Abstract
In this work we describe several portable sequential and parallel algorithms for solving the inverse eigenproblem for Real Symmetric Toeplitz matrices. The algorithms are based on Newton’s method (and some variations), for solving nonlinear systems. We exploit the structure and some properties of Toeplitz matrices to reduce the cost, and use Finite Difference techniques to approximate the Jacobian matrix. With this approach, the storage cost is considerably reduced, compared with parallel algorithms proposed by other authors. Furthermore, all the algorithms are efficient in computational cost terms. We have implemented the parallel algorithms using the parallel numerical linear algebra library SCALAPACK based on the MPI environment. Experimental results have been obtained using two different architectures: a shared memory multiprocessor, the SGI PowerChallenge, and a cluster of Pentium II PC’s connected through a Myrinet network. The algorithms obtained show a good scalability in most cases.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Anderson E., Bai Z., Bischof C., Demmel J., Dongarra J., Du Croz J., Greenbaum A., Hammarling S., Mackenney A., Ostrouchov S., Sorensen D. (1995). LAPACK Users’ Guide. Second edition. SIAM Publications, Philadelphia.
Badía J.M., Vidal A.M. (1997). On the bisection Method for the Computation of the Eigenvalues of Symmetric Tridiagonal Matrices. Technical Report. DI 01-02/97. Departamento de Inform’atica. Universidad Jaime I. Castellón.
Badía J.M., Vidal A.M. (1999). Solving the Inverse Toeplitz Using SCALAPACK and MPI. Lecture Notes in Computer Science, 1697 (1999), pp. 372–379. Ed. Springer.Verlag.
Blackford L.S., Choi J., Cleary A., D’Azevedo E., Demmel J. Dhillon I., Dongarra J., Hammarling S., Henry G., Petitet A., Stanley K., Walker D., Whaley R.C. (1997). SCALAPACK Users’ Guide. SIAM Publications, Philadelphia.
Cantoni, A. and Butler, F. (1976). Eigenvalues and eigenvectors of Symmetric Centrosymmetric Matrices. Lin. Alg. Appl., no. 13. pp. 275–288.
Chu M.T. (1998). Inverse Eigenvalue Problems. SIAM Rev., vol. 40, no.1, pp. 1–39.
Demmel, J., MacKenney, A. (1992). A Test Matrix Generation Suite, LAPACK Working Note 9. Courant Institute, New York.
Dongarra J.J., Dunigan T. (1995). Message-Passing Performance of Various Computers. Technical Report UT-CS-95-299. Departtment of Computer Science. University of Tenesse.
Golub G. H., Van Loan C. F. (1996). Matrix Computations (third edition). John Hopkins University Press.
Kelley C.T. (1995). Iterative Methods for Linear and Nonlinear Equations. Frontiers in Applied Mathematics, SIAM Publications, Philadelphia.
Kumar R., Grama A., Gupta A., Karypis G. (1994). Introduction to Parallel Computing: Design and Analysis of Algorithms. The Benjamin Cumimngs Publishing Company.
Laurie D.P. (1988). A numerical approach to the inverse Toeplitz eigenproblem matrix, J. Sci. Statist.Comput., 9 (1988), pp.401–405.
Ly T.Y., Zeng Z. (1993). The Laguerre iteration in solving the symmetric tridiagonal eigenproblem, revisited. SIAM J. Sci. Statist. Compt., vol. 13, no. 5, pp 1145–1173.
Myrinet. Myrinet overview. (Online). http://www.myri.com/myrinet/overview/index.html.
Moré J.J., Garbow B.S., Hillstrom K.E, (1980). User Guide for MINPACK-1. Technical report. ANL-80-74. Argonne National Laboratory.
Peinado J, Vidal A.M. (1999). A new Parallel approach to the Toeplitz Inverse Eigenproblem using the Newton’s Method and Finite Difference techniques. Technical Report. DSIC-II/15/99. Departamento de Sistemas Informáticos y Computación. Universidad Politécnica de Valencia.
Sun, X.H. (1995). The relation of Scalability and Execution Time. Internal Report. 95-62, ICASE NASA Langley Research Center, Hampton, VA, 1995.
Skjellum A. (1994). Using MPI: Portable Programming with Message-Passing Interface. MIT Press Group.
Trench, W.F. (1997). Numerical Solution of the Inverse Eigenvalue Problem for Real Symmetric Toeplitz Matrices. SIAM J. Sci. Comput., vol. 18, no. 6. pp. 1722–1736.
Whaley R.C. (1994). Basic Linear Algebra Communication Subprograms (BLACS): Analysis and Implementation Across Multiple Parallel Architectures. Computer Science Dept. Technical Report CS 94-234, University of Tennesee, Knoxville.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Peinado, J., Vidal, A.M. (2001). A New Parallel Approach to the Toeplitz Inverse Eigenproblem Using Newton-like Methods. In: Palma, J.M.L.M., Dongarra, J., Hernández, V. (eds) Vector and Parallel Processing — VECPAR 2000. VECPAR 2000. Lecture Notes in Computer Science, vol 1981. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44942-6_29
Download citation
DOI: https://doi.org/10.1007/3-540-44942-6_29
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-41999-0
Online ISBN: 978-3-540-44942-3
eBook Packages: Springer Book Archive