Document Type : Original Article

Authors

School of Mathematics and Statistics, Central South University, Changsha 410083, Hunan P.R. China.

Abstract

This paper proposes two modified generalized Gauss-Seidel iteration techniques to determine the Absolute Value Equations (AVEs). Convergence of the new techniques is established under some appropriate conditions lastly; several numerical examples verify the significance of the techniques. 

Keywords

  • Mangasarian, O. L. (2014). Linear complementarity as absolute value equation solution. Optimization letters8(4), 1529-1534. https://doi.org/10.1007/s11590-013-0656-z
  • Prokopyev, O. (2009). On equivalent reformulations for absolute value equations. Computational optimization and applications44(3), 363-372. https://doi.org/10.1007/s10589-007-9158-1
  • Mezzadri, F. (2020). On the solution of general absolute value equations. Applied mathematics letters107, 106462. https://doi.org/10.1016/j.aml.2020.106462
  • Rohn*, J. (2004). A theorem of the alternatives for the equation Ax+ B|x|= b. Linear and multilinear algebra52(6), 421-426. https://doi.org/10.1080/0308108042000220686
  • Mangasarian, O. L., & Meyer, R. R. (2006). Absolute value equations. Linear algebra and its applications419(2-3), 359-367. https://doi.org/10.1016/j.laa.2006.05.004
  • Ali, R., Khan, M. R., Abidi, A., Rasheed, S., & Galal, A. M. (2021). Application of PEST and PEHF in magneto-Williamson nanofluid depending on the suction/injection. Case studies in thermal engineering27, 101329. https://doi.org/10.1016/j.csite.2021.101329
  • Mao, X., Wang, X., Edalatpanah, S. A., & Fallah, M. (2019). The monomial preconditioned SSOR method for linear complementarity problem. IEEE access7, 73649-73655. DOI:1109/ACCESS.2019.2920485
  • Najafi, H. S., & Edalatpanah, S. A. (2012). A kind of symmetrical iterative methods to solve special class of LCP (M, q). International journal of applied mathematics and applications4(2), 183-189.
  • Ullah, I., Ali, R., Nawab, H., Uddin, I., Muhammad, T., Khan, I., & Nisar, K. S. (2022). Theoretical analysis of activation energy effect on Prandtl–Eyring nanoliquid flow subject to melting condition. Journal of non-equilibrium thermodynamics47(1), 1-12. https://doi.org/10.1515/jnet-2020-0092
  • Najafi, H. S., & Edalatpanah, S. A. (2013). SOR-like methods for non-Hermitian positive definite linear complementarity problems. Advanced modeling and optimization15(3), 697-704. https://camo.ici.ro/journal/vol15/v15c10.pdf
  • Salkuyeh, D. K. (2014). The Picard–HSS iteration method for absolute value equations. Optimization letters8(8), 2191-2202. https://doi.org/10.1007/s11590-014-0727-9
  • Ke, Y. F., & Ma, C. F. (2017). SOR-like iteration method for solving absolute value equations. Applied mathematics and computation311, 195-202. https://doi.org/10.1016/j.amc.2017.05.035
  • Chen, C., Yu, D., & Han, D. (2020). Optimal parameter for the SOR-like iteration method for solving the system of absolute value equations. Retrieved from https://doi.org/10.48550/arXiv.2001.05781
  • Edalatpour, V., Hezari, D., & Salkuyeh, D. K. (2017). A generalization of the Gauss–Seidel iteration method for solving absolute value equations. Applied mathematics and computation293, 156-167. https://doi.org/10.1016/j.amc.2016.08.020
  • Wu, S., & Li, C. (2020). A special shift splitting iteration method for absolute value equation. AIMS mathematics5(5), 5171-5183. https://www.aimspress.com/fileOther/PDF/Math/math-05-05-332.pdf
  • Moosaei, H., Ketabchi, S., & Jafari, H. (2015). Minimum norm solution of the absolute value equations via simulated annealing algorithm. Afrika matematika26(7), 1221-1228. https://doi.org/10.1007/s13370-014-0281-8
  • Fakharzadeh, A., & Shams, N. N. (2020). An efficient algorithm for solving absolute value equations. Journal of mathematical extension15. https://ijmex.com/index.php/ijmex/article/view/1393
  • Ali, R., Kejia, P., & Ali, A. (2020). Two generalized successive overrelaxation methods for solving absolute value equations. Mathematics, 40(4), 44-55.
  • Ali, R., Khan, I., Ali, A., & Mohamed, A. (2022). Two new generalized iteration methods for solving absolute value equations using M-matrix. AIMS mathematics7(5), 8176-8187. http://www.aimspress.com/aimspress-data/math/2022/5/PDF/math-07-05-455.pdf
  • Ali, R. (2022). The solution of absolute value equations using two new generalized gauss-seidel iteration methods. Computational and mathematical methods2022. https://doi.org/10.1155/2022/4266576
  • Ali, R., Ali, A., Alam, M. M., & Mohamed, A. (2022). Numerical solution of the absolute value equations using two matrix splitting fixed point iteration methods. Journal of function spaces2022. https://doi.org/10.1155/2022/7934796
  • Ali, R., Ali, A., & Iqbal, S. (2022). Iterative methods for solving absolute value equations. The journal of mathematics and computer science26, 322-329.
  • Ali, R. (2022). Numerical solution of the absolute value equation using modified iteration methods. Computational and mathematical methods2022. https://downloads.hindawi.com/journals/cmm/2022/2828457.pdf
  • Khan, A., Iqbal, J., Akgül, A., Ali, R., Du, Y., Hussain, A., ... & Vijayakumar, V. (2022). A newton-type technique for solving absolute value equations. Alexandria engineering journal. https://doi.org/10.1016/j.aej.2022.08.052
  • Li, C. X. (2017). A preconditioned AOR iterative method for the absolute value equations. International journal of computational methods14(02), 1750016. https://doi.org/10.1142/S0219876217500165
  • Guo, P., Wu, S. L., & Li, C. X. (2019). On the SOR-like iteration method for solving absolute value equations. Applied mathematics letters97, 107-113. https://doi.org/10.1016/j.aml.2019.03.033