How to prove that a preconditioner cannot be superlinear

Mathematics of Computation

Volumn 72, Issue 243, 2003, Pages 1305-1316

  Serra Capizzano, S ^a   Tyrtyshnikov, E ^b  

^a UNIVERSITY OF INSUBRIA   (Italy)

^b INSTITUTE OF NUMERICAL MATHEMATICS   (Russian Federation)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0038129707     PISSN: 00255718     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0025-5718-03-01506-0     Document Type: Article

References (28)

1. O. Axelsson and G. Lindskög, The rate of convergence of the preconditioned conjugate gradient method, Numer. Math., 52 (1986), pp. 499-523. MR 88a:65037b
2. D. Bini and M. Capovani, Spectral and computational properties of band symmetric Toeplitz matrices, Linear Algebra Appl., 52/53 (1983), pp. 99-126. MR 85k:15008
3. D. Bini and P. Favati, On a matrix algebra related to the discrete Hartley transform, SIAM J. Matrix Anal. Appl., 14 (1993), pp. 500-507. MR 94h:65026
4. R. Chan and M. Ng, Conjugate gradient methods for Toeplitz matrices, SIAM Revue. 38 (1996), pp. 427-482. MR 971:65048
5. R. Chan, Q. Chang, and H. Sun, Multigrid methods for ill-conditioned symmetric Toeplitz systems, SIAM J. Sci. Comp., 19-2 (1998), pp. 516-529. MR 99b:65030
6. P. Davis, Circulant Matrices. John Wiley and Sons, New York, 1979. MR 81a:15003
7. F. Di Benedetto and S. Serra Capizzano, A unifying approach to abstract matrix algebra preconditioning, Numer. Math., 82-1 (1999), pp. 57-90. MR 2000b:65084
8. F. Di Benedetto and S. Serra Capizzano, Optimal multilevel matrix algebra operators, Linear Multilin. Algebra., 48 (2000), pp. 35-66. MR 2001k:47048
9. G. Fiorentino and S. Serra, Multigrid methods for symmetric positive definite block Toeplitz matrices with nonnegative generating functions, SIAM J. Sci. Comp., 17-5 (1996), pp. 1068-1081. MR 97h:65039
10. W. Gautschi, The condition of Vandermonde-like matrices involving orthogonal polynomials, Linear Algebra Appl., 52/53 (1983), pp. 293-300. MR 84i:65043
11. U. Grenander and G. Szegö, Toeplitz Forms and Their Applications. Second Edition, Chelsea, New York, 1984. MR 88b:42031
12. G. Heinig and K. Rost, Representation of Toeplitz-plus-Hankel matrices using trigonometric transformations with applications to fast matrix-vector multiplication, Linear Algebra Appl., 275/276 (1998), pp. 225-248. MR 99d:65091
13. T. Kailath and V. Olshevsky, Displacement structure approach to discrete-trigonometric-transform based preconditioners of G. Strang type and T. Chan type, Proc. "Workshop on Toeplitz matrices", Cortona (Italy), September 1996. Calcolo, 33 (1996), pp. 191-208.
14. T. Kailath and A Sayed, Displacement structure: theory and applications, SIAM Revue, 37 (1995), pp. 297-386. MR 96h:15015
15. J. Kamm and J. Nagy, Optimal Kronecker product approximation of block Toeplitz matrices, SIAM J. Matrix Anal. Appl., 22-1 (2000), pp. 155-172. MR 2001m:65061
16. D. Noutsos, S. Serra Capizzano and P. Vassalos, Spectral equivalence and matrix algebra preconditioners for multilevel Toeplitz systems: a negative result, Contemp. Math., in press.
17. S. Serra, Preconditioning strategie for asymptotically ill-conditioned block Toeplitz systems, BIT, 34 (1994), pp. 579-594. MR 98a:65052
18. S. Serra Capizzano, Toeplitz preconditioners constructed from linear approximation processes, SIAM J. Matrix Anal. Appl., 20-2 (1998), pp. 446-465. MR 99j:65051
19. S. Serra Capizzano, Korovkin theorems and linear positive Gram matrix algebras approximation of Toeplitz matrices, Linear Algebra Appl., 284 (1998), pp. 307-334. MR 99h:65088
20. S. Serra, A Korovkin-type theory for finite Toeplitz operators via matrix algebras, Numer. Math., 82-1 (1999), pp. 117-142. MR 2000b:65091
21. S. Serra Capizzano and E. Tyrtyshnikov, Any circulant-like preconditioner for multilevel matrices is not superlinear, SIAM J. Matrix Anal. Appl., 21-2 (1999), pp. 431-439. MR 2000i:65049
22. S. Serra Capizzano, Matrix algebra preconditioners for multilevel Toeplitz matrices are not superlinear, Linear Algebra Appl., 343/344 (2002), pp. 303-319.
23. A. van der Sluis and H.A. van der Vorst, The rate of convergence of conjugate gradients, Numer. Math. 48 (1986), pp. 543-560. MR 87h:65061
24. E. Tyrtyshnikov, Circulant preconditioners with unbounded inverses, Linear Algebra Appl., 216 (1995), pp. 1-23. MR 95m:15019
25. E. Tyrtyshnikov, A unifying approach to some old and new theorems on distribution and clustering, Linear Algebra Appl., 232 (1996), pp. 1-43. MR 96m:15018
26. E. Tyrtyshnikov and N. Zamarashkin, Spectra of multilevel Toeplitz matrices: advanced theory via simple matrix relationships, Linear Algebra Appl 270 (1998), pp. 15-27. MR 98i:65030
27. E. Tyrtyshnikov, A Brief Introduction to Numerical Analysis, Birkhauser, Boston, 1997. MR 97m:65005
28. E. Tyrtyshnikov, Distributions and clusters. In: Matrix Methods and Algorithms, Institute of Numerical Mathematics of the Russian Academy of Sciences, 1993, pp. 124-166. MR 98m:65061