Bounds for eigenvalues of nonsingular H-tensor

2015 ◽  
Vol 29 ◽  
pp. 3-16 ◽  
Author(s):  
Xue-Zhong Wang ◽  
Yimin Wei
Keyword(s):  
Spectral Radius ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Numerical Examples

The bounds for the Z-spectral radius of nonsingular H -tensor, the upper and lower bounds for the minimum H-eigenvalue of nonsingular (strong) M -tensor are studied in this paper. The sharper bounds are obtained. Numerical examples illustrate that our bounds give tighter bounds.

scholarly journals Bounds for spectral radius of nonnegative tensors using matrix-digragh-based approach

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Guimin Liu ◽  
Hongbin Lv
Keyword(s):  
Spectral Radius ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Nonnegative Tensor ◽  
Related Literature ◽  
Numerical Examples ◽  
Nonnegative Tensors

<p style='text-indent:20px;'>We obtain the improved results of the upper and lower bounds for the spectral radius of a nonnegative tensor by its majorization matrix's digraph. Numerical examples are also given to show that our results are significantly superior to the results of related literature.</p>

scholarly journals A modified S-type eigenvalue localization set of tensors applications

Filomat ◽  
2018 ◽  
Vol 32 (18) ◽  
pp. 6395-6416
Author(s):  
Zhengge Huang ◽  
Ligong Wang ◽  
Zhong Xu ◽  
Jingjing Cui
Keyword(s):  
Spectral Radius ◽  
Linear Algebra ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Numerical Examples ◽  
Nonnegative Tensors ◽  
Eigenvalue Localization ◽  
Localization Set ◽  
Weakly Irreducible ◽  
Eigenvalues Of Tensors

Based on the S-type eigenvalue localization set developed by Li et al. (Linear Algebra Appl. 493 (2016) 469-483) for tensors, a modified S-type eigenvalue localization set for tensors is established in this paper by excluding some sets from the existing S-type eigenvalue localization set developed by Huang et al. (arXiv: 1602.07568v1, 2016). The proposed set containing all eigenvalues of tensors is much sharper compared with that employed by Li et al. and Huang et al. As its applications, a criteria, which can be utilized for identifying the nonsingularity of tensors, is developed. In addition, we provide new upper and lower bounds for the spectral radius of nonnegative tensors and the minimum H-eigenvalue of weakly irreducible strong M-tensors. These bounds are superior to some previous results, which is illustrated by some numerical examples.

On the two sided barrier problem

1965 ◽  
Vol 2 (01) ◽  
pp. 79-87
Author(s):  
Masanobu Shinozuka
Keyword(s):  
Random Process ◽  
Ground Motion ◽  
Lower Bounds ◽  
Present Method ◽  
Structural Response ◽  
Random Processes ◽  
Upper And Lower Bounds ◽  
Numerical Examples ◽  
Long Run ◽  
Barrier Problem

Upper and lower bounds are given for the probability that a separable random process X(t) will take values outside the interval (— λ 1, λ 2) for 0 ≦ t ≦ T, where λ 1 and λ 2 are positive constants. The random process needs to be neither stationary, Gaussian nor purely random (white noise). In engineering applications, X(t) is usually a random process decaying with time at least in the long run such as the structural response to the acceleration of ground motion due to earthquake. Numerical examples show that the present method estimates the probability between the upper and lower bounds which are sufficiently close to be useful when the random processes decay with time.

The accuracy of Rayleigh’s method of calculating the natural frequencies of vibrating systems

1952 ◽  
Vol 211 (1105) ◽  
pp. 204-224 ◽  
Keyword(s):  
Lower Bounds ◽  
Natural Frequencies ◽  
Hermitian Operator ◽  
Normal Modes ◽  
Comparison Theorems ◽  
Upper And Lower Bounds ◽  
Subsequent Paper ◽  
Numerical Examples ◽  
Modes Of Vibration ◽  
Vibrating Systems

In this paper a theorem of Kato (1949) which provides upper and lower bounds for the eigenvalues of a Hermitian operator is modified and generalized so as to give upper and lower bounds for the normal frequencies of oscillation of a conservative dynamical system. The method given here is directly applicable to a system specified by generalized co-ordinates with both elastic and inertial couplings. It can be applied to any one of the normal modes of vibration of the system. The bounds obtained are much closer than those given by Rayleigh’s comparison theorems in which the inertia or elasticity of the system is changed, and they are in fact the ‘best possible’ bounds. The principles of the computation of upper and lower bounds is explained in this paper and will be illustrated by some numerical examples in a subsequent paper.

On the distance and distance signless Laplacian eigenvalues of graphs and the smallest Gersgorin disc

2018 ◽  
Vol 34 ◽  
pp. 191-204 ◽  
Author(s):  
Fouzul Atik ◽  
Pratima Panigrahi
Keyword(s):  
Spectral Radius ◽  
Lower Bounds ◽  
Nonnegative Matrix ◽  
Laplacian Matrix ◽  
Upper And Lower Bounds ◽  
Laplacian Spectral Radius ◽  
Laplacian Eigenvalues ◽  
Signless Laplacian Matrix ◽  
Signless Laplacian Spectral Radius ◽  
Signless Laplacian

The \emph{distance matrix} of a simple connected graph $G$ is $D(G)=(d_{ij})$, where $d_{ij}$ is the distance between the $i$th and $j$th vertices of $G$. The \emph{distance signless Laplacian matrix} of the graph $G$ is $D_Q(G)=D(G)+Tr(G)$, where $Tr(G)$ is a diagonal matrix whose $i$th diagonal entry is the transmission of the vertex $i$ in $G$. In this paper, first, upper and lower bounds for the spectral radius of a nonnegative matrix are constructed. Applying this result, upper and lower bounds for the distance and distance signless Laplacian spectral radius of graphs are given, and the extremal graphs for these bounds are obtained. Also, upper bounds for the modulus of all distance (respectively, distance signless Laplacian) eigenvalues other than the distance (respectively, distance signless Laplacian) spectral radius of graphs are given. These bounds are probably first of their kind as the authors do not find in the literature any bound for these eigenvalues. Finally, for some classes of graphs, it is shown that all distance (respectively, distance signless Laplacian) eigenvalues other than the distance (respectively, distance signless Laplacian) spectral radius lie in the smallest Ger\^sgorin disc of the distance (respectively, distance signless Laplacian) matrix.

On the two sided barrier problem

1965 ◽  
Vol 2 (1) ◽  
pp. 79-87 ◽  
Author(s):  
Masanobu Shinozuka
Keyword(s):  
Random Process ◽  
Ground Motion ◽  
Lower Bounds ◽  
Present Method ◽  
Structural Response ◽  
Random Processes ◽  
Upper And Lower Bounds ◽  
Numerical Examples ◽  
Long Run ◽  
Barrier Problem

Upper and lower bounds are given for the probability that a separable random process X(t) will take values outside the interval (— λ1, λ2) for 0 ≦ t ≦ T, where λ1 and λ2 are positive constants.The random process needs to be neither stationary, Gaussian nor purely random (white noise).In engineering applications, X(t) is usually a random process decaying with time at least in the long run such as the structural response to the acceleration of ground motion due to earthquake.Numerical examples show that the present method estimates the probability between the upper and lower bounds which are sufficiently close to be useful when the random processes decay with time.

scholarly journals Some Refinements and Generalizations of I. Schur Type Inequalities

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Xian-Ming Gu ◽  
Ting-Zhu Huang ◽  
Wei-Ru Xu ◽  
Hou-Biao Li ◽  
Liang Li ◽  
...  
Keyword(s):  
Lower Bounds ◽  
Recent Literature ◽  
Structural Characteristics ◽  
Variable Parameter ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Numerical Examples ◽  
Schur Inequality

Recently, extensive researches on estimating the value ofehave been studied. In this paper, the structural characteristics of I. Schur type inequalities are exploited to generalize the corresponding inequalities by variable parameter techniques. Some novel upper and lower bounds for the I. Schur inequality have also been obtained and the upper bounds may be obtained with the help ofMapleand automated proving package (Bottema). Numerical examples are employed to demonstrate the reliability of the approximation of these new upper and lower bounds, which improve some known results in the recent literature.

scholarly journals On the Dα spectral radius of strongly connected digraphs

Filomat ◽  
2021 ◽  
Vol 35 (4) ◽  
pp. 1289-1304
Author(s):  
Weige Xi
Keyword(s):  
Lower Bound ◽  
Spectral Radius ◽  
Lower Bounds ◽  
Distance Matrix ◽  
Upper And Lower Bounds ◽  
Strongly Connected Digraph ◽  
Connected Digraph ◽  
The Matrix ◽  
Strongly Connected ◽  
The Difference

Let G be a strongly connected digraph with distance matrix D(G) and let Tr(G) be the diagonal matrix with vertex transmissions of G. For any real ? ? [0, 1], define the matrix D?(G) as D?(G) = ?Tr(G) + (1-?)D(G). The D? spectral radius of G is the spectral radius of D?(G). In this paper, we first give some upper and lower bounds for the D? spectral radius of G and characterize the extremal digraphs. Moreover, for digraphs that are not transmission regular, we give a lower bound on the difference between the maximum vertex transmission and the D? spectral radius. Finally, we obtain the D? eigenvalues of the join of certain regular digraphs.

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