Efficient solution of symmetric/Hermitian eigenvalue problems on parallel arrays of vector processors

Research Report MRR97-053

Efficient solution of symmetric/Hermitian eigenvalue problems on parallel arrays of vector processors

David L. Harrar II and Margaret H. Kahn

Abstract: We consider the solution of the eigenvalue problem (EVP) $A u = \lambda u$ , where A is large, dense, and symmetric or Hermitian. These problems arise in a great many applications, perhaps most notably in structural dynamics calculations. The methods presented here are suitable for determining the complete eigendecomposition of A or for computing only selected eigenvalues and, optionally, eigenvectors.

The overall algorithm proceeds in four basic steps: (1) Householder reduction of the dense symmetric or Hermitian matrix A to real tridiagonal form T, (2) calculation of eigenvalues of T via a multisection Sturm-count procedure, (3) computation of eigenvectors of T via inverse iteration, and (4) recovery of eigenvectors of A from those of T. The implementation is targeted at parallel arrays of powerful vector processors, for example the Fujitsu VPP series of supercomputers.

Performance comparisons of our routines with those in the popular LAPACK and ScaLAPACK libraries are made using an application from quantum chemistry and illustrate the efficient vectorization and parallelization of our algorithm and its implementation.

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