1,720,998 research outputs found
From convergence in measure to convergence of matrix-sequences through concave functions and singular values
No full textEstimates for the minimum eigenvalue and the condition number of Hermitian (block) Toeplitz matrices
No full textTopological foundations of an asymptotic approximation theory for sequences of matrices with increasing size
No full textOn the asymptotic spectral distribution of increasing size matrices: test functions, spectral clustering, and asymptotic estimates of outliers
No full textSpectral distribution of PDE discretization matrices from isogeometric analysis: The case of L1 coefficients and non-regular geometry
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The theory of locally Toeplitz sequences: A review, an extension, and a few representative applications
No full textThe theory of locally Toeplitz (LT) sequences is a powerful apparatus for computing the asymptotic singular value and eigenvalue distribution of the discretization matrices An arising from the numerical approximation of partial differential equations (PDEs). Indeed, when the discretization parameter n tends to infinity, the matrices An give rise to a sequence {An}n, which often can be expressed as a finite sum of LT sequences. In this work, we review and extend the theory of LT sequences, which dates back to the pioneering work by Tilli in 1998 and was partially developed by the second author during the last decade.We also present some applications of the theory to the finite difference and finite element approximation of PDEs
A matrix-less and parallel interpolation–extrapolation algorithm for computing the eigenvalues of preconditioned banded symmetric Toeplitz matrices
Full text linkIn the past few years, Bogoya, Boettcher, 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
Block generalized locally Toeplitz sequences: The case of matrix functions and an engineering application
No full textThe theory of block generalized locally Toeplitz (GLT) sequences is a powerful apparatus for computing the spectral distribution of block-structured matrices arising from the discretization of differential problems, with a special reference to systems of differential equations (DEs) and to the higher-order finite element or discontinuous Galerkin approximation of both scalar and vectorial DEs. In the present paper, the theory of block GLT sequences is extended by proving that {f(An)}n is a block GLT sequence as long as f is continuous and {An}n is a block GLT sequence formed by Hermitian matrices. It is also provided a relevant application of this result to the computation of the distribution of the numerical eigenvalues obtained from the higher-order isogeometric Galerkin discretization of second-order variable-coefficient differential eigenvalue problems (a topic of interest not only in numerical analysis but also in engineering)
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