Circulant preconditioners for toeplitz matrices with positive continuous generating functions

Raymond H. CHAN; Man Chung YEUNG

doi:10.1090/S0025-5718-1992-1106960-1

Circulant preconditioners for toeplitz matrices with positive continuous generating functions

Raymond H. CHAN^*
, Man Chung YEUNG

^*Corresponding author for this work

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

31 Citations (Scopus)

Abstract

We consider the solution of n-by-n Toeplitz systems A_nx = b by the preconditioned conjugate gradient method. The preconditioner C_n is the circulant matrix that minimizes ∥B_n − A_n∥_F over all circulant matrices B_n. We show that if the generating function f is a positive 2π-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.

Original language	English
Pages (from-to)	233-240
Number of pages	8
Journal	Mathematics of Computation
Volume	58
Issue number	197
DOIs	https://doi.org/10.1090/S0025-5718-1992-1106960-1
Publication status	Published - Jan 1992
Externally published	Yes

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title = "Circulant preconditioners for toeplitz matrices with positive continuous generating functions",

abstract = "We consider the solution of n-by-n Toeplitz systems Anx = b by the preconditioned conjugate gradient method. The preconditioner Cn is the circulant matrix that minimizes ∥Bn − An∥F over all circulant matrices Bn. We show that if the generating function f is a positive 2π-periodic continuous function, then the spectrum of the preconditioned system Cn−1 An 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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AU - CHAN, Raymond H.

AU - YEUNG, Man Chung

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N2 - We consider the solution of n-by-n Toeplitz systems Anx = b by the preconditioned conjugate gradient method. The preconditioner Cn is the circulant matrix that minimizes ∥Bn − An∥F over all circulant matrices Bn. We show that if the generating function f is a positive 2π-periodic continuous function, then the spectrum of the preconditioned system Cn−1 An 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.

AB - We consider the solution of n-by-n Toeplitz systems Anx = b by the preconditioned conjugate gradient method. The preconditioner Cn is the circulant matrix that minimizes ∥Bn − An∥F over all circulant matrices Bn. We show that if the generating function f is a positive 2π-periodic continuous function, then the spectrum of the preconditioned system Cn−1 An 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.

UR - https://www.scopus.com/pages/publications/84968515314

U2 - 10.1090/S0025-5718-1992-1106960-1

DO - 10.1090/S0025-5718-1992-1106960-1

M3 - Journal Article (refereed)

AN - SCOPUS:84968515314

SN - 0025-5718

VL - 58

SP - 233

EP - 240

JO - Mathematics of Computation

JF - Mathematics of Computation

IS - 197

ER -

Circulant preconditioners for toeplitz matrices with positive continuous generating functions

Abstract

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