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

International Journal of Applied Mathematics and Computer Science

Volumn 16, Issue 2, 2006, Pages 241-249

  Guarracino, Mario Rosario ^a   Perla, Francesca ^b   Zanetti, Paolo ^b  

^a CNR   (Italy)

^b UNIVERSITY OF NAPLES PARTHENOPE   (Italy)

Author keywords

Cluster architecture; Parallel eigensolver; Sparse matrices; Symmetric block Lanczos algorithm

Indexed keywords

ALGORITHMS; EIGENVALUES AND EIGENFUNCTIONS; MATHEMATICAL MODELS; MATRIX ALGEBRA; PARALLEL PROCESSING SYSTEMS;

DISTRIBUTED MEMORY MULTICOMPUTERS; HIGH PERFORMANCE EIGENVALUES COMPUTATION; PARALLEL EIGENSOLVER; SYMMETRIC BLOCK LANCZOS ALGORITHM;

DISTRIBUTED COMPUTER SYSTEMS;

EID: 33745449474     PISSN: 1641876X     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (29)

1. Baglama J., Calvetti D. and Reichel L. Irbleigs (2003): A MATLAB program for computing a few eigenpairs of a large sparse Hermitian matrix. - ACM TOMS, Vol. 29, No. 5, pp. 337-348.
2. Bai Z. (1994): Error analysis of the Lanczos algorithm for the nonsymmetric eigenvalues problem. - Math. Comp., Vol. 62, No. 205, pp. 209-226.
3. Boisvert R.F., Pozo R., Remington K., Barrett R. and Dongarra J. (1997): The Matrix Market: A web resource for test matrix collections, In: Quality of Numerical Software, Assessment and Enhancement (Ronald F. Boisvert, Ed.). - London: Chapman & Hall, pp. 125-137.
4. Choi J., Demmel J., Dhillon I., Dongarra J., Ostrouchov S., Petitet A., Stanley K., Walker D. and Whaley R.C., ScaLAPACK (1996): A portable linear algebra library for distributed memory computers: Design and performance. - Comput. Phys. Comm., Vol. 97, No. 1-2, pp. 1-15.
5. Choi J., Dongarra J., Ostrouchov S., Petitet A., Walker D. and Whaley R.C. (1995): A proposal for a set of parallel basic linear algebra subprograms. - Tech. Rep. UT-CS-95-292, University of Tennessee, Knoxville.
6. Cline A.K., Golub G.H. and Platzman G.W. (1976): Calculations of normal modes of oceans using a Lanczos method, In: Sparse Matrix Computations (J.R. Bunch and D.J. Rose, Eds.). - London: Academic Press.
7. Cullum J.K. and Willoughby R.A. (2002): Lanczos Algorithms for Large Symmetric Eigenvalue Computations, Vol.1: Theory. - Phildelphia: SIAM.
8. Dongarra J., Du Croz J., Hammarling S. and Duff I. (1990): A set of level 3 basic linear algebra subprograms. - ACM Trans. Math. Software, Vol. 16, No. 1, pp. 1-17.
9. Dongarra J. and Whaley R.C. (1995): A user's guide to the BLACS vl.1. - Tech. Rep. UT-CS-95-281, University of Tennessee, Knoxville.
10. Filippone S. and Colajanni M. (2000): PSBLAS: A library for parallel linear algebra computation on sparse matrices. - ACM Trans. Math. Software, Vol. 26, No. 4, pp. 527-550.
11. Golub G.H. and Van Loan C.F. (1989): Matrix Computations, 2nd Ed. - Baltimore, MD: Johns Hopkins Univ. Press.
12. Gropp W., Lusk E. and Skjellum A. (1999): Using MPI - 2nd Edition: Portable Parallel Programming with the Message Passing Interface. - Cambridge, MA: MIT Press.
13. Guarracino M.R. and Perla F. (1994): A parallel block Lanczos algorithm for distributed memory architectures. - J. Par. Alg. Appl., Vol. 4, No. 1-2, pp. 211-221.
14. Guarracino M.R. and Perla F. (1995a): A parallel modified block Lanczos' algorithm for distributed memory architectures, In: Proc. Euromicro Workshop Parallel and Distributed Processing, San Remo, Italy, pp. 424-431.
15. Guarracino M.R. and Perla F. (1995b): A sparse, symmetric eigensolver for distributed memory architectures: Parallel implementation and numerical stability. - Tech. Rep. 11, Center for Research on Parallel Computing and Supercomputers, Naples, Italy.
16. Hernandez V., Roman J.E. and Vidal V. (2003): SLEPc: Scalable library for eigenvalue problem computations. - Lect. Notes Comput. Sci., Vol. 2565, pp. 377-391.
17. Higham N.J. (1995): The test matrix toolbox for MATLAB (version 3.0) - Tech. Rep. Vol. 276, Manchester Centre for Computational Mathematics.
18. Jones M.T. and Patrick M.L. (1989): The use of Lanczos's method to solve the large generalized symmetric definite eigenvalue problem. - Tech. Rep. ICASE 89-67, Langley Research Center.
19. Lanczos C. (1950): An iteration method for the solution of the eigenvalue problem of linear differential and integral operators. - J. Res. Nat. Bur. Stand., Vol. 45, No. 4, pp. 225-281.
20. Lehoucq R.B., Sorensen D.C. and Yang C. (1998): ARPACK Users' Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods. - Phildelphia: SIAM.
21. Paige C. (1976): Error analysis of the Lanczos algorithm for tridiagonalizing a symmetric matrix. - J. Inst. Math. Appl., Vol. 18, No. 3, pp. 341-349.
22. Parlett B.N. (1980): The Symmetric Eigenvalues Problem. - Englewood Cliffs, NJ: Prentice-Hall.
23. Parlett B.N. and Scott D.S. (1979): The Lanczos algorithm with selective orthogonalization. - Math. Comp., Vol. 33, No. 145, pp. 217-238.
24. Saad Y. (1992): Numerical Methods for Large Eigenvalues Problems. - Manchester. Manchester Univ. Press, Halsted Press.
25. Saad Y. and Malevsky A.V., P-SPARSLIB (1995): A portable library of distributed memory sparse iterative solvers, In: Proceedings of Parallel Computings Technologies (V.E. Malyshkin et. al., Eds.) (PaCT-95), 3rd International Conference, St. Petersburg. - Heidelberg: Springer.
26. Webster F. and Lo G. (1996): Projective block Lanczos algorithm for dense, Hermitian eigensystems. - J. Comput. Phys., Vol. 124, No. 1, pp. 146-161.
27. Wilkinson J.H. (1965): The Algebraic Eigenvalue Problem. - Oxford: Clarendon Press.
28. Wu K. and Simon H. (1999a):PArallel efficiency of the Lanczos method for eigenvalue problems. - Tech. Rep. LBNL-42828.
29. Wu K. and Simon H. (1999b): TRLAN user guide. - Tech. Rep. Lawrence Berkeley National Laboratory, LBNL-42953.