Numerical Methods and Mathematics for Structured Eigenvalue and Inverse Eigenvalue Problems

研究者将设计工程和科学计算中出现的结构特征值问题的数值方法。目标包括改进数值稳定性和改进具有高级架构的计算机的性能。为此，研究人员将完成一小套算法，作为结构保留算法配方中的“成分”。可能的“成分”包括撕裂方法，如分而治之，同时对角化，以及变体QR和Jacobi算法。第二个研究领域是分配特征值的方法，通过状态和输出反馈的复平面区域。区域的鲁棒分配通过使线性控制系统不太可能由于数据中的扰动或不确定性而失败来提高线性控制系统的可靠性。研究方向是选择结构化目标函数和设计专门的优化方法。第三个研究领域是计算测量从可控对到最近的不可控对的距离。这里的成功，希望将推广到一种手段，估计距离的一般矩阵铅笔代数品种的非一般铅笔和一般条件估计。

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.