The circulant operator in the banach algebra of matrices

Raymond H. CHAN; Xiao Qing JIN; Man Chung YEUNG

doi:10.1016/0024-3795(91)90324-P

The circulant operator in the banach algebra of matrices

Raymond H. CHAN^*
, Xiao Qing JIN
, Man Chung YEUNG

^*Corresponding author for this work

Research output: Journal Publications › Journal Article (refereed) › peer-review

65 Citations (Scopus)

Abstract

We study an operator c which maps every n-by-n matrix A_n to a circulant matrix c(A_n) that minimizes the Frobenius norm {norm of matrix}A_n - C_n{norm of matrix}F over all n-by-n circulant matrices C_n. The circulant matrix c(A_n), called the optimal circulant preconditioner, has proved to be a good preconditioner for a general class of Toeplitz systems. In this paper, we give different formulations of the operator, discuss its algebraic and geometric properties, and compute its operator norms in different Banach algebras of matrices. Using these results, we are able to give an efficient algorithm for finding the superoptimal circulant preconditioner which is defined to be the minimizer of {norm of matrix}I - C_n^-1A_n{norm of matrix}_F over all nonsingular circulant matrices C_n.

Original language	English
Pages (from-to)	41-53
Number of pages	13
Journal	Linear Algebra and Its Applications
Volume	149
DOIs	https://doi.org/10.1016/0024-3795(91)90324-P
Publication status	Published - 15 Apr 1991
Externally published	Yes

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title = "The circulant operator in the banach algebra of matrices",

abstract = "We study an operator c which maps every n-by-n matrix An to a circulant matrix c(An) that minimizes the Frobenius norm \{norm of matrix\}An - Cn\{norm of matrix\}F over all n-by-n circulant matrices Cn. The circulant matrix c(An), called the optimal circulant preconditioner, has proved to be a good preconditioner for a general class of Toeplitz systems. In this paper, we give different formulations of the operator, discuss its algebraic and geometric properties, and compute its operator norms in different Banach algebras of matrices. Using these results, we are able to give an efficient algorithm for finding the superoptimal circulant preconditioner which is defined to be the minimizer of \{norm of matrix\}I - Cn-1An\{norm of matrix\}F over all nonsingular circulant matrices Cn.",

author = "CHAN, \{Raymond H.\} and JIN, \{Xiao Qing\} and YEUNG, \{Man Chung\}",

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doi = "10.1016/0024-3795(91)90324-P",

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T1 - The circulant operator in the banach algebra of matrices

AU - CHAN, Raymond H.

AU - JIN, Xiao Qing

AU - YEUNG, Man Chung

PY - 1991/4/15

Y1 - 1991/4/15

N2 - We study an operator c which maps every n-by-n matrix An to a circulant matrix c(An) that minimizes the Frobenius norm {norm of matrix}An - Cn{norm of matrix}F over all n-by-n circulant matrices Cn. The circulant matrix c(An), called the optimal circulant preconditioner, has proved to be a good preconditioner for a general class of Toeplitz systems. In this paper, we give different formulations of the operator, discuss its algebraic and geometric properties, and compute its operator norms in different Banach algebras of matrices. Using these results, we are able to give an efficient algorithm for finding the superoptimal circulant preconditioner which is defined to be the minimizer of {norm of matrix}I - Cn-1An{norm of matrix}F over all nonsingular circulant matrices Cn.

AB - We study an operator c which maps every n-by-n matrix An to a circulant matrix c(An) that minimizes the Frobenius norm {norm of matrix}An - Cn{norm of matrix}F over all n-by-n circulant matrices Cn. The circulant matrix c(An), called the optimal circulant preconditioner, has proved to be a good preconditioner for a general class of Toeplitz systems. In this paper, we give different formulations of the operator, discuss its algebraic and geometric properties, and compute its operator norms in different Banach algebras of matrices. Using these results, we are able to give an efficient algorithm for finding the superoptimal circulant preconditioner which is defined to be the minimizer of {norm of matrix}I - Cn-1An{norm of matrix}F over all nonsingular circulant matrices Cn.

UR - https://www.scopus.com/pages/publications/0002520370

U2 - 10.1016/0024-3795(91)90324-P

DO - 10.1016/0024-3795(91)90324-P

M3 - Journal Article (refereed)

AN - SCOPUS:0002520370

SN - 0024-3795

VL - 149

SP - 41

EP - 53

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

ER -

The circulant operator in the banach algebra of matrices

Abstract

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