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.
