Algebraic Eigenvalue Research Articles

Regions in the Complex Plane Containing the Eigenvalues of a Matrix

The Gersgorin circle theorem gives a region in the complex plane which contains all the eigenvalues of a square complex matrix. It is one of those rare instances of a theorem which is elegant and useful and which has a short, elementary proof. It is surprising that it hasn't made its way into many introductory texts on linear algebra. One reason for this neglect may be the fact that many (even most) mathematicians still regard linear algebra as being only about algebra. But modern linear algebra is more than algebra. It's linear systems, matrix theory and analysis, geometry, applications, numerics and, of course, algebra. The first course in linear algebra at most colleges and universities is to a great extent a service course for future scientists. Gersgorin's theorem is not an algebraic theorem. It is a simple analytic theorem involving elementary inequalities derived from the basic algebraic eigenvalue/ eigenvector equation. If A = [aij] is a complex matrix of order n and A is an eigenvalue of A, then there exists a nonzero vector x = (x1, x2,. . ., xn)T in Cn such that

Efficient computation of simultaneous radiation-trapping and particle diffusion in an atomic vapor

We give an efficient algorithm for the computation of simultaneous radiation trapping and particle diffusion in an atomic vapor. We combine the solutions of the trapping-without-diffusion and the diffusion-without-trapping problems by solving an algebraic eigenvalue problem. As an example for the use of our new method, we show a computation of the fraction of wall-quenched atoms.

Numerical methods for eighth-, tenth- and twelfth-order eigenvalue problems arising in thermal instability

Second-order finite-difference methods are developed for the numerical solutions of the eighth-, tenth- and twelfth-order eigenvalue problems arising in the study of the effect of rotation on a horizontal layer of fluid heated from below. Instability setting-in as overstability may be modelled by an eighth-order ordinary differential equation. When a uniform magnetic field also acts across the fluid in the same direction as gravity, instability setting-in as ordinary convection may be modelled by a tenth-order differential equation, while instability setting-in as overstability may be modelled by a twelfth-order differential equation. The numerical methods are developed by making direct replacements of the derivatives in the differential equations and then by computing the eigenvalues, which may incorporate Rayleigh number, horizontal wave speed and a time constant, from the resulting algebraic eigenvalue problem. The eigenvalues are also computed by writing the differential equations as systems of second-order differential equations and then using second- and fourth-order methods to obtain the eigenvalues. Numerical results obtained using the two approaches are compared with estimates appearing in the literature.

Heat conduction on graphs

A mathematical model for heat conduction in a graph-like object provides a setting for interpreting an algebraic eigenvalue problem associated with a graph. Applications include bounds for eigenvalues and a way to find eigenvalues and eigenvectors for a subdivision graph.

A symmetric and positive definite variational BEM for 2-D free vibration analysis

A general BEM model for structural dynamics is derived by using a symmetric and positive definite variational formulation. The functional employed involves domain displacements and boundary tractions and displacements. These variables are taken to be independent of one another. The boundary variables are expressed in terms of their nodal values while the domain displacement field is approximated by a linear combination of static fundamental solutions. The source point of the latter is located outside the domain. The resolving system is a linear system and for free vibration a classic linear algebraic eigenvalue problem is inferred. The stiffness and mass matrices are symmetric and positive definite and the domain integral, when associated with the inertial term, can be transformed into a boundary integral. Numerical results are presented to prove the efficiency of the method.

Non-axisymmetric modes in viscoelastic taylor-couette flow

In this work, the linear stability analysis of the viscoelastic Taylor-Couette flow against non-axisymmetric disturbances is investigated. A pseudospectrally generated, generalized algebraic eigenvalue problem is constructed from the linearized set of the three-dimensional governing equations around the steady-state azimuthal solution. Numerical evaluation of the critical eigenvalues shows that for an upper-convected Maxwell model and for the specific set of geometric and kinematic parameters examined in this work, the azimuthal Couette (base) flow becomes unstable against non-axisymmetric time periodic disturbances before it does so for axisymmetric ones, provided the elasticity number ϵ ( De/ Re) is larger than some non-zero but small value (ϵ⪢ 0.01). In addition, as ϵ increases, different families of eigensolutions become responsible for the onset of instability. In particular, the azimuthal wavenumber of the critical eigensolution has been found to change from 1 to 2 to 3 and then back to 2 as ϵ increases from 0.01 to infinity (inertialess flow). In an analogous fashion to the axisymmetric viscoelastic Taylor-Couette flow, two possible patterns of time-dependent solutions (limit cycles) can emerge after the onset of instability: ribbons and spirals, corresponding to azimuthal and traveling waves, respectively. These patterns are dictated solely by the symmetry of the primary flow and have already been observed in conjunction with experiments involving Newtonian fluids but with the two cylinders counter-rotatng instead of co-rotating as considered here. Inclusion of a non-zero solvent viscosity (Oldroyd-B model) has been found to affect the results quantitatively but not qualitatively. These theoretical predictions are of particular importance for the interpretation of the experimental data obtained in a Taylor-Couette flow using highly elastic viscoelastic fluids.

Rayleigh-Taylor instability of two superposed fluids under imposed horizontal and parallel rotation and magnetic fields

We present numerical and analytical results for the linear Rayleigh-Taylor instability of a two-layer nonviscous magnetofluid under uniform horizontal magnetic and rotation fields. The parameters are the non-dimensional density difference A, rotation field F, magnetic field B, and angle of propagation of perturbation θ. When rotation is present alone, an increase in F may stabilize the system in some range of values, after which instability occurs again. The effect of a magnetic field in this system is investigated. An increase in B diminishes the stable angular area and decreases the critical value of F for stability. The critical values of F for stability, as functions of B and θ, were obtained by means of an approximation of the nonlinear algebraic eigenvalue equation. Moreover, exact solutions, independent of A, for the critical values of F for instability were obtained.

A Galerkin finite element technique for stochastic field problems

A new finite element method is developed to analyse the structures with more than one parameter behaving in a stochastic manner. The Galerkin weighted residual method is used. As a generalization, this paper treats the random eigenvalue problems arising when the material property values of the structural systems are having stochastic fluctuations i.e. the measurement errors can be accounted for when probabilistic modelling is used. The free vibration problem of a stochastic beam whose Young's modulus and mass density are distributed stochastically is considered. Using Galerkin's weighted residual procedure, a stochastic finite element method is developed and implemented to arrive at a random algebraic eigenvalue problem. The stochastic characteristics of eigensolutions are derived in terms of the stochastic material property variations. Numerical examples are given. It is demonstrated that through this formulation the finite element discretization need not be dependent on the characteristics of stochastic fields describing the fluctuations in the material property values. Further, calculation of covariances between stiffness elements, mass elements, etc., is carried out in a relatively simple manner and is more general in nature.

Non-conservatively loaded stochastic columns

A new finite element method is developed to analyse non-conservative structures with more than one parameter behaving in a stochastic manner. As a generalization, this paper treats the subsequent non-self-adjoint random eigenvalue problem that arises when the material property values of the non-conservative structural system have stochastic fluctuations resulting from manufacturing and measurement errors. The free vibration problems of stochastic Beck's column and stochastic Leipholz column whose Young's modulus and mass density are distributed stochastically are considered. The stochastic finite element method that is developed, is implemented to arrive at a random non-self-adjoint algebraic eigenvalue problem. The stochastic characteristics of eigensolutions are derived in terms of the stochastic material property variations. Numerical examples are given. It is demonstrated that, through this formulation, the finite element discretization need not be dependent on the characteristics of stochastic processes of the fluctuations in material property value.

Deferred Shifting Schemes for Parallel QR Methods

Parallel implementation of the QR algorithm for solving the symmetric eigenvalue problem requires more than a straightforward transcription of sequential code to parallel code. Experimental adjustments include new shifting techniques and omission of the initial reducing to upper Hessenberg form. In this paper, the theory of the convergence of algorithms of decomposition type for the algebraic eigenvalue problem developed by Watkins and Elsner is generalized. The results are extended to deferred shifting schemes, which allow pipelining of iterations in parallel implementations, and analyzing the deterioration of the convergence rate. Furthermore, it is shown that eigenvalues need not be simple to obtain quadratic convergence for nondefective nonsymmetric matrices and cubic convergence for symmetric matrices.

Solution to the wave equation in the numerical simulation of buried heterostructure lasers

In the numerical simulation of buried heterostructure (BH) lasers a fast, accurate method is essential to solve the wave equation. The paper compares the effective index method with the weighted index method and it is found that if modes other than the fundamental are to be considered the effective index method becomes inaccurate. The authors investigate several methods for solving the algebraic eigenvalue equation which results from the effective and weighted index methods. It is found that a method based on inverse iteration with successive eigenvalue refinement is highly efficient, with good initial approximations to the eigenvalue and eigenvector obtained using a Sturm sequence and inversion.

A novel algebraic solution to Schrödinger equations

We obtain approximations to some bound-state solutions of Schrödinger equations with a variety of central potentials V( r) without any direct calculation of the matrix elements of V( r). The method involves solutions of two algebraic eigenvalue problems, one for a real tridiagonal matrix, and one for a more general real symmetric matrix. The eigenvalues, calculated from matrices of dimension less than 64, generally converge rapidly towards their correct limiting values.

Nucleation field of the infinite ferromagnetic circular cylinder at high temperature

Near, but below Curie's temperature T c , the magnetization increases with applied field above saturation. Therefore, when we approach T c in a ferromagnet it becomes no longer possible to neglect the change in magnitude of the local magnetization due to magnetic fields. For the purpose of our problem the Brown's equations are extended by using a variational procedure. The equations derived are used to study the problem of the nucleation field of the infinite circular ferromagnetic cylinder. The regular part of the solution of the linearized equations is given in terms of Bessel functions and the resulting algebraic eigenvalue problem is solved numerically. The dependence of the nucleation field from the various parameters of the problem is discussed as well as the size of the single domain particle considered.

Quantum efficiency and signal bandwidth of thallium atomic line filters

We derive a new, simplified method for the computation of trapping effects in coupled three-level atomic systems. The linearity of the Holstein radiation trapping equation allows the reduction of the general rate equation to the solution of the basic two-level Holstein equation and to an algebraic eigenvalue problem. The method is applied to the numerical simulation of the quantum efficiency and the signal bandwidth of thallium atomic line filters. With a pumping scheme that achieves population inversion between the ground state and the metastable state, 90% quantum efficiency and 10 MHz signal bandwidth can be achieved, while in noninverting pumping schemes, even quite high pump intensities result in less than 60% quantum efficiency and 8 MHz signal bandwidth. A tradeoff between quantum efficiency, signal bandwidth and pump power is possible.

A new computational technique for the stability analysis of slender rods

A new computational technique for the stability analysis of slender rods with variable cross-sections under general loading conditions is presented. In this approach, the dependent variable and the variable coefficients appearing in the governing equations are expanded in a finite series of Chebyshev polynomials. The main feature of this technique is that the original boundary value problem associated with the differential equation is reduced to an algebraic eigenvalue problem. The proposed technique is applied to study the static buckling of Euler column and the flutter behavior of a cantilever column subjected to uniformly distributed tangential loading. The numerical results from the suggested technique are found to be extremely accurate when compared to other techniques available in literature. It is shown that this approach can also be employed in a symbolic form. The merits of the present method in comparison to the standard solution procedures like finite difference and Galerkin methods are discussed.

Nonlinear vibrations of orthotropic shallow shells of revolution

A set of nonlinearly coupled algebraic and differential eigenvalue equations of nonlinear axisymmetric free vibration of orthotropic shallow thin spherical and conical shells are formulated following an assumed time-mode approach suggested in this paper. Analytic solutions are presented and an asymptotic relation for the amplitude-frequency response of the shells is derived. The effects of geometrical and material parameters on vibrations of the shells are investigated.

A study on basis conjugate matrices

The purpose of this paper is to discuss some important ideas to construct a basis conjugate matrix which can be used to solve algebraic eigenvalue equations.

Relativistic calculations for atoms: self-consistent treatment of Breit interaction and nuclear volume effect

Estimates of the ground state energies of several closed-shell atoms and ions with atomic numbers 1<or=Z<or=102 are presented. A relativistic formulation has been employed: energies are obtained as expectation values of the appropriate combination of the Dirac, Coulomb and Breit Hamiltonians with respect to the corresponding Slater determinants of four-component spherical spinor orbitals. Atomic nuclei have been modelled as uniform spherical charge distributions. An expansion method using a kinetically-balanced basis constructed from Gaussian type functions ( approximately rne- alpha ir2, where n is an integer and the alpha i form a geometric progression) has been used to formulate the variational problem for the optimal orbital radial functions in each case. The resulting sets of coupled algebraic eigenvalue problems were solved approximately using the self-consistent-field iterative method. The authors' results are compared with those of other workers and with finite-difference calculations carried out specifically for this purpose. They estimate the accuracy of their work to be within a few parts in 108 in most cases.

A variational approach to the magneto-elastic buckling problem of an arbitrary number of superconducting beams

Based upon a variational principle and the associated theory derived in three preceding papers, an expression for the magneto-elastic buckling value for a system of an arbitrary number of parallel superconducting beams is given. The total current is supposed to be equal both in magnitude and direction for all beams, and the cross-sections are circular. The expression for the buckling value is formulated more explicitly in terms of the so-called buckling amplitudes, the latter following from an algebraic eigenvalue problem. The pertinent matrix is formulated in terms of complex functions, which are replaced by real potentials. The matrix elements are calculated by a numerical method, solving a set of intergral equations with regular kernels. Apart from the buckling value(s) the buckling modes are also obtained. Finally, our results are compared with the results of a mathematically less complicated theory, i.e. the method of Biot and Savart.

On a correction of Numerov-like eigenvalue approximations for Sturm-Liouville problems

For computing approximations for the eigenvalues of Sturm-Liouville problems, Numerov's method and other finite-difference methods are often used, giving rise to an algebraic eigenvalue problem of which the smallest eigenvalues approximate the fundamental eigenvalue and the first few harmonics of the original problem. Recently the authors have derived a modified Numerov method with step-dependent coefficients, which integrates exactly the functions 1, x, x 2, x 3, sin kx and cos kx. The parameter k is calculated by minimizing the error term associated to the above modified multistep method equation. We shall show that this modified Numerov method delivers much more accurate eigenvalues than the classical Numerov method which is based on a same grid pattern. Several numerical experiments will be presented.