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Mathematics of Computation

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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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Keywords: Toeplitz matrix, circulant matrix, preconditioned conjugate gradient method, generating function
Article copyright: © Copyright 1992 American Mathematical Society