Mathematical Sciences: Analyzing Failures in Algorithms forGeneralized Eigenproblems Using Singularity Theory

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The Principal Investigator, together with senior researcher A. Edelman, plan to apply geometric techniques of singularity theory to analyze and explain the behavior of certain numerical algorithms widely used in eigensystem analysis and systems and control. These algorithms include such apparently diverse applications as computing the Jordan canonical form of a matrix, determining the distance from a control system to the nearest uncontrollable system, and "clustering" the zeros of a polynomial. Although there are algorithms for these problems that often work in practice, there are certain classes of problems for which they perform quite poorly. These algorithms have been widely studied and their failures duly noted, but no unifying explanation of their behavior has yet emerged. Numerical Linear Algebra is an area of importance in its own right and has major implications in many areas of computing.