Numerical analysis of nonlinear matrix equations

拟议的研究是在应用数学领域。它的目的是推进非线性矩阵方程的研究，在理论和数值方面。 * 非线性矩阵方程出现在各种应用中。例如，他们在纳米研究中发挥重要作用，在经济模型和自动控制问题的研究中，以及在电信，计算机性能和库存控制中出现的随机模型中。对于每种情况，实际感兴趣的解决方案是满足矩阵方程的特定矩阵。我将在一个更一般的环境中研究这些非线性矩阵方程，但要考虑到特定的应用。我的总体目标是设计有效和可靠的算法，以找到所需的解决方案，提供一个仔细的数学分析，这些方程和设计的算法，并把这些算法变成计算机代码，可以很容易地使用的研究人员在各个应用领域。我将讨论的一个特殊问题是如何定量或定性地确定解对方程中参数的依赖性。我建议的研究还包括研究矩阵p次根，这是非线性矩阵方程的解决方案。例如，矩阵的p次根被用在某些金融应用和光学中。我的主要目标是建立一些理论结果，将提供有用的信息，一些数值方法计算矩阵p次根。* 我提出的研究结果将对非线性矩阵方程的理论和应用作出重大贡献，并将有助于提高加拿大在这一重要领域的研究的国际地位。 *** **

The proposed research is in the area of applied mathematics. It aims to advance the study of nonlinear matrix equations, in both theoretical and numerical aspects.  ***Nonlinear matrix equations arise in a variety of applications. For example, they play important roles in nano research, in the study of economic models and problems in automatic control, as well as in stochastic models for queueing problems arising in telecommunications, computer performance and inventory control. For each case, the solution of practical interest is a particular matrix satisfying the matrix equation. I will study these nonlinear matrix equations in a more general setting, but with particular applications in mind. My general objectives are to design efficient and reliable algorithms for finding the desired solutions, to provide a careful mathematical analysis of these equations and the algorithms designed, and to turn these algorithms into computer code that can be readily used by researchers in various application areas. One particular question that I will address is how to determine, quantitatively or qualitatively, the dependence of the solutions on the parameters in the equations. My proposed research also includes the study of matrix pth roots, which are solutions of nonlinear matrix equations. Matrix pth roots are used, for example, in some financial applications and in optics. My main objective is to establish some theoretical results that will provide useful information about some numerical methods for computing the matrix pth roots. ****The results of my proposed research will be a significant contribution to the theory and applications of nonlinear matrix equations, and will help to raise the international status of Canadian research in this important area.  *** **