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

Research output: Journal Publications and Reviews › RGC 21 - Publication in refereed journal › 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⁻¹A_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
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

Research Keywords

Toeplitz matrix
circulant matrix
preconditioned conjugate gradient method
generating function

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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−1An 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.",

keywords = "Toeplitz matrix, circulant matrix, preconditioned conjugate gradient method, generating function",

author = "CHAN, \{Raymond H.\} and Man-Chung YEUNG",

year = "1992",

month = jan,

doi = "10.1090/S0025-5718-1992-1106960-1",

language = "English",

volume = "58",

pages = "233--240",

journal = "Mathematics of Computation",

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TY - JOUR

T1 - CIRCULANT PRECONDITIONERS FOR TOEPLITZ MATRICES WITH POSITIVE CONTINUOUS GENERATING FUNCTIONS

AU - CHAN, Raymond H.

AU - YEUNG, Man-Chung

PY - 1992/1

Y1 - 1992/1

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−1An 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−1An 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.

KW - Toeplitz matrix

KW - circulant matrix

KW - preconditioned conjugate gradient method

KW - generating function

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

UR - https://www.scopus.com/record/pubmetrics.uri?eid=2-s2.0-84968515314&origin=recordpage

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

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

M3 - RGC 21 - Publication in refereed journal

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

Research Keywords

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