A PARALLEL IMPLEMENTATION OF A GENERALIZED LANCZOS PROCEDURE FOR STRUCTURAL DYNAMIC ANALYSIS

The Lanczos method has rapidly become the preferred method of solution for the generalized eigenvalue problems. The recent emergence of para llel computers has aroused much interest in the practical implementati on of the Lanczos algorithm on these high performance computers. This paper describes an implementation of a generalized Lanczos algorithm o n a distributed memory parallel computer, with specific application to structural dynamic analysis. One major cost in the parallel implement ation of the generalized Lanczos procedure is the factorization of the (shifted) stiffness matrix and the forward and backward solution of t riangular systems. In this paper, we review a parallel sparse matrix f actorization scheme and propose a strategy for inverting the principal block submatrix factors to facilitate the forward and backward soluti on of triangular systems on distributed memory parallel computers. We also discuss the different strategies in the implementation of mass-ma trix-vector multiplication and how they are used in the implementation of the Lanczos procedure. The Lanczos procedure implemented includes partial and external selective reorthogonalizations. Spectral shifts a re introduced when memory space is not sufficient for storing the Lanc zos vectors. The tradeoffs between spectral shifts and Lanczos iterati ons are discussed. Numerical results on Intel's parallel computers, th e iPSC/860 hypercube and the Paragon machines will be presented to ill ustrate the effectiveness and scalability of the parallel generalized Lanczos procedure.