Eigenvalues and eigenvectors of tau matrices with applications to Markov processes and economics

Ekström, Sven‐Erik; Garoni, Carlo; Jozefiak, Adam; Perla, Jesse

doi:10.1016/j.laa.2021.06.005

Cited by 6 publications

(5 citation statements)

References 23 publications

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“…We can associate to this matrix a function g of the form g(θ) = a + 2b cos θ. For some values of ε and ϕ, the exact eigenvalues of T n, ,ϕ (g) are given by a sampling with specific grids; for detailed examples see [7] and [21] for asymptotic results.…”

Section: Etna Kent State University and Johann Radon Institute (Ricam)mentioning

confidence: 99%

A note on the spectral analysis of matrix sequences via GLT momentary symbols: from all-at-once solution of parabolic problems to distributed fractional order matrices

Bolten¹,

Ekström²,

Furci³

et al. 2023

etna

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The first focus of this paper is the characterization of the spectrum and the singular values of the coefficient matrix stemming from the discretization of a parabolic diffusion problem using a space-time grid and secondly from the approximation of distributed-order fractional equations. For this purpose we use the classical GLT theory and the new concept of GLT momentary symbols. The first permits us to describe the singular value or eigenvalue asymptotic distribution of the sequence of the coefficient matrices. The latter permits us to derive a function that describes the singular value or eigenvalue distribution of the matrix of the sequence, even for small matrix sizes, but under given assumptions. The paper is concluded with a list of open problems, including the use of our machinery in the study of iteration matrices, especially those concerning multigrid-type techniques.

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Section: Etna Kent State University and Johann Radon Institute (Ricam)mentioning

confidence: 99%

A note on the spectral analysis of matrix sequences via GLT momentary symbols: from all-at-once solution of parabolic problems to distributed fractional order matrices

Bolten¹,

Ekström²,

Furci³

et al. 2023

etna

View full text Add to dashboard Cite

show abstract

“…We can associate to this matrix a function g of the form g(θ) = a + 2b cos θ. For some values of ε and ϕ the exact eigenvalues of T n,ǫ,ϕ (g) are given by sampling with specific grids; for detailed examples see [6] and [18] for asymptotic results.…”

Section: Matrix Algebrasmentioning

confidence: 99%

A note on the spectral analysis of matrix sequences via GLT momentary symbols: from all-at-once solution of parabolic problems to distributed fractional order matrices

Bolten¹,

Ekström²,

Furci³

et al. 2022

Preprint

Self Cite

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The first focus of this paper is the characterization of the spectrum and the singular values of the coefficient matrix stemming from the discretization with space-time grid for a parabolic diffusion problem and from the approximation of distributed order fractional equations. For this purpose we will use the classical GLT theory and the new concept of GLT momentary symbols. The first permits to describe the singular value or eigenvalue asymptotic distribution of the sequence of the coefficient matrices, the latter permits to derive a function, which describes the singular value or eigenvalue distribution of the matrix of the sequence, even for small matrix-sizes but under given assumptions. The note is concluded with a list of open problems, including the use of our machinery in the study of iteration matrices, especially those concerning multigrid-type techniques.

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“…We notice that any matrix X n belonging to the τ class [1,2,9] is diagonalized by the sine transform S n reported in (8), whose columns v…”

Section: Eigenvaluesmentioning

confidence: 99%

“…Table 2: [Ordering of eigenvalues (non-monotone symmetric symbol, f (θ) = cos(θ) + cos(2θ)))] Ordering of the eigenvalues using permutation Π−1 n (j), where two eigenvalues have switched places, does not give a missmatch of (−1) j+1 λ Π−1 n (j) (T n (f )) and λ ρ(j) (H n (f )) (as in Table 1). A closely related matrix, to the Toplitz matrix T n (f ) in Remark 3, is the Toeplitz-like matrix T n,0,0 = T n (f )−R n where R n is a low-rank matrix with 1/2 in the top left and bottom right corners; T n,ε,ϕ (f ) is the generated matrix by the symbol f belonging to the τ ε,ϕ -algebra; e.g., see [1,2,9]. For all matrices T n,ε,ϕ (f ), where ε, ϕ ∈ {−1, 0, 1} we know the full eigendecomposition and "perfect grids" ξ j,n ; e.g., λ j (T j,0,0 (f )) = f (θ j,n ) and Q n is the discrete sine transform, DST.…”

Section: Eigenvaluesmentioning

confidence: 99%

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Theoretical results for eigenvalues, singular values, and eigenvectors of (flipped) Toeplitz matrices and related computational proposals

Barbarino¹,

Ekström²,

Serra‐Capizzano³

et al. 2022

Preprint

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In a series of recent papers the spectral behavior of the matrix sequence {YnTn(f )} is studied in the sense of the spectral distribution, where Yn is the main antidiagonal (or flip matrix) and Tn(f ) is the Toeplitz matrix generated by the function f , with f being Lebesgue integrable and with real Fourier coefficients. This kind of study is also motivated by computational purposes for the solution of the related large linear systems using the (preconditioned) MINRES algorithm. Here we complement the spectral study with more results holding both asymptotically and for a fixed dimension n, and with regard to eigenvalues and singular values: the final target is the design of ad hoc procedures for the computation of the related spectra via matrix-less algorithms, with a cost being linear in the number of computed eigenvalues. We emphasize that the challenge of the case of non-monotone generating functions is considered in this note, for which the previous matrix-less algorithms fail. Numerical experiments are reported and commented, with the aim of showing in a visual way the theoretical analysis.

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Eigenvalues and eigenvectors of tau matrices with applications to Markov processes and economics

Cited by 6 publications

References 23 publications

A note on the spectral analysis of matrix sequences via GLT momentary symbols: from all-at-once solution of parabolic problems to distributed fractional order matrices

A note on the spectral analysis of matrix sequences via GLT momentary symbols: from all-at-once solution of parabolic problems to distributed fractional order matrices

A note on the spectral analysis of matrix sequences via GLT momentary symbols: from all-at-once solution of parabolic problems to distributed fractional order matrices

Theoretical results for eigenvalues, singular values, and eigenvectors of (flipped) Toeplitz matrices and related computational proposals

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