TY - JOUR
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 - http://www.scopus.com/inward/record.url?scp=0002520370&partnerID=8YFLogxK
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 -