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

1. Introduction

In recent years, much attention has been given in the literature to solve matrix equations (Erfanifar and Hajarian, 2023, 2024a, b, d; Erfanifar et al., 2023). Consider the quadratic matrix equation(1.1)$XAX=AXA,$where $A,X\in {\mathbb{C}}^{n\times n}$, is called the nonlinear Yang-Baxter-like matrix equation (YBLME). The term “Yang-Baxter equation” actually stems from the work of statistical physicists C.N. Yang and R.J. Baxter in the 1970s. The equation that they introduced is a fundamental concept in theoretical physics, especially in the context of integrable systems, statistical mechanics, and quantum groups. The Yang-Baxter equation in mathematics and physics has many applications, including in statistical mechanics, quantum physics, knot theory, and quantum computation (Tian, 2016; Dong and Ding, 2016; Mansour et al., 2017).

The YBLME possesses significant relevance in the realms of completely integrable quantum and classical systems, as well as in exactly solvable models within statistical physics. In recent years, this equation has garnered substantial attention, undergoing intensive study due to its implications and applications. Additionally, its profound connections with areas of mathematics such as group theory and algebraic geometry have become increasingly evident (Kumar et al., 2022; Baxter, 1972).

Integrable quantum theory is a broad field with applications spanning condensed matter and statistical physics, establishing numerous connections to various branches of mathematics. It presents a distinctive avenue to explore the intricacies of quantum field theory, enabling a more profound understanding of relativistic particle theories by treating them as scaling limits of quantum chains and classical lattice systems. This approach offers deeper insights beyond conventional methods. Acquiring knowledge in integrable quantum theory equips theoretical physicists with valuable tools and contributes to a broader comprehension of subjects that are of interest to mathematicians.

Over the last 3 decades, the study of YBLME has extended beyond statistical physics and garnered interest from mathematicians due to its profound connections to various branches of pure mathematics. Notably, recent advancements in knot theory have revealed the formulation of knot invariants using statistical mechanics methodologies. While this novel approach to knot invariants can be explored in diverse contexts, a particularly insightful perspective is provided by statistical mechanics, with YBLME at its core. YBLME’s significance manifests in...