Mathematical Sciences: Eigensystem Computations

沃特金斯 研究人员研究改进的算法，适用于高性能并行计算解决特征值和不变子空间问题。 通过数值实验和理论分析研究了QR算法的新的可并行化公式。 新的移位策略，允许许多迭代一次进行，并详细研究了QR算法的迭代中的移位的传播过程。 另一个目的是研究各种特征系统算法之间的关系，经典的真实的分析和近似理论的相关领域，以及在信号处理和控制中的应用。 这一活动的目的是澄清和统一理论。 特征系统计算--矩阵的特征值和不变子空间的计算--是不同领域许多问题的核心，包括结构（建筑物、桥梁、飞机等）的振动和稳定性， 所有类型的控制系统（机器人，工厂，汽车，飞机）和信号处理（通信，地震石油勘探）。 因此，掌握各种可靠、有效的本征系统计算方法是至关重要的。 本计画的目的是研究和发展分析本征系统的新计算方法。 今天和不久的将来的高性能计算机的组织方式与旧计算机非常不同，需要设计新方法并重新组织旧方法，以有效利用新机器的处理能力。 研究人员研究并开发合适的方法。 另一个目标是组织和统一本征系统计算背后的理论，以造福未来的计算科学家和工程师。

Watkins The investigator studies improved algorithms that are suitable for high-performance parallel computing for solving eigenvalue and invariant subspace problems. New parallelizable formulations of the QR algorithm are investigated through both numerical experiments and theoretical analysis. New shift strategies that allow many iterations to proceed at once are implemented, and the process of propagation of shifts in iterations of the QR algorithm is studied in detail. An additional aim is to study the relationships between various eigensystem algorithms, the related areas of classical real analysis and approximation theory, and applications in signal processing and control. The point of this activity is to clarify and unify the theory. Eigensystem computations --- the calculation of eigenvalues and invariant subspaces of matrices --- lie at the heart of numerous problems in diverse areas, including vibrations and stability of structures (buildings, bridges, aircraft, etc.), control systems of all types (robots, factories, automobiles, aircraft), and signal processing (communications, seismic oil exploration). It is therefore vital to have at hand a variety of reliable, efficient methods for computing eigensystems. The aim of this project is to study and develop new computational methods for analyzing eigensystems. The high-performance computers of today and the near future are organized very differently from older computers and require that new methods be devised and old methods be reorganized to make efficient use of the processing power of the new machines. The investigator studies and develops suitable methods. An additional aim is to organize and unify the theory behind eigensystem computations for the benefit of the computational scientists and engineers of tomorrow.