A new preconditioner for Toeplitz matrices

IEEE Signal Processing Letters

Volumn 16, Issue 9, 2009, Pages 758-761

  Domínguez Jiménez, María Elena ^a   Ferreira, Paulo J S G ^b  

^a UNIVERSIDAD POLITÉCNICA DE MADRID   (Spain)

^b UNIVERSITY OF AVEIRO   (Portugal)

Author keywords

PCG; Preconditioners; Toeplitz matrices

Indexed keywords

CIRCULANTS; EIGENVALUES; PCG; POSITIVE DEFINITE; PRECONDITIONED CONJUGATE GRADIENT METHOD; PRECONDITIONED MATRIX; PRECONDITIONERS; RAPID CONVERGENCE; SPECTRAL PROPERTIES; SUFFICIENT CONDITIONS; TOEPLITZ; TOEPLITZ MATRICES;

CONJUGATE GRADIENT METHOD; EIGENVALUES AND EIGENFUNCTIONS;

MATRIX ALGEBRA;

EID: 77955420128     PISSN: 10709908     EISSN: None     Source Type: Journal    
DOI: 10.1109/LSP.2009.2024735     Document Type: Article

References (20)

1. K. Chen,Matrix Preconditioning Techniques and Applications. Cambridge, U.K.: Cambridge Univ. Press, 2005.
2. G. H. Golub and C. F. Van Loan, Matrix Computations. Baltimore, MD: The Johns Hopkins Univ. Press, 1996.
3. D. Bini and V. Y. Pan, Polynomial and Matrix Computations. Boston, MA: Birkhäuser, 1994.
4. M. R. Hestenes and E. Stiefel, "Methods of conjugate gradients for solving linear systems," J. Res. Nat. Bureau Stand., vol. 49, no. 6, pp. 409-436, Dec. 1952.
5. T.-K. Ku and C.-C. J. Kuo, "Design and analysis of Toeplitz preconditioners," IEEE Trans. Signal Process., vol. 40, pp. 129-141, Jan. 1992.
6. R. H. Chan and K.-P. Ng, "Toeplitz preconditioners for hermitian Toeplitz-systems," Linear Algebra Applicat., vol. 190, pp. 181-208, 1993.
7. T. Huckle, "Some aspects of circulant preconditioners," SIAM J. Sci. Comput., vol. 14, no. 3, pp. 531-541, May 1993.
8. R. H. Chan and M. K. Ng, "Conjugate gradient methods for Toeplitz systems," SIAM Rev., vol. 38, no. 3, pp. 427-482, Sep. 1996. (Pubitemid 126599620)
9. T. Huckle, "Iterative methods for ill-conditioned Toeplitz matrices," Calcolo, vol. 33, pp. 177-190, 1996. (Pubitemid 126175969)
10. S. Serra, "Optimal, quasi-optimal and superlinear band-Toeplitz preconditioners for asymptotically ill-conditioned positive definite Toeplitz systems," Math. Comput., vol. 66, no. 218, pp. 651-665, Apr. 1997. (Pubitemid 127383116)
11. D. Potts and G. Steidl, "Preconditioners for ill-conditioned Toeplitz matrices," BIT, vol. 39, no. 3, pp. 513-533, 1999.
12. R. H. Chan, A. M. Yip, and M. K. Ng, "The best circulant preconditioners for hermitian Toeplitz systems," SIAM J. Numer. Anal., vol. 38, no. 3, pp. 876-896, 2000. (Pubitemid 33778942)
13. R. H. Chan, M. K. Ng, and A. M. Yip, "The best circulant preconditioners for hermitian Toeplitz systems II: The multiple-zero case," Numer. Math., vol. 92, pp. 17-40, 2002. (Pubitemid 36161518)
14. D. G. Luenberger, Linear and Nonlinear Programming. Reading, MA: Addison-Wesley, 1984.
15. T. Kailath and A. H. Sayed, Fast Reliable Algorithms for MatricesWith Structure. Philadelphia, PA: SIAM, 1999.
16. S. S. Capizzano and E. Tyrtyshnikov, "Any circulant-like preconditioner for multilevel matrices is not superlinear," SIAM J. Matrix Anal. Appl., vol. 21, no. 2, pp. 431-439, 1999.
17. P. J. S. G. Ferreira and M. E. Domínguez, "Trading-off matrix size and matrix structure: Handling Toeplitz Equations by embedding on a larger circulant set,", submitted for publication.
18. P. J. S. G. Ferreira, M. Blanke and T. Söderström, Eds., "Localization of the eigenvalues of Toeplitz matrices using additive decomposition, embedding in circulants, and the Fourier transform," in Proc. SysID'94, 10th IFAC Symposium on System Identification, Copenhagen, Denmark, Jul. 1994, vol. III, pp. 271-276.
19. U. Grenander and G. Szegö, Toeplitz Forms and Their Applications. Los Angeles, CA: Chelsea, 1958.
20. M. Oudin and J. P. Delmas, "Asymptotic generalized eigenvalue distribution of block multilevel Toeplitz matrices," IEEE Trans. Signal Process., vol. 57, pp. 382-387, Jan. 2009.