Which circulant preconditioner is better?

Mathematics of Computation

Volumn 65, Issue 213, 1996, Pages 137-150

  Strela, V V ^a   Tyrtyshnikov, E E ^b  

^a MASSACHUSETTS INSTITUTE OF TECHNOLOGY   (United States)

^b INSTITUTE OF NUMERICAL MATHEMATICS   (Russian Federation)

Author keywords

Circulants; Eigenvalue clustering; Fourier series; Preconditioning; Toeplitz matrices

Indexed keywords

CIRCULANT MATRIX; EIGENVALUE CLUSTERING; FOURIER SERIES; MATRIX DECOMPOSITION; PRECONDITIONING; TOEPLITZ MATRIX;

HARMONIC ANALYSIS;

MATRIX ALGEBRA;

EID: 0030356143     PISSN: 00255718     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0025-5718-96-00682-5     Document Type: Article

References (16)

1. O. Axelsson and G. Lindskog, On the rate of convergence of the preconditioned conjugate gradient method, Numer. Math. 48 (1986), 499-523. MR 88a:65037b
2. R. H. Chan, The spectrum of a family of circulant preconditioned Toeplitz systems, SIAM J. Numer. Anal. 26 (1989), 503-506. MR 90f:65048
3. R. H. Chan, X-Q. Jin, and M.-C. Yeung, The spectra of super-optimal circulant preconditioned Toeplitz systems. SIAM J. Numer. Anal. 28 (1991), 871-879. MR 92a:65099
4. R. H. Chan and G. Strang, Toeplitz equations by conjugate gradients with circulant preconditioner, SIAM J. Sci. Statist. Comput. 10 (1989), 104-119. MR 90d:65069
5. T. Chan, An optimal circulant preconditioner for Toeplitz systems, SIAM J. Sci. Statist. Comput. 9 (1988), 766-771. MR 89e:65046
6. I. Gohberg, T. Kailath, I. Koltracht, and T. Lancaster, Efficient solution of linear systems of equations with recursive structure, Linear Algebra Appl. 80 (1986), 80-113. MR 87i:65058
7. T. Kailath, S. Kung, M. Morf, Displacement ranks of matrices and linear equations, J. Math. Anal. Appl. 68 (1979), 395-407. MR 80k:65029
8. G. Strang, A proposal for Toeplitz matrix calculations, Stud. Appl. Math. 74 (1986), 171-176.
9. V. Strela, Exploration of circulant preconditioning properties. Matrix Methods and Algorithms, IVM RAN, Moscow, 1993, pp. 9-46.
10. M. Tismenetsky, A decomposition of Toeplitz matrices and optimal circulant preconditioning, Linear Algebra Appl. 154/156 (1991), 105-121. MR 92c:65056
11. E. Tyrtyshnikov, Toeplitz matrices, some of their analogues and applications, Dept. Numer. Math., USSR Acad. of Sci., Moscow (1989) (in Russian).
12. _, Optimal and superoptimal circulant preconditioners, SIAM J. Matrix Anal. Appl. 13 (1992), 459-453. MR 92k:65062
13. _, A unifying approach to some old and new theorems on distribution and clustering, Linear Algebra Appl. (to appear).
14. _, Circulant preconditioner s with unbounded inverses, Linear Algebra Appl. 216 (1995), 1-23.
15. _, Fast and parallel inertia finder for Toeplitz expanded matrices, to appear.
16. M. Yeung and R. Chan, Circulant preconditioners for Toeplitz matrices with piecewise continuous generating functions, Math. Comp. 61 (1993), 701-718. MR 94a:65024