Convergence theory of efficient parametric iterative methods for solving the Yang-Baxter-like matrix equation

Purpose In this study, we present a novel parametric iterative method for computing the polar decomposition and determining the matrix sign function. Design/methodology/approach This method demonstrates exceptional efficiency, requiring only two matrix-by-matrix multiplications and one matrix inversion per iteration. Additionally, we establish that the convergence order of the proposed method is three and four, and confirm that it is asymptotically stable. Findings In conclusion, we extend the iterative method to solve the Yang-Baxter-like matrix equation. The efficiency indices of the proposed methods are shown to be superior compared to previous approaches. Originality/value The efficiency and accuracy of our proposed methods are demonstrated through various high-dimensional numerical examples, highlighting their superiority over established methods.