The best circulant preconditioners for Hermitian Toeplitz systems

SIAM Journal on Numerical Analysis

Volumn 38, Issue 3, 2001, Pages 876-896

  Chan, Raymond H ^a   Yip, Andy M ^b   Ng, Michael K ^b  

^a CHINESE UNIVERSITY OF HONG KONG   (Hong Kong)

^b UNIVERSITY OF HONG KONG   (Hong Kong)

Author keywords

Circulant preconditioner; Kernel functions; Preconditioned conjugate gradient method; Toeplitz systems

Indexed keywords

EID: 0034437059     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036142999354083     Document Type: Article

References (23)

1. G. S. AMMAR AND W. B. GRAGG, Superfast solution of real positive definite Toeplitz systems, SIAM J. Matrix Anal. Appl., 9 (1988), pp. 61-76.
2. O. AXELSSON AND V. BARKER, Finite Element Solution of Boundary Value Problems, Theory and Computation, Academic Press, Orlando, 1984.
3. F. DI BENEDETTO, Analysis of preconditioning techniques for ill-conditioned Toeplitz matrices, SIAM J. Sci. Comput., 16 (1995), pp. 682-697.
4. F. DI BENEDETTO, G. FIORENTINO, AND S. SERRA, C. G. preconditioning for Toeplitz matrices, Comput. Math. Appl., 25 (1993), pp. 35-45.
5. R. CHAN, Circulant preconditioned for Hermitian Toeplitz systems, SIAM J. Matrix Anal. Appl., 10 (1989), pp. 542-550.
6. R. CHAN, Toeplitz preconditioned for Toeplitz systems with nonnegative generating functions, IMA J. Numer. Anal., 11 (1991), pp. 333-345.
7. R. H. CHAN AND M. K. NG, Conjugate gradient methods for Toeplitz systems, SIAM Rev., 38 (1996), pp. 427-482.
8. R. CHAN, T. TSO, AND H. SUN, Circulant preconditioners from B-splines, in Proceedings of the SPIE Symposium on Advanced Signal Processing: Algorithms, Architectures, and Implementations, Vol. 3162, San Diego, 1997, pp. 338-347.
9. R. H. CHAN AND M.-C. YEUNG, Circulant preconditioners constructed from kernels, SIAM J. Numer. Anal., 29 (1992), pp. 1093-1103.
10. R. CHAN, A. YIP, C. LEUNG, AND M. NG, Circulant preconditioners for ill-conditioned Hermitian Toeplitz matrices, in Proceedings of the International Congress of Chinese Mathematicians, Beijing, 1998, pp. 165-174.
11. T. F. CHAN, An optimal circulant preconditioner for Toeplitz systems, SIAM J. Sci. Stat. Comput., 9 (1988), pp. 766-771.
12. L. ELDÉN, An algorithm for the regularization of ill-conditioned, banded least squares problems, SIAM J. Sci. Stat. Comput., 5 (1984). pp. 237-254.
13. R. HORN AND C. JOHNSON, Matrix Analysis, Cambridge University Press, Cambridge, UK, 1985.
14. C. GROETSCH, The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind, Pitman, Boston, 1984.
15. G. LORENTZ, Approximation of Functions, Holt, Rinehart and Winston, New York, 1966.
16. D. POTTS AND G. STEIDL, Preconditioners for ill-conditioned Toeplitz matrices, BIT, 39 (1999), pp. 513-533.
17. D. POTTS AND G. STEIDL, Preconditioned for III-Conditioned Toeplitz Systems Constructed from Positive Kernels, preprint, 1999.
18. S. SERRA, Preconditioning strategies for Hermitian Toeplitz systems with nondefinite generating functions, SIAM J. Matrix Anal. Appl., 17 (1996), pp. 1007-1019.
19. S. SERRA, On the extreme eigenvalues of Hermitian (block) Toeplitz matrices, Linear Algebra Appl., 270 (1998), pp. 109-129.
20. G. STRANG, Introduction to Applied Mathematics, Wellesley-Cambridge Press, Wellesley, MA, 1986.
21. G. STRANG, A proposal for Toeplitz matrix calculations, Stud. Appl. Math., 74 (1986), pp. 171-176.
22. W. F. TRENCH, An algorithm for the inversion of finite Toeplitz matrices, J. Soc. Indust. Appl. Math., 12 (1964), pp. 515-522.
23. E. TYRTYSHNIKOV, Circulant preconditioned with unbounded inverses, Linear Algebra Appl., 216 (1995), pp. 1-24.