Abstract
In this short paper, a new technology-based two-sweep iteration method for the absolute value equations is proposed, and, by constructing a novel comparison theorem about the norm size of these two matrices A and |A|, the convergence of the above method is given on the premise that the included parameters meet some appropriate conditions. Numerical simulation experiments are presented to verify that our method is more effective and practical than other popular methods.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Rohn J (2004) A theorem of the alternatives for the equation \(Ax+B|x|=b\). Linear Multilinear Algebr 52:421–426
Ketabchi S, Moosaei H (2012) An efficient method for optimal correcting of absolute value equations by minimal changes in the right hand side. Comput Math Appl 64:1882–1885
Ketabchi S, Moosaei H (2012) Minimum norm solution to the absolute value equation in the convex case. J Optim Theory Appl 154:1080–1087
Ketabchi S, Moosaei H, Fallhi S (2013) Optimal error correction of the absolute value equations using a genetic algorithm. Comput Math Model 57:2339–2342
Mangasarian OL, Meyer RR (2006) Absolute value equations. Linear Algebra Appl 419:359–367
Noor MA, Iqbal J, Al-Said E (2012) Residual iterative method for solving absolute value equations. In: Abstr. Appl Anal
Huang BH, Ma CF (2019) Convergent conditions of the generalized Newton method for absolute value equation over second order cones. Appl Math Lett 92:151–157
Lian YY, Li CX, Wu SL (2018) Weaker convergent results of the generalized Newton method for the generalized absolute value equations. J Comput Appl Math 338:221–226
Guo P, Wu SL, Li CX (2019) On the SOR-like iteration method for solving absolute value equations. Appl Math Lett 97:107–113
Cottle RW, Pang JS, Stone RE (1992) The linear complementarity problem. Academic Press, New York
Tseng P (1995) On linear convergence of iterative methods for the variational inequality problem. J Comput Appl Math 60:237–252
Cottle RW, Dantzig GB (1968) Complementary pivot theory of mathematical programming. Linear Algebra Appl 1:103–125
Hadjidimos A, Lapidakis M, Tzoumas M (2011) On iterative solution for linear complementarity problem with an \(H_{+}\)-matrix. SIAM J Matrix Anal Appl 33:97–110
Zheng N, Yin JF (2014) Convergence of accelerated modulus-based matrix splitting iteration methods for linear complementarity problem with an \(H_+\)-matrix. J Comput Appl Math 260:281–293
Mangasarian OL (2009) A generalized Newton method for absolute value equations. Optim Lett 3:101–108
Li CX (2016) A modified generalized Newton method for absolute value equations. J Optim Theory Appl 170:1055–1059
Cvetkovió L (1997) Two-sweep iterative methods. Nonlinear Anal 30:25–30
Woźnicki ZI (1993) Estimation of the optimum relaxation factors in partial factorization iterative methods. SIAM J Matrix Anal Appl 14:59–73
Golub GH, Varga RS (1961) Chebyshev semi-iterative methods, successive overrrelaxation iterative methods, and second order Richardson iterative methods-I. Numer Math 3:147–156
Golub GH, Varga RS (1961) Chebyshev semi-iterative methods, successive overrrelaxation iterative methods, and second order Richardson iterative methods-II. Numer Math 3:157–168
Wu SL, Li CX (2016) Two-sweep modulus-based matrix splitting iteration methods for linear complementarity problems. J Comput Appl Math 302:327–339
Shen SQ, Huang TZ (2006) Convergence and comparison theorems for double splittings of matrices. Comput Math Appl 51:1751–1760
Davis T (2020) University of Florida sparse matrix collection, University of Florida, Gainesville. http://www.cise.ufl.edu/research/sparse/matrices/
Ke YF, Ma CF (2017) SOR-like iteration method for solving absolute value equations. Appl Math Comput 311:195–202
Caccetta L, Qu B, Zhou GL (2011) A globally and quadratically convergent method for absolute value equations. Comput Optim Appl 48:45–58
Bai ZZ (2010) Modulus-based matrix splitting iteration methods for linear complementarity problems. Numer Linear Algebra Appl 17:917–933
Zheng N, Yin JF (2013) Accelerated modulus-based matrix splitting iteration methods for linear complementarity problems. Numer Algor 64:245–261
Zhang LL (2011) Two-step modulus-based matrix splitting iteration method for linear complementarity problems. Numer Algor 57:83–99
Mezzadri F (2020) On the solution of general absolute value equations. Appl Math Lett 107:106462
Ke YF (2020) The new iteration algorithm for absolute value equation. Appl Math Lett 99:105990
Wu SL, Li CX (2020) A note on unique solvability of the absolute value equation. Optim Lett 14:1957–1960
Bu F, Ma CF (2020) The tensor splitting methods for solving tensor absolute value equation. Comput Appl Math 39(3):1–11
Lv CQ, Ma CF (2017) Picard splitting method and Picard CG method for solving the absolute value equation. J Nonlinear Sci Appl 10:3643–3654
Acknowledgements
The authors would like to thank the anonymous referees for providing helpful suggestions, which greatly improved the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The work is supported by the National Natural Science Foundation of China (nos. 11701456, 11801452, 11671060), Fundamental Research Project of Natural Science in Shaanxi Province General Project (Youth) (nos. 2019JQ-415, 2019JQ-196), Fundamental Research Funds for the Central Universities (nos. 2452017219, 2452018017), and the Natural Science Foundation Project of CQ CSTC (no. cstc2019jcyj-msxmX0267)
Rights and permissions
About this article
Cite this article
Zhang, H., Zhang, Y., Li, Y. et al. An improved two-sweep iteration method for absolute value equations. Comp. Appl. Math. 41, 122 (2022). https://doi.org/10.1007/s40314-022-01832-3
Received:
Revised:
Accepted:
Published:
Version of record:
DOI: https://doi.org/10.1007/s40314-022-01832-3