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

数值线性代数在各种科学和工程应用中有几个重要的应用。广义特征值问题Kx= λ Mx具有重要的实际意义，特别是在结构工程中，它是振动和屈曲问题。为了解决这些问题，正在开发基于Lanczos方法的新软件LANZ。LANZ使用一种称为动态移位的技术来提高Lanczos算法的效率和可靠性。研究了用Lanczos方法求解对称不定线性系统和求三对角矩阵特征值的改进方法。对Parlett和Scott的选择性正交化算法的修改正在测试中。利用LANZ在凸C-220、Cray 2和Cray Y-MP上的实现来研究改进方法的性能，并将其与结构工程师使用的子空间迭代代码进行比较。提出了在具有大量处理器的MIMD并行系统上高效、鲁棒地实现LANZ的研究。

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.