Abstract
The general Lanczos method overcomes the breakdown and the block reduction problems which appear in the block nonsymmetric Lanczos method. In the general method we exploit the relation between Lanczos and the two-sided Gram-Schmidt processes. This paper presents a parallel implementation of the general Lanczos method on the massively parallel processor CRAY T3D, where good efficiencies and performances can be obtained.
Supported by the ESPRIT III Basic Research Programm of the EC under contract num. 9072. (Project GEPPCOM).
Supported by the R+D Project GV-1076/93 of the Valencian Community.
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References
J.I. Aliaga (1995), Algoritmos paralelos basados en el método de Lanczos. Aplicaciones en problemas de control, Tesis Doctoral, Departamento de Sistemas Informáticos y Computación, Universidad Politécnica de Valencia.
J.I. Aliaga, D.L. Boley, R.W. Freund and V. Hernández (1996), A Lanczos-type method for multiple starting vectors, TR-96-059, Dept. Comp. Science, Univ. Minnesota, (also submitted to Mathematics of Computation).
D.L. Boley and G.H. Golub (1991), The nonsymmetric Lanczos algorithm and controllability, Systems & Controls Letters, Vol. 16, pp. 97–105, North-Holland.
CRAY Research (1994a), Cray MPP Fortran programmers manual.
CRAY Research (1994b), PVM and HeNCE Programmer's Manual.
J.K. Cullum and R.A. Willoughby (1985), Lanczos algorithms for large symmetric eigenvalue computations, Vol. I Theory and Vol. II Programs, Birkhauser.
I.S. Duff, R. G. Grimes, J.H. Lewis (1989), Sparse matrix test problems, ACM Transactions on Mathematical Software, Vol. 15, pp. 1–14.
R.W. Freund, M.H. Gutknecht and N.M. Nachtigal (1993), An implementation of the look-ahead Lanczos algorithm for non-Hermitian matrices, SIAM Journal of Scientific Computation, Vol. 14, n∘ 14, pp. 137–158.
G.H. Golub and R. Underwood (1977), The block Lanczos method for computing eigenvalues, in Mathematical Soft. III, J.H. Rice Ed., Acad. Press, pp. 364–377.
M.H. Gutknecht (1992), A completed theory of the unsymmetric Lanczos process and related algorithms, part I, SIAM Journal on Matrix Analysis and Applications, Vol. 13, num. 2, pp. 594–639.
H.M. Kim and R.R. Craig Jr (1988), Structural dynamics analysis using an unsymmetric block Lanczos algorithm, International Journal for Numerical Method in Engineering, num. 26, pp. 2305–2318.
C. Lanczos (1950), An iteration method for the solution of the eigenvalue problem of linear differential and integral operators, Journal of Research of the National Bureau of Standards, Vol. 45, pp. 255–282.
B.N. Parlett (1992), Reduction to tridiagonal form and minimal realizations, SIAM Journal on Matrix Analysis and Applications, Vol. 13, num. 2, pp. 567–593.
B.N. Parlett, D.R. Taylor and Z.A. Liu (1985), A look-ahead Lanczos algorithm for unsymmetric matrices, Mathem. of Comput., Vol. 44, num. 169, pp. 105–124.
B.N. Parlett and D.S. Scott (1979), The Lanczos algorithm with selective orthogonalization, Mathem. of Computation, Vol. 33, num. 149, pp. 217–238.
J.L. Pérez, V. Hernández and J. Aliaga (1996), Nucleos Computacionales de Métodos Iterativos de Resolución de Sistemas Lineales y Cálcula de Valores y Vectores Propios. Análisis e Implementación sobre el Multicomputador Masivamente Paralelo Cray T3D, DSIC-II-28/96, DSIC., Univ. Polit. Valencia.
H.D.Simon (1984), The Lanczos algorithm with partial reorthogonalization, Mathematics of Computation, Vol. 42, num. 165, pp.115–142.
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© 1997 Springer-Verlag Berlin Heidelberg
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Aliagal, J.I., Hernández, V., Pérez, J.L. (1997). A Parallel implementation of the general Lanczos method on the CRAY T3D. In: Palma, J.M.L.M., Dongarra, J. (eds) Vector and Parallel Processing — VECPAR'96. VECPAR 1996. Lecture Notes in Computer Science, vol 1215. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-62828-2_119
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DOI: https://doi.org/10.1007/3-540-62828-2_119
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