Abstract
We consider a parallel method for solving generalized eigenvalue problems that arise from molecular orbital computations. We use a moment-based method that finds several eigenvalues and their corresponding eigenvectors in a given domain, which is suitable for master-worker type parallel programming models. The computation of eigenvalues 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.
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References
Arnoldi, W.E.: The principle of minimized iteration in the solution of the matrix eigenproblem. Quarterly of Appl. Math. 9, 17–29 (1951)
Bai, Z.: Krylov subspace techniques for reduced-order modeling of large-scale dynamical systems. Appl. Numer. Math. 43, 9–44 (2002)
Inadomi, Y., Nakano, T., Kitaura, K., Nagashima, U.: Definition of molecular orbitals in fragment molecular orbital method. Chem. Phys. Letters 364, 139–143 (2002)
Kravanja, P., Sakurai, T., Van Barel, M.: On locating clusters of zeros of analytic functions. BIT 39, 646–682 (1999)
Kravanja, P., Sakurai, T., Sugiura, H., Van Barel, M.: A perturbation result for generalized eigenvalue problems and its application to error estimation in a quadrature method for computing zeros of analytic functions. J. Comput. Appl. Math. 161, 339–347 (2003)
Magolu, M.M.M.: Incomplete factorization-based preconditionings for solving the Helmholtz equation. Int. J. Numer. Meth. Engng. 50, 1088–1101 (2001)
Ninf-G: http://ninf.apgrid.org/
Ruhe, A.: Rational Krylov algorithms for nonsymmetric eigenvalue problems II: matrix pairs. Lin. Alg. Appl. 197, 283–295 (1984)
Saad, Y.: Iterative Methods for Large Eigenvalue Problems. Manchester University Press, Manchester (1992)
Sakurai, T., Kravanja, P., Sugiura, H., Van Barel, M.: An error analysis of two related quadrature methods for computing zeros of analytic functions. J. Comput. Appl. Math. 152, 467–480 (2003)
Sakurai, T., Sugiura, H.: A projection method for generalized eigenvalue problems. J. Comput. Appl. Math. 159, 119–128 (2003)
Sakurai, T., Hayakawa, K., Sato, M., Takahashi, D.: A parallel method for large sparse generalized eigenvalue problems by OmniRPC in a grid environment. In: Dongarra, J.J., Madsen, K., Waśniewski, J. (eds.) PARA 2004. LNCS, vol. 3732, pp. 1151–1158. Springer, Heidelberg (2006)
Sakurai, T., Kodaki, Y., Umeda, H., Inadomi, Y., Watanabe, T., Nagashima, U.: A hybrid parallel method for large sparse eigenvalue problems on a grid computing environment using Ninf-G/MPI. In: Lirkov, I., Margenov, S., Waśniewski, J. (eds.) LSSC 2005. LNCS, vol. 3743, pp. 438–445. Springer, Heidelberg (2006)
Sakurai, T.: A Rayleigh-Ritz type method for large-scale generalized eigenvalue problems. In: Abstract of 1st China-Japan-Korea Joint Conference on Computational Mathematics, Sapporo, pp. 50–52 (2006)
van der Vorst, H.A., Melissen, J.B.M.: A Petrov-Galerkin type method for solving A x = b, where A is a symmetric complex matrix. IEEE Trans. on Magnetics 26, 706–708 (1990)
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Sakurai, T. et al. (2007). A Master-Worker Type Eigensolver for Molecular Orbital Computations. In: Kågström, B., Elmroth, E., Dongarra, J., Waśniewski, J. (eds) Applied Parallel Computing. State of the Art in Scientific Computing. PARA 2006. Lecture Notes in Computer Science, vol 4699. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-75755-9_74
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DOI: https://doi.org/10.1007/978-3-540-75755-9_74
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