A Master-Worker Type Eigensolver for Molecular Orbital Computations Tetsuya Sakurai 1,3 , Yoshihisa Kodaki 2 , Hiroto Tadano 3 , Hiroaki Umeda 3,4 , Yuichi Inadomi 5 , Toshio Watanabe 3,4 , and Umpei Nagashima 3,4 1 Department of Computer Science, University of Tsukuba, Tsukuba 305-8573, Japan sakurai@cs.tsukuba.ac.jp 2 Graduate School of Systems and Information Engineering, University of Tsukuba, Tsukuba 305-8573, Japan 3 Core Research for Evolutional Science and Technology, Japan Science and Technology Agency, Japan 4 Computational Science Research Division, National Institute of Advanced Industrial Science and Technology 5 Computing and Communications Center, Kyushu University, Fukuoka 812-8581, Japan Abstract. We consider a parallel method for solving generalized eigen- value problems that arise from molecular orbital computations. We use a moment-based method that finds several eigenvalues and their corre- sponding eigenvectors in a given domain, which is suitable for master- worker type parallel programming models. The computation of eigen- values using explicit moments is sometimes numerically unstable. We show that a Rayleigh-Ritz procedure can be used to avoid the use of explicit moments. As a test problem, we use the matrices that arise in the calculation of molecular orbitals. We report the performance of the application of the proposed method with several PC clusters connected through a hybrid MPI and GridRPC system. 1 Introduction Generalized eigenvalue problems arise in many scientific and engineering appli- cations. Several methods for such eigenvalue problems are building sequences of subspaces that contain the desired eigenvectors. Krylov subspace based tech- niques are powerful tools for large-scale eigenvalue problems [1,2,8,9]. The rela- tions among Krylov subspace methods, moment-matching approach and Pad´ e approximation are shown in [2]. In this paper we consider a parallel method for finding several eigenvalues and eigenvectors of generalized eigenvalue problems in a grid computing environment. A master-worker type algorithm is efficient to obtain a good performance with distributed computing resources. However, it is not easy to adjust eigensolvers for such type algorithms because of iterative processes of solvers. We use a method using a contour integral proposed in [11] to find eigenvalues that lie inside a given domain. In this method, a small matrix pencil that has B. K˚ agstr¨om et al. (Eds.): PARA 2006, LNCS 4699, pp. 617–625, 2007. c  Springer-Verlag Berlin Heidelberg 2007