We explore the condition numbers of the nonlinear matrix equation − * = . Explicit expressions for the normwise, mixed, and componentwise condition numbers are derived. The upper bounds for the mixed and componentwise condition numbers are obtained. The numerical result favors the fact that our estimations are fairly sharp. Also, the relative upper perturbation bounds give satisfactory results for small perturbations in the input data.

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.

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In this paper, we propose the inversion free iterative method to find symmetric solution of thenonlinear matrix equation 𝑿 − 𝑨∗𝑿𝒒𝑨 = 𝑰 (𝒒 ≥ 𝟐), where 𝑋 is an unknown symmetricsolution, 𝐴 is a given Hermitian matrix and 𝑞 is a positive integer. The convergence of theproposed method is derived. Numerical examples demonstrate that the proposed iterative methodis quite efficient and converges well when the initial guess is sufficiently close to the approximatesolution. Keywords: Symmetric solution, nonlinear matrix equation, inversion free, iterative method