Bounds for the extremal eigenvalues of a class of symmetric tridiagonal matrices with applications

We consider a class of symmetric tridiagonal matrices which may be viewed as perturbations of Toeplitz matrices. The Toeplitz structure is destroyed since two elements on each off-diagonal are perturbed. Based on a careful analysis, we derive sharp bounds for the extremal eigenvalues of this class of matrices in terms of the original data of the given matrix. In this way, we also obtain a lower bound for the smallest singular value of certain matrices. Some numerical results indicate that our bounds are extremely good.     

Posterior two-sided estimates for eigenvalues of the singular Sturm-Liouville problem

Based on three-point difference and variational-difference schemes for auxiliary nonsingular spectral problems providing for a two-sided approximation of eigenvalues of the singular Sturm-Liouville problem, posterior upper and lower estimates for eigenvalues of the input singular problem are obtained. Bibliography: 3 titles. Translated fromObchyslyuval’na ta Prykladna Matematyka, No. 76, 1992, pp. 39–49.     

Landesman–Lazer conditions for difference equations involving sublinear perturbations

We study the existence and uniqueness for discrete Neumann and periodic problems. We consider both ordinary and partial difference equations involving sublinear perturbations. All the proofs are based on reformulating these discrete problems as a general singular algebraic system. Firstly, we use variational techniques (specifically, the Saddle Point Theorem) and prove the existence result based on a type of Landesman–Lazer condition. Then we show that for a certain class of bounded nonlinearities this condition is even necessary and therefore, we specify also the cases in which there does not exist any solution. Finally, the uniqueness is discussed.     

The solution methods for the largest eigenvalue (singular value) of nonnegative tensors and convergence analysis

In this paper we study two solution methods for finding the largest eigenvalue (singular value) of general square (rectangular) nonnegative tensors. For a positive tensor, one can find the largest eigenvalue (singular value) based on the properties of the positive tensor and the power-type method. While for a general nonnegative tensor, we use a series of decreasing positive perturbations of the original tensor and repeatedly recall power-type method for finding the largest eigenvalue (singular value) of a positive tensor with an inexact strategy. We prove the convergence of the method for the general nonnegative tensor. Under a certain assumption, the computing complexity of the method is established. Motivated by the interior-point method for the convex optimization, we put forward a one-step inner iteration power-type method, whose convergence is also established under certain assumption. Additionally, by using embedding technique, we show the relationship between the singular values of the rectangular tensor and the eigenvalues of related square tensor, which suggests another way for finding the largest singular value of nonnegative rectangular tensor besides direct power-type method for this problem. Finally, numerical examples of our algorithms are reported, which demonstrate the convergence behaviors of our methods and show that the algorithms presented are promising.     

Singular left-definite Sturm-Liouville problems

We study singular left-definite Sturm-Liouville problems with an indefinite weight function. The existence of eigenvalues is established based on the existence of eigenvalues of corresponding right-definite problems. Furthermore, for each singular left-definite problem with limit-circle non-oscillatory endpoints we construct a regular left-definite problem with the same eigenvalues and use it to obtain properties of eigenvalues and eigenfunctions. Inequalities among eigenvalues recently established for regular left-definite problems are extended to the singular case.     

Lower bounds of error estimates for singular perturbation problems with Jacobi-Spectral approximations

With weighted orthogonal Jacobi polynomials, we study spectral approximations for singular perturbation problems on an interval. The singular parameters of the model are included in the basis functions, and then its stiff matrix is diagonal. Considering the estimations for weighted orthogonal coefficients, a special technique is proposed to investigate the a posteriori error estimates. In view of the difficulty of a posteriori error estimates for spectral approximations, we employ a truncation projection to study lower bounds for the models. Specially, we present the lower bounds of a posteriori error estimates with two different weighted norms in details.     

Singular point perturbation of an odd operator in ℤ2-Graded space

In the paper we study supersymmetric models for point interaction perturbations of operators of Dirac type and their spectral properties. Such models are considered in the class of odd self-adjoint operators in ℤ₂-graded Pontryagin space. We present in detail the previously considered realization method of strongly singular perturbation by means of their embedding into the theory of self-adjoint extensions. We describe odd self-adjoint extensions of odd symmetric operators with deficiency indices (1,1) in ℤ₂-graded Pontryagin space and squares of such extensions using Krein’s formula for the resolvent. The results obtained are refined in application to singular perturbations of odd self-adjoint differential operators. Translated fromMatematicheskie Zametki, Vol. 66, No. 6, pp. 924–940, December, 1999.     

Singular perturbations of some nonlinear problems

We deal with singular perturbations of nonlinear problems depending on a small parameter ε > 0. First we consider the abstract theory of singular perturbations of variational inequalities involving some nonlinear operators, defined in Banach spaces, and describe the asymptotic behavior of these solutions as ε → 0. Then these abstract results are applied to some boundary value problems. Bibliography: 15 titles.     

半线性系统的Robin边值问题的奇摄动

本文利用微分不等式的方法与技巧研究半线性系统的Robin边值问题的奇摄动.我们假定相应的退化系统至少有一个Ⅰ_q-稳定的解.这种依分量Ⅰ_q-稳定性的条件将允许我们去得到解的每一个分量的估计.     

Spectral convergence and nonlinear dynamics of reaction-diffusion equations under perturbations of the domain

In this paper we obtain the continuity of attractors for semilinear parabolic problems with Neumann boundary conditions relatively to perturbations of the domain. We show that, if the perturbations on the domain are such that the convergence of eigenvalues and eigenfunctions of the Neumann Laplacian is granted then, we obtain the upper semicontinuity of the attractors. If, moreover, every equilibrium of the unperturbed problem is hyperbolic we also obtain the continuity of attractors. We also give necessary and sufficient conditions for the spectral convergence of Neumann problems under perturbations of the domain.     

W2,2 interior convergence for some class of elliptic anisotropic singular perturbations problems

In this paper, we deal with anisotropic singular perturbations of some class of elliptic problems. We study the asymptotic behavior of the solution in a certain second-order pseudo Sobolev space.     

Eigenvalue bifurcation in multiparameter families of non-self-adjoint operator matrices

We consider two-point non-self-adjoint boundary eigenvalue problems for linear matrix differential operators. The coefficient matrices in the differential expressions and the matrix boundary conditions are assumed to depend analytically on the complex spectral parameter λ and on the vector of real physical parameters p. We study perturbations of semi-simple multiple eigenvalues as well as perturbations of non-derogatory eigenvalues under small variations of p. Explicit formulae describing the bifurcation of the eigenvalues are derived. Application to the problem of excitation of unstable modes in rotating continua such as spherically symmetric MHD α ²-dynamo and circular string demonstrates the efficiency and applicability of the approach.     

Coefficient Stability of the Solution of a Difference Scheme Approximating a Mixed Problem for a Semilinear Parabolic Equation

We study the coefficient stability of a difference scheme approximating a mixed problem for a one-dimensional semilinear parabolic equation. We obtain sufficient conditions on the input data under which the solutions of the differential and difference problems are bounded. We also obtain estimates of perturbations of the solution of a linearized difference scheme with respect to perturbations of the coefficients; these estimates agree with the estimates for the differential problem.     

Explicit multistep method for the numerical solution of stiff differential equations

An explicit multistep method of variable order for integrating stiff systems with high accuracy and low computational costs is examined. To stabilize the computational scheme, componentwise estimates are used for the eigenvalues of the Jacobian matrix having the greatest moduli. These estimates are obtained at preliminary stages of the integration step. Examples are given to demonstrate that, for certain stiff problems, the method proposed is as efficient as the best implicit methods.     

Square roots of elliptic second order divergence operators on strongly Lipschitz domains: $L^p$ theory

We study estimates for square roots of second order elliptic non necessarily selfadjoint operators in divergence form on Lipschitz domains subject to Dirichlet or to Neumann boundary conditions, pursuing our work [4] where we considered operators on . We obtain among other things for all if L is real symmetric and the domain bounded, which is new for . We also obtain similar results for perturbations of constant coefficients operators. Our methods rely on a singular integral representation, Calderón-Zygmund theory and quadratic estimates. A feature of this study is the use of a commutator between the resolvent of the Laplacian (Dirichlet and Neumann) and partial derivatives which carries the geometry of the boundary. Received: 12 January 2000 / Published online: 4 May 2001     

Weyl numbers and eigenvalues of abstract summing operators

We estimate Weyl numbers and eigenvalues of operators via studying their abstract summing norms. In particular we prove estimates of these summing norms for abstract interpolation Lorentz spaces. For this we combine factorization theorems with estimates of concavity constants. Finally we apply our general eigenvalue results to integral operators with kernels of weakly singular type. We obtain asymptotically optimal estimates which extend the well-known classical results.     

Computation of exact inertia and inclusions of eigenvalues (singular values) of tridiagonal (bidiagonal) matrices

This report may be considered as a non-trivial extension of an unpublished report by William Kahan (Accurate Eigenvalues of a symmetric tri-diagonal matrix, Technical Report CS 41, Computer Science Department, Stanford University, 1966). His interplay between matrix theory and computer arithmetic led to the development of algorithms for computing accurate eigenvalues and singular values. His report is generally considered as the precursor for the development of IEEE standard 754 for binary arithmetic. This standard has been universally adopted by virtually all PC, workstation and midrange hardware manufactures and tens of billions of such machines have been produced. Now we use the features in this standard to improve the original algorithm.In this paper, we describe an algorithm in floating-point arithmetic to compute the exact inertia of a real symmetric (shifted) tridiagonal matrix. The inertia, denoted by the integer triplet (π, ν, ζ), is defined as the number of positive, negative and zero eigenvalues of a real symmetric (or complex Hermitian) matrix and the adjective exact refers to the eigenvalues computed in exact arithmetic. This requires the floating-point computation of the diagonal matrix D of the LDL^t factorization of the shifted tridiagonal matrix T − τI with +∞ and −∞ rounding modes defined in IEEE 754 standard. We are not aware of any other algorithm which gives the exact answer to a numerical problem when implemented in floating-point arithmetic in standard working precisions. The guaranteed intervals for eigenvalues are obtained by bisection or multisection with this exact inertia information. Similarly, using the Golub-Kahan form, guaranteed intervals for singular values of bidiagonal matrices can be computed. The diameter of the eigenvalue (singular value) intervals depends on the number of shifts with inconsistent inertia in two rounding modes. Our algorithm not only guarantees the accuracy of the solutions but is also consistent across different IEEE 754 standard compliant architectures. The unprecedented accuracy provided by our algorithms could be also used to debug and validate standard floating-point algorithms for computation of eigenvalues (singular values). Accurate eigenvalues (singular values) are also required by certain algorithms to compute accurate eigenvectors (singular vectors).We demonstrate the accuracy of our algorithms by using standard matrix examples. For the Wilkinson matrix, the eigenvalues (in IEEE double precision) are very accurate with an (open) interval diameter of 6 ulps (units of the last place held of the mantissa) for one of the eigenvalues and lesser (down to 2 ulps) for others. These results are consistent across many architectures including Intel, AMD, SGI and DEC Alpha. However, by enabling IEEE double extended precision arithmetic in Intel/AMD 32-bit architectures at no extra computational cost, the (open) interval diameters were reduced to one ulp, which is the best possible solution for this problem. We have also computed the eigenvalues of a tridiagonal matrix which manifests in Gauss-Laguerre quadrature and the results are extremely good in double extended precision but less so in double precision. To demonstrate the accuracy of computed singular values, we have also computed the eigenvalues of the Kac₃₀ matrix, which is the Golub-Kahan form of a bidiagonal matrix. The tridiagonal matrix has known integer eigenvalues. The bidiagonal Cholesky factor of the Gauss-Laguerre tridiagonal is also included in the singular value study.     

Eigenvalue bifurcation in multiparameter families of non-self-adjoint operator matrices

We consider two-point non-self-adjoint boundary eigenvalue problems for linear matrix differential operators. The coefficient matrices in the differential expressions and the matrix boundary conditions are assumed to depend analytically on the complex spectral parameter λ and on the vector of real physical parameters p. We study perturbations of semi-simple multiple eigenvalues as well as perturbations of non-derogatory eigenvalues under small variations of p. Explicit formulae describing the bifurcation of the eigenvalues are derived. Application to the problem of excitation of unstable modes in rotating continua such as spherically symmetric MHD α ²-dynamo and circular string demonstrates the efficiency and applicability of the approach.     

Some local eigenvalue estimates involving curvatures

We first establish a local Faber–Krahn isoperimetric comparison in terms of scalar curvature pinching. Secondly we derive estimates of Cheeger constants related to the Dirichlet and Neumann problems via the (relative) isoperimetric profiles which allow us to obtain, in particular, lower bounds for first non-zero eigenvalues of the problem of Dirichlet and Neumann. These estimates involve scalar curvature and mean curvature respectively.