Which Circulant Preconditioner is Better?

Authors:
V. V. Strela and E. E. Tyrtyshnikov

Journal:
Math. Comp. **65** (1996), 137-150

MSC (1991):
Primary 15A18, 15A57, 65F15; Secondary 42A16, 15A23

DOI:
https://doi.org/10.1090/S0025-5718-96-00682-5

MathSciNet review:
1325875

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Abstract | References | Similar Articles | Additional Information

Abstract: The eigenvalue clustering of matrices and is experimentally studied, where , and respectively are Toeplitz matrices, Strang, and optimal circulant preconditioners generated by the Fourier expansion of a function . Some illustrations are given to show how the clustering depends on the smoothness of and which preconditioner is preferable. An original technique for experimental exploration of the clustering rate is presented. This technique is based on the bisection idea and on the Toeplitz decomposition of a three-matrix product , where is a Toeplitz matrix and is a circulant. In particular, it is proved that the Toeplitz (displacement) rank of is not greater than 4, provided that and are symmetric.

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Additional Information

**V. V. Strela**

Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

Email:
strela@math.mit.edu

**E. E. Tyrtyshnikov**

Affiliation:
Institute of Numerical Mathematics, Russian Academy of Sciences, Leninskij Prosp., 32–A, 117334, Moscow, Russia

Email:
tee@adonis.iasnet.com

DOI:
https://doi.org/10.1090/S0025-5718-96-00682-5

Keywords:
Preconditioning,
eigenvalue clustering,
circulants,
Toeplitz matrices,
Fourier series

Received by editor(s):
December 28, 1993

Received by editor(s) in revised form:
August 3, 1994

Article copyright:
© Copyright 1996
American Mathematical Society