Preconditioning strategies for non‐Hermitian Toeplitz linear systems

Huckle, Thomas; Serra‐Capizzano, Stefano; Possio, Cristina Tablino

doi:10.1002/nla.396

Cited by 18 publications

(22 citation statements)

References 22 publications

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“…The idea goes back to early 1990s [6,8] in the simpler definite case and has the further nice feature of being naturally extendible, very naturally and effectively, to the multi-level setting [24], both in terms of efficiency and of convergence speed. Furthermore, quite recently, in [32,17,18] the same approach has been adapted for handling the more involved indefinite-Hermitian and non-Hermitian Toeplitz structures. A popular alternative is the use of fast transform based preconditioners (see [7,19,21] and references there reported): their advantages are pronounced in the unilevel case, but unfortunately the related good convergence features cannot be translated in a multi-level setting, see [22,28] and references therein for negative results and the related precise statements.…”

Section: (Band) Toeplitz Preconditioning For Dense Toeplitz Matricesmentioning

confidence: 99%

Stability of the notion of approximating class of sequences and applications

Serra‐Capizzano

Sundqvist

2008

Journal of Computational and Applied Mathematics

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Given an approximating class of sequences { { Bn, m }n }m for { An }n, we prove that { { Bn, m + }n }m (X+ being the pseudo-inverse of Moore-Penrose) is an approximating class of sequences for { An + }n, where { An }n is a sparsely vanishing sequence of matrices An of size dn with dk > dq for k > q, k, q ∈ N. As a consequence, we extend distributional spectral results on the algebra generated by Toeplitz sequences, by including the (pseudo) inversion operation, in the case where the sequences that are (pseudo) inverted are distributed as sparsely vanishing symbols. Applications to preconditioning and a potential use in image/signal restoration problems are presented

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Section: (Band) Toeplitz Preconditioning For Dense Toeplitz Matricesmentioning

confidence: 99%

Stability of the notion of approximating class of sequences and applications

Serra‐Capizzano

Sundqvist

2008

Journal of Computational and Applied Mathematics

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“…Matrices of the form P n (f, g) are important for the fast solution of large Toeplitz linear systems (in connection with the preconditioned conjugate gradient method [9][10][11]18] or of more general preconditioned Krylov methods [15,16]). Furthermore, up to low rank corrections, they appear in the context of the spectral approximation of differential operators in which a low rank correction of T n (g) is the mass matrix and a low rank correction of T n (f ) is the stiffness matrix.…”

Section: Introductionmentioning

confidence: 99%

Are the eigenvalues of preconditioned banded symmetric Toeplitz matrices known in almost closed form?

et al. 2017

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Bogoya, Böttcher, Grudsky, and Maximenko have recently obtained the precise asymptotic expansion for the eigenvalues of a sequence of Toeplitz matrices {T n (f )}, under suitable assumptions on the associated generating function f . In this paper, we provide numerical evidence that some of these assumptions can be relaxed and extended to the case of a sequence of preconditioned Toeplitz matrices {T −1 n (g)T n (f )}, for f trigonometric polynomial, g nonnegative, not identically zero trigonometric polynomial, r = f/g, and where the ratio r plays the same role as f Numer Algor in the nonpreconditioned case. Moreover, based on the eigenvalue asymptotics, we devise an extrapolation algorithm for computing the eigenvalues of preconditioned banded symmetric Toeplitz matrices with a high level of accuracy, with a relatively low computational cost, and with potential application to the computation of the spectrum of differential operators.

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“…The solution of non-Hermitian Toeplitz systems has been investigated by a number of researchers [4,8,9,11,12,17,20,22,23]. Circulant preconditioners and skew-circulant preconditioners [8,9], trigonometric preconditioners [12,22], Toeplitz-circulant preconditioners [4,11] and banded Toeplitz preconditioners [17] for (1.1) have been proposed. For generating function with zeros, Chan and Ching [4] consider decomposing f (θ ) as…”

Section: Introductionmentioning

confidence: 99%

Inverse product Toeplitz preconditioners for non-Hermitian Toeplitz systems

Lin

2009

Numer Algor

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In this paper, we first propose product Toeplitz preconditioners (in an inverse form) for non-Hermitian Toeplitz matrices generated by functions with zeros. Our inverse product-type preconditioner is of the formwhere T F , T L , and T U are full, band lower triangular, and band upper triangular Toeplitz matrices, respectively. Our basic idea is to decompose the generating function properly such that all factors T F , T L , and T U of the preconditioner are as well-conditioned as possible. We prove that under certain conditions, the preconditioned matrix has eigenvalues and singular values clustered around 1. Then we use a similar idea to modify the preconditioner proposed in Ku and Kuo (SIAM J Sci Stat Comput 13: [1470][1471][1472][1473][1474][1475][1476][1477][1478][1479][1480][1481][1482][1483][1484][1485][1486][1487] 1992) to handle the zeros in rational generating functions. Numerical results, including applications to the computation of the stationary probability distribution of Markovian queuing models with batch arrival, are given to illustrate the good performance of the proposed preconditioners.

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Preconditioning strategies for non‐Hermitian Toeplitz linear systems

Abstract: In this paper, we propose and analyse preconditioning strategies for non-Hermitian Toeplitz linear systems. These techniques used in connection with the GMRES algorithm allow to solve in an optimal way the corresponding linear systems

Cited by 18 publications

References 22 publications

Stability of the notion of approximating class of sequences and applications

Stability of the notion of approximating class of sequences and applications

Are the eigenvalues of preconditioned banded symmetric Toeplitz matrices known in almost closed form?

Inverse product Toeplitz preconditioners for non-Hermitian Toeplitz systems

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