Tetsuya Sakurai
, Kentaro Hayakawa
,
Mitsuhisa Sato and Daisuke Takahashi
Affiliation:
Institute of Information Sciences and Electronics,
University of Tsukuba
Master's Program in Science and Engineering,
University of Tsukuba
1-1-1 Tennodai, Tsukuba 305-8573, Japan
E-mail: sakurai@is.tsukuba.ac.jp
In this paper we present a parallel method for finding several
eigenvalues and eigenvectors of a generalized eigenvalue problem
, where 
and 
are large sparse matrices.
A moment-based method to find all the eigenvalues that lie inside a given domain is used. In this method, a small matrix pencil that has only the desired eigenvalues is derived by solving large sparse linear equations constructed from A and B. These equations can be solved independently, thus we solve them on remote nodes in parallel. In this approach, we don't need to exchange data between remote nodes. Therefore the presented method is suitable for master-worker programming models. Moreover, it has a good load balancing property.
We have implemented and tested the method in a grid environment by using a grid RPC (remote procedure call) system called OmniRPC. OmniRPC is a grid PRC system which allows seamless parallel programming in both cluster and gird environments. It supports remote hosts with "ssh" as well as a grid environment with Globus. The program uses automatic-initializable remote module facility of OmniRPC to keep an initial data in a remote executable, when invoked in a remote host.
The performance of the presented method on PC clusters that were used over wide-area network was evaluated. As a test problem, we used the matrices that arise in calculation of the electronic state of molecular hydrogen. The results show that the presented method is efficient in a grid environment.