Calcolo degli autovalori di una equazione integrale di fredholm di 2a specie a nucleo simmetrico del «ciclo chiuso»

An eigenvalue problem concerning a Fredholm integral equation of second kind, with a symmetric kernel of «closed cycle» is considered. The Rayleigh Ritz procedure and the orthogonal invariants method are applied for determining rigorous bounds for the eigenvalues.     

On the efficient and accurate solution of the skew-symmetric eigenvalue problem: An arrangement of new and already known algorithmic formulations

Algorithms are presented to solve the special eigenvalue problem $\text{AZ}\text{=}\text{Zλ}$, where $\text{A}$ is skew-symmetric. The effective use of Householder's method, the bisection method and inverse iteration for solving the complete eigen-value problem are described in some detail. Simultaneous vector iteration is formulated for skew-symmetric matrices. The amount of work for the skew-symmetric Jacobi algorithm and the simultaneous vector iteration may be reduced by using the solution of a simplified eigenvalue problem. For Hermitian matrices also quadratic eigenvalue bounds for groups of eigenvalues and linear bounds for groups of eigenvectors are derived. The case where the set of calculated eigenvectors is not orthonormal is considered in some detail. In principle, the skew-symmetric eigenvalue problem may be easily transformed into a symmetric eigenvalue problem; but such a procedure has the following disadvantages: first, the results are in general less accurate, and, second, the eigenvectors which belong to well separated eigenvalues are not uniquely determined.     

A High Accuracy Post-processing Algorithm for the Eigenvalues of Elliptic Operators

In a very recent paper (Hu et al., The lower bounds for eigenvalues of elliptic operators by nonconforming finite element methods, Preprint, 2010), we prove that the eigenvalues by the nonconforming finite element methods are smaller than the exact ones for the elliptic operators. It is well-known that the conforming finite element methods produce the eigenvalues above to the exact ones. In this paper, we combine these two aspects and derive a new post-processing algorithm to approximate the eigenvalues of elliptic operators. We implement this algorithm and find that it actually yields very high accuracy approximation on very coarser mesh. The numerical results demonstrate that the high accuracy herein is of two fold: the much higher accuracy approximation on the very coarser mesh and the much higher convergence rate than a single lower/upper bound approximation. Moreover, we propose some acceleration technique for the algorithm of the discrete eigenvalue problem based on the solution of the discrete eigenvalue problem which yields the upper bound of the eigenvalue. With this acceleration technique we only need several iterations (two iterations in our example) to find the numerical solution of the discrete eigenvalue problem which produces the lower bound of the eigenvalue. Therefore we only need to solve essentially one discrete eigenvalue problem.     

A Rayleigh–Ritz style method for large-scale discriminant analysis

Linear Discriminant Analysis (LDA) is one of the most popular approaches for supervised feature extraction and dimension reduction. However, the computation of LDA involves dense matrices eigendecomposition, which is time-consuming for large-scale problems. In this paper, we present a novel algorithm called Rayleigh–Ritz Discriminant Analysis (RRDA) for efficiently solving LDA. While much of the prior research focus on transforming the generalized eigenvalue problem into a least squares formulation, our method is instead based on the well-established Rayleigh–Ritz framework for general eigenvalue problems and seeks to directly solve the generalized eigenvalue problem of LDA. By exploiting the structures in LDA problems, we are able to design customized and highly efficient subspace expansion and extraction strategy for the Rayleigh–Ritz procedure. To reduce the storage requirement and computational complexity of RRDA for high dimensional, low sample size data, we also establish an equivalent reduced model of RRDA. Practical implementations and the convergence result of our method are also discussed. Our experimental results on several real world data sets indicate the performance of the proposed algorithm.     

解大规模非对称矩阵特征问题的精化Arnoldi方法的一种变形

§1.引言 传统的投影类方法是计算大规模非对称矩阵特征问题Ax=λx部分特征对的主要方法,它们包括Arnoldi方法、块Arnoldi方法、同时迭代法、Davidson方法和Jacobi-Davidson方法,贾提出的精化投影类方法目前被公认为是另一类重要     

Residuals of refined projection methods for large matrix eigenproblems

In terms of the deviation of the eigenvector ϕ from a projection subspace E, a priori error bounds are established for the residual norm of an approximate eigenpair obtained by a refined projection method. It is shown that the residual converges to zero as the deviation tends to zero. Finally, how to efficiently compute refined Ritz vectors is discussed in the refined symmetric Lanczos method.     

Difetti ed eccessi degli autovalori del classico problema di «buckling»

The classical buckling eigenvalue problem for a square plate clamped along its boundary is here considered. By using the Rayleigh-Ritz method and the method of orthogonal invariants, we obtain upper and lower bounds for the first 60 eigenvalues. Numerical tables are given. The multiplicity of the first eigenvalues and the symmetries of the corresponding eigenfunctions are also studied.      

Assessing the stability of linear time-invariant continuous interval dynamic systems

It is known that stability analysis of linear time-invariant dynamic systems under parameter uncertainties can be equated to estimating the range of the eigenvalues of matrices whose elements are intervals. In this note, first the problem of finding tight outer bounds on the eigenvalue ranges is considered. A method for computing such bounds is suggested which consists, essentially, of setting up and solving a system of n mildly nonlinear algebraic equations, n being the size of the interval matrix investigated. The main result of the note, however, is a method for determining the right end-point of the exact eigenvalue ranges. The latter makes use of the outer bounds. It is applicable if certain computationally verifiable monotonicity conditions are fulfilled. The methods suggested can be applied for robust stability analysis of both continuous- and discrete-time systems. Numerical examples illustrating the applicability of the new methods are also provided.     

Projection and deflation method for partial pole assignment inlinear state feedback

Two projection methods are proposed for partial pole placement in linear control systems. These methods are of interest when the system is very large and only a few of its poles must be assigned. The first method is based on computing an orthonormal basis of the left invariant subspace associated with the eigenvalues to be assigned and then solving a small inverse eigenvalue problem resulting from projecting the initial problem into that subspace. The second method can be regarded as a variant of the Weilandt deflation technique used in eigenvalue methods     

Semiclassical spectral confinement for the sine-Gordon equation

The inverse scattering method for solving the sine-Gordon equation in laboratory coordinates requires the analysis of the Faddeev–Takhtajan eigenvalue problem. This problem is not self-adjoint and the eigenvalues may lie anywhere in the complex plane, so it is of interest to determine conditions on the initial data that restrict where the eigenvalues can be. We establish bounds on the eigenvalues for a broad class of zero-charge initial data that are applicable in the semiclassical or zero-dispersion limit. It is shown that no point off the coordinate axes or turning point curve can be an eigenvalue if the dispersion parameter is sufficiently small.     

A Riccati-type algorithm for solving generalized Hermitian eigenvalue problems

The paper describes a heuristic algorithm for solving a generalized Hermitian eigenvalue problem fast. The algorithm searches a subspace for an approximate solution of the problem. If the approximate solution is unacceptable, the subspace is expanded to a larger one, and then, in the expanded subspace a possibly better approximated solution is computed. The algorithm iterates these two steps alternately. Thus, the speed of the convergence of the algorithm depends on how to generate a subspace. In this paper, we derive a Riccati equation whose solution can correct the approximate solution of a generalized Hermitian eigenvalue problem to the exact one. In other words, the solution of the eigenvalue problem can be found if a subspace is expanded by the solution of the Riccati equation. This is a feature the existing algorithms such as the Krylov subspace algorithm implemented in the MATLAB and the Jacobi–Davidson algorithm do not have. However, similar to solving the eigenvalue problem, solving the Riccati equation is time-consuming. We consider solving the Riccati equation with low accuracy and use its approximate solution to expand a subspace. The implementation of this heuristic algorithm is discussed so that the computational cost of the algorithm can be saved. Some experimental results show that the heuristic algorithm converges within fewer iterations and thus requires lesser computational time comparing with the existing algorithms.

     

A strategy for detecting extreme eigenvalues bounding gaps in the discrete spectrum of self-adjoint operators

For a self-adjoint linear operator with a discrete spectrum or a Hermitian matrix, the “extreme” eigenvalues define the boundaries of clusters in the spectrum of real eigenvalues. The outer extreme ones are the largest and the smallest eigenvalues. If there are extended intervals in the spectrum in which no eigenvalues are present, the eigenvalues bounding these gaps are the inner extreme eigenvalues.We will describe a procedure for detecting the extreme eigenvalues that relies on the relationship between the acceleration rate of polynomial acceleration iteration and the norm of the matrix via the spectral theorem, applicable to normal matrices. The strategy makes use of the fast growth rate of Chebyshev polynomials to distinguish ranges in the spectrum of the matrix which are devoid of eigenvalues.The method is numerically stable with regard to the dimension of the matrix problem and is thus capable of handling matrices of large dimension. The overall computational cost is quadratic in the size of a dense matrix; linear in the size of a sparse matrix. We verify computationally that the algorithm is accurate and efficient, even on large matrices.     

Application of the Feynman-Kac path integral method in finding excited states of quantum systems

Group theory considerations and properties of a continuous path are used to define a failure tree procedure for finding eigenvalues of the Schrödinger equation using stochastic methods. The procedure is used to calculate the lowest excited state eigenvalues of eigenfunctions possessing anti-symmetric nodal regions in configuration space using the Feynman-Kac path integral method. Within this method the solution of the imaginary time Schrödinger equation is approximated by random walk simulations on a discrete grid constrained only by symmetry considerations of the Hamiltonian. The required symmetry constraints on random walk simulations are associated with a given irreducible representation and are found by identifying the eigenvalues for the irreducible representation corresponding to symmetric or antisymmetric eigenfunctions for each group operator. The method provides exact eigenvalues of excited states in the limit of infinitesimal step size and infinite time. The numerical method is applied to compute the eigenvalues of the lowest excited states of the hydrogenic atom that transform as Γ² and Γ⁴ irreducible representations. Numerical results are compared with exact analytical results.     

On the stability of some high-resolution beamforming methods

Orthogonal beamforming is the name of certain high-resolution methods for estimating the spectra of a wave field received by an array of sensors. The methods use the eigenvalues and eigenvectors of the spectral matrix of the sensor outputs. The problem is to predict the behavior of such methods when only an estimate of the matrix is known. The sensor outputs may consist of sensor noise, ambient noise and noise from a finite set of discrete sources. The properties of the eigensystem of the spectral matrix in the case of weak ambient noise motivate the methods of orthogonal beamforming, for example Pisarenko's nonlinear peak estimates and the projection estimates of Owsley. If the spectral matrix is estimated by one of the classical methods, some asymptotic distributional properties of the matrix estimate and its eigensystem are well known. They can be used to determine asymptotic expressions, e.g. for the first and second moments of the peak estimators and to approximate the distributions. The parameters, however, cannot be calculated in applications, since the eigensystem of the exact spectral matrix is required. Therefore, we have developed bounds for the deviation of the peak estimates which only use weak knowledge about the matrix. We have applied some results on perturbations of hermitian operators. The asymptotic behavior of the bounds for the projection estimator is investigated and possibilities for their estimation are indicated. Finally, we report on extensive simulations with random matrices to evaluate the new bounds. As a result, we have found that the projection estimator behaves stably and there are tight bounds if the eigenvalues of interest are sufficiently separated from the rest.     

Three-dimensional vibration analysis of thick, circular and annular plates with nonlinear thickness variation

A three-dimensional (3-D) method of analysis is presented for determining the free vibration frequencies and mode shapes of thick, circular and annular plates with nonlinear thickness variation along the radial direction. Unlike conventional plate theories, which are mathematically two-dimensional (2-D), the present method is based upon the 3-D dynamic equations of elasticity. Displacement components u_s, u_z, and u_θ in the radial, thickness, and circumferential directions, respectively, are taken to be sinusoidal in time, periodic in θ, and algebraic polynomials in the s and z directions. Potential (strain) and kinetic energies of the plates are formulated, and the Ritz method is used to solve the eigenvalue problem, thus yielding upper bound values of the frequencies by minimizing the frequencies. As the degree of the polynomials is increased, frequencies converge to the exact values. Convergence to four-digit exactitude is demonstrated for the first five frequencies of the plates. Numerical results are presented for completely free, annular and circular plates with uniform, linear, and quadratic variations in thickness. Comparisons are also made between results obtained from the present 3-D and previously published thin plate (2-D) data.     

Efficient Two-Sided Estimates for the Spectrum of Some Elliptic Operators

Using the principle of maximum, we establish the upper and lower bounds for the spectrum of some elliptic operators and their grid analogs. More accurate estimates of the spectrum of differential operators are obtained from the exact formulas for the error of the eigenvalues by the finite-difference method. Two-sided estimates of the eigenvalues of difference analogs of spectral problems give a majorant and a minorant for the error of the phase velocities of grid waves in vibration problems for various objects.

     

Fehlerabschätzung reeller Eigenwerte und Eigenvektoren von Matrizen

Zusammenfassung Reelle, einfache Eigenwerte und die dazugehörigen reellen Eigenvektoren einer quadratischen Matrix werden mit Hilfe einer Intervallarithmetik unter Berücksichtigung aller Rundungsfehler abgeschätzt. Für diese Fehlerabschätzung benötigt man als Daten die Näherungswerte eines Eigenwertes und des entsprechenden Eigenvektors, welche man sich nach irgendeinem numerischen Verfahren verschafft hat. Als Ergebnis erhält man ein Intervall und einen Intervallvektor, welche den zugehörigen exakten Eigenwert bzw. den exakten Eigenvektor enthalten. Gleichzeitig ergibt sich daraus auch die Existenz eines reellen Eigenwertes.

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Error estimates for real eigenvalues and eigenvectors of matrices

Summary Bounds including round off errors are given for real simple eigenvalues and corresponding eigenvectors of a square matrix by using interval-arithmetics. For this error estimate approximate values are needed for an eigenvalue and the corresponding eigenvector which can be gained by an arbitrary numerical method. The result is an interval and an intervalvector containing the exact eigenvalue and eigenvector respectively. Also the existence of a real eigenvalue is verified.

     

Efficient computation of high index Sturm-Liouville eigenvalues for problems in physics

Finding the eigenvalues of a Sturm-Liouville problem can be a computationally challenging task, especially when a large set of eigenvalues is computed, or just when particularly large eigenvalues are sought. This is a consequence of the highly oscillatory behavior of the solutions corresponding to high eigenvalues, which forces a naive integrator to take increasingly smaller steps. We will discuss the most used approaches to the numerical solution of the Sturm-Liouville problem: finite differences and variational methods, both leading to a matrix eigenvalue problem; shooting methods using an initial-value solver; and coefficient approximation methods. Special attention will be paid to techniques that yield uniform approximation over the whole eigenvalue spectrum and that allow large steps even for high eigenvalues.     

Prüfer method for periodic and semi-periodic sturm-liouville eigenvalue problems ∗

In reference [19], the authors developed a shooting algorithm for Sturm-Liouville eigenvalue problems associated with periodic and semi-periodic boundary conditions. The technique is based on the application of the Floquet theory, and it has proven to be efficient for computing eigenvalues. However, the performance of this technique depends upon the choice of the starting eigenvalues. In the present paper, we continue our study and employ the Prüfer method. An attractive property of this method is that eigenvalues can usually be accurately computed even when no information on the eigenvalue distribution is provided. Sufficient conditions for convergence, error bounds and a procedure to improve the stability are discussed. Some numerical examples are given to illustrate the effectiveness of the proposed method.