On upper bounds for Laplacian graph eigenvalues

In this paper, we obtain the following upper bound for the largest Laplacian graph eigenvalue λ(G):

     

Error bounds for computed eigenvalues and eigenvectors

Summary On the basis of an existence theorem for solutions of nonlinear systems, a method is given for finding rigorous error bounds for computed eigenvalues and eigenvectors of real matrices. It does not require the usual assumption that the true eigenvectors span the whole space. Further, a priori error estimates for eigenpairs corrected by an iterative method are given. Finally the results are illustrated with numerical examples.Dedicated to Professor Yoshikazu Nakai on his sixtieth birthday     

Sharp bounds for the first non-zero Stekloff eigenvalues

Let (M,,) be an n(2)-dimensional compact Riemannian manifold with boundary and non-negative Ricci curvature. Consider the following two Stekloff eigenvalue problems

where Δ is the Laplacian operator on M and ν denotes the outward unit normal on ∂M. The first non-zero eigenvalues of the above problems will be denoted by p₁ and q₁, respectively. In the present paper, we prove that if the principle curvatures of the second fundamental form of ∂M are bounded below by a positive constant c, then with equality holding if and only if Ω is isometric to an n-dimensional Euclidean ball of radius , here λ₁ denotes the first non-zero eigenvalue of the Laplacian of ∂M. We also show that if the mean curvature of ∂M is bounded below by a positive constant c then q₁nc with equality holding if and only if M is isometric to an n-dimensional Euclidean ball of radius . Finally, we show that q₁A/V and that if the equality holds and if there is a point x₀∂M such that the mean curvature of ∂M at x₀ is no less than A/{nV}, then M is isometric to an n-dimensional Euclidean ball, being A and V the area of ∂M and the volume of M, respectively.     

Two perturbation bounds for singular values and eigenvalues

The relative error in as an approximation to α is measured by

Universal bounds for eigenvalues of the biharmonic operator

In this paper, we study the eigenvalues of the clamped plate problem:

     

Lower bounds for eigenvalues of the Dirac-Witten operator

We get optimal lower bounds for the eigenvalues of the Dirac-Witten operator of compact(with or without boundary) spacelike hypersurfaces of Lorentian manifold satisfying certain conditions,just in terms of the mean curvature and the scalar curvature and the spinor energy-momentum tensor. In the limiting case,the spacelike hypersurface is either maximal and Einstein manifold with positive scalar curvature or Ricci-flat manifold with nonzero constant mean curvature.     

Upper bounds for the eigenvalues of Hessian equations

In this paper, we prove some upper bounds for the Dirichlet eigenvalues of a class of fully nonlinear elliptic equations, namely the Hessian equations.     

On bounds for eigenvalues of real symmetric matrices

Let A=(a_ij) be a real symmetric matrix of order n. We characterize all nonnegative vectors x=(x₁,...,x_n) and y=(y₁,...,y_n) such that any real symmetric matrix B=(b_ij), with b_ij=a_ij, i≠jhas its eigenvalues in the union of the intervals [b_ij?y_i, b_ij+ x_i]. Moreover, given such a set of intervals, we derive better bounds for the eigenvalues of B using the 2n quantities {b_ii?y, b_ii+x_i}, i=1,..., n.     

Leighton's bounds for Sturm-Liouville eigenvalues

An error is pointed out in a method of W. Leighton for computing two-sided bounds for the eigenvalues of the Sturm-Liouville problem (ry′)′ + λpy = 0, y(a) = y(b) = 0. The error is corrected, the underlying theory is examined and the method is generalized.     

More bounds for elgenvalues using traces

Let the n × n complex matrix A have complex eigenvalues λ₁,λ₂,…λ_n. Upper and lower bounds for Σ(Reλ_i)² are obtained, extending similar bounds for Σ|λ_i|² obtained by Eberlein (1965), Henrici (1962), and Kress, de Vries, and Wegmann (1974). These bounds involve the traces of A¹A, B², C², and D², where $\text{B}\text{=}\text{1}\text{2}\text{(}\text{A}\text{+}\text{A}{}^1\text{)}$, $\text{C}\text{=}\text{1}\text{2}\text{(}\text{A}\text{?}\text{A}{}^1\text{) /i}$, and $\text{D}\text{=}\text{AA}{}^1\text{?}\text{A}{}^1\text{A}$, and strengthen some of the results in our earlier paper “Bounds for eigenvalues using traces” in Linear Algebra and Appl. [12].     

Error bounds for computed eigenvalues and eigenvectors. II

Summary In this paper, motivated by Symm-Wilkinson's paper [5], we describe a method which finds the rigorous error bounds for a computed eigenvalue ⁽⁰⁾ and a computed eigenvectorx ⁽⁰⁾ of any matrix A. The assumption in a previous paper [6] that ⁽⁰⁾,x ⁽⁰⁾ andA are real is not necessary in this paper. In connection with this method, Symm-Wilkinson's procedure is discussed, too.     

Error bounds and estimates for eigenvalues of integral equations

Summary Approximate solutions of the linear integral equation eigenvalue problem can be obtained by the replacement of the integral by a numerical quadrature formula and then collocation to obtain a linear algebraic eigenvalue problem. This method is often called the Nyström method and its convergence was discussed in [7]. In this paper computable error bounds and dominant error terms are derived for the approximation of simple eigenvalues of nonsymmetric kernels.     

Extrinsic eigenvalues upper bounds for submanifolds in weighted manifolds

Annals of Global Analysis and Geometry - We prove Reilly-type upper bounds for divergence-type operators of the second order as well as for Steklov problems on submanifolds of Riemannian manifolds...     

Lower bounds for generalized eigenvalues of the quasilinear systems

In this paper, we establish several new Lyapunov-type inequalities for two classes of one-dimensional quasilinear elliptic systems of resonant type, which generalize or improve all related existing ones. Then we use the Lyapunov-type inequalities obtained in this paper to derive a better lower bound for the generalized eigenvalues of the one-dimensional quasilinear elliptic system with the Dirichlet boundary conditions.     

Differential inequalities for Riesz means and Weyl-type bounds for eigenvalues

We derive differential inequalities and difference inequalities for Riesz means of eigenvalues of the Dirichlet Laplacian,

     

Perturbation bounds for means of eigenvalues and invariant subspaces

When a matrix is close to a matrix with a multiple eigenvalue, the arithmetic mean of a group of eigenvalues is a good approximation to this multiple eigenvalue. A theorem of Gershgorin type for means of eigenvalues is proved and applied as a perturbation theorem for a degenerate matrix.For a multiple eigenvalue we derive bounds for computed bases of subspaces of eigenvectors and principal vectors, relating them to the spaces spanned by the last singular vectors of corresponding powers of the matrix. These bounds assure that, provided the dimensionalities are chosen appropriately, the angles of rotation of the subspaces are of the same order of magnitude as the perturbation of the matrix.A numerical example is given.