Design and analysis of Toeplitz preconditioners

Ku, Ta-Kang; Kuo, Chien-Nan

doi:10.1109/78.157188

Cited by 60 publications

(23 citation statements)

References 19 publications

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“…Clearly, R-1T has a smaller condition number and a better clustering feature than T. Consequently, the PCG method performs better than the CG method. This phenomenon is closely related to the point Toeplitz result [10], where we found that although there are asymptotically 2x + 1 distinct eigenvalues, the PCG method converges asymptotically in 7x + 1 iterations. This can be easily explained by the analysis given in 3.…”

supporting

confidence: 72%

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On the Spectrum of a Family of Preconditioned Block Toeplitz Matrices

Ku¹,

Kuo²

1992

SIAM J. Sci. and Stat. Comput.

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Research on preconditioning Toeplitz matrices with circulant matrices has been active recently. The preconditioning technique can be easily generalized to block Toeplitz matrices.That is, for a block Toeplitz matrix T consisting of N N blocks with M M elements per block, a block circulant matrix R is used with the same block structure as its preconditioner. In this research, the spectral clustering property of the preconditioned matrix R-1T with T generated by two-dimensional rational functions T(z,,zy) of order (p:r,q:,pu,qv) is examined. It is shown that the eigenvalues of R-1T are clustered around unity except at most O(M/u + N"/) outliers, where max(p, q) and max(p, qy). Furthermore, if T is separable, the outliers are clustered together such that R-1T has at most (2/x + 1)(2 + 1) asymptotic distinct eigenvalues. The superior convergence behavior of the preconditioned conjugate gradient (PCG) method over the conjugate gradient (CG) method is explained by a smaller condition number and a better clustering property of the spectrum of the preconditioned matrix R-1T.

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supporting

confidence: 72%

“…First, for every point Toeplitz matrix Tn, Inl _< N-1, we construct a circulant preconditioner T with (2.9) t0,n, t--l,n -[-tM--l,n, t-2,n -+-tM--2,n, "'', tl--M,n tl,n as the first row [10]. By assuming the Dirichlet and periodic boundary conditions, we obtain block Toeplitz and circulant matrices, respectively.…”

mentioning

confidence: 99%

On the Spectrum of a Family of Preconditioned Block Toeplitz Matrices

Ku¹,

Kuo²

1992

SIAM J. Sci. and Stat. Comput.

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“…A preconditioner C, which can be inverted easily, is used to reduce the eigenvalue spread of C-IR. Several approaches for designing such preconditioner have been proposed by Chan [11], Strang [12], Ku and Kuo [13] for Toeplitz matrix. However, if the matrix R is not Toeplitz, these approaches fall.…”

Section: Preconditioningmentioning

confidence: 99%

Transform domain conjugate gradient algorithm for adaptive filtering

Zou

Chan

et al. 2000

J. of Electron.(China)

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This paper proposed a new normalized transform domain conjugate gradient algorithm (NT-CGA), which applies the data independent normalized orthogonal transform technique to approximately whiten the input signal and utilises the modified conjugate gradient method to perform sample-by-sample updating of the filter weights more efficiently. Simulation results illustrated that the proposed algorithm has the ability to provide a fast convergence speed and lower steady-error compared to that of traditional least mean square algorithm (LMSA), normalized transform domain least mean square algorithm (NT-LMSA), Quasi-Newton least mean square algorithm (Q-LMSA) and time domain conjugate gradient algorithm (TD-CGA) when the input signal is heavily coloured.

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“…In the second method [12∼14] both the preconditioner P and its inverse P −1 are represented as the sum of many ω-circulant matrices. The third is just the embedding method [15] , based on which the ω-circulant preconditioner P [ω] proposed by Mei [5] can be considered as a general way of the above method for constructing the preconditioner. Theoretical analysis and actual data processing [8,16∼18] have proved that the PCG method often leads to fast convergence for many Toeplitz equation sets.…”

Section: Introductionmentioning

confidence: 99%

Preconditioned Equation Sets and Their Applications

Mei¹,

Hong²

2004

Chinese J of Geophysics

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We establish the w‐circulant preconditioned equations set PN[w]Xw = B corresponding to the general Toeplitz equation set ANX = B. By choosing different criteria for constructing preconditioners, we obtain different preconditioned equation sets PN[w]Xw = B and an optimal value of w. Theoretical analysis and practical calculation prove that the inversed results by using this method are much better than those by the traditional Fourier method.

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Design and analysis of Toeplitz preconditioners

Cited by 60 publications

References 19 publications

On the Spectrum of a Family of Preconditioned Block Toeplitz Matrices

On the Spectrum of a Family of Preconditioned Block Toeplitz Matrices

Transform domain conjugate gradient algorithm for adaptive filtering

Preconditioned Equation Sets and Their Applications

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