Computation of the smallest positive eigenvalue of a quadratic λ-matrix

Summary The computation of the smallest positive eigenvalue ^* of a quadratic -matrix is used to determine the stability of structures. In addition to existence results we derive two efficient and reliable methods to calculate ^*. Both methods are based on shift techniques which are discussed thoroughly with respect to convergence.     

On the differential properties of the support function of the∈-subdifferential of a convex function

We study differential properties of the support function of the-subdifferential of a convex function; applications in algorithmics are also given.     

On the optimality and sharpness of Laguerre’s lower bound on the smallest eigenvalue of a symmetric positive definite matrix

Lower bounds on the smallest eigenvalue of a symmetric positive definite matrix A ∈ R ^m×m play an important role in condition number estimation and in iterative methods for singular value computation. In particular, the bounds based on Tr(A ^?1) and Tr(A ^?2) have attracted attention recently, because they can be computed in O(m) operations when A is tridiagonal. In this paper, we focus on these bounds and investigate their properties in detail. First, we consider the problem of finding the optimal bound that can be computed solely from Tr(A ^?1) and Tr(A ^?2) and show that the so called Laguerre’s lower bound is the optimal one in terms of sharpness. Next, we study the gap between the Laguerre bound and the smallest eigenvalue. We characterize the situation in which the gap becomes largest in terms of the eigenvalue distribution of A and show that the gap becomes smallest when {Tr(A ^?1)}²/Tr(A ^?2) approaches 1 or m. These results will be useful, for example, in designing efficient shift strategies for singular value computation algorithms.     

A note on the second largest eigenvalue of the laplacian matrix of a graph ∗

In this note, a lower bound for the second largest eigenvalue of the Laplacian matrix of a graph is given in terms of the second largest degree of the graph.     

On a form of the scattering matrix for ρ-perturbations of an abstract wave equations

We present the definition of ρ-perturbations of an abstract wave equation. As a special case, this definition involves perturbations with compact support for the classical wave equation. We construct the scattering matrix for equations of such a type. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 4, pp. 445–457, April, 1999.     

The eigenvalue problem for solitary waves of a boussinesq equation,via a generalization of the rouché theorem

We consider the study of an eigenvalue problem obtained by linearizing about solitary wave solutions of a Boussinesq equation. Instead of using the technique of Evans functions as done by Pego and Weinstein in [R. Pego and M. Weinstein, Convective Linear Stability of Solitary Waves for Boussinesq equation. AMS, 99, 311–375] for this particular problem, we perform Fourier analysis to characterize solutions of the eigenvalue problem in terms of a multiplier operator and use the strong relationship between the eigenvalue problem for the linearized Boussinesq equation and the eigenvalue problem associated with the linearization about solitary wave solutions of a special form of the KdV equation. By using a generalization of the Rouché Theorem and the asymptotic behavior of the Fourier symbol corresponding to the eigenvalues problem for the Boussinesq equation and the Fourier symbol corresponding to the eigenvalues problem for the KdV equation, we show nonexistence of eigenvalues with respect to weighted space in a planar region containing the right-half plane.     

Takagi’s decomposition of a symmetric unitary matrix as a finite algorithm

Takagi’s decomposition is an analog (for complex symmetric matrices and for unitary similarities replaced by unitary congruences) of the eigenvalue decomposition of Hermitian matrices. It is shown that, if a complex matrix is not only symmetric but is also unitary, then its Takagi decomposition can be found by quadratic radicals, that is, by means of a finite algorithm that involves arithmetic operations and quadratic radicals. A similar fact is valid for the eigenvalue decomposition of reflections, which are Hermitian unitary matrices.     

Estimation of the location parameter of the ℓ1-norm symmetric matrix variate distributions

An identity of integrals for the ℓ₁-norm symmetric matrix variate distributions with unknown common location parameter and unknown and possibly unequal scale parameters of the columns is established. An unbiased estimator for the location parameter is obtained and is shown to dominate the maximum likelihood estimator under the squared error loss. Under certain conditions this unbiased estimator is the uniformly minimum variance unbiased estimator.     

Closed convex hull of multivalent symmetric close-to-convex functions of order β with the Montel normalization

We obtain a one-to-one tranformation of the family of multivalent symmetric close-to-convex functions of order β, onto the family of multivalent symmetric close-to-convex functions of order β with the Montel normalization. Using this transformation we determine the closed convex hull and extreme points of the latter class for β≥1. The present investigation is partially supported by National Board for Higher Mathematics, Department of Atomic Energy, Government of India under Grant No. 48/2/2003-R & D II.     

Closed convex hull of multivalent symmetric close-to-convex functions of order β with the Montel normalization

We obtain a one-to-one tranformation of the family of multivalent symmetric close-to-convex functions of order β, onto the family of multivalent symmetric close-to-convex functions of order β with the Montel normalization. Using this transformation we determine the closed convex hull and extreme points of the latter class for β≥1.     

Convergence analysis of a modified BFGS method on convex minimizations

A modified BFGS method is proposed for unconstrained optimization. The global convergence and the superlinear convergence of the convex functions are established under suitable assumptions. Numerical results show that this method is interesting.     

Limits as p → ∞ of p-laplacian eigenvalue problems perturbed with a concave or convex term

We investigate the asymptotic behaviour as p → ∞ of sequences of positive weak solutions of the equation $$\left\{\begin{array}{l}-\Delta_p u = \lambda\,u^{p-1}+ u^{q(p)-1}\quad {\rm in}\quad \Omega,\\ u = 0 \quad {\rm on}\quad \partial\Omega,\end{array} \right.$$ where λ > 0 and either 1 < q(p) < p or p < q(p), with ${{\lim_{p\to\infty}{q(p)}/{p}=Q\neq1}}$ . Uniform limits are characterized as positive viscosity solutions of the problem $$\left\{\begin{array}{l}\min\left\{|\nabla u (x)| - \max\{\Lambda\,u (x),u ^Q(x)\}, -\Delta_{\infty}u (x)\right\} = 0 \quad {\rm in} \quad \Omega,\\ u = 0\quad {\rm on}\quad \partial\Omega.\end{array}\right.$$ for appropriate values of Λ > 0. Due to the decoupling of the nonlinearity under the limit process, the limit problem exhibits an intermediate behavior between an eigenvalue problem and a problem with a power-like right-hand side. Existence and non-existence results for both the original and the limit problems are obtained.     

Solution of the partial eigenvalue problem for a regular matrix pencilD(λ) =A − λB

One presents the ALGOL procedures which implement the algorithm for the determination of the group of smallest (greatest) eigenvalues and their corresponding eigenvectors for a matrix pencil where A and B are real square matrices of simple structure. From the initial pencil one constructs a matrix C, whose eigenvalues are taken as the initial approximations to the eigenvalues from the group of the smallest (greatest) eigenvalues of the pencil. The refinement of the eigen-values is performed on the basis of the theory of perturbations. Then one determines the eigen-vectors and one computes the infinite norm of the residual. One gives ALGOL programs and test examples.     

Symmetries of quadratic form classes and of quadratic surd continued fractions. Part I: A Poincaré tiling of the de Sitter world

The problem of classifying the indefinite binary quadratic forms with integer coefficients is solved by introducing a special partition of the de Sitter world, where the coefficients of the forms lie, into separate domains. Under the action of the special linear group acting on the integer plane lattice, each class of indefinite forms has a well-defined finite number of representatives inside each such domain. In the second part, we will show how to obtain the symmetry type of a class and also the number of its points in all domains from a single representative of that class.