The Experts below are selected from a list of 318 Experts worldwide ranked by ideXlab platform
Heinrich Voss - One of the best experts on this subject based on the ideXlab platform.
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A hybrid method for computing a smallest eigenvalue of a symmetric and positive definite Toeplitz Matrix
2020Co-Authors: Aleksandra Kostić, Heinrich VossAbstract:In this paper we suggest a hybrid method for computing the smallest eigenvalue of a symmetric and positive definite Toeplitz Matrix which takes advantage of two types of methods, Newton’s method for the characteristic polynomial and projection methods based on rational interpolation of the secular equation.
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Recurrence relations for the even and odd characteristic polynomials of a symmetric Toeplitz Matrix
Journal of Computational and Applied Mathematics, 2005Co-Authors: Aleksandra Kostić, Heinrich VossAbstract:In a recent paper (Computation of the smallest even and odd eigenvalues of a symmetric positive-definite Toeplitz Matrix, SIAM J. Matrix Anal. Appl. 25 (2004) 949-963) Melman proved a recurrence relation of the even and odd characteristic polynomials of a real symmetric Toeplitz Matrix T on which a symmetry exploiting method for computing the smallest eigenvalue of T can be based. In this note, we present a proof of the recurrence relation which is less technical and more transparent.
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Symmetric schemes for computing the minimum eigenvalue of a symmetric Toeplitz Matrix
Linear Algebra and its Applications, 1999Co-Authors: Heinrich VossAbstract:Abstract Mackens and Voss [8,9] presented two generalizations of a method of Cybenko and Van Loan [4] for computing the smallest eigenvalue of a symmetric, positive-definite Toeplitz Matrix. Taking advantage of the symmetry or skew-symmetry of the corresponding eigenvector both methods are improved considerably.
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Bounds for the minimum eigenvalue of a symmetric Toeplitz Matrix
1998Co-Authors: Heinrich VossAbstract:In a recent paper Melman derived upper bounds for the smallest eigen value of a real symmetric Toeplitz Matrix in terms of the smallest roots of rational and polynomial approximations of the secular equation f the best of which being constructed by the Pad e approximation of f In this paper we prove that this bound is the smallest eigenvalue of the projection of the given eigenvalue problem onto a Krylov space of T n of dimension This interpretation of the bound suggests enhanced bounds of increasing ac curacy They can be substantially improved further by exploiting symmetry properties of the principal eigenvector of Tn
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Newton type methods for computing the smallest eigenvalue of a symmetric Toeplitz Matrix
Journal of Applied Mathematics and Mechanics, 1998Co-Authors: Heinrich Voss, Wolfgang MackensAbstract:Several methods for computing the smallest eigenvalue of a symmetric positive de nite Toeplitz Matrix are presented They converge from the left to the minimum eigenvalue and they rely on Newton s method and interpolation of the characteristic polynomial with no need for introductory bisection steps The methods are conceptually much simpler than the ones introduced by the same authors based on rational interpolation of the secular equation
M. Van Barel - One of the best experts on this subject based on the ideXlab platform.
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A fast algorithm for computing the smallest eigenvalue of a symmetric positive‐definite Toeplitz Matrix
Numerical Linear Algebra With Applications, 2020Co-Authors: Nicola Mastronardi, M. Van Barel, Raf VandebrilAbstract:Recent progress in signal processing and estimation has generated considerable interest in the problem of computing the smallest eigenvalue of a symmetric positive-definite (SPD) Toeplitz Matrix. An algorithm for computing upper and lower bounds to the smallest eigenvalue of a SPD Toeplitz Matrix has been recently derived (Linear Algebra Appl. 2007; DOI: 10.1016/j.laa.2007.05.008). The algorithm relies on the computation of the R factor of the QR factorization of the Toeplitz Matrix and the inverse of R. The simultaneous computation of R and R−1 is efficiently accomplished by the generalized Schur algorithm. In this paper, exploiting the properties of the latter algorithm, a numerical method to compute the smallest eigenvalue and the corresponding eigenvector of SPD Toeplitz matrices in an accurate way is proposed. Copyright © 2008 John Wiley & Sons, Ltd.
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a schur based algorithm for computing bounds to the smallest eigenvalue of a symmetric positive definite Toeplitz Matrix
Linear Algebra and its Applications, 2008Co-Authors: Nicola Mastronardi, M. Van Barel, Raf VandebrilAbstract:Abstract Recent progress in signal processing and estimation has generated c onsiderable interest in the problem of computing the smallest eigenvalue of symmetric positive definite Toeplitz matrices. Several algorithms have been proposed in the literature. They compute the smallest eigenvalue in an iterative fashion, many of them relying on the Levinson–Durbin solution of sequences of Yule–Walker systems. Exploiting the properties of two algorithms recently developed for estimating a lower and an upper bound of the smallest singular value of upper triangular matrices, respectively, an algorithm for computing bounds to the smallest eigenvalue of a symmetric positive definite Toeplitz Matrix is derived. The algorithm relies on the computation of the R factor of the QR -factorization of the Toeplitz Matrix and the inverse of R . The simultaneous computation of R and R −1 is efficiently accomplished by the generalized Schur algorithm.
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Computing a Lower Bound of the Smallest Eigenvalue of a Symmetric Positive-Definite Toeplitz Matrix
IEEE Transactions on Information Theory, 2008Co-Authors: Teresa Laudadio, Nicola Mastronardi, M. Van BarelAbstract:In this correspondence, several algorithms to compute a lower bound of the smallest eigenvalue of a symmetric positive-definite Toeplitz Matrix are described and compared in terms of accuracy and computational efficiency. Exploiting the Toeplitz structure of the considered Matrix, new theoretical insights are derived and an efficient implementation of some of the aforementioned algorithms is provided.
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A Schur-based algorithm for computing the smallest eigenvalue of a symmetric positive definite Toeplitz Matrix
2006Co-Authors: Nicola Mastronardi, M. Van Barel, Raf VandebrilAbstract:Recent progress in signal processing and estimation has generated considerable interest in the problem of computing the smallest eigenvalue of symmetric positive definite Toeplitz matrices. Several algorithms have been proposed in the literature. Many of them compute the smallest eigenvalue in an iterative fashion, relying on the Levinson–Durbin solution of sequences of Yule–Walker systems. Exploiting the properties of two algorithms recently developed for estimating a lower and an upper bound of the smallest singular value of upper triangular matrices, respectively, an algorithm for computing the smallest eigenvalue of a symmetric positive definite Toeplitz Matrix is derived. The algorithm relies on the computation of the R factor of the QR–factorization of the Toeplitz Matrix and the inverse of R. The latter computation is efficiently accomplished by the generalized Schur algorithm.
Nicola Mastronardi - One of the best experts on this subject based on the ideXlab platform.
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A fast algorithm for computing the smallest eigenvalue of a symmetric positive‐definite Toeplitz Matrix
Numerical Linear Algebra With Applications, 2020Co-Authors: Nicola Mastronardi, M. Van Barel, Raf VandebrilAbstract:Recent progress in signal processing and estimation has generated considerable interest in the problem of computing the smallest eigenvalue of a symmetric positive-definite (SPD) Toeplitz Matrix. An algorithm for computing upper and lower bounds to the smallest eigenvalue of a SPD Toeplitz Matrix has been recently derived (Linear Algebra Appl. 2007; DOI: 10.1016/j.laa.2007.05.008). The algorithm relies on the computation of the R factor of the QR factorization of the Toeplitz Matrix and the inverse of R. The simultaneous computation of R and R−1 is efficiently accomplished by the generalized Schur algorithm. In this paper, exploiting the properties of the latter algorithm, a numerical method to compute the smallest eigenvalue and the corresponding eigenvector of SPD Toeplitz matrices in an accurate way is proposed. Copyright © 2008 John Wiley & Sons, Ltd.
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a schur based algorithm for computing bounds to the smallest eigenvalue of a symmetric positive definite Toeplitz Matrix
Linear Algebra and its Applications, 2008Co-Authors: Nicola Mastronardi, M. Van Barel, Raf VandebrilAbstract:Abstract Recent progress in signal processing and estimation has generated c onsiderable interest in the problem of computing the smallest eigenvalue of symmetric positive definite Toeplitz matrices. Several algorithms have been proposed in the literature. They compute the smallest eigenvalue in an iterative fashion, many of them relying on the Levinson–Durbin solution of sequences of Yule–Walker systems. Exploiting the properties of two algorithms recently developed for estimating a lower and an upper bound of the smallest singular value of upper triangular matrices, respectively, an algorithm for computing bounds to the smallest eigenvalue of a symmetric positive definite Toeplitz Matrix is derived. The algorithm relies on the computation of the R factor of the QR -factorization of the Toeplitz Matrix and the inverse of R . The simultaneous computation of R and R −1 is efficiently accomplished by the generalized Schur algorithm.
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Computing a Lower Bound of the Smallest Eigenvalue of a Symmetric Positive-Definite Toeplitz Matrix
IEEE Transactions on Information Theory, 2008Co-Authors: Teresa Laudadio, Nicola Mastronardi, M. Van BarelAbstract:In this correspondence, several algorithms to compute a lower bound of the smallest eigenvalue of a symmetric positive-definite Toeplitz Matrix are described and compared in terms of accuracy and computational efficiency. Exploiting the Toeplitz structure of the considered Matrix, new theoretical insights are derived and an efficient implementation of some of the aforementioned algorithms is provided.
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A Schur-based algorithm for computing the smallest eigenvalue of a symmetric positive definite Toeplitz Matrix
2006Co-Authors: Nicola Mastronardi, M. Van Barel, Raf VandebrilAbstract:Recent progress in signal processing and estimation has generated considerable interest in the problem of computing the smallest eigenvalue of symmetric positive definite Toeplitz matrices. Several algorithms have been proposed in the literature. Many of them compute the smallest eigenvalue in an iterative fashion, relying on the Levinson–Durbin solution of sequences of Yule–Walker systems. Exploiting the properties of two algorithms recently developed for estimating a lower and an upper bound of the smallest singular value of upper triangular matrices, respectively, an algorithm for computing the smallest eigenvalue of a symmetric positive definite Toeplitz Matrix is derived. The algorithm relies on the computation of the R factor of the QR–factorization of the Toeplitz Matrix and the inverse of R. The latter computation is efficiently accomplished by the generalized Schur algorithm.
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computing the smallest eigenpair of a symmetric positive definite Toeplitz Matrix
SIAM Journal on Scientific Computing, 1999Co-Authors: Nicola Mastronardi, Daniel BoleyAbstract:An algorithm for computing the smallest eigenvalue of a symmetric positive definite Toeplitz Matrix is presented. The eigenvalue is approximated from below by Newton's method applied to the characteristic polynomial of the Matrix. The Newton's step is calculated by a Levinson--Durbin type recursion. Simultaneously, this recursion produces a realistic error bound of the actual approximation without additional computing effort as well as a simple and efficient way to compute the associated eigenvector.
Raf Vandebril - One of the best experts on this subject based on the ideXlab platform.
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A fast algorithm for computing the smallest eigenvalue of a symmetric positive‐definite Toeplitz Matrix
Numerical Linear Algebra With Applications, 2020Co-Authors: Nicola Mastronardi, M. Van Barel, Raf VandebrilAbstract:Recent progress in signal processing and estimation has generated considerable interest in the problem of computing the smallest eigenvalue of a symmetric positive-definite (SPD) Toeplitz Matrix. An algorithm for computing upper and lower bounds to the smallest eigenvalue of a SPD Toeplitz Matrix has been recently derived (Linear Algebra Appl. 2007; DOI: 10.1016/j.laa.2007.05.008). The algorithm relies on the computation of the R factor of the QR factorization of the Toeplitz Matrix and the inverse of R. The simultaneous computation of R and R−1 is efficiently accomplished by the generalized Schur algorithm. In this paper, exploiting the properties of the latter algorithm, a numerical method to compute the smallest eigenvalue and the corresponding eigenvector of SPD Toeplitz matrices in an accurate way is proposed. Copyright © 2008 John Wiley & Sons, Ltd.
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a schur based algorithm for computing bounds to the smallest eigenvalue of a symmetric positive definite Toeplitz Matrix
Linear Algebra and its Applications, 2008Co-Authors: Nicola Mastronardi, M. Van Barel, Raf VandebrilAbstract:Abstract Recent progress in signal processing and estimation has generated c onsiderable interest in the problem of computing the smallest eigenvalue of symmetric positive definite Toeplitz matrices. Several algorithms have been proposed in the literature. They compute the smallest eigenvalue in an iterative fashion, many of them relying on the Levinson–Durbin solution of sequences of Yule–Walker systems. Exploiting the properties of two algorithms recently developed for estimating a lower and an upper bound of the smallest singular value of upper triangular matrices, respectively, an algorithm for computing bounds to the smallest eigenvalue of a symmetric positive definite Toeplitz Matrix is derived. The algorithm relies on the computation of the R factor of the QR -factorization of the Toeplitz Matrix and the inverse of R . The simultaneous computation of R and R −1 is efficiently accomplished by the generalized Schur algorithm.
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A Schur-based algorithm for computing the smallest eigenvalue of a symmetric positive definite Toeplitz Matrix
2006Co-Authors: Nicola Mastronardi, M. Van Barel, Raf VandebrilAbstract:Recent progress in signal processing and estimation has generated considerable interest in the problem of computing the smallest eigenvalue of symmetric positive definite Toeplitz matrices. Several algorithms have been proposed in the literature. Many of them compute the smallest eigenvalue in an iterative fashion, relying on the Levinson–Durbin solution of sequences of Yule–Walker systems. Exploiting the properties of two algorithms recently developed for estimating a lower and an upper bound of the smallest singular value of upper triangular matrices, respectively, an algorithm for computing the smallest eigenvalue of a symmetric positive definite Toeplitz Matrix is derived. The algorithm relies on the computation of the R factor of the QR–factorization of the Toeplitz Matrix and the inverse of R. The latter computation is efficiently accomplished by the generalized Schur algorithm.
Teresa Laudadio - One of the best experts on this subject based on the ideXlab platform.
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Computing a Lower Bound of the Smallest Eigenvalue of a Symmetric Positive-Definite Toeplitz Matrix
IEEE Transactions on Information Theory, 2008Co-Authors: Teresa Laudadio, Nicola Mastronardi, M. Van BarelAbstract:In this correspondence, several algorithms to compute a lower bound of the smallest eigenvalue of a symmetric positive-definite Toeplitz Matrix are described and compared in terms of accuracy and computational efficiency. Exploiting the Toeplitz structure of the considered Matrix, new theoretical insights are derived and an efficient implementation of some of the aforementioned algorithms is provided.
