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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

Circulant preconditioners for Toeplitz matrices with positive continuous generating functions


Authors: Raymond H. Chan and Man-Chung Yeung
Journal: Math. Comp. 58 (1992), 233-240
MSC: Primary 65F10; Secondary 65F15
MathSciNet review: 1106960
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Abstract: We consider the solution of n-by-n Toeplitz systems $ {A_n}x = b$ by the preconditioned conjugate gradient method. The preconditioner $ {C_n}$ is the circulant matrix that minimizes $ {\left\Vert {{B_n} - {A_n}} \right\Vert _F}$ over all circulant matrices $ {B_n}$. We show that if the generating function f is a positive $ 2\pi $-periodic continuous function, then the spectrum of the preconditioned system $ C_n^{ - 1}{A_n}$ will be clustered around one. In particular, if the preconditioned conjugate gradient method is applied to solve the preconditioned system, the convergence rate is superlinear.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1992-1106960-1
PII: S 0025-5718(1992)1106960-1
Keywords: Toeplitz matrix, circulant matrix, preconditioned conjugate gradient method, generating function
Article copyright: © Copyright 1992 American Mathematical Society