Bounds on the Extreme Eigenvalues of Real Symmetric Toeplitz Matrices

Melman, Aaron

doi:10.1137/s0895479898339621

Cited by 7 publications

(4 citation statements)

References 26 publications

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“…Bounds on the spectrum of finite Toeplitz matrices are of interest in many applications [5,14,19,26]. When a real symmetric Toeplitz operator (or matrix) is generated by a positive sequence, the Gershgorin circle theorem [40, §3.3] often gives a satisfactory upper bound on its spectral radius or the largest eigenvalue.…”

Section: Properties Of Heat Potentialsmentioning

confidence: 99%

“…It is clear from (26) that to get a stability bound we need to control the gap between C 1 (T ) and 1 2 . For T ≥ 1, this turns out to shrink only polynomially in T :…”

Section: The Dirichlet Problem In One Dimensionmentioning

confidence: 99%

“…Proof. We use a technique that will recur throughout this paper: we take the inner product of (26) with…”

Section: The Dirichlet Problem In One Dimensionmentioning

confidence: 99%

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Explicit unconditionally stable methods for the heat equation via potential theory

Barnett

Epstein

Greengard

et al. 2019

Pure Appl. Analysis

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We study the stability properties of explicit marching schemes for second-kind Volterra integral equations that arise when solving boundary value problems for the heat equation by means of potential theory. It is well known that explicit finite difference or finite element schemes for the heat equation are stable only if the time step ∆t is of the order O(∆x 2 ), where ∆x is the finest spatial grid spacing. In contrast, for the Dirichlet and Neumann problems on the unit ball in all dimensions d ≥ 1, we show that the simplest Volterra marching scheme, i.e., the forward Euler scheme, is unconditionally stable. Our proof is based on an explicit spectral radius bound of the marching matrix, leading to an estimate that an L 2 -norm of the solution to the integral equation is bounded by c d T d/2 times the norm of the right hand side. For the Robin problem on the half space in any dimension, with constant Robin (heat transfer) coefficient κ, we exhibit a constant C such that the forward Euler scheme is stable if ∆t < C/κ 2 , independent of any spatial discretization. This relies on new lower bounds on the spectrum of real symmetric Toeplitz matrices defined by convex sequences. Finally, we show that the forward Euler scheme is unconditionally stable for the Dirichlet problem on any smooth convex domain in any dimension, in L ∞ -norm.

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Section: Properties Of Heat Potentialsmentioning

confidence: 99%

“…It is clear from (26) that to get a stability bound we need to control the gap between C 1 (T ) and 1 2 . For T ≥ 1, this turns out to shrink only polynomially in T :…”

Section: The Dirichlet Problem In One Dimensionmentioning

confidence: 99%

See 1 more Smart Citation

Explicit unconditionally stable methods for the heat equation via potential theory

Barnett

Epstein

Greengard

et al. 2019

Pure Appl. Analysis

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“…For example, expressions for the exact (Akansu & Torun, 2012) or approximate (e.g., outer bounds; Hertz, 1992;Melman, 2000) eigenvalues of real symmetric Toeplitz matrices have been described in many sources. Here, we describe new bounds for the extreme eigenvalues of an AR(1) Toeplitz correlation matrix using simple functions of the generating parameter, r. For any matrix, T, with structure given in (32), it can be shown that…”

Section: Evaluating Penalized Regression Models With Pd Fungible Corrmentioning

confidence: 99%

Fungible Correlation Matrices: A Method for Generating Nonsingular, Singular, and Improper Correlation Matrices for Monte Carlo Research

Waller

2016

Multivariate Behavioral Research

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For a fixed set of standardized regression coefficients and a fixed coefficient of determination (R-squared), an infinite number of predictor correlation matrices will satisfy the implied quadratic form. I call such matrices fungible correlation matrices. In this article, I describe an algorithm for generating positive definite (PD), positive semidefinite (PSD), or indefinite (ID) fungible correlation matrices that have a random or fixed smallest eigenvalue. The underlying equations of this algorithm are reviewed from both algebraic and geometric perspectives. Two simulation studies illustrate that fungible correlation matrices can be profitably used in Monte Carlo research. The first study uses PD fungible correlation matrices to compare penalized regression algorithms. The second study uses ID fungible correlation matrices to compare matrix-smoothing algorithms. R code for generating fungible correlation matrices is presented in the supplemental materials.

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Voß

2000

Numerical Algorithms

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Bounds on the Extreme Eigenvalues of Real Symmetric Toeplitz Matrices

Cited by 7 publications

References 26 publications

Explicit unconditionally stable methods for the heat equation via potential theory

Explicit unconditionally stable methods for the heat equation via potential theory

Fungible Correlation Matrices: A Method for Generating Nonsingular, Singular, and Improper Correlation Matrices for Monte Carlo Research

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