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

首席研究员和高级研究员 A.埃德尔曼，计划运用奇点的几何技术 理论来分析和解释某些数值的行为， 算法广泛应用于特征系统分析和系统， 控制 这些算法包含了明显不同的 计算矩阵的Jordan标准形的应用， 确定从控制系统到最近的 不可控的系统，并“集群”的零， 多项式 虽然有算法可以解决这些问题 在实践中经常有效， 他们表现不佳的问题。 这些算法 已经被广泛研究，他们的失败也得到了应有的注意，但没有 对它们行为的统一解释还没有出现。 数值线性代数是一个重要的领域， 对，并且在计算的许多领域都有重要的影响。

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.