The absolute value equation (AVE) is particularly important in numerical mathematics, mathematical programming, and optimization. Due to its close connection to linear complementarity problems, significant research efforts have been devoted to developing efficient numerical solvers for AVEs. In this study, we presented a novel iterative methodology that combines Newton's method with fixed-point iteration, known as the Newton technique fixed-point iteration (NTFPI) method, to solve AVEs. The proposed solution integrated the optimal elements of Newton methods and fixed-point iterations to enhance the convergence speed and stabilize calculations. A theoretical examination was performed to determine the convergence characteristics of the approach. In addition, numerical assessments were performed to determine the efficacy of the NTFPI method in relation to other recognized Newton and fixed-point iterative techniques. The findings indicated that the proposed method enhanced accuracy and reduced repetitions, particularly for complex and large-scale tasks. These findings improved the numerical approaches for AVEs and demonstrate their potential applications in engineering, computational economics, and numerical optimization.
Citation: Nifatamah Makaje, Asma Yafad, Aniruth Phon-On. A new two step iterative method based on Newton and fixed point for solving absolute value equations[J]. AIMS Mathematics, 2026, 11(5): 14302-14322. doi: 10.3934/math.2026587
The absolute value equation (AVE) is particularly important in numerical mathematics, mathematical programming, and optimization. Due to its close connection to linear complementarity problems, significant research efforts have been devoted to developing efficient numerical solvers for AVEs. In this study, we presented a novel iterative methodology that combines Newton's method with fixed-point iteration, known as the Newton technique fixed-point iteration (NTFPI) method, to solve AVEs. The proposed solution integrated the optimal elements of Newton methods and fixed-point iterations to enhance the convergence speed and stabilize calculations. A theoretical examination was performed to determine the convergence characteristics of the approach. In addition, numerical assessments were performed to determine the efficacy of the NTFPI method in relation to other recognized Newton and fixed-point iterative techniques. The findings indicated that the proposed method enhanced accuracy and reduced repetitions, particularly for complex and large-scale tasks. These findings improved the numerical approaches for AVEs and demonstrate their potential applications in engineering, computational economics, and numerical optimization.
| [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] |
X. H. Shao, W. C. Zhao, Relaxed modified Newton-based iteration method for generalized absolute value equations, AIMS Math., 8 (2023), 4714–4725. https://doi.org/10.3934/math.2023233 doi: 10.3934/math.2023233
|
| [3] |
S. L. Wu, S. Q. Shen, On the unique solution of the generalized absolute value equation, Optim. Lett., 15 (2021), 2017–2024. https://doi.org/10.1007/s11590-020-01672-2 doi: 10.1007/s11590-020-01672-2
|
| [4] | O. L. Mangasarian, R. R. Meyer, Absolute value equation, Linear Algebra Appl., 419 (2006), 359–367. |
| [5] | R. W. Cottle, G. Dantzig, Complementary pivot theory of mathematical programming, Linear Algebra Appl., 1 (1968), 103–125. |
| [6] |
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
|
| [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] |
O. A. Prokopyev, On equivalent reformulations for absolute value equations, Comput. Optim. Appl., 44 (2009), 363–372. https://doi.org/10.1007/s10589-007-9158-1 doi: 10.1007/s10589-007-9158-1
|
| [9] |
C. X. Li, A modified generalized Newton method for absolute value equations, J. Optim. Theory Appl., 170 (2016), 1055–1059. https://doi.org/10.1007/s10957-016-0956-4 doi: 10.1007/s10957-016-0956-4
|
| [10] |
C. X. Li, A preconditioned AOR iterative method for the absolute value equations, Int. J. Comput. Methods, 14 (2017), 1750016. https://doi.org/10.1142/S0219876217500165 doi: 10.1142/S0219876217500165
|
| [11] | L. Abdallah, M. Haddou, T. Migot, Solving absolute value equation using complementarity and smoothing functions, J. Comput. Appl. Math., 327 (2018), 196–207. |
| [12] | A. J. Fakharzadeh, N. N. Shams, An efficient algorithm for solving absolute value equations, J. Math. Extension, 15 (2021), 1–23. |
| [13] |
R. Ali, Z. Zhang, F. A. Awwad, The study of new fixed-point iteration schemes for solving absolute value equations, Heliyon, 10 (2024), e34505. https://doi.org/10.1016/j.heliyon.2024.e34505 doi: 10.1016/j.heliyon.2024.e34505
|
| [14] |
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
|
| [15] |
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
|
| [16] |
J. Feng, S. 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
|
| [17] |
M. Dehghan, A. Shirilord, Matrix multisplitting Picard-iterative method for solving generalized absolute value matrix equation, Appl. Numer. Math., 158 (2020), 425–438. https://doi.org/10.1016/j.apnum.2020.08.001 doi: 10.1016/j.apnum.2020.08.001
|
| [18] |
A. Khan, J. Iqbal, A. Akgül, R. Ali, Y. Du, A. Hussain, et al., A Newton-type technique for solving absolute value equations, Alex. Eng. J., 64 (2023), 291–296. https://doi.org/10.1016/j.aej.2022.08.052 doi: 10.1016/j.aej.2022.08.052
|
| [19] |
R. Ali, A. Ali, M. M. Alam, A. Mohamed, Numerical solution of the absolute value equations using two matrix splitting fixed point iteration methods, J. Funct. Spaces, 2022 (2022), 7934796. https://doi.org/10.1155/2022/7934796 doi: 10.1155/2022/7934796
|
| [20] |
R. Ali, F. A. Awwad, E. A. A. Ismail, The development of new efficient iterative methods for the solution of absolute value equations, AIMS Math., 9 (2024), 22565–22577. https://doi.org/10.3934/math.20241098 doi: 10.3934/math.20241098
|
| [21] |
R. Ali, K. Pan, A. Ali, Two new iteration methods with optimal parameters for solving absolute value equations, Int. J. Appl. Comput. Math., 8 (2022), 123. https://doi.org/10.1007/s40819-022-01324-2 doi: 10.1007/s40819-022-01324-2
|
| [22] |
R. Ali, K. Pan, Two new fixed point iterative schemes for absolute value equations, Jpn. J. Ind. Appl. Math., 40 (2023), 303–314. https://doi.org/10.1007/s13160-022-00526-x doi: 10.1007/s13160-022-00526-x
|
| [23] |
M. Guo, Q. Wu, Two effective inexact iteration methods for solving the generalized absolute value equations, AIMS Math., 7 (2022), 18675–18689. https://doi.org/10.3934/math.20221027 doi: 10.3934/math.20221027
|
| [24] |
S. X. Miao, X. T. Xiong, J. Wen, On Picard-SHSS iteration method for absolute value equation, AIMS Math., 6 (2021), 1743–1753. https://doi.org/10.3934/math.2021104 doi: 10.3934/math.2021104
|
| [25] |
H. Zhou, S. L. Wu, On the unique solution of a class of absolute value equations $Ax - B|Cx| = d$, AIMS Math., 6 (2021), 8912–8919. https://doi.org/10.3934/math.2021517 doi: 10.3934/math.2021517
|
| [26] |
D. Yu, C. Chen, D. Han, A modified fixed point iteration method for solving the system of absolute value equations, Optimization, 71 (2020), 449–461. https://doi.org/10.1080/02331934.2020.1804568 doi: 10.1080/02331934.2020.1804568
|
| [27] | A. Yafad, N. Makaje, A. Phon-On, Numerical solution of the absolute value equations using two step iterative method, Proceedings 50th International Congress on Science, Technology and Technology-based Innovation (STT 50), 2024. |
| [28] |
W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc., 4 (1953), 506–510. https://doi.org/10.2307/2032162 doi: 10.2307/2032162
|
| [29] |
M. Achache, N. Anane, On unique solvability and Picard's iterative method for absolute value equations, Bull. Transilv. Univ. Brasov Ser. III Math. Comput. Sci., 13 (2021), 13–26. https://doi.org/10.31926/but.mif.2021.1.63.1.2 doi: 10.31926/but.mif.2021.1.63.1.2
|
| [30] |
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
|
| [31] |
N. Anane, M. Achache, Preconditioned conjugate gradient methods for absolute value equations, J. Numer. Anal. Approx. Theory, 49 (2020), 3–14. https://doi.org/10.33993/jnaat491-1197 doi: 10.33993/jnaat491-1197
|
| [32] |
M. Kostadinov, M. Krstić, K. Rajković, M. D. Petković, Adaptive coefficients iterative method for computing matrix inverse, Linear Algebra Appl., 731 (2026), 277–305. https://doi.org/10.1016/j.laa.2025.11.016 doi: 10.1016/j.laa.2025.11.016
|
| [33] |
M. D. Petković, M. S. Petković, Hyper-power methods for the computation of outer inverses, J. Comput. Appl. Math., 278 (2015), 110–118. https://doi.org/10.1016/j.cam.2014.09.024 doi: 10.1016/j.cam.2014.09.024
|
| [34] |
M. D. Petković, M. A. Krstić, K. P. Rajković, Rapid generalized Schultz iterative methods for the computation of outer inverses, J. Comput. Appl. Math., 344 (2018), 572–584. https://doi.org/10.1016/j.cam.2018.05.048 doi: 10.1016/j.cam.2018.05.048
|
Nifatamah Makaje, Asma Yafad, Aniruth Phon-On. A new two step iterative method based on Newton and fixed point for solving absolute value equations[J]. AIMS Mathematics, 2026, 11(5): 14302-14322. doi: 10.3934/math.2026587
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