Numerical analysis of nonlinear matrix equations

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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.  *** **