两类非线性矩阵方程解的研究与应用

本项目研究一类基于格林函数方法处理纳米器件的量子输运的非线性矩阵方程和一类杨—巴克斯特矩阵方程。对于第一类方程，由于参数的变化会导致矩阵方程没有稳定的解，在这种情况下感兴趣的解是有一个特殊的弱稳定复对称解。如何找到这个解、如何创建高精度数值算法将是本项目的研究重点。同时将刻画解的特性、解区间估计以及解的扰动分析。这将使这类非线性矩阵方程的理论和算法研究更为系统和深入。..在物理学中，杨-巴克斯特方程是首个引入统计力学领域的相容方程。简化后的自由参数的杨-巴克斯特方程AXA=XAX叫做杨-巴克斯特矩阵方程。本项目将系统、深入地解决杨-巴克斯特矩阵方程的数值求解问题，并给出扰动分析和收敛性分析。

In this project, a class of nonlinear matrix equation arising from Green's function approach for treating quantum transport in nano devices and a class of Yang-Baxter matrix equation are studied. For the first class equation, one is mainly interested in those values of the parameter for which the matrix equation has no stabilizing solutions. The solution of interest in this case is a special weakly stabilizing complex symmetric solution. How to compute the solution and how to construct high precision numerical algorithm are the focus of research in this project. The property of the solution will be analyzed. The estimation of the solution interval and the perturbation analysis will be established. This project will help to further enrich and expand the theory and algorithms of the nonlinear matrix equation problem more systematic and deepen. ..In physics, the Yang–Baxter equation is a consistency equation which was first introduced in the field of statistical mechanics. 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. This project will be systematic and in depth solve the numerical solutions of the Yang—Baxter matrix equation. The perturbation analysis and convergence analysis will be established.