An efficient iterative algorithm for finding a nontrivial symmetric solution of the Yang-Baxter-like matrix equation
Chacha Stephen Chacha
- Mathematics department, Mkwawa University College of Education, (A constituent College of the University of Dar es salaam), P. O. Box 2513, Iringa, Tanzania.
- Mathematics department, Pusan National University, Busan, 46241, Republic of Korea.
This paper presents an efficient iterative method to obtain a nontrivial symmetric solution of the Yang-Baxter-like matrix equation \(AXA=XAX \). Necessary conditions for the convergence of the propounded iterative method are derived. Finally, three numerical examples to illustrate the efficiency of the proposed method and the preciseness of our theoretical results are provided.
- Yang-Baxter matrix equation
- iterative method
- nontrivial solution
- Newton's method
D. Bachiller, F. Cedó, A family of solutions of the Yang-Baxter equation, J. Algebra, 412 (2014), 218–229.
R. J. Baxter, Partition function of the eight–vertex lattice model, Ann. Physics, 70 (1972), 193–228.
F. Cedó, E. Jespers, J. Okniski, Retractability of set theoretic solutions of the Yang-Baxter equation , Adv. Math., 224 (2010), 2472–2484.
J. Ding, N. H. Rhee, A nontrivial solution to a stochastic matrix equation, East Asian J. Appl. Math., 2 (2012), 277–284.
J. Ding, N. H. Rhee , Computing solutions of the Yang-Baxter-like matrix equation for diagonalisable matrices, East Asian J. Appl. Math., 5 (2015), 75–84.
J. Ding, N. H. Rhee, Spectral solutions of the Yang–Baxter matrix equation , J. Math. Anal. Appl., 402 (2013), 567–573. 1
J. Ding, H. Tian , Solving the YangBaxter–like matrix equation for a class of elementary matrices, Comput. Math. Appl., 72 (2016), 1541–1548.
J. Ding, C. Zhang, On the structure of the spectral solutions of the Yang–Baxter matrix equation , Appl. Math. Lett., 35 (2014), 86–89.
J. Ding, C. Zhang, N. H. Rhee, Further solutions of a Yang-Baxter–like matrix equation, East Asian J. Appl. Math., 2 (2013), 352–362.
J. Ding, C. Zhang, N. H. Rhee, Commuting solutions of the Yang-Baxter matrix equation, Appl. Math. Lett., 44 (2015), 1–4.
J. Ding, A. Zhou, Eigenvalues of rank-one updated matrices with some applications, Appl. Math. Lett., 20 (2007), 1223– 1226.
Q. Dong, J. Ding, Complete commuting solutions of the Yang-Baxter-like matrix equation for diagonalizable matrices, Comput. Math. Appl., 72 (2016), 194–201. 1
F. Felix , Nonlinear Equations, Quantum Groups and Duality Theorems: A Primer on the Yang-Baxter Equation , VDM Verlag, Germany (2009)
T. Gateva–Ivanova, Quadratic algebras, Yang-Baxter equation, and Artin–Schelter regularity, Adv. Math., 230 (2012), 2153–2175.
J. Hietarinta , All solutions to the constant quantum Yang-Baxter equation in two dimensions, Phys. Lett. A, 165 (1992), 245–251.
M. Jimbo, Yang-Baxter Equation in Integrable Systems , World Scientific Publishing Co., Teaneck (1989)
P. P. Kulish, E. K. Sklyanin , Solutions of the Yang-Baxter equation, in: Yang-Baxter Equation in Integrable Systems, Adv. Series in Math. Phys., 10 (1990), 172–198.
S. I. A. Mansour, J. Ding, Q. L. Huang , Explicit solutions of the Yang–Baxter–like matrix equation for an idempotent matrix, Appl. Math. Lett., 63 (2017), 71–76.
H. Tian, All solutions of the Yang–Baxter–like matrix equation for rank-one matrices , Appl. Math. Lett., 51 (2016), 55–59.
C. N. Yang, Some exact results for the many–body problem in one dimension with repulsive delta–function interaction, Phys. Rev. Lett., 19 (1967), 1312–1315.
C. Yang, M. Ge , Braid Group, Knot Theory, and Statistical Mechanics, World Scientific Publishing Co., Teaneck (1989)