Available online at www.isr-publications.com/jnsa J. Nonlinear Sci. Appl., 12 (2019), 21–29 Research Article ISSN: 2008-1898 Journal Homepage: www.isr-publications.com/jnsa An efficient iterative algorithm for finding a nontrivial sym- metric solution of the Yang-Baxter-like matrix equation Chacha Stephen Chacha a,b,∗ , Hyun-Min Kim b,∗ a Mathematics department, Mkwawa University College of Education (A constituent College of the University of Dar es salaam), P. O. Box 2513, Iringa, Tanzania. b Mathematics department, Pusan National University, Busan, 46241, Republic of Korea. Abstract 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. Keywords: Yang-Baxter matrix equation, iterative method, nontrivial solution, Newton’s method. 2010 MSC: 15A24, 65F10, 65H10. c 2019 All rights reserved. 1. Introduction We consider the nonlinear Yang-Baxter matrix equation (YBME) AXA = XAX, (1.1) where A is a given n × n real matrix. The term Yang–Baxter equation was coined a result of independent work by Yang and Baxter in the late 1970s. YBME arises in real life applications in many fields such braid group, quantum field theory, statistical mechanics, quantum groups, differential equations, knot theory, and other disciplines (see [13, 20, 21]). In some current works, YBME has been termed as the Yang-Baxter-like matrix equation ([7, 9, 10, 18]). The study of YBME has continued to be a hot topic of research for the past few decades as well as in recent years (see [1, 3, 4, 14, 16]). With some restrictions on the matrix A, many authors have been striving to exploit all nontrivial solutions (1.1). For instance, Ding and Tian [7] investigated all solutions of YBME by restricting the matrix A = I - v T u, where u and v are two n-dimensional vectors in the complex field such that v T u = 0. Mansour et al. [18] derived an explicit expression for obtaining all solutions of (1.1) ∗ Corresponding authors Email addresses: chchstephen@yahoo.com (Chacha Stephen Chacha), hyunmin@pusan.ac.kr (Hyun-Min Kim) doi: 10.22436/jnsa.012.01.02 Received: 2018-06-14 Revised: 2018-08-08 Accepted: 2018-08-16