Toeplitz preconditioners for toeplitz systems with nonnegative generating functions

Raymond H. CHAN

doi:10.1093/imanum/11.3.333

Toeplitz preconditioners for toeplitz systems with nonnegative generating functions

Raymond H. CHAN^*

^*Corresponding author for this work

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

125 Citations (Scopus)

Abstract

The preconditioned conjugate gradient method is employed to solve Toeplitz systems T_nx = b where the generating functions of the n-by-n Toeplitz matrices T_n are continuous nonnegative periodic functions defined in [-π,π]. The preconditioned C_n are band Toeplitz matrices with band-widths independent of n. We prove that the spectra of C_n^-1T_n are uniformly bounded by constants independent of n. In particular, we show that the solutions of T_nx = b can be obtained in O(nlogn) operations.

Original language	English
Pages (from-to)	333-345
Number of pages	13
Journal	IMA Journal of Numerical Analysis
Volume	11
Issue number	3
DOIs	https://doi.org/10.1093/imanum/11.3.333
Publication status	Published - Jul 1991
Externally published	Yes

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title = "Toeplitz preconditioners for toeplitz systems with nonnegative generating functions",

abstract = "The preconditioned conjugate gradient method is employed to solve Toeplitz systems Tnx = b where the generating functions of the n-by-n Toeplitz matrices Tn are continuous nonnegative periodic functions defined in [-π,π]. The preconditioned Cn are band Toeplitz matrices with band-widths independent of n. We prove that the spectra of Cn-1Tn are uniformly bounded by constants independent of n. In particular, we show that the solutions of Tnx = b can be obtained in O(nlogn) operations.",

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N2 - The preconditioned conjugate gradient method is employed to solve Toeplitz systems Tnx = b where the generating functions of the n-by-n Toeplitz matrices Tn are continuous nonnegative periodic functions defined in [-π,π]. The preconditioned Cn are band Toeplitz matrices with band-widths independent of n. We prove that the spectra of Cn-1Tn are uniformly bounded by constants independent of n. In particular, we show that the solutions of Tnx = b can be obtained in O(nlogn) operations.

AB - The preconditioned conjugate gradient method is employed to solve Toeplitz systems Tnx = b where the generating functions of the n-by-n Toeplitz matrices Tn are continuous nonnegative periodic functions defined in [-π,π]. The preconditioned Cn are band Toeplitz matrices with band-widths independent of n. We prove that the spectra of Cn-1Tn are uniformly bounded by constants independent of n. In particular, we show that the solutions of Tnx = b can be obtained in O(nlogn) operations.

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DO - 10.1093/imanum/11.3.333

M3 - Journal Article (refereed)

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EP - 345

JO - IMA Journal of Numerical Analysis

JF - IMA Journal of Numerical Analysis

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ER -

Toeplitz preconditioners for toeplitz systems with nonnegative generating functions

Abstract

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