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41.
Albrecht Bttcher Sergei M. Grudsky Egor A. Maksimenko 《Linear algebra and its applications》2008,429(1):346-366
The Szegö and Avram–Parter theorems give the limit of the arithmetic mean of the values of certain test functions at the eigenvalues of Hermitian Toeplitz matrices and the singular values of arbitrary Toeplitz matrices, respectively, as the matrix dimension goes to infinity. The question on whether these theorems are true whenever they make sense is essentially the question on whether they are valid for all continuous, nonnegative, and monotonously increasing test functions. We show that, surprisingly, the answer to this question is negative. On the other hand, we prove the two theorems in a general form which includes all versions known so far. 相似文献
42.
Aliki D. Muradova 《Advances in Computational Mathematics》2008,29(2):179-206
In this paper a spectral method and a numerical continuation algorithm for solving eigenvalue problems for the rectangular
von Kármán plate with different boundary conditions (simply supported, partially or totally clamped) and physical parameters
are introduced. The solution of these problems has a postbuckling behaviour. The spectral method is based on a variational
principle (Galerkin’s approach) with a choice of global basis functions which are combinations of trigonometric functions.
Convergence results of this method are proved and the rate of convergence is estimated. The discretized nonlinear model is
treated by Newton’s iterative scheme and numerical continuation. Branches of eigenfunctions found by the algorithm are traced.
Numerical results of solving the problems for polygonal and ferroconcrete plates are presented.
Communicated by A. Zhou. 相似文献
43.
Florent Renac 《Journal of computational physics》2011,230(14):5739-5752
An algorithm for stabilizing linear iterative schemes is developed in this study. The recursive projection method is applied in order to stabilize divergent numerical algorithms. A criterion for selecting the divergent subspace of the iteration matrix with an approximate eigenvalue problem is introduced. The performance of the present algorithm is investigated in terms of storage requirements and CPU costs and is compared to the original Krylov criterion. Theoretical results on the divergent subspace selection accuracy are established. The method is then applied to the resolution of the linear advection–diffusion equation and to a sensitivity analysis for a turbulent transonic flow in the context of aerodynamic shape optimization. Numerical experiments demonstrate better robustness and faster convergence properties of the stabilization algorithm with the new criterion based on the approximate eigenvalue problem. This criterion requires only slight additional operations and memory which vanish in the limit of large linear systems. 相似文献
44.
In this paper we prove general inequalities involving the weighted mean curvature of compact submanifolds immersed in weighted manifolds. As a consequence we obtain a relative linear isoperimetric inequality for such submanifolds. We also prove an extrinsic upper bound to the first non-zero eigenvalue of the drift Laplacian on closed submanifolds of weighted manifolds. 相似文献
45.
In 2011, the fundamental gap conjecture for Schrödinger operators was proven. This can be used to estimate the ground state energy of the time-independent Schrödinger equation with a convex potential and relative error ε. Classical deterministic algorithms solving this problem have cost exponential in the number of its degrees of freedom d. We show a quantum algorithm, that is based on a perturbation method, for estimating the ground state energy with relative error ε. The cost of the algorithm is polynomial in d and ε−1, while the number of qubits is polynomial in d and logε−1. In addition, we present an algorithm for preparing a quantum state that overlaps within 1−δ,δ∈(0,1), with the ground state eigenvector of the discretized Hamiltonian. This algorithm also approximates the ground state with relative error ε. The cost of the algorithm is polynomial in d, ε−1 and δ−1, while the number of qubits is polynomial in d, logε−1 and logδ−1. 相似文献
46.
47.
n this paper, we present an inexact inverse subspace iteration method for computing a few eigenpairs of the generalized eigenvalue problem Ax=λBx. We first formulate a version of inexact inverse subspace iteration in which the approximation from one step is used as an initial approximation for the next step. We then analyze the convergence property, which relates the accuracy in the inner iteration to the convergence rate of the outer iteration. In particular, the linear convergence property of the inverse subspace iteration is preserved. Numerical examples are given to demonstrate the theoretical results. 相似文献
48.
L. Gemignani 《Applied mathematics and computation》2011,217(19):7573-7578
Diagonal plus semiseparable matrices are constructed, the eigenvalues of which are algebraic numbers expressed by simple closed trigonometric formulas. 相似文献
49.
This paper studies the existence of nontrivial solutions for a class of higher-order m-point singular boundary value problems based on the topological degree of a completely continuous field, the first eigenvalue and its corresponding eigenfunction of a special linear operator. The nonlinear term in the boundary value problems is sign-changing and may be unbounded from below. 相似文献
50.
For the eigenvalue problem—λΔu = q(x)u in IRd, with the weight function q changing sign, conditions are discussed for existence of eigenvalues with positive decaying eigenfunctions. 相似文献