Extremal eigenvalue intervals of symmetric tridiagonal interval matrices

Computing the extremal eigenvalue bounds of interval matrices is non‐deterministic polynomial‐time (NP)‐hard. We investigate bounds on real eigenvalues of real symmetric tridiagonal interval matrices and prove that for a given real symmetric tridiagonal interval matrices, we can achieve its exact range of the smallest and largest eigenvalues just by computing extremal eigenvalues of four symmetric tridiagonal matrices.     

A property of eigenvalue bounds for a class of symmetric tridiagonal interval matrices

The eigenvalue bounds of interval matrices are often required in some mechanical and engineering fields. In this paper, we consider an interval eigenvalue problem with symmetric tridiagonal matrices. A theoretical result is obtained that under certain assumptions the upper and lower bounds of interval eigenvalues of the problem must be achieved just at some vertex matrices of the interval matrix. Then a sufficient condition is provided to guarantee the assumption to be satisfied. The conclusion is illustrated also by a numerical example. Copyright © 2010 John Wiley & Sons, Ltd.     

Some counting questions for matrices with restricted entries

We obtain upper bounds for the number of arbitrary and symmetric matrices with integer entries in a given box (in an arbitrary location) and a given determinant. We then apply these bounds to estimate the number of matrices in such boxes which have an integer eigenvalues. Finally, we outline some open questions.     

The inverse eigenvalue problem for linear trees

We prove the sufficiency of the Linear Superposition Principle for linear trees, which characterizes the spectra achievable by a real symmetric matrix whose underlying graph is a linear tree. The necessity was previously proven in [10]. This is the most general class of trees for which the inverse eigenvalue problem has been solved. We explore many consequences, including the Degree Conjecture for possible spectra, upper bounds for the minimum number of eigenvalues of multiplicity 1, and the equality of the diameter of a linear tree and its minimum number of distinct eigenvalues.     

Exact bounds for the sensitivity analysis of structures with uncertain-but-bounded parameters

Based on interval mathematical theory, the interval analysis method for the sensitivity analysis of the structure is advanced in this paper. The interval analysis method deals with the upper and lower bounds on eigenvalues of structures with uncertain-but-bounded (or interval) parameters. The stiffness matrix and the mass matrix of the structure, whose elements have the initial errors, are unknown except for the fact that they belong to given bounded matrix sets. The set of possible matrices can be described by the interval matrix. In terms of structural parameters, the stiffness matrix and the mass matrix take the non-negative decomposition. By means of interval extension, the generalized interval eigenvalue problem of structures with uncertain-but-bounded parameters can be divided into two generalized eigenvalue problems of a pair of real symmetric matrix pair by the real analysis method. Unlike normal sensitivity analysis method, the interval analysis method obtains informations on the response of structures with structural parameters (or design variables) changing and without any partial differential operation. Low computational effort and wide application rang are the characteristic of the proposed method. Two illustrative numerical examples illustrate the efficiency of the interval analysis.     

The spread of matrices and polynomials

The spread of a matrix (or polynomial) is the maximum distance between any two of its eigenvalues (or its zeros). E. Deutsch has recently given upper bounds for the spread of matrices and polynomials. We obtain sharper, simpler upper bounds and observe that they are also upper bounds for the sum of the absolute values of the two largest eigenvalues (or zeros).     

The Stability Analysis of the Solutions of Inverse Eigenvalue Problems

This paper gives perturbation bounds of some solutions of the classical additive and multiplicative inverse eigenvalue problems for real symmetric matrices.     

On the accuracy of the Gerschgorin circle theorem for bounding the spread of a real symmetric matrix

The spread of a matrix with real eigenvalues is the difference between its largest and smallest eigenvalues. The Gerschgorin circle theorem can be used to bound the extreme eigenvalues of the matrix and hence its spread. For nonsymmetric matrices the Gerschgorin bound on the spread may be larger by an arbitrary factor than the actual spread even if the matrix is symmetrizable. This is not true for real symmetric matrices. It is shown that for real symmetric matrices (or complex Hermitian matrices) the ratio between the bound and the spread is bounded by $\text{p+1}$, where p is the maximum number of off diagonal nonzeros in any row of the matrix. For full matrices this is just $\text{n}$. This bound is not quite sharp for n greater than 2, but examples with ratios of $\text{n?1}$ for all n are given. For banded matrices with m nonzero bands the maximum ratio is bounded by $\text{m}$ independent of the size of n. This bound is sharp provided only that n is at least 2m. For sparse matrices, p may be quite small and the Gerschgorin bound may be surprisingly accurate.     

Upper bounds for eigenvalues of Cauchy-Hankel tensors

We present upper bounds of eigenvalues for finite and infinite dimensional Cauchy-Hankel tensors. It is proved that an m-order infinite dimensional Cauchy-Hankel tensor defines a bounded and positively (m－1)-homogeneous operator from l¹ into l^p (1<p<∞); and two upper bounds of corresponding positively homogeneous operator norms are given. Moreover, for a fourth-order real partially symmetric Cauchy-Hankel tensor, suffcient and necessary conditions of M-positive definiteness are obtained, and an upper bound of M-eigenvalue is also shown.     

On the spread of matrices and polynomials

The spread of a matrix (polynomial) is defined as the maximum distance between two of its eigenvalues (zeros). In this note upper bounds are found for the spread of matrices and polynomials.     

Some New Bounds for Singular Values and Eigenvalues of Matrix Products

For two Hermitian matrices A and B, at least one of which is positive semidefinite, we give upper and lower bounds for each eigenvalue of AB in terms of the eigenvalues of A and B. For two complex matrices A,B with known singular values, upper and lower bounds are deduced for each singular value of AB.     

On large-scale diagonalization techniques for the Anderson model of localization

We propose efficient preconditioning algorithms for an eigenvalue problem arising in quantum physics, namely the computation of a few interior eigenvalues and their associated eigenvectors for large-scale sparse real and symmetric indefinite matrices of the Anderson model of localization. Our preconditioning approaches for the shift-and-invert symmetric indefinite linear system are based on maximum weighted matchings and algebraic multi-level incomplete LDL^T factorizations. These techniques can be seen as a complement to the alternative idea of using more complete pivoting techniques for the highly ill-conditioned symmetric indefinite Anderson matrices. Our numerical examples reveal that recent algebraic multi-level preconditioning solvers can accelerate the computation of a large-scale eigenvalue problem corresponding to the Anderson model of localization by several orders of magnitude. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A filtering method for the interval eigenvalue problem

We consider the general problem of computing intervals that contain the real eigenvalues of interval matrices. Given an outer approximation (superset) of the real eigenvalue set of an interval matrix, we propose a filtering method that iteratively improves the approximation. Even though our method is based on a sufficient regularity condition, it is very efficient in practice and our experimental results suggest that it improves, in general, significantly the initial outer approximation. The proposed method works for general, as well as for symmetric interval matrices.     

On the asymptotics of all eigenvalues of Hermitian Toeplitz band matrices

While extreme eigenvalues of large Hermitian Toeplitz matrices have been studied in detail for a long time, much less is known about individual inner eigenvalues. This paper explores the behavior of the jth eigenvalue of an n-by-n banded Hermitian Toeplitz matrix as n goes to infinity and provides asymptotic formulas that are uniform in j for 1 ≤ j ≤ n. The real-valued generating function of the matrices is assumed to increase strictly from its minimum to its maximum and then to decrease strictly back from the maximum to the minimum, having nonzero second derivatives at the minimum and the maximum.     

Inertia‐based spectrum slicing for symmetric quadratic eigenvalue problems

In the quadratic eigenvalue problem (QEP) with all coefficient matrices symmetric, there can be complex eigenvalues. However, some applications need to compute real eigenvalues only. We propose a Lanczos‐based method for computing all real eigenvalues contained in a given interval of large‐scale symmetric QEPs. The method uses matrix inertias of the quadratic polynomial evaluated at different shift values. In this way, for hyperbolic problems, it is possible to make sure that all eigenvalues in the interval have been computed. We also discuss the general nonhyperbolic case. Our implementation is memory‐efficient by representing the computed pseudo‐Lanczos basis in a compact tensor product representation. We show results of computational experiments with a parallel implementation in the SLEPc library.     

Homotopy method for the eigenvalues of symmetric tridiagonal matrices

We will present the homotopy method for finding eigenvalues of symmetric, tridiagonal matrices. This method finds eigenvalues separately, which can be a large advantage on systems with parallel processors. We will introduce the method and establish some bounds that justify the use of Newton’s method in constructing the homotopy curves.     

Über das Stekloffsche Eigenwertproblem: Isoperimetrische Ungleichungen für symmetrische Gebiete

Summary Isoperimetric inequalities ofPólya [6] for symmetric membranes are extended to the Stekloff problem. The given symmetric domainG _z is mapped conformally onto a circle; some (harmonic) eigenfunctions of the circle are transplanted ontoG _z ; application of Rayleigh's and Poincaré's principles to the transplanted functions gives upper bounds for a number of eigenvalues ofG _z which depends on the order of symmetry of the domain.     

矩阵特征值的几个不等式

研究矩阵特征值的上、下界以及特征值的实部、虚部的不等式 ,给出了特征值一些新的上界和下界     

Computing eigenspaces with specified eigenvalues of a regular matrix pair (A,B) and condition estimation: theory,algorithms and software

Theory, algorithms and LAPACK-style software for computing a pair of deflating subspaces with specified eigenvalues of a regular matrix pair (A, B) and error bounds for computed quantities (eigenvalues and eigenspaces) are presented. Thereordering of specified eigenvalues is performed with a direct orthogonal transformation method with guaranteed numerical stability. Each swap of two adjacent diagonal blocks in the real generalized Schur form, where at least one of them corresponds to a complex conjugate pair of eigenvalues, involves solving a generalized Sylvester equation and the construction of two orthogonal transformation matrices from certain eigenspaces associated with the diagonal blocks. The swapping of two 1×1 blocks is performed using orthogonal (unitary) Givens rotations. Theerror bounds are based on estimates of condition numbers for eigenvalues and eigenspaces. The software computes reciprocal values of a condition number for an individual eigenvalue (or a cluster of eigenvalues), a condition number for an eigenvector (or eigenspace), and spectral projectors onto a selected cluster. By computing reciprocal values we avoid overflow. Changes in eigenvectors and eigenspaces are measured by their change in angle. The condition numbers yield bothasymptotic andglobal error bounds. The asymptotic bounds are only accurate for small perturbations (E, F) of (A, B), while the global bounds work for all (E, F.) up to a certain bound, whose size is determined by the conditioning of the problem. It is also shown how these upper bounds can be estimated. Fortran 77software that implements our algorithms for reordering eigenvalues, computing (left and right) deflating subspaces with specified eigenvalues and condition number estimation are presented. Computational experiments that illustrate the accuracy, efficiency and reliability of our software are also described.     

Eigenvalues for infinite matrices

This paper is concerned with the problem of determining the location of eigenvalues for diagonally dominant infinite matrices; upper and lower bounds for eigenvalues are established. For tridiagonal matrices, a numerical procedure for improving the bounds is given, and the approximation of the eigenvectors is also discussed. The techniques are illustrated for the solution of the well-known Mathieu's equation.