A parallel block Lanczos algorithm and its implementation for the evaluation of some eigenvalues of large sparse symmetric matrices on multicomputers

In the present work we describe HPEC (High Performance Eigenvalues Computation), a parallel software package for the evaluation of some eigenvalues of a large sparse symmetric matrix. It implements an efficient and portable Block Lanczos algorithm for distributed memory multicomputers. HPEC is based on basic linear algebra operations for sparse and dense matrices, some of which have been derived by ScaLAPACK library modules. Numerical experiments have been carried out to evaluate HPEC performance on a cluster of workstations with test matrices from Matrix Market and Higham’s collections. A comparison with a PARPACK routine is also detailed. Finally, parallel performance is evaluated on random matrices, using standard parameters.

[1]  B. Parlett,et al.  The Lanczos algorithm with selective orthogonalization , 1979 .

[2]  Mario Rosario Guarracino,et al.  A Parallel Block Lanczos Algorithm for Distributed Memory Architectures , 1994, Parallel Algorithms Appl..

[3]  Yousef Saad,et al.  P-Sparslib: A Portable Library Of Distributed Memory Sparse Iterative Solvers , 1995, PACT 1995.

[4]  Kesheng Wu,et al.  Parallel Efficiency of the Lanczos Method for Eigenvalue Problems , 1998, PPSC.

[5]  J. H. Wilkinson.  The algebraic eigenvalue problem , 1966 .

[6]  Jack J. Dongarra,et al.  A set of level 3 basic linear algebra subprograms , 1990, TOMS.

[7]  Michele Colajanni,et al.  PSBLAS: a library for parallel linear algebra computation on sparse matrices , 2000, TOMS.

[8]  Chao Yang,et al.  ARPACK users' guide - solution of large-scale eigenvalue problems with implicitly restarted Arnoldi methods , 1998, Software, environments, tools.

[9]  Lothar Reichel,et al.  Algorithm 827: irbleigs: A MATLAB program for computing a few eigenpairs of a large sparse Hermitian matrix , 2003, TOMS.

[10]  James Demmel,et al.  ScaLAPACK: A Portable Linear Algebra Library for Distributed Memory Computers - Design Issues and Performance , 1995, Proceedings of the 1996 ACM/IEEE Conference on Supercomputing.

[11]  Zhaojun Bai,et al.  Error analysis of the Lanczos algorithm for the nonsymmetric eigenvalue problem , 1994 .

[12]  H. Simon,et al.  TRLAN User Guide , 1999 .

[13]  Mario Rosario Guarracino,et al.  A parallel modified block Lanczos' algorithm for distributed memory architectures , 1995, Proceedings Euromicro Workshop on Parallel and Distributed Processing.

[14]  Jaeyoung Choi,et al.  A Proposal for a Set of Parallel Basic Linear Algebra Subprograms , 1995, PARA.

[15]  F. Webster,et al.  Projective Block Lanczos Algorithm for Dense, Hermitian Eigensystems , 1996 .

[16]  Merrell L. Patrick,et al.  The use of Lanczos's method to solve the large generalized symmetric definite eigenvalue problem , 1989 .

[17]  J. Cullum,et al.  Lanczos Algorithms for Large Symmetric Eigenvalue Computations, Vol. 1 , 2002 .

[18]  C. Paige.  Error Analysis of the Lanczos Algorithm for Tridiagonalizing a Symmetric Matrix , 1976 .

[19]  C. Lanczos.  An iteration method for the solution of the eigenvalue problem of linear differential and integral operators , 1950 .

[20]  Vicente Hernández,et al.  SLEPc: Scalable Library for Eigenvalue Problem Computations , 2002, VECPAR.

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

[22]  Richard F. Barrett,et al.  Matrix Market: a web resource for test matrix collections , 1996, Quality of Numerical Software.

[23]  J. Cullum,et al.  Lanczos Algorithms for Large Symmetric Eigenvalue Computations Vol. I Theory , 1984 .

[24]  Gene H. Golub,et al.  CALCULATION OF NORMAL MODES OF OCEANS USING A LANCZOS METHOD , 1976 .