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2006 | 16 | 2 | 241-249
Tytuł artykułu

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

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Rocznik
Tom
16
Numer
2
Strony
241-249
Opis fizyczny
Daty
wydano
2006
otrzymano
2005-04-13
poprawiono
2005-10-16
Twórcy
  • Institute for High Performance Computing and Networking - ICAR-CNR, Via P. Castellino, 111-80131 Naples, Italy
  • University of Naples Parthenope, Via Medina, 40-80133 Naples, Italy
  • University of Naples Parthenope, Via Medina, 40-80133 Naples, Italy
Bibliografia
  • Baglama J., Calvetti D. and Reichel L. Irbleigs (2003): A MATLAB program for computing a few eigenpairs of a large sparsei Hermitian matrix. - ACM TOMS, Vol. 29, No. 5, pp. 337-348.
  • Bai Z. (1994): Error analysis of the Lanczos algorithm for the nonsymmetric eigenvalues problem. - Math. Comp., Vol. 62, No. 205, pp. 209-226.
  • 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 and Hall, pp. 125-137.
  • Choi J., Demmel J., Dhillon I., Dongarra J., Ostrouchov S., Petitet A., Stanley K., Walker D. and Whaley R.C., ScaLAPACK (1996): Aportable linear algebra library for distributed memory computers: Design and performance. - Comput. Phys. Comm., Vol. 97, No. 1-2, pp. 1-15.
  • 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.
  • Cline A.K., Golub G.H. and Platzman G.W. (1976): Calculationsof normal modes of oceans using a Lanczos method, In: Sparse Matrix Computations (J.R. Bunch and D.J. Rose, Eds.). - London: Academic Press.
  • Cullum J.K. and Willoughby R.A. (2002): Lanczos Algorithmsfor Large Symmetric Eigenvalue Computations, Vol.I: Theory. - Phildelphia: SIAM.
  • Dongarra J., Du Croz J., Hammarling S. and Duff I. (1990): A set of level 3 basic linear algebra subprograms. - ACMi Trans. Math. Software, Vol. 16, No. 1, pp. 1-17.
  • Dongarra J. and Whaley R.C. (1995): A user's guide to the BLACS v1.1. - Tech. Rep. UT-CS-95-281, University of Tennessee, Knoxville.
  • 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.
  • Golub G.H. and Van Loan C.F. (1989): Matrix Computations, 2nd Ed. - Baltimore, MD: Johns Hopkins Univ. Press.
  • Gropp W., Lusk E. and Skjellum A. (1999): Using MPI - 2nd Edition: Portable Parallel Programming with the Message Passing Interface. - Cambridge, MA: MIT Press.
  • 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.
  • 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.
  • 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.
  • 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.
  • Higham N.J. (1995): The test matrix toolbox for MATLAB(version 3.0) - Tech. Rep. Vol. 276, Manchester Centre for Computational Mathematics.
  • 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. ICASE89-67, Langley Research Center.
  • 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.
  • 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.
  • 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.
  • Parlett B.N. (1980): The Symmetric Eigenvalues Problem. - Englewood Cliffs, NJ: Prentice-Hall.
  • Parlett B.N. and Scott D.S. (1979): The Lanczos algorithm with selective orthogonalization. - Math. Comp., Vol. 33, No. 145, pp. 217-238.
  • Saad Y. (1992): Numerical Methods for Large Eigenvalues Problems. - Manchester: Manchester Univ. Press, Halsted Press.
  • 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.
  • Webster F. and Lo G. (1996): Projective block Lanczos algorithm for dense, Hermitianeigensystems. - J. Comput. Phys., Vol. 124, No. 1, pp. 146-161.
  • Wilkinson J.H. (1965): The Algebraic Eigenvalue Problem. - Oxford: Clarendon Press.
  • Wu K. and Simon H. (1999a): Parallel efficiency of the Lanczos method for eigenvalue problems. - Tech. Rep. LBNL-42828.
  • Wu K. and Simon H. (1999b): TRLAN user guide. - Tech. Rep. Lawrence Berkeley National Laboratory, LBNL-42953.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-amcv16i2p241bwm
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