Parallel solution of partial symmetric eigenvalue problems from electronic structure calculations

Authors:
T. Auckenthaler;V. Blum;H. -J. Bungartz;T. Huckle;R. Johanni;L. Krämer;B. Lang;H. Lederer;P. R. Willems
Affiliations:
Fakultät für Informatik, Technische Universität München, D-85748 Garching, Germany;Fritz-Haber-Institut der Max-Planck-Gesellschaft, D-14195 Berlin, Germany;Fakultät für Informatik, Technische Universität München, D-85748 Garching, Germany;Fakultät für Informatik, Technische Universität München, D-85748 Garching, Germany;Rechenzentrum Garching der Max-Planck-Gesellschaft am Max-Planck-Institut für Plasmaphysik, D-85748 Garching, Germany;Fachbereich C, Bergische Universität Wuppertal, D-42097 Wuppertal, Germany;Fachbereich C, Bergische Universität Wuppertal, D-42097 Wuppertal, Germany;Rechenzentrum Garching der Max-Planck-Gesellschaft am Max-Planck-Institut für Plasmaphysik, D-85748 Garching, Germany;Fachbereich C, Bergische Universität Wuppertal, D-42097 Wuppertal, Germany
Venue:
Parallel Computing
Year:
2011

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Abstract

The computation of selected eigenvalues and eigenvectors of a symmetric (Hermitian) matrix is an important subtask in many contexts, for example in electronic structure calculations. If a significant portion of the eigensystem is required then typically direct eigensolvers are used. The central three steps are: reduce the matrix to tridiagonal form, compute the eigenpairs of the tridiagonal matrix, and transform the eigenvectors back. To better utilize memory hierarchies, the reduction may be effected in two stages: full to banded, and banded to tridiagonal. Then the back transformation of the eigenvectors also involves two stages. For large problems, the eigensystem calculations can be the computational bottleneck, in particular with large numbers of processors. In this paper we discuss variants of the tridiagonal-to-banded back transformation, improving the parallel efficiency for large numbers of processors as well as the per-processor utilization. We also modify the divide-and-conquer algorithm for symmetric tridiagonal matrices such that it can compute a subset of the eigenpairs at reduced cost. The effectiveness of our modifications is demonstrated with numerical experiments.