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## Reliable eigenvalues of symmetric tridiagonals

Ralha, Rui
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
66%
For the eigenvalues of a symmetric tridiagonal matrix T, the most accurate algorithms deliver approximations which are the exact eigenvalues of a matrix whose entries differ from the corresponding entries of T by small relative perturbations. However, for matrices with eigenvalues of different magnitudes, the number of correct digits in the computed approximations for eigenvalues of size smaller than ‖T‖₂ depends on how well such eigenvalues are defined by the data. Some classes of matrices are known to define their eigenvalues to high relative accuracy but, in general, there is no simple way to estimate well the number of correct digits in the approximations. To remedy this, we propose a method that provides sharp bounds for the eigenvalues of T. We present some numerical examples to illustrate the usefulness of our method.; FEDER (Programa Operacional Factores de Competitividade); FCT (Projecto PEst-C/MAT/UI0013/2011

## Bounds for the signless laplacian energy

Abreu, N.; Cardoso, D.M.; Gutman, I.; Martins, E.A.; Robbiano, M.
Tipo: Artigo de Revista Científica
ENG
Relevância na Pesquisa
56.03%
The energy of a graph G is the sum of the absolute values of the eigenvalues of the adjacency matrix of G. The Laplacian (respectively, the signless Laplacian) energy of G is the sum of the absolute values of the differences between the eigenvalues of the Laplacian (respectively, signless Laplacian) matrix and the arithmetic mean of the vertex degrees of the graph. In this paper, among some results which relate these energies, we point out some bounds to them using the energy of the line graph of G. Most of these bounds are valid for both energies, Laplacian and signless Laplacian. However, we present two new upper bounds on the signless Laplacian which are not upper bounds for the Laplacian energy. © 2010 Elsevier Inc. All rights reserved.; FCT; FEDER/POCI 2010; CNPq; PQ-305016/2006–2007; Serbian Ministry of Science; No. 144015G; Mecesup 2 UCN 0605; Fondecyt-IC Project 11090211

## Upper Bounds for Randic Spread

Gomes, Helena; Martins, Enide; Robbiano, María; San Martín, Bernardo
Fonte: Mathematical Chemistry Monographs Publicador: Mathematical Chemistry Monographs
Tipo: Artigo de Revista Científica
ENG
Relevância na Pesquisa
55.75%
The Randi´c spread of a simple undirected graph G, sprR(G), is equal to the maximal difference between two eigenvalues of the Randi´c matrix, disregarding the spectral radius [Gomes et al., MATCH Commun. Math. Comput. Chem. 72 (2014) 249–266]. Using a rank-one perturbation on the Randi´c matrix of G it is obtained a new matrix whose matricial spread coincide with sprR(G). By means of this result, upper bounds for sprR(G) are obtained.

## Lower bounds for eigenvalues of self-adjoint problems

Gundersen, Gary G.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
55.87%
The equation y″ + [λ - q(x)]y = 0 on (0, ∞) or (-∞, ∞), in which q(x) → ∞ as x → ∞ or x → ± ∞, has a complete set of eigenfunctions with discrete eigenvalues {λn}n=0∞. We derive an inequality that contains λn, by using a quick and elementary method that does not employ a comparison theorem or assume anything special. Explicit lower bounds for λn can often be easily obtained, and three examples are given. The method also gives respectable lower bounds for λn in the classical Sturm—Liouville case.

## Desigualdades universais para autovalores do operador poli-harmônico; Universal bounds for eigenvalues of the polyharmonic operator

PEREIRA, Rosane Gomes
Fonte: Universidade Federal de Goiás; BR; UFG; Mestrado em Matemática; Ciências Exatas e da Terra Publicador: Universidade Federal de Goiás; BR; UFG; Mestrado em Matemática; Ciências Exatas e da Terra
Tipo: Dissertação Formato: application/pdf
POR
Relevância na Pesquisa
65.97%
In this work, we study eigenvalues of polyharmonic operators on compact Riemannian manifolds with boundary (possibly empty). Here, we bring in a universal inequality for the eigenvalues of the polyharmonic operator on compact domains in an Euclidean space Rn. This inequality controls the kth eigenvalue by the lower eigenvalues, independently of the particular geometry of the domain. Besides, a inequality we present covers the important Yang inequality on eigenvalues of the Dirichlet Laplacian. Finally, we introduce universal inequalities for eigenvalues of polyharmonic operator on compact domains in a unit n-sphere Sn. NOTE: Programs do not copy or copy errors with certain symbols, formulas, formatting, etc ..., n of Rn and Sn are overwritten. View all content by clicking pdf - dissertation at the bottom of the screen.; Neste trabalho, estudamos autovalores do operador poli-harmônico em variedades Riemannianas compactas com fronteira ( possivelmente vazia ). Aqui, apresentamos uma desigualdade universal para os autovalores do operador poliharmônico em domínios compactos no Espaço Euclidiano Rn. Esta desigualdade controla o k-ésimo autovalor pelos autovalores menores, independentemente da geometria particular do domínio. Além disso...

## Relative perturbation theory for diagonally dominant matrices

Dailey, Megan; Martínez Dopico, Froilán C.; Ye, Qiang
Fonte: Society for Industrial and Applied Mathematics Publicador: Society for Industrial and Applied Mathematics
Tipo: info:eu-repo/semantics/publishedVersion; info:eu-repo/semantics/article
Relevância na Pesquisa
55.97%
In this paper, strong relative perturbation bounds are developed for a number of linear algebra problems involving diagonally dominant matrices. The key point is to parameterize diagonally dominant matrices using their off-diagonal entries and diagonally dominant parts and to consider small relative componentwise perturbations of these parameters. This allows us to obtain new relative perturbation bounds for the inverse, the solution to linear systems, the symmetric indefinite eigenvalue problem, the singular value problem, and the nonsymmetric eigenvalue problem. These bounds are much stronger than traditional perturbation results, since they are independent of either the standard condition number or the magnitude of eigenvalues/singular values. Together with previously derived perturbation bounds for the LDU factorization and the symmetric positive definite eigenvalue problem, this paper presents a complete and detailed account of relative structured perturbation theory for diagonally dominant matrices.; This research was partially supported by the Ministerio de Economía y Competitividad of Spain under grant MTM2012-32542.

## Sharp Bounds for Eigenvalues and Multiplicities on Surfaces of Revolution

Engman, Martin
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
55.82%
We find sharp upper bounds for the multiplicities and the numerical values of all the distinct eigenvalues on a surface of revolution diffeomorphic to the sphere.; Comment: LaTeX2e, 8 pages

## New Probabilistic Bounds on Eigenvalues and Eigenvectors of Random Kernel Matrices

Reyhani, Nima; Hino, Hideitsu; Vigario, Ricardo
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
56.04%
Kernel methods are successful approaches for different machine learning problems. This success is mainly rooted in using feature maps and kernel matrices. Some methods rely on the eigenvalues/eigenvectors of the kernel matrix, while for other methods the spectral information can be used to estimate the excess risk. An important question remains on how close the sample eigenvalues/eigenvectors are to the population values. In this paper, we improve earlier results on concentration bounds for eigenvalues of general kernel matrices. For distance and inner product kernel functions, e.g. radial basis functions, we provide new concentration bounds, which are characterized by the eigenvalues of the sample covariance matrix. Meanwhile, the obstacles for sharper bounds are accounted for and partially addressed. As a case study, we derive a concentration inequality for sample kernel target-alignment.

## Lower bound estimates for eigenvalues of the Laplacian

Cheng, Qing-Ming; Qi, Xuerong
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
55.78%
For an $n$-dimensional polytope $\Omega$ in $\mathbb{R}^{n}$, we study lower bounds for eigenvalues of the Dirichlet eigenvalue problem of the Laplacian. In the asymptotic formula on the average of the first $k$ eigenvalues, Li and Yau (1983) obtained the first term with the order $k^{\frac2n}$, which is optimal. The next landmark goal is to give the second term with the order $k^{\frac1n}$ in the asymptotic formula. For this purpose, Kova\v{r}\'{\i}k, Vugalter and Weidl (2009) have made an important breakthrough in the case of dimension 2. It is our purpose to study the $n$-dimensional case for arbitrary dimension $n$. We obtain the second term in the asymptotic sense.; Comment: 15 pages

## Two-sided bounds for eigenvalues of differential operators with applications to Friedrichs', Poincar\'e, trace, and similar constants

Šebestová, Ivana; Vejchodský, Tomáš
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
55.91%
We present a general numerical method for computing guaranteed two-sided bounds for principal eigenvalues of symmetric linear elliptic differential operators. The approach is based on the Galerkin method, on the method of a priori-a posteriori inequalities, and on a complementarity technique. The two-sided bounds are formulated in a general Hilbert space setting and as a byproduct we prove an abstract inequality of Friedrichs'-Poincar\'e type. The abstract results are then applied to Friedrichs', Poincar\'e, and trace inequalities and fully computable two-sided bounds on the optimal constants in these inequalities are obtained. Accuracy of the method is illustrated on numerical examples.; Comment: Extended numerical experiments and minor corrections of the previous version. This version has been accepted for publication by SIAM J. Numer. Anal

## Extrinsic Bounds for Eigenvalues of the Dirac Operator

Baer, Christian
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
55.9%
We derive upper eigenvalue bounds for the Dirac operator of a closed hypersurface in a manifold with Killing spinors such as Euclidean space, spheres or hyperbolic space. The bounds involve the Willmore functional. Relations with the Willmore inequality are briefly discussed. In higher codimension we obtain bounds on the eigenvalues of the Dirac operator of the submanifold twisted with the spinor bundle of the normal bundle.; Comment: 24 pages, LaTeX2e. to appear in Ann. Glob. Anal. Geom

## Universal bounds for eigenvalues of a buckling problem II

Cheng, Qing-Ming; Yang, Hongcang
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
55.79%
In this paper, we investigate universal estimates for eigenvalues of a buckling problem. For a bounded domain in a Euclidean space, we give a positive contribution for obtaining a sharp universal inequality for eigenvalues of the buckling problem. For a domain in the unit sphere, we give an important improvement on the results of Wang and Xia [J. Funct. Anal. 245(2007), 334-352].; Comment: 21 pages, a final version to appear in Trans. Amer. Math. Soc

## Upper and lower bounds for eigenvalues of the clamped plate problem

Cheng, Qing-Ming; Wei, Guoxin
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
55.82%
In this paper, we study estimates for eigenvalues of the clamped plate problem. A sharp upper bound for eigenvalues is given and the lower bound for eigenvalues in [10] is improved.; Comment: 16 pages

## Guaranteed Lower and upper bounds for eigenvalues of second order elliptic operators in any dimension

Hu, Jun; Ma, Rui
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
66.04%
In this paper, a new method is proposed to produce guaranteed lower bounds for eigenvalues of general second order elliptic operators in any dimension. Unlike most methods in the literature, the proposed method only needs to solve one discrete eigenvalue problem but not involves any base or intermediate eigenvalue problems, and does not need any a priori information concerning exact eigenvalues either. Moreover, it just assumes basic regularity of exact eigenfunctions. This method is defined by a novel generalized Crouzeix-Raviart element which is proved to yield asymptotic lower bounds for eigenvalues of general second order elliptic operators, and a simple post-processing method. As a byproduct, a simple and cheap method is also proposed to obtain guaranteed upper bounds for eigenvalues, which is based on generalized Crouzeix-Raviart element approximate eigenfunctions, an averaging interpolation from the the generalized Crouzeix-Raviart element space to the conforming linear element space, and an usual Rayleigh-Ritz procedure. The ingredients for the analysis consist of a crucial projection property of the canonical interpolation operator of the generalized Crouzeix-Raviart element, explicitly computable constants for two interpolation operators. Numerics are provided to demonstrate the theoretical results.

## The Bounds for Eigenvalues of Normalized and Signless Laplacian Matrices

Büyükköse, Şerife; Eski, Şehri Gülčiček
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
55.93%
In this paper, we obtain the bounds of the extreme eigenvalues of a normalized and signless Laplacian matrices using by their traces. In addition, we determine the bounds for k-th eigenvalues of normalized and signless Laplacian matrices.; Comment: 5 pages

## Note on Bounds for Eigenvalues using Traces

Sharma, R.; Kumar, R.; Saini, R.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
55.81%
We show that various old and new bounds involving eigenvalues of a complex n x n matrix are immediate consequences of the inequalities involving variance of real and complex numbers.; Comment: 13 pages

## Perturbation bounds of eigenvalues of Hermitian matrices with block structures

Nakatsukasa, Yuji
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
55.88%
We derive new perturbation bounds for eigenvalues of Hermitian matrices with block structures. The structures we consider range from a standard 2-by-2 block form to block tridiagonal and tridigaonal forms. The main idea is the observation that an eigenvalue is insensitive to componentwise perturbations if the corresponding eigenvector components are small. We show that the same idea can be used to explain two well-known phenomena, one concerning extremal eigenvalues of Wilkinson's matrices and another concerning the efficiency of aggressive early deflation applied to the symmetric tridiagonal QR algorithm.; Comment: 12 pages

## Nonpositive Eigenvalues of the Adjacency Matrix and Lower Bounds for Laplacian Eigenvalues

Charles, Zachary B.; Farber, Miriam; Johnson, Charles R.; Kennedy-Shaffer, Lee
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
55.92%
Let $NPO(k)$ be the smallest number $n$ such that the adjacency matrix of any undirected graph with $n$ vertices or more has at least $k$ nonpositive eigenvalues. We show that $NPO(k)$ is well-defined and prove that the values of $NPO(k)$ for $k=1,2,3,4,5$ are $1,3,6,10,16$ respectively. In addition, we prove that for all $k \geq 5$, $R(k,k+1) \ge NPO(k) > T_k$, in which $R(k,k+1)$ is the Ramsey number for $k$ and $k+1$, and $T_k$ is the $k^{th}$ triangular number. This implies new lower bounds for eigenvalues of Laplacian matrices: the $k$-th largest eigenvalue is bounded from below by the $NPO(k)$-th largest degree, which generalizes some prior results.; Comment: 23 pages, 12 figures

## The Lower Bounds for Eigenvalues of Elliptic Operators --By Nonconforming Finite Element Methods

Hu, Jun; Huang, Yunqing; Lin, Qun
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
66.08%
The aim of the paper is to introduce a new systematic method that can produce lower bounds for eigenvalues. The main idea is to use nonconforming finite element methods. The general conclusion herein is that if local approximation properties of nonconforming finite element spaces $V_h$ are better than global continuity properties of $V_h$, corresponding methods will produce lower bounds for eigenvalues. More precisely, under three conditions on continuity and approximation properties of nonconforming finite element spaces we first show abstract error estimates of approximate eigenvalues and eigenfunctions. Subsequently, we propose one more condition and prove that it is sufficient to guarantee nonconforming finite element methods to produce lower bounds for eigenvalues of symmetric elliptic operators. As one application, we show that this condition hold for most nonconforming elements in literature. As another important application, this condition provides a guidance to modify known nonconforming elements in literature and to propose new nonconforming elements. In fact, we enrich locally the Crouzeix-Raviart element such that the new element satisfies the condition; we propose a new nonconforming element for second order elliptic operators and prove that it will yield lower bounds for eigenvalues. Finally...

## Universal Bounds for Eigenvalues of the Polyharmonic Operators

Jost, Jürgen; Li-Jost, Xianqing; Wang, Qiaoling; Xia, Changyu
We study eigenvalues of polyharmonic operators on compact Riemannian manifolds with boundary (possibly empty). In particular, we prove a universal inequality for the eigenvalues of the polyharmonic operators on compact domains in a Euclidean space. This inequality controls the $k$th eigenvalue by the lower eigenvalues, independently of the particular geometry of the domain. Our inequality is sharper than the known Payne-P\'olya-Weinberg type inequality and also covers the important Yang inequality on eigenvalues of the Dirichlet Laplacian. We also prove universal inequalities for the lower order eigenvalues of the polyharmonic operator on compact domains in a Euclidean space which in the case of the biharmonic operator and the buckling problem strengthen the estimates obtained by Ashbaugh. Finally, we prove universal inequalities for eigenvalues of polyharmonic operators of any order on compact domains in the sphere.; Comment: 30 pages