http://dx.doi.org/10.1137/080720280">
 

Document Type

Journal Article

Department/Unit

Department of Mathematics

Title

Approximate inverse circulant-plus-diagonal preconditioners for toeplitz-plus-diagonal matrices

Language

English

Abstract

We consider the solutions of Hermitian positive definite Toeplitz-plus-diagonal systems (T +D)x = b, where T is a Toeplitz matrix and D is diagonal and positive. However, unlike the case of Toeplitz systems, no fast direct solvers have been developed for solving them. In this paper, we employ the preconditioned conjugate gradient method with approximate inverse circulant-plusdiagonal preconditioners to solving such systems. The proposed preconditioner can be constructed and implemented efficiently using fast Fourier transforms. We show that if the entries of T decay away exponentially from the main diagonals, the preconditioned conjugate gradient method applied to the preconditioned system converges very quickly. Numerical examples including spatial regularization for image deconvolution application are given to illustrate the effectiveness of the proposed preconditioner. © 2010 Society for Industrial and Applied Mathematics.

Keywords

Approximate inverse, Circulant matrices, Convergence analysis, Toeplitz-plus-diagonal matrices

Publication Date

2010

Source Publication Title

SIAM Journal on Scientific Computing

Volume

32

Issue

3

Start Page

1442

End Page

1464

Publisher

Society for Industrial and Applied Mathematics

ISSN (print)

10648275

ISSN (electronic)

10957197

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