Eigenvalue Problems

This paper describes new matrix transformations suited to the eficient calculation of critical eigenvalues of large scale power system dpamic models. The key advantage of these methods is their ability to converge to the critical eigenvalues (unstable or low damped) of the system almost independently of the given initial estimate. Matrix transforms such as inverse iteration and S-matrix can be thought as special cases of the described method. These transforms can also be used to inhibit convergence to a known eigenvalue, yielding better overall eficiency when finding several eigenvalues.

Recently, a class of Boundary Value Methods (BVMs) has been introduced for the estimation of the eigenvalues of Sturm-Liouville problems with Dirichlet boundary conditions. The aim of this paper is to extend the application of such BVMs to problems with boundary conditions of general form and to compare the approximations obtained with those given by the corrected Numerov method.

According to the great mathematician Henri Lebesgue, making direct comparisons of objects with regard to a property is a fundamental mathematical process for deriving measurements. Measuring objects by using a known scale first then comparing the measurements works well for properties for which scales of measurement exist. The theme of this paper is that direct comparisons are necessary to establish measurements for intangible properties that have no scales of measurement. In that case the value derived for each element depends on what other elements it is compared with. We show how relative scales can be derived by making pairwise comparisons using numerical judgments from an absolute scale of numbers. Such measurements, when used to represent comparisons can be related and combined to define a cardinal scale of absolute numbers that is stronger than a ratio scale. They are necessary to use when intangible factors need to be added and multiplied among themselves and with tangible factors. To derive and synthesize relative scales systematically, the factors are arranged in a hierarchic or a network structure and measured according to the criteria represented within these structures. The process of making comparisons to derive scales of measurement is illustrated in two types of practical real life decisions, the Iran nuclear show-down with the West in this decade and building a Disney park in Hong Kong in 2005. It is then generalized to the case of making a continuum of comparisons by using Fredholm's equation of the second kind whose solution gives rise to a functional equation. The Fourier transform of the solution of this equation in the complex domain is a sum of Dirac distributions demonstrating that proportionate response to stimuli is a process of firing and synthesis of firings as neurons in the brain do. The Fourier transform of the solution of the equation in the real domain leads to nearly inverse square responses to natural influences. Various generalizations and critiques of the approach are included.

The exponential of a matrix and the spectral decomposition of a matrix can be computed knowing nothing more than the eigenvalues of the matrix and the Cayley-Hamilton theorem. The arrangement of the ideas in this paper is simple enough to be taught to beginning students of ODEs.

It was observed long ago that the obstruction to the accurate computation of eigenvalues of large non-self-adjoint matrices is inherent in the problem. The basic idea is that the resolvent of a highly non-normal operator can be very large far away from the spectrum. This leads to an easily observable fact that algorithms for locating eigenvalues will typically find some "false eigenvalues". These false eigenvalues also explain one of the most surprising phenomena in linear PDEs, namely the fact (discovered by Hans Lewy in 1957, in Berkeley) that one cannot always locally solve the PDE P u = f. Almost immediately after that discovery, Hörmander provided an explanation of Lewy's example showing that almost all operators with non-constanct complex valued coefficients are not locally solvable. In modern language, that was done by considering the essentially dual problem of existence of non-propagating singularities. The purpose of this article is to review this work in the context of "almost eigenvalues" and from the point of view of semi-classical analysis.

Fisher variance ratio tests are developed for determining (1) the number of statisticaliy significant abstract factors responsible for a data matrix and (2) the significance of target vectors projected into the abstract factor space. F-tests, developed from the viewpoint of vector distributions, are applied to various data sets taken from the chemical literature.

We investigate the recursive inversion of matrices with circulant blocks. Matrices of this type appear in several applications of Computational Electromagnetics and in the numerical solution of integral equations with the boundary-element method. The inversion is based on the diagonalization of each circulant block by means of the discrete Fourier transform and the application of a recursive algorithm for the inversion of the matrix with diagonal blocks, determined by the eigenvalues of each block. The efficiency of the recursive inversion is exhibited by determining its computational complexity. An implementation of the algorithm in MATLAB is given and numerical results are presented to demonstrate the efficiency in terms of CPU time of our approach.

""This monograph gives a complete overview on the subject of nonconservative stability from the modern point of view. Relevant mathematical concepts are presented, as well as rigorous stability results and numerous classical and contemporary examples from mechanics and physics.

It deals with both finite- and infinite-dimensional nonconservative systems and covers the fundamentals of the theory, including such topics as Lyapunov stability and linear stability analysis, Hamiltonian and gyroscopic systems, reversible and circulatory systems, influence of structure of forces on stability, and dissipation-induced instabilities, as well as concrete physical problems, including perturbative techniques for nonself-adjoint boundary eigenvalue problems, theory of the destabilization paradox due to small damping in continuous circulatory systems, Krein-space related perturbation theory for the MHD kinematic mean field α²-dynamo, analysis of Campbell diagrams and friction-induced flutter in gyroscopic continua, non-Hermitian perturbation of Hermitian matrices with applications to optics, and magnetorotational instability and the Velikhov-Chandrasekhar paradox.

The book serves present and prospective specialists providing the current state of knowledge in the actively developing field of nonconservative stability theory. Its understanding is vital for many areas of technology, ranging from such traditional ones as rotor dynamics, aeroelasticity and structural mechanics to modern problems of hydro- and magnetohydrodynamics and celestial mechanics.
""

Bu çalışmada "ASME Section VIII Division 1, 2001 Edition" a göre, belirli deprem yükleri altında tasarlanmış ince cidarlı silindirik kolon türü basınçlı kapların, yeni çalışma şartları ve artan deprem yüklerine göre doğrusal olmayan statik gerilme analizi, özdeğer (eigenvalue) yöntemi ile burkulma analizi, cevap spektrum yüklemesi kullanılarak mod birleştirme yöntemi ile dinamik analiz uygulanması sonucunda en uygun takviyelendirme yönteminin seçilmesi amaçlanmış ve bu yöntem kullanılarak gerçek bir basınçlı kolonun takviyelendirilmesi anlatılmıştır.

Problemas auto-valores generalizados s ̃ao problemas de valores e vetores pr ́oprios do seguinte tipo:
Achar um par (v, λ) satisfazendo A v = λ B v
sendo v uma matriz coluna n ̃ao nula e λ um nu ́mero complexo. A e B s ̃ao matrizes retangulares arbitr ́arias.
Denominamos ao λ auto-valor generalizado e v auto-vetor generalizado relativo `a λ. Relacionamos em seguida, alguns problemas que podem ser equacionados como problemas de
auto-valores generalizados, na  ́area de Mecˆanica.
1. C ́alculo estrutural
Estudo de vibra ̧c ̃oes livres de uma estrutura, com ou sem pr ́e-tens ̃ao, n ̃ao amortecida e sem
movimento de rota ̧c ̃ao.
Procura-se os menores auto-valores ou aqueles que est ̃ao num determinado intervalo, para saber se uma forc ̧a excitadora pode gerar ressonˆancia, ou n ̃ao.
Nesse caso A  ́e a matriz de rigidez (K), eventualmente aumentada pela matriz de rigidez geom ́etrica, notada Kg, se a estrutura tem pr ́e-tens ̃ao. B corresponde `a matriz de massa (M ).
O problema de auto valores generalizados neste caso  ́e
(K + Kg) v = ω2M v (ω  ́e igual a 2πf, sendo f a frequˆencia pr ́opria ).
2. Determina ̧c ̃ao do modo de flambagem linear
Neste caso a aproxima ̧c ̃ao que fornece a flambagem linear  ́e talque v  ́e o modo de flambagem
associado `a carga cr ́ıtica λ.
O problema  ́e: K v = −λKg v
com K matriz de rigidez, Kg matriz de rigidez geom ́etrica gerada pelo campo de pr ́e-tens ̃ao devido `a carregamentos.
3. C ́alculo das velocidades cr ́ıticas de um sistema em rota ̧c ̃ao
Neste caso A  ́e a matriz de rigidez (K), B  ́e a matriz de rigidez `a rotac ̧ ̃ao Kr . A velocidade
cr ́ıtica de rotac ̧ ̃ao procurada, λ,  ́e soluc ̧ ̃ao do problema: K v = −λ2Kr v .
4. Problemas acoplados
Problemas de comportamento dinˆamico de uma estrutura que interage com um meio externo
ou interno (pode ser um fluido, um material piezo-el ́etrico, etc).
Estas Notas tem como objetivo descrever de forma sistem ́atica este assunto, tanto no seu as- pecto te ́orico como computacional. Descrevemos os principais resultados e indicamos algumas das principais aplicac ̧ ̃oes, que poder ́a ser de utilidade para pessoas que trabalham com este tipo de problemas, como por exemplo, aqueles que se dedicam a vibrac ̧ ̃oes e controle.

For past few decades the study about phononic crystals are getting more popular. Understanding and controlling the phononic properties of the phononic crystals provides opportunities to reduce environmental noise. In this thesis there is a brief introduction to phononic crystals.
My research topic of the thesis is about vibration reduction of beams connected to spring resonators with one degree of freedom, with the ultimate purpose to find bandgaps. In the bandgaps of the frequency ranges, flexural wave cannot propagate in the structures. I have done two analysis to find bandgaps. The first one is finding flexural bandgaps through Eigenfrequencies. And the second one is finding band gaps through frequency response analysis. Two methods also shows us the range of band gaps.
All the simulation works have been done in COMSOL Multiphysics. COMSOL Multiphysics is a very suitable tool for analysis of wave propagation in phononic crystals. In the thesis, I give the steps for two kind of simulations in COMSOL Multiphysics.

KEY WORDS：Phononic Crystals, FEM, Eigenfrequencies, Frequency Response Analysis, Band Gap, Euler-Bernoulli Beam, Spring Resonator.

Lattice structures can be developed for operating in high temperature environments. Ceramic-metal FGMs (Functionally graded materials) can be considered for manufacturing lattice plates. In the current study, an exact material modeling using Heaviside distribution functions is performed for lattice FGM plates. Using the classical theory, motion governing equations for lattice FGM plates under uniform temperature field has been derived. Free vibration problem of lattice FGM plates subjected to initial thermal loading and considering simply-supported boundary conditions has been solved using closed form Galerkin method. The analytical results are verified through comparison by those obtained from ABAQUS FEM (finite element method) commercial software. The solution presented here could be treated as a benchmark for future studies about FGM stiffened plates.

We present a method for the determination of eigenvalues of a symmetric tridiagonal matrix which combines Givens' Sturm bisection [4, 5-] with interpolation, to accelerate convergence in high precision cases. By using an appropriate root of the absolute value of the determinant to derive the interpolation weight, results are obtained which compare favorably with the Barth, Martin, Wilkinson algorithm [1].

A layerwise (zigzag) finite element formulation is developed for the buckling analysis of stiffened laminated plates. The laminated plate is discretized into layers along the thickness direction. Each layer of the laminated plate is modeled by the degenerated shell elements, and the ...

The problem about a body in a three dimensional infinite channel is considered in the framework of the theory of linear water-waves. The body has a rough surface characterized by a small parameter ε > 0 while the distance of the body to the water surface is also of order ε. Under a certain symmetry assumption, the accumulation effect for trapped mode frequencies is established, namely, it is proved that, for any given d > 0 and integer N > 0, there exists ε(d, N ) > 0 such that the problem has at least N eigenvalues in the interval (0, d) of the continuous spectrum in the case ε ∈ (0, ε(d, N )). The corresponding eigenfunctions decay exponentially at infinity, have finite energy, and imply trapped modes.

En esta investigación se presenta un estudio analítico de la variabilidad de la respuesta dinámica de pórticos planos linealmente elásticos, la cual se deriva de las eigensoluciones a la vibración libre sin amortiguamiento del método matricial de Stodola-Vianello, del método de Holzer-Rayleigh y del método de superposición modal. Se han establecido 6 pórticos planos de acero estructural de constitución uniforme como muestra. Para la estructuración de los modelos dinámicos se ha impuesto la idealización del edificio simple. Entre los resultados obtenidos se destaca que el método de Holzer-Rayleigh es el método iterativo más exacto para aproximar la respuesta dinámica de pórticos planos, pero es invariablemente dependiente del modelo de edificio simple. Por otra parte, el método matricial de Stodola-Vianello resulta eficiente para aproximar la respuesta dinámica del último modo de pórticos planos con 2 y 3 grados de libertad. Por tanto, puede ser combinado con el método de Holzer-Rayleigh para aproximar, de manera más exacta, la respuesta dinámica de los modos primarios de estos sistemas discretos. La eigensolución del método de superposición modal, generada por la función eig de MATLAB 7.10.0.499 (R2010a), constituye la asíntota de convergencia de la eigensolución de los métodos iterativos, por lo que proporciona la aproximación más exacta a las respuestas dinámicas de pórticos planos.

In this paper a relation between graph distance matrices of the star graph and its generalizations and Euclidean distance matrices is considered. It is proven that distance matrices of certain families of graphs are circum Euclidean. Their spectrum and generating points are given in a closed form.

The eigenvalue problem of holomorphic functions on the unit disc for
the third boundary condition with general coefficient is studied using Fourier analysis.
With a general anti-polynomial coefficient a variable number of additional boundary
conditions need to be imposed to determine the eigenvalue uniquely. An additional
boundary condition is required to obtain a unique eigenvalue when the coefficient includes
an essential singularity rather than a pole. In either case explicit solutions are derived.

Eigenvalue computations for structured rank matrices are the subject of many investigations nowadays. There exist methods for transforming matrices into structured rank form, QR-algorithms for semiseparable and semiseparable plus diagonal form, methods for reducing structured rank matrices efficiently to Hessenberg form and so forth. Eigenvalue computations for the symmetric case, involving semiseparable and semiseparable plus diagonal matrices have been thoroughly explored. A first attempt for computing the eigenvalues of nonsymmetric matrices via intermediate Hessenberg-like matrices (i.e. a matrix having all subblocks in the lower triangular part of rank at most one) was restricted to the single shift strategy. Unfortunately this leads in general to the use of complex shifts switching thereby from real to complex operations. This paper will explain a general multishift implementation for Hessenberg-like matrices (semiseparable matrices are a special case and hence also admit this approach). Besides a general multishift QR-step, this will also admit restriction to real computations when computing the eigenvalues of arbitrary real matrices. Details on the implementation are provided as well as numerical experiments proving the viability of the presented approach.

In this paper we present a method for transient analysis of availability and survivability of a system with the identical components and identical repairmen. The considered system is supposed to consist of series of k-out-of-n or parallel components. We employed the Markov models, eigen vectors and eigenvalues for analyzing the transient availability and survivability of the system. The method is implemented through an algorithm which is tested in MATLAB programming environment. The new method enjoys a stronger mathematical foundation and more flexibility for analyzing the transient availability and survivability of the system.

We consider Dirichlet Laplacian in a thin curved three-dimensional rod. The rod is finite. Its cross-section is constant and small, and rotates along the reference curve in an arbitrary way. We find a two-parametric set of the eigenvalues of such operator and construct their complete asymptotic expansions. We show that this two-parametric set contains any prescribed number of the first eigenvalues of the considered operator. We obtain the complete asymptotic expansions for the eigenfunctions associated with these first eigenvalues.

A hardware solution is presented to obtain the eigenvalues and eigenvectors of a real and symmetrical matrix using field-programmable gate arrays (FPGAs). Currently, this system is used to compute the eigenvalues and eigenvectors in covariance matrices for applications in digital image processing that make use of the principal component analysis (PCA) technique. The proposed solution in this paper is based on the Jacobi method, but in comparison with other related works, it presents a different architecture that remarkably improves execution time, while reducing the number of consumed resources of the FPGA.

This paper gives a new way of showing that certain constant degree graphs are graph expanders. This is done by giving new proofs of expansion for three permutations of the Gabber-Galil expander. Our results give an expansion factor of 3 16 for subgraphs of these three-regular graphs with (p − 1) 2 inputs for p prime. The proofs are not based on eigenvalue methods or higher algebra. The same methods show the expected number of probes for unsuccessful search in double hashing is bounded by 1 1−α , where α is the load factor. This assumes a double hashing scheme in which two hash functions are randomly and independently chosen from a specified uniform distribution. The result is valid regardless of the distribution of the inputs. This is analogous to Carter and Wegman's result for hashing with chaining. This paper concludes by elaborating on how any sufficiently sized subset of inputs in any distribution expands in the subgraph of the Gabber-Galil graph expander of focus. This is related to any key distribution having expected 1 1−α probes for unsuccessful search for double hashing given the initial random, independent, and uniform choice of two universal hash functions.

Lattice plates have been extensively used as lightweight structures in many industries. In the present research, an exact material modeling using Heaviside distribution functions is presented for lattice composite plates. Using CLT (classical lamination theory), equilibrium equations for lattice composite plates under uniform temperature field are derived. The equations are solved by performing closed form Galerkin method. The thermal buckling modes are obtained by solving the eigenvalue problem. The general solution presented in this paper includes laminates with arbitrary stacking sequences. The accuracy of theoretical model is checked through comparisons with finite element results obtained by simulation in ABAQUS and good agreement is achieved.

Estimating the energies and splitting of the 1s2s singlet and triplet states of helium is a classic
exercise in quantum perturbation theory but yields only qualitatively correct results. Using a six-line
computer program, the 1s2s energies calculated by matrix diagonalization using a seven-state
basis improve the results to 0.4% error or better. This is an effective and practical illustration of the
quantitative power of quantum mechanics, at a level accessible to undergraduate students.

An iterative, CFD-based approach for aeroelastic computations in the frequency domain is presented. The method relies on a linearized formulation of the aeroelastic problem and a fixed-point iteration approach and enables the computation of the
eigenproperties of each of the wet aeroelastic eigenmodes. Numerical experiments on the aeroelastic analysis and design optimization of two wing configurations illustrate the capability of the method for the fast and accurate aeroelastic analysis of aircraft configurations and its advantage over classical time-domain approaches.

We report the results of a study on the spectral properties of Laplace and Stokes operators modified with a volume penalization term designed to approximate Dirichlet conditions in the limit when a penalization parameter, η, tends to zero. The eigenvalues and eigenfunctions are determined either analytically or numerically as functions of η, both in the continuous case and after applying Fourier or finite difference discretization schemes. For fixed η, we find that only the part of the spectrum corresponding to eigenvalues λ≲η−1 approaches Dirichlet boundary conditions, while the remainder of the spectrum is made of uncontrolled, spurious wall modes. The penalization error for the controlled eigenfunctions is estimated as a function of η and λ. Surprisingly, in the Stokes case, we show that the eigenfunctions approximately satisfy, with a precision O(η), Navier slip boundary conditions with slip length equal to √η. Moreover, for a given discretization, we show that there exists a value of η, corresponding to a balance between penalization and discretization errors, below which no further gain in precision is achieved. These results shed light on the behavior of volume penalization schemes when solving the Navier–Stokes equations, outline the limitations of the method, and give indications on how to choose the penalization parameter in practical cases.

We investigate one-parameter families of two-dimensional bright spinning solitons (ring vortices) in dispersive media combining cubic self-focusing and quintic self-defocusing nonlinearities. In direct simulations, the spinning solitons display a symmetry-breaking azimuthal instability, which leads to breakup of a soliton into a set of fragments, each being a stable nonspinning soliton. The fragments fly out tangentially to the circular crest of the original vortex ring. If the soliton's energy is large enough, the instability develops so slowly that the spinning solitons may be regarded as virtually stable ones, in accord with earlier published results. Growth rates of perturbation eigenmodes with different azimuthal "quantum numbers" are calculated as a function of the soliton's propagation constant κ from a numerical solution of the linearized equations. As a result, a narrow (in terms of κ) stability window is found for extremely broad solitons with values of the "spin" s = 1 and 2. However, analytical consideration of a special perturbation mode in the form of a spontaneous shift of the soliton's central "bubble" (core of the vortex embedded in a broad soliton) demonstrates that even extremely broad solitons are subject to an exponentially weak instability against this mode. In actual simulations, a manifestation of this instability is found in a three-dimensional soliton with s = 1. In the case when the two-dimensional spinning solitons are subject to tangible azimuthal instability, the number of the nonspinning fragments into which the soliton splits is usually, but not always, equal to the azimuthal number of the instability eigenmode with the largest growth rate.

In this work we study the asymptotic distribution of eigenvalues in one-dimensional open sets. The method of proof is rather elementary, based on the Dirichlet lattice points problem, which enable us to consider sets with infinite measure. Also, we derive some estimates for the the spectral counting function of the Laplace operator on unbounded two-dimensional domains.

Given an n-by-n Hermitian matrix A and a real number λ, index i is said to be Parter (resp. neutral, downer) if the multiplicity of λ as an eigenvalue of A(i) is one more (resp. the same, one less) than that in A. In case the multiplicity of λ in A is at least 2 and the graph of A is a tree, there are always Parter vertices. Our purpose here is to advance the classification of vertices and, in particular, to relate classification to the combinatorial structure of eigenspaces. Some general results are given and then used to deduce some rather specific facts, not otherwise easily observed. Examples are given.

This paper presents a different numerical solution to compute eigenvalues of the Schrödinger equation with the potentials in graphene structures [1]. The research subjects include the Schrödinger equation and the exchange-correlation energy of the graphene structures in Grachev's article. Specifically, we used the pseudospectral method basing on the Chebyshev-Gauss-Lobatto grid to determine the approximate numerical results of the problem. The results are the discrete energy spectra and the corresponding eigenfunctions of the nonlinear spin waves in the graphene structure. Additionally, these results can be applied to create the nonlinear spin waves in the graphene structures.

The task of discovering natural groupings of input patterns, or clustering, is an important aspect machine learning and pattern analysis. In this paper, we study the widely-used spectral clustering algorithm which clusters data using eigenvectors of a similarity/affinity matrix derived from a data set. In particular, we aim to solve two critical issues in spectral clustering: (1) How to automatically determine the number of clusters? and (2) How to perform effective clustering given noisy and sparse data? An analysis of the characteristics of eigenspace is carried out which shows that (a) Not every eigenvectors of a data affinity matrix is informative and relevant for clustering; (b) Eigenvector selection is critical because using uninformative/irrelevant eigenvectors could lead to poor clustering results; and (c) The corresponding eigenvalues cannot be used for relevant eigenvector selection given a realistic data set.

The finite annual rate of population increase (l) is a fundamental demographic parameter that characterizes the relative annual change in animal numbers. Uncertainty in the estimation of l from demographic population viability analyses (PVAs) has been largely limited to sensitivity analysis, calculating a pseudo-distribution for b l using Monte Carlo methods, or by use of bootstrap methods. The delta method has been used and suggested by several researchers, but no one has provided the computational means to implement it. In this paper, we present Mathematica code to calculate l and its variance based on eigenvalue calculations of a Leslie transition matrix. We demonstrate the procedure using data from a Hawaiian hawk (Buteo solitarius) study.

A maximum principle is proved for the weak solutions u ∈ L ∞ (R × T 3 ) of the telegraph equation u tt − ∆ x u + cu t + λu = f (t, x), in space dimension three, when c > 0, λ ∈ (0, c 2 /4] and f ∈ L ∞ (R × T 3 ) (Theorem 1). The result is extended to a solution and a forcing belonging to a suitable space of bounded measures (Theorem 2). Those results provide a method of upper and lower solutions for the semilinear equation

Although upwind discretization of convection will lead to a diagonally dominant coefficient matrix, on arbitrary grids the latter is not necessarily positive real, i.e. its symmetric part need not be positive definite ('negative diffusion'). Especially on contractingexpanding grids this property can be lost. The paper discusses a conservative (finite-volume) upwind variant for which the latter property is guaranteed to hold, irrespective of grid (ir)regularity. Further, empirically it is found that often its global discretization error is smaller than that of the 'traditional' (finite-difference) upwind method. Finally, it is shown that in many situations its extremal eigenvalues at the outer side of the spectrum move towards the imaginary axis, thus enhancing the stability of explicit time-integration methods.

This study analyzes the effects of link flexibility on the dynamic stability of a force-controlled flexible manipulator. The closed-loop dynamic equation for a l-link manipulator is first derived explicitly using the modal representation technique. Stability analysis is then carried out by computing the system eigenvalues. Results show that the link flexibility is one of the causes of dynamic instability of motion of the flexible robotic arm.

An energy-dependent partitioning scheme is explored for extracting a small number of eigenvalues of a real symmetric matrix with the help of genetic algorithm. The proposed method is tested with matrices of different sizes (30 × 30 to 1000 × 1000). Comparison is made with Löwdin's strategy for solving the problem. The relative advantages and disadvantages of the GA-based method are analyzed.