| Citation: |
Changfeng Ma, Yuhong Wu, Yajun Xie. THE NEWTON-TYPE SPLITTING ITERATIVE METHOD FOR A CLASS OF COUPLED SYLVESTER-LIKE ABSOLUTE VALUE EQUATION[J]. Journal of Applied Analysis & Computation, 2024, 14(6): 3306-3331. doi: 10.11948/20240006
|
THE NEWTON-TYPE SPLITTING ITERATIVE METHOD FOR A CLASS OF COUPLED SYLVESTER-LIKE ABSOLUTE VALUE EQUATION
-
Abstract
In this paper, the Newton-type splitting iterative method for a class of coupled Sylvester-like absolute value equation is proposed. Some sufficient conditions for the existence of the unique solution of the coupled Sylvester-like absolute value equation are given and sufficient conditions for the nonexistence of solution is discussed. The Newton-base bimatrix splitting iteration method, the Newton-base generalized Gauss-Seidel bimatrix splitting iteration method and the inexact relaxed generalized Newton bimatrix splitting method are proposed to solve the coupled Sylvester-like absolute value equation. Numerical experiments confirm the conclusions proposed in this paper.
-
-
References
[1] L. Abdallah, M. Haddou and T. Migot, Solving absolute value equation using complementarity and smoothing functions, J. Comput. Appl. Math., 2018, 327, 196–207. doi: 10.1016/j.cam.2017.06.019 [2] R. W. Cottle, J. S. Pang and R. E. Stone, The Linear Complementarity Problem, Academic, San Diego, 1992. [3] J. Y. B. Cruz, O. P. Ferreira and L. F. Prudente, On the global convergence of the inexact semi-smooth Newton method for absolute value equation, Comput. Optim. Appl., 2016, 65, 93–108. doi: 10.1007/s10589-016-9837-x [4] B. Hashemi, Sufficient conditions for the solvability of a Sylvester-like absolute value matrix equation, Appl. Math. Lett., 2021, 112, 3. [5] B. H. Huang and C. F. Ma, Convergent conditions of the generalized Newton method for absolute value equation over second order cones, Appl. Math. Let., 2019, 92, 1–6. doi: 10.1016/j.aml.2018.12.021 [6] Y. F. Ke, The new iteration algorithm for absolute value equation, Appl. Math. Lett., 2020, 99, 105990. doi: 10.1016/j.aml.2019.07.021 [7] Y. F. Ke and C. F. Ma, An alternating direction method for nonnegative solutions of the matrix equation $AX+YB=C$, Comput. Appl. Math., 2017, 36, 5. [8] C. F. Ma, Y. F. Ke, J. Tang and B. G. Chen, Numerical Linear Algebra and Algorithms, National Defence Industry Press, Beijing, 2017. [9] O. L. Mangasarian, A generalized Newton method for absolute value equations, Optim. Lett., 2009, 3, 101–108. doi: 10.1007/s11590-008-0094-5 [10] O. L. Mangasarian, Absolute value programming, Comput. Optim. Appli., 2007, 36, 43–53. doi: 10.1007/s10589-006-0395-5 [11] O. L. Mangasarian and R. R. Meyer, Absolute value equations, Linear Algebra Appli., 2006, 419, 359–367. doi: 10.1016/j.laa.2006.05.004 [12] O. L. Mangasarian and R. R. Meyer, Absolute value equations, Linear Algebra Appl., 2006, 419, 359–367. doi: 10.1016/j.laa.2006.05.004 [13] A. Mansoori and M. Erfanian, A dynamic model to solve the absolute value equations, J. Comput. Appl. Math., 2018, 333, 28–35. doi: 10.1016/j.cam.2017.09.032 [14] F. Mezzadri, On the solution of general absolute value equations, Appl. Math. Lett., 2020, 107, 7. [15] O. Prokopyev, On equivalent reformulations for absolute value equations, Comput. Optim. Appli., 2009, 44, 363–372. doi: 10.1007/s10589-007-9158-1 [16] J. Rohn, Systems of linear interval equations, Linear Algebra Appl., 1989, 126, 39–78. doi: 10.1016/0024-3795(89)90004-9 [17] J. Rohn, On unique solvability of the absolute value equation, Optim. Lett., 2009, 3, 603–606. doi: 10.1007/s11590-009-0129-6 [18] J. Rohn, An algorithm for solving the absolute value equation, Electron. J. Linear Algebra, 2009, 18, 589–599. [19] J. Rohn, A theorem of the alternatives for the equation $Ax + B|x|= b$, Linear Multilinear Algebra, 2004, 52, 421–426. doi: 10.1080/0308108042000220686 [20] J. Y. Tang and J. C. Zhou, A quadratically convergent descent method for the absolute value equation $Ax+B|x|=b$, Oper. Res. Lett., 2019, 47, 229–234. doi: 10.1016/j.orl.2019.03.014 [21] A. Wang, Y. Cao, J. X. Chen, et al., Modified Newton-type iteration methods for generalized absolute value equations, J. Optim. Theory Appl., 2019, 181, 216–230. doi: 10.1007/s10957-018-1439-6 [22] S. L. Wu, The unique solution of a class of the new generalized absolute value equation, Appl. Math. Lett., 2021, 116, 107029. doi: 10.1016/j.aml.2021.107029 [23] S. L. Wu, T. Z. Huang and X. L. Zhao, A modified SSOR iterative method for augmented systems, Comput. Appl. Math., 2009, 228, 424–433. doi: 10.1016/j.cam.2008.10.006 [24] S. L. Wu and C. X. Li, A note on unique solvability of the absolute value equation, Optim. Lett., 2020, 14, 1957–1960. [25] S. L. Wu and S. Q. Shen, On the unique solution of the generalized absolute value equation, Optim. Lett., 2020, 6, 1–8. [26] S. F. Xu, Matrix Calculation in Cybernetics, Higher Education Press, Beijing, 2011. -
-
Figures(2) / Tables(2)
Export File
Citation
Changfeng Ma, Yuhong Wu, Yajun Xie. THE NEWTON-TYPE SPLITTING ITERATIVE METHOD FOR A CLASS OF COUPLED SYLVESTER-LIKE ABSOLUTE VALUE EQUATION[J]. Journal of Applied Analysis & Computation, 2024, 14(6): 3306-3331. doi: 10.11948/20240006
Format
Content
DownLoad:
-
Figure 1.
Convergence effect for Example 5.2. When n=49 and n=64, the iteration counts of the SI method and the NHSBSI method are so high that they are not reflected in the figure.
-
Figure 2.
Convergence effect for Example 5.3