Fully parallel algorithm for the symmetric eigenvalue problem
In this paper we present a parallel algorithm for the symmetric algebraic eigenvalue problem. The algorithm is based upon a divide and conquer scheme suggested by Cuppen for computing the eigensystem of a symmetric tridiagonal matrix. We extend this idea to obtain a parallel algorithm that retains a number of active parallel processes that is greater than or equal to the initial number throughout the course of the computation. The algorithm does not suffer from loss of parallelism associated with a typical divide and conquer dependency graph. We give a new deflation technique which together with a robust root finding technique will assure computation of an eigensystem to full accuracy in the residuals and in the orthogonality of eigenvectors. A brief analysis of the numerical properties and sensitivity to round off error is presented to indicate where numerical difficulties may occur. We also suggest a way to combine the tridiagonal method with a parallelization of the initial reduction to tridiagonal form to obtain a fully parallel algorithm for the symmetric eigenvalue problems. The algorithm is able to exploit parallelism at all levels of the computation and is well suited to a variety of architectures. We also touch upon implementation and portability issues with particular attention paid to the role of the algorithm as a library subroutine. We discuss a technique used to implement the algorithm which allows the code to be transported to a significant variety of existing parallel parallel computers. Computational results are presented for several machines to demonstrate this. These results are very encouraging with respect to both accuracy and speedup. A surprising result is that the parallel algorithm, even when run in serial mode, can be significantly faster than the best sequential algorithm on large problems, and is effective on moderate size problems when run in serial mode.
- Research Organization:
- Argonne National Lab., IL (USA)
- DOE Contract Number:
- W-31-109-ENG-38; AC05-84OR21400; FG02-85ER25001
- OSTI ID:
- 5870703
- Report Number(s):
- ANL/MCS-TM-62; CONF-8511169-2; CSRD-542; ON: DE86007553
- Resource Relation:
- Conference: 2. Society for Industrial and Applied Mathematics conference on parallel processing for scientific computing, Norfolk, VA, USA, 18 Nov 1985; Other Information: Portions of this document are illegible in microfiche products. Original copy available until stock is exhausted
- Country of Publication:
- United States
- Language:
- English
Similar Records
Implementation of a fully parallel algorithm for the symmetric eigenvalue problem
Parallel solution of the symmetric tridiagonal eigenproblem. Research report