Yang-Baxter矩阵方程解的研究与应用

由于矩阵方程AXA=XAX与自由参数Yang—Baxter方程形式上相似，所以这个方程叫做Yang—Baxter矩阵方程。Yang—Baxter方程在统计力学、纽结理论、C*-代数、辫子群和量子理论中是一个研究热点。但即使是这样一个简单的矩阵方程，在矩阵论中却还没有被进行很好的研究。一个原因是要找到所有的解是特别困难的。不管矩阵A是怎样的形式，方程都有 X = 0 和 X = A 这两个平凡解，因此只需求解非平凡解。.本项目将系统、深入地解决Yang—Baxter矩阵方程AXA=XAX的求解问题。给出方程解的解析解、解的性质；运用牛顿迭代算法，Halley方法，Chebyshev方法和super-Halley方法等方法创建方程的数值解，并给出收敛性分析。并为工程和技术人员解决相关应用问题提供科学依据。

The matrix equation AXA=XAX is called here the Yang–Baxter matrix equation since it is similar in format to the one in the parameter-free Yang–Baxter equation. In the fields of statistical mechanics, knot theory, C*-algebra, braid groups, and quantum theory, the Yang–Baxter equation has been a hot research topic. However, to our knowledge even the relevant simple matrix equation has not been seriously explored in matrix theory. One reason is the fact that finding all the solutions is a difficult task. There are two trivial solutions, X = 0 and X = A, no matter what A is, so we seek only nontrivial solutions..This project will be systematic and in depth solve the solutions of the Yang—Baxter matrix equation. The analytic solutions will be given. The property of the solutions will be analyzed. Newton iterative algorithm, Halley algorithm, Chebyshev algorithm, and super-Halley algorithm will be designed. Convergence analysis will be established. The project will provide a scientific basis for engineering and technical personnel to solve the related application problems.