Numerical Methods and Mathematics for Structured Eigenvalue and Inverse Eigenvalue Problems

研究人员将为工程和科学计算中出现的结构化特征值问题设计数值方法。目标包括提高数值稳定性和在具有先进体系结构的计算机上提高性能。为此，研究人员将完成一小组算法，这些算法充当结构保持算法配方中的“成分”。可能的“成分”包括撕裂方法，如分割和征服，同时对角化，以及不同的QR和Jacobi算法。第二个研究领域是通过状态和输出反馈将特征值分配给复杂平面区域的方法。通过使线性控制系统不太可能因数据中的扰动或不确定性而失效，对区域的稳健分配提高了线性控制系统的可靠性。研究将朝着选择结构化目标函数和设计专门的优化方法的方向进行。第三个研究领域是计算从可控对到最近的不可控对的距离。这里的成功有望推广到估计一般矩阵铅笔到非一般铅笔的代数种类的距离的方法和一般条件估计器。

The investigator will design numerical methods for structured eigenvalue problems that arise in Engineering and Scientific computation. Goals include improved numerical stability and improved performance on computers with advanced architectures. Toward this end, the investigator will complete a small set of algorithms that serve as "ingredients" in recipes for structure preserving algorithms. Possible "ingredients" include tearing methods like Divide and Conquer, simultaneous diagonalization, and variant QR and Jacobi algorithms. The second area of study is methods for assigning eigenvalues by state and output feedback to regions of the complex plane. Robust assignment to regions improves reliability of linear control systems by making them less likely to fail due to perturbations or uncertainty in the data. Research will be in the direction of selecting structured objective functions and designing specialized optimization methods. The third area of study is computational measure of the distance from a controllable pair to the nearest uncontrollable pair. Success here hopefully will generalize to a means of estimating the distance of a generic matrix pencil to an algebraic variety of nongeneric pencils and to general condition estimators.