A modified generalized Newton method applied to solving absolute value equations

Shuang Liu; Xiaofeng Wang; Shuang Liu; Xiaofeng Wang

doi:10.3934/era.2026238

Electronic Research Archive

2026, Volume 34, Issue 8: 5352-5366. doi: 10.3934/era.2026238

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A modified generalized Newton method applied to solving absolute value equations

Shuang Liu ,
Xiaofeng Wang ^,

School of Mathematical Sciences, Bohai University, Liaoning 121000, China

Received: 28 March 2026 Revised: 21 May 2026 Accepted: 09 June 2026 Published: 23 June 2026

In this paper, we present a modified generalized Newton method (MGNM) for solving absolute value equation (AVEs) $ Ax - |x| = b $. Some known iterative schemes are special cases of our method. We prove that the generated sequence of iterates is well defined and converges linearly under condition $ \|A^{-1}\| < \frac{1}{3(2t_0+t_1)+1} $ ($ t_0 $ and $ t_1 $ are constants greater than or equal to 0, and not both zero). Compared to Newton's method, the MGNM can guarantee the convergence of the iterative method under weaker conditions. Numerical experiments on various types of problems are conducted to illustrate the superior efficiency and accuracy of the proposed method, showing its superiority over some existing methods.
- absolute value equations,
- modified generalized Newton method,
- convergence
Citation: Shuang Liu, Xiaofeng Wang. A modified generalized Newton method applied to solving absolute value equations[J]. Electronic Research Archive, 2026, 34(8): 5352-5366. doi: 10.3934/era.2026238

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Abstract

In this paper, we present a modified generalized Newton method (MGNM) for solving absolute value equation (AVEs) $ Ax - |x| = b $. Some known iterative schemes are special cases of our method. We prove that the generated sequence of iterates is well defined and converges linearly under condition $ \|A^{-1}\| < \frac{1}{3(2t_0+t_1)+1} $ ($ t_0 $ and $ t_1 $ are constants greater than or equal to 0, and not both zero). Compared to Newton's method, the MGNM can guarantee the convergence of the iterative method under weaker conditions. Numerical experiments on various types of problems are conducted to illustrate the superior efficiency and accuracy of the proposed method, showing its superiority over some existing methods.

References

[1]	J. Rohn, A theorem of the alternatives for the equation $Ax + B\|x\| = b$, Linear Multilinear Algebra, 52 (2004), 421–426. https://doi.org/10.1080/0308108042000220686 doi: 10.1080/0308108042000220686
[2]	O. L. Mangasarian, Absolute value equation solution via concave minimization, Optim. Lett., 1 (2007), 3–8. https://doi.org/10.1007/s11590-006-0005-6 doi: 10.1007/s11590-006-0005-6
[3]	S. J. Chung, NP-completeness of the linear complementarity problem, J. Optim. Theory Appl., 60 (1989), 393–399. https://doi.org/10.1007/BF00940344 doi: 10.1007/BF00940344
[4]	J. M. Ortega, W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, New York: Academic Press, 1970. https://doi.org/10.1137/1.9780898719468
[5]	O. L. Mangasarian, A generalized Newton method for absolute value equations, Optim. Lett., 3 (2009), 101–108. https://doi.org/10.1007/s11590-008-0094-5 doi: 10.1007/s11590-008-0094-5
[6]	J. F. Traub, Iterative Methods for the Solution of Equations, Prentice Hall, New York, 1964. https://doi.org/10.2307/2316079
[7]	F. K. Haghani, On generalized Traub's method for absolute value equations, J. Optim. Theory Appl., 166 (2015), 619–625. https://doi.org/10.1007/s10957-015-0712-1 doi: 10.1007/s10957-015-0712-1
[8]	J. M. Feng, S. Y. Liu, A new two-step iterative method for solving absolute value equations, J. Inequal. Appl., 2019 (2019), 39. https://doi.org/10.1186/s13660-019-1969-y doi: 10.1186/s13660-019-1969-y
[9]	Q. F. Su, A unified model for solving a system of nonlinear equations, Appl. Math. Comput., 290 (2016), 46–55. https://doi.org/10.1016/j.amc.2016.05.047 doi: 10.1016/j.amc.2016.05.047
[10]	N. Gul, H. B. Chen, J. Iqbal, R. Shah, A new two-step iterative technique for efficiently solving absolute value equations, Eng. Comput., 41 (2024), 1272–1284. https://doi.org/10.1108/EC-11-2023-0754 doi: 10.1108/EC-11-2023-0754
[11]	A. Khan, J. Iqbal, A. Akgül, R. Ali, Y. Du, A. Hussain, et al., A Newton-type technique for solving absolute value equations, Alexandria Eng. J., 64 (2023), 291–296. https://doi.org/10.1016/j.aej.2022.08.052 doi: 10.1016/j.aej.2022.08.052
[12]	L. Shi, J. Iqbal, M. Arif, A. Khan, A two-step Newton-type method for solving system of absolute value equations, Math. Probl. Eng., 2020 (2020), 1–7. https://doi.org/10.1155/2020/2798080 doi: 10.1155/2020/2798080
[13]	X. F. Wang, S. N. Guo, Dynamic analysis of a family of iterative methods with fifth-order convergence, Fractal Fract., 9 (2025), 783. https://doi.org/10.3390/fractalfract9120783 doi: 10.3390/fractalfract9120783
[14]	J. Yu, X. F. Wang, A single parameter fourth-order Jarrat-type iterative method for solving nonlinear systems, AIMS Math., 10 (2025), 7847–7863. https://doi.org/10.3934/math.2025360 doi: 10.3934/math.2025360
[15]	Y. T. Guo, Q. S. Wu, X. F. Wang, An extension of high-order Kou's method for solving nonlinear systems and its stability analysis, Electron. Res. Arch., 33 (2025), 1566–1588. https://doi.org/10.3934/era.2025074 doi: 10.3934/era.2025074
[16]	Y. F. Ke, C. F. Ma, SOR-like iteration method for solving absolute value equations, Appl. Math. Comput., 311 (2017), 195–202. https://doi.org/10.1016/j.amc.2017.05.035 doi: 10.1016/j.amc.2017.05.035
[17]	N. Shang, X. F. Wang, Y. Wang, Local convergence analysis of a novel Kurchatov-type derivative-free methods with and without memory for solving nonlinear systems, Int. J. Comput. Methods, 23 (2026), 2550033. https://doi.org/10.1142/S0219876225500331 doi: 10.1142/S0219876225500331
[18]	X. Wang, J. Yu, High-efficiency multi-step derivative-free iterative methods with and without memory for solving nonlinear systems, Numerical Algorithms, 2026. https://doi.org/10.1007/s11075-026-02347-2
[19]	D. D. Ruan, X. F. Wang, Local and semilocal convergences of a fourth-order derivative-free method in Banach spaces and its applications, Eng. Comput., 43 (2026), 1219–1237. https://doi.org/10.1108/EC-06-2025-0639 doi: 10.1108/EC-06-2025-0639
[20]	X. M. Gu, T. Z. Huang, H. B. Li, S. F. Wang, L. Li, Two CSCS-based iteration methods for solving absolute value equations, J. Appl. Anal. Comput., 7 (2017), 1336–1356. https://doi.org/10.11948/2017082 doi: 10.11948/2017082
[21]	O. L. Mangasarian, Absolute value equation solution via dual complementarity, Optim. Lett., 7 (2013), 625–630. https://doi.org/10.1007/s11590-012-0469-5 doi: 10.1007/s11590-012-0469-5
[22]	J. Feng, S. Liu, An improved generalized Newton method for absolute value equations, SpringerPlus, 5 (2016), 1042. https://doi.org/10.1186/s40064-016-2720-5 doi: 10.1186/s40064-016-2720-5
[23]	V. Edalatpour, D. Hezari, D. K. Salkuyeh, A generalization of the Gauss–Seidel iteration method for solving absolute value equations, Appl. Math. Comput., 293 (2017), 156–167. https://doi.org/10.1016/j.amc.2016.08.020 doi: 10.1016/j.amc.2016.08.020
[24]	J. M. Feng, S. Y. Liu, A three-step iterative method for solving absolute value equations, J. Math., 2020 (2020), 1–7. https://doi.org/10.1155/2020/8531403 doi: 10.1155/2020/8531403
[25]	Z. S. Yu, L. Li, Y. Yuan, A modified multivariate spectral gradient algorithm for solving absolute value equations, Appl. Math. Lett., 121 (2021), 107461. https://doi.org/10.1016/j.aml.2021.107461 doi: 10.1016/j.aml.2021.107461
[26]	A. Khan, J. Iqbal, R. Shah, A new efficient two-step iterative method for solving absolute value equations, Eng. Comput., 41 (2024), 597–610. https://doi.org/10.1108/EC-11-2023-0781 doi: 10.1108/EC-11-2023-0781
[27]	P. Guo, J. Iqbal, S. M. Ghufran, M. Arif, R. K. Alhefthi, L. Shi, A new efficient method for absolute value equations, Mathematics, 11 (2023), 3356. https://doi.org/10.3390/math11153356 doi: 10.3390/math11153356
[28]	N. Gul, H. B. Chen, R. Shah, A. Ali, Simpson's three-eighths approach for computing solutions of absolute value equations numerically, J. Appl. Anal. Comput., 15 (2025), 862–875. https://doi.org/10.11948/20240179 doi: 10.11948/20240179
[29]	B. Saheya, C. H. Yu, J. S. Chen, Numerical comparisons based on four smoothing functions for absolute value equation, J. Appl. Math. Comput., 56 (2018), 131–149. https://doi.org/10.1007/s12190-016-1065-0 doi: 10.1007/s12190-016-1065-0
[30]	M. A. Noor, J. Iqbal, K. I. Noor, E. Al-Said, On an iterative method for solving absolute value equations, Optim. Lett., 6 (2012), 1027–1033. https://doi.org/10.1007/s11590-011-0332-0 doi: 10.1007/s11590-011-0332-0

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