Spectral properties of flipped Toeplitz matrices and related preconditioning

Mazza, Mariarosa; Pestana, Jennifer

doi:10.1007/s10543-018-0740-y

Cited by 16 publications

(16 citation statements)

References 21 publications

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“…A consequence of Theorem 3.5 is that the eigenvalues of A n (|f |) −1 Y n A n lie in [−1− , 1+ ], where for large enough n the parameter is small. Although eigenvalues may be close to the origin, most cluster at −1 and 1, in line with Theorem 3.4 in [23].…”

Section: Theorem 34 Characterizes the Eigenvalues Ofsupporting

confidence: 67%

Preconditioners for Symmetrized Toeplitz and Multilevel Toeplitz Matrices

Pestana¹

2019

SIAM J. Matrix Anal. Appl.

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When solving linear systems with nonsymmetric Toeplitz or multilevel Toeplitz matrices using Krylov subspace methods, the coefficient matrix may be symmetrized. The preconditioned MINRES method can then be applied to this symmetrized system, which allows rigorous upper bounds on the number of MINRES iterations to be obtained. However, effective preconditioners for symmetrized (multilevel) Toeplitz matrices are lacking. Here, we propose novel ideal preconditioners, and investigate the spectra of the preconditioned matrices. We show how these preconditioners can be approximated and demonstrate their effectiveness via numerical experiments.

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Section: Theorem 34 Characterizes the Eigenvalues Ofsupporting

confidence: 67%

Preconditioners for Symmetrized Toeplitz and Multilevel Toeplitz Matrices

Pestana¹

2019

SIAM J. Matrix Anal. Appl.

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“…A series of numerical examples concerning different generating functions and circulant preconditioners have also been provided to support our theoretical results. We acknowledge that similar results are given in [12] by using different techniques: while our approach is based on the notion of approximating class of sequences, the derivations in [12] are obtained by using the powerful *-algebra structure of the GLT sequences.…”

Section: Discussionmentioning

confidence: 78%

“…Figure 6 shows the sampling ψ |f | (θ j,n ) approximates the eigenvalues of the matrix (Y n ⊗ I s )T n,s [f ] well. We observe the four branches of eigenvalues [−12, −8] ∪ [−3, −1] ∪ [1,3] ∪ [8,12] as described by Theorem 3.4.…”

Section: Numerical Experiments On the Spectral Distribution Of {Y N Tmentioning

confidence: 98%

The Eigenvalue Distribution of Special 2-by-2 Block Matrix-Sequences with Applications to the Case of Symmetrized Toeplitz Structures

Furci

Hon²,

Mursaleen

et al. 2019

SIAM J. Matrix Anal. Appl.

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Given a Lebesgue integrable function f over [0, 2π], we consider the sequence of matrices {Y n T n [f ]} n , where T n [f ] is the n-by-n Toeplitz matrix generated by f and Y n is the flip permutation matrix, also called the anti-identity matrix. Because of the unitary character of Y n , the singular values of T n [f ] and Y n T n [f ] coincide. However, the eigenvalues are affected substantially by the action of the matrix Y n . Under the assumption that the Fourier coefficients are real, we prove that {Y n T n [f ]} n is distributed in the eigenvalue sense aswith g(θ) = |f (θ)|. We also consider the preconditioning introduced by Pestana and Wathen and, by using the same arguments, we prove that the preconditioned sequence is distributed in the eigenvalue sense as φ 1 , under the mild assumption that f is sparsely vanishing. We emphasize that the mathematical tools introduced in this setting have a general character and in fact can be potentially used in different contexts. A number of numerical experiments are provided and critically discussed.

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“…This well-known fact has been exploited to develop solvers that are typically faster than GMRES applied to the original system, thanks to the short-term recurrences of MINRES [364][365][366]. We stress that Y does not, in general, improve the convergence rate [367,368]; a second, symmetric positive definite, preconditioner is almost always required. However, in contrast to the nonsymmetric system, the preconditioner choice can be guided by MINRES convergence theory, and mathematical guarantees of fast convergence can be obtained.…”

Section: Preconditioners With "Nonstandard" Goalsmentioning

confidence: 99%

Preconditioners for Krylov subspace methods: An overview

Pearson

Pestana

2020

GAMM-Mitteilungen

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When simulating a mechanism from science or engineering, or an industrial process, one is frequently required to construct a mathematical model, and then resolve this model numerically. If accurate numerical solutions are necessary or desirable, this can involve solving large-scale systems of equations. One major class of solution methods is that of preconditioned iterative methods, involving preconditioners which are computationally cheap to apply while also capturing information contained in the linear system. In this article, we give a short survey of the field of preconditioning. We introduce a range of preconditioners for partial differential equations, followed by optimization problems, before discussing preconditioners constructed with less standard objectives in mind.

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Spectral properties of flipped Toeplitz matrices and related preconditioning

Cited by 16 publications

References 21 publications

Preconditioners for Symmetrized Toeplitz and Multilevel Toeplitz Matrices

Preconditioners for Symmetrized Toeplitz and Multilevel Toeplitz Matrices

The Eigenvalue Distribution of Special 2-by-2 Block Matrix-Sequences with Applications to the Case of Symmetrized Toeplitz Structures

Preconditioners for Krylov subspace methods: An overview

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