The Use of Lanczos' Method to Solve the Large Generalized Symmetric Eigenvalue Problem in Parallel

数值线性代数有几个重要的应用 在各种科学和工程应用中。 广义特征值问题Kx= λ Mx， 重要的实际意义，特别是在结构 当振动和屈曲时产生的工程 问题. 基于Lanczos方法的新软件LANZ正在 为解决这些问题而开发。 LANZ使用一种技术 称为动态换档，以提高效率和可靠性 Lanczos算法 一种改进的求解对称 不确定线性系统和寻找特征值的 使用Lanczos方法时出现的三对角矩阵是 被研究。 Parlett和Scott的选择性 正交化算法正在测试中。 的实现 LANZ on a Convex C-220、Cray 2和Cray Y-MP被用于 研究了改进方法的性能，并与 结构工程师使用的子空间迭代代码。 研究 从而在MIMD上实现高效、健壮的LANZ 提出了具有大量处理器的并行系统。

Numerical linear algebra has several important applications in a wide variety of scientific and engineering applications. The generalized eigenvalue problem, Kx= lamda Mx, is of significant practical importance, especially in structural engineering where it arises as the vibration and buckling problem. New software, LANZ, based on Lanczos' method is being developed for solving these problems. LANZ uses a technique called dynamic shifting to improve the efficiency and reliability of the Lanczos algorithm. Improved method for solving symmetric indefinite linear systems and for finding eigenvalues of the tridiagonal matrices that arise when using Lanczos' method are being studied. A modification of Parlett and Scott's selective orthogonalization algorithm is being tested. Implementations of LANZ on a Convex C-220, Cray 2 and Cray Y-MP are being used to study the performance of the improved method and compare it with a subspace iteration code used by structural engineers. Research leading to an efficient, robust implementation of LANZ on MIMD parallel systems with a large number of processors is proposed.