson, Z.-Z. Bai and S.-X. Qiu, A class of nested iteration schemes for linear systems with a coefficient matrix with a dominant positive definite symmetric part, Numer. Algorithms 35 (2004), no. 2–4, 351–372. 10.1023/B:NUMA.0000021766.70028.66Search in Google Scholar
[2] Z.-Z. Bai, A class of two-stage iterative methods for systems of weakly nonlinear equations, Numer. Algorithms 14 (1997), no. 4, 295–319. 10.1023/A:1019125332723Search in Google Scholar
[3] Z.-Z. Bai, On the convergence of the multisplitting methods for the linear complementarity problem, SIAM J. Matrix Anal. Appl. 21 (1999), no. 1, 67–78. 10.1137/S0895479897324032Search in Google Scholar
[4] Z.-Z. Bai, Modulus-based matrix splitting iteration methods for linear complementarity problems, Numer. Linear Algebra Appl. 17 (2010), no. 6, 917–933. 10.1002/nla.680Search in Google Scholar
[5] Z.-Z. Bai, G. H. Golub and M. K. Ng, Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, SIAM J. Matrix Anal. Appl. 24 (2003), no. 3, 603–626. 10.1137/S0895479801395458Search in Google Scholar
[6] Z.-Z. Bai and X.-P. Guo, On Newton-HSS methods for systems of nonlinear equations with positive-definite Jacobian matrices, J. Comput. Math. 28 (2010), no. 2, 235–260. 10.4208/jcm.2009.10-m2836Search in Google Scholar
[7] Z.-Z. Bai and A. Hadjidimos, Optimization of extrapolated Cayley transform with non-Hermitian positive definite matrix, Linear Algebra Appl. 463 (2014), 322–339. 10.1016/j.laa.2014.08.021Search in Google Scholar
[8] Z.-Z. Bai and X. Yang, On HSS-based iteration methods for weakly nonlinear systems, Appl. Numer. Math. 59 (2009), no. 12, 2923–2936. 10.1016/j.apnum.2009.06.005Search in Google Scholar
[9] Z.-Z. Bai, J.-F. Yin and Y.-F. Su, A shift-splitting preconditioner for non-Hermitian positive definite matrices, J. Comput. Math. 24 (2006), no. 4, 539–552. Search in Google Scholar
[10] M. Benzi, A generalization of the Hermitian and skew-Hermitian splitting iteration, SIAM J. Matrix Anal. Appl. 31 (2009), no. 2, 360–374. 10.1137/080723181Search in Google Scholar
[11] R. W. Cottle, J.-S. Pang and R. E. Stone, The Linear Complementarity Problem, Academic Press, Boston, 1992. Search in Google Scholar
[12] A. Frommer and G. Mayer, Convergence of relaxed parallel multisplitting methods, Linear Algebra Appl. 119 (1989), 141–152. 10.1016/0024-3795(89)90074-8Search in Google Scholar
[13] X.-P. Guo and I. S. Duff, Semilocal and global convergence of the Newton-HSS method for systems of nonlinear equations, Numer. Linear Algebra Appl. 18 (2011), no. 3, 299–315. 10.1002/nla.713Search in Google Scholar
[14] S.-L. Hu, Z.-H. Huang and Q. Zhang, A generalized Newton method for absolute value equations associated with second order cones, J. Comput. Appl. Math. 235 (2011), no. 5, 1490–1501. 10.1016/j.cam.2010.08.036Search in Google Scholar
[15] O. L. Mangasarian, Absolute value programming, Comput. Optim. Appl. 36 (2007), no. 1, 43–53. 10.1007/s10589-006-0395-5Search in Google Scholar
[16] O. L. Mangasarian, A generalized Newton method for absolute value equations, Optim. Lett. 3 (2009), no. 1, 101–108. 10.1007/s11590-008-0094-5Search in Google Scholar
[17] O. L. Mangasarian, A hybrid algorithm for solving the absolute value equation, Optim. Lett. 9 (2015), no. 7, 1469–1474. 10.1007/s11590-015-0893-4Search in Google Scholar
[18] O. L. Mangasarian and R. R. Meyer, Absolute value equations, Linear Algebra Appl. 419 (2006), no. 2–3, 359–367. 10.1016/j.laa.2006.05.004Search in Google Scholar
[19] C. T. Nguyen, B. Saheya, Y.-L. Chang and J.-S. Chen, Unified smoothing functions for absolute value equation associated with second-order cone, Appl. Numer. Math. 135 (2019), 206–227. 10.1016/j.apnum.2018.08.019Search in Google Scholar
[20]
J. Rohn,
A theorem of the alternatives for the equation
[21] J. Rohn, An algorithm for solving the absolute value equation, Electron. J. Linear Algebra 18 (2009), 589–599. 10.13001/1081-3810.1332Search in Google Scholar
[22] J. Rohn, V. Hooshyarbakhsh and R. Farhadsefat, An iterative method for solving absolute value equations and sufficient conditions for unique solvability, Optim. Lett. 8 (2014), no. 1, 35–44. 10.1007/s11590-012-0560-ySearch in Google Scholar
[23] D. K. Salkuyeh, The Picard-HSS iteration method for absolute value equations, Optim. Lett. 8 (2014), no. 8, 2191–2202. 10.1007/s11590-014-0727-9Search in Google Scholar
[24] S.-L. Wu and P. Guo, Modulus-based matrix splitting algorithms for the quasi-complementarity problems, Appl. Numer. Math. 132 (2018), 127–137. 10.1016/j.apnum.2018.05.017Search in Google Scholar
[25] M.-Z. Zhu and Y.-E. Qi, The nonlinear HSS-like iteration method for absolute value equations, preprint (2018), https://arxiv.org/abs/1403.7013v4. Search in Google Scholar
© 2021 Walter de Gruyter GmbH, Berlin/Boston
