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A note on the fast algorithm for block Toeplitz systems with tensor structure
Kou K.-I.; Sin V.-K.; Jin X.-Q.
2002-03-10
Source PublicationApplied Mathematics and Computation
ISSN00963003
Volume126Issue:2-3Pages:187-197
Abstract

We study the solutions of block Toeplitz systems Tx = b by using the preconditioned conjugate gradient (PCG) method. Here T = T ⊗ T and T,i = m,n are Toeplitz matrices. In [X. Jin, Appl. Math. Comput. 73 (1995) 115-124], Jin introduced a fast algorithm for these systems by applying the PCG method. This fast algorithm allows a tensor problem to be reduced to a one-dimensional problem. It was proved that if the mn-by-mn system is well conditioned, then the PCG method converges superlinearly and only O(mnlogmn) operations are required in solving the preconditioned system. However, only well-conditioned systems were considered in Jin, 1995. In this paper, we apply this fast algorithm with the {ω}-circulant preconditioners proposed in [D. Potts, G. Steidl, Preconditioners for Ill-Conditioned toeplitz matrices, BIT, to be appeared] to solve the ill-conditioned systems. Numerical results are included to illustrate the effectiveness of the algorithm for solving the preconditioned systems by using the PCG method. An application in image restoration is also given. © 2002 Elsevier Science Inc. All rights reserved.

KeywordBlock Toeplitz Matrix Image Restoration Pcg Method Preconditioner
DOI10.1016/S0096-3003(00)00150-8
URLView the original
Indexed BySCI
Language英语
WOS Research AreaMathematics
WOS SubjectMathematics, Applied
WOS IDWOS:000173667100004
PublisherELSEVIER SCIENCE INC
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Citation statistics
Cited Times [WOS]:4   [WOS Record]     [Related Records in WOS]
Document TypeJournal article
CollectionDEPARTMENT OF MATHEMATICS
DEPARTMENT OF ELECTROMECHANICAL ENGINEERING
AffiliationUniversidade de Macau
Recommended Citation
GB/T 7714
Kou K.-I.,Sin V.-K.,Jin X.-Q.. A note on the fast algorithm for block Toeplitz systems with tensor structure[J]. Applied Mathematics and Computation,2002,126(2-3):187-197.
APA Kou K.-I.,Sin V.-K.,&Jin X.-Q..(2002).A note on the fast algorithm for block Toeplitz systems with tensor structure.Applied Mathematics and Computation,126(2-3),187-197.
MLA Kou K.-I.,et al."A note on the fast algorithm for block Toeplitz systems with tensor structure".Applied Mathematics and Computation 126.2-3(2002):187-197.
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