Eigenvalues of Hermitian Toeplitz Matrices Generated by Simple-loop Symbols with Relaxed Smoothness

Bogoya, J. M.; Grudsky, Sergei M.; Maximenko, Egor A.

doi:10.1007/978-3-319-49182-0_11

Cited by 30 publications

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“…Bogoya et al [4][5][6] 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 have shown numerical evidence that some of these assumptions can be relaxed and extended to the case of a sequence of preconditioned Toeplitz matrices {P n (f, g) = 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 in the nonpreconditioned case.…”

Section: Discussionmentioning

confidence: 99%

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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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Section: Discussionmentioning

confidence: 99%

“…In the pure Toeplitz case, that is for g = 1 identically, so that P n (f, g) = T n (f ) and r = f , the result is proven in [4][5][6], if the RCTP f is monotone and satisfies certain additional assumptions, which include the requirements that f (θ) = 0 for θ ∈ (0, π) and f (θ) = 0 for θ ∈ {0, π}. The symbols…”

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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“…is the remainder (the error), which satisfies the inequality |E (q) j,n,α | ≤ C α h α+1 for some constant C α depending only on α and f. We note that in the scalar-valued case s = 1, several theoretical and computational results are available in support of the above expansion [2,5,6,9,[14][15][16], including also extensions to preconditioned matrices and matrices arising in a differential context [1,13].…”

Section: Mathematics Subject Classificationmentioning

confidence: 99%

“…, s} such that j = k. This version of the global condition is of course much simpler to verify. Moreover, in the case s = 1 it reduces to the monotonicity condition already used in the literature; see [2,5,6,9,15,16] and references therein.…”

Section: Mathematics Subject Classificationmentioning

confidence: 99%

Exact formulae and matrix-less eigensolvers for block banded symmetric Toeplitz matrices

2018

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Precise asymptotic expansions for the eigenvalues of a Toeplitz matrix T n ( f ), as the matrix size n tends to infinity, have recently been obtained, under suitable assumptions on the associated generating function f . A restriction is that f has to be polynomial, monotone, and scalar-valued. In this paper we focus on the case where f is an s × s matrix-valued trigonometric polynomial with s ≥ 1, and T n (f) is the block Toeplitz matrix generated by f, whose size is N (n, s) = sn. The case s = 1 corresponds to that already treated in the literature. We numerically derive conditions which ensure the existence of an asymptotic expansion for the eigenvalues. Such conditions generalize those known for the scalar-valued setting. Furthermore, following a proposal in the scalar-valued case by the first author, Garoni, and the third author, we devise an extrapolation algorithm for computing the eigenvalues of banded symmetric block Toeplitz matrices with a high level of accuracy and a low computational cost. The resulting algorithm is an eigensolver that does not need to store the original matrix, does not need to perform matrix-vector products, and for this reason is called matrixless. We use the asymptotic expansion for the efficient computation of the spectrum of special block Toeplitz structures and we provide exact formulae for the eigenvalues of the matrices coming from the Q p Lagrangian Finite Element approximation of a second order elliptic differential problem. Numerical results are presented and critically discussed.

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“…In the case where u = 1 identically and v satisfies some additional assumptions, Conjecture 1 was formally proved by Bogoya, Böttcher, Grudsky, and Maximenko in a sequence of recent papers [6,8,10]. Assuming Conjecture 1, in Section 2 of this paper, we describe and analyze a new algorithm for computing the eigenvalues of X n = T n (u) −1 T n (v); and in Section 3, we illustrate its performance through numerical experiments.…”

Section: Conjecturementioning

confidence: 99%

A matrix-less and parallel interpolation–extrapolation algorithm for computing the eigenvalues of preconditioned banded symmetric Toeplitz matrices

Ekström

Garoni

2018

Numer Algor

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In the past few years, Bogoya, Böttcher, Grudsky, and Maximenko obtained the precise asymptotic expansion for the eigenvalues of a Toeplitz matrix T n (f ), under suitable assumptions on the generating function f , as the matrix size n goes to infinity. On the basis of several numerical experiments, it was conjectured by Serra-Capizzano that a completely analogous expansion also holds for the eigenvalues of the preconditioned Toeplitz matrix T n (u) −1 T n (v), provided f = v/u is monotone and further conditions on u and v are satisfied. Based on this expansion, we here propose and analyze an interpolation-extrapolation algorithm for computing the eigenvalues of T n (u) −1 T n (v). The algorithm is suited for parallel implementation and it may be called "matrix-less" as it does not need to store the entries of the matrix. We illustrate the performance of the algorithm through numerical experiments and we also present its generalization to the case where f = v/u is non-monotone.

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Eigenvalues of Hermitian Toeplitz Matrices Generated by Simple-loop Symbols with Relaxed Smoothness

Cited by 30 publications

References 0 publications

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

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

Exact formulae and matrix-less eigensolvers for block banded symmetric Toeplitz matrices

A matrix-less and parallel interpolation–extrapolation algorithm for computing the eigenvalues of preconditioned banded symmetric Toeplitz matrices

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