This article explores the application of exact line search and successive over-relaxation techniques to enhance the convergence of Newton method in solving the Yang-Baxter matrix equation for nontrivial numerical solutions. Moreover, the normwise, mixed, and componentwise condition numbers are derived to assess the sensitivity of solutions. Numerical experiments demonstrate that the exact line search method significantly improves convergence speed, particularly for larger matrices, by reducing both the number of iterations and residuals more effectively than the successive over-relaxation technique. Furthermore, the mixed and componentwise condition numbers consistently yield values close to one, indicating that the Yang-Baxter equation is well-conditioned. In contrast, the relatively high normwise condition numbers suggest an increased sensitivity to perturbations.