Solving Eigenproblems on Multicomputers: Two Different Approaches

Abstract The authors analyze two direct methods for solving the symmetric eigenproblem of large and sparse matrices in terms of their parallel implementations. Their proposals provide efficient parallel solutions for problems with large computational requirements. The direct solutions were decomposed in the following phases: (1) structuring the input matrix — the Lanczos method with complete reorthogonalization was implemented for this stage; (2) solving the eigenproblem of a structured matrix — two methods were applied to solve this stage, bisection and inverse iteration methods, and the divide-and-conquer method; (3) computing the eigenvectors of the input matrix, carried out by a matrix–matrix product. These methods have been implemented on a multiprocessor system and their performance evaluations carried out on a Cray T3E system with up to 32 nodes. Results show tha t the management of the memory hierarchy improves substantially as the number of processors increases, and that this is one of the reasons why superlinear speedups are obtained.

[1]  J. Bunch,et al.  Rank-one modification of the symmetric eigenproblem , 1978 .

[2]  John G. Lewis,et al.  Sparse matrix test problems , 1982, SGNM.

[3]  H. Bernstein.  An accelerated bisection method for the calculation of eigenvalues of a symmetric tridiagonal matrix , 2015 .

[4]  H. Simon.  Analysis of the symmetric Lanczos algorithm with reorthogonalization methods , 1984 .

[5]  J. Cullum,et al.  Lanczos algorithms for large symmetric eigenvalue computations , 1985 .

[6]  Jack J. Dongarra,et al.  A fully parallel algorithm for the symmetric eigenvalue problem , 1985, PPSC.

[7]  John L. Gustafson,et al.  Reevaluating Amdahl's law , 1988, CACM.

[8]  Ilse C. F. Ipsen,et al.  Solving the Symmetric Tridiagonal Eigenvalue Problem on the Hypercube , 1990, SIAM J. Sci. Comput..

[9]  Peter Weidner,et al.  A parallel algorithm for determining all eigenvalues of large real symmetric tridiagonal matrices , 1992, Parallel Comput..

[10]  Michael W. Berry,et al.  Large-Scale Sparse Singular Value Computations , 1992 .

[11]  Mitsuhisa Sato,et al.  An Experience with Super-Linear Speedup Achieved by Parallel Computing on a Workstation Cluster: Parallel Calculation of Density of States of Large Scale Cyclic Polyacenes , 1995, Parallel Comput..

[12]  Inmaculada García,et al.  Parallel implementation of the Lanczos method for sparse matrices: analysis of data distributions , 1996, ICS '96.

[13]  Inmaculada García,et al.  Evaluation of the Work Load Balance in Irregular Problems using Value Based Data Distributions , 1997, Euro-PDS.

[14]  Inmaculada García,et al.  A Parallel Implementation of the Eigenproblem for Large, Symmetric and Sparse Matrices , 1999, PVM/MPI.

[15]  Jack J. Dongarra,et al.  A Parallel Divide and Conquer Algorithm for the Symmetric Eigenvalue Problem on Distributed Memory Architectures , 1999, SIAM J. Sci. Comput..

[16]  Inmaculada García,et al.  Parallel implementation for large and sparse eigenproblems , 2001, Acta Cybern..