Eigensystem Analysis Research Articles

Eigen-System Analysis of X-ray Diffraction Profile Deconvolution Methods Explains Ill-Conditioning

Several deconvolution methods common in X-ray diffraction profile studies have been examined using an eigen-system analysis in which an error-bound function is used to represent the maximum difference between the solution and true specimen profiles. This approach quantifies the sources of misfitting and ill-conditioning that appear in the solution profile and expresses them as a function of the control parameter for a particular method. A simulation of an instrument-broadened profile overlaid with random noise and background signals was used to evaluate the error-bound function for the iterative and constrained deconvolution methods, and the properties of the error-bound function were related to the features of the solution profile for each method. This analysis quantifies the terms that contribute to the ill-conditioning of the solution profile. It shows that, even for optimum values of the control parameters, positivity is not preserved and spurious oscillations are present in the solution profile.

Estimating 3-D rigid body transformations: a comparison of four major algorithms

A common need in machine vision is to compute the 3-D rigid body transformation that aligns two sets of points for which correspondence is known. A comparative analysis is presented here of four popular and efficient algorithms, each of which computes the translational and rotational components of the transform in closed form, as the solution to a least squares formulation of the problem. They differ in terms of the transformation representation used and the mathematical derivation of the solution, using respectively singular value decomposition or eigensystem computation based on the standard $[ \vec{R}, \vec{T} ]$ representation, and the eigensystem analysis of matrices derived from unit and dual quaternion forms of the transform. This comparison presents both qualitative and quantitative results of several experiments designed to determine (1) the accuracy and robustness of each algorithm in the presence of different levels of noise, (2) the stability with respect to degenerate data sets, and (3) relative computation time of each approach under different conditions. The results indicate that under “ideal” data conditions (no noise) certain distinctions in accuracy and stability can be seen. But for “typical, real-world” noise levels, there is no difference in the robustness of the final solutions (contrary to certain previously published results). Efficiency, in terms of execution time, is found to be highly dependent on the computer system setup.

On the eigenvalues basic to the analytical solution of convective heat transfer with axial diffusion effects

AbstractThe integral transform method is employed in the analytical solution of the non‐classical eigenvalue problem that appears in connection with the analysis of forced convection in ducts, including the effect of fluid heat diffusion in the axial direction. The related eigenfunctions are expanded in terms of eigenfunctions from a simpler auxiliary eigenvalue problem, and the original eigenvalues are determined from the associated matrix eigensystem analysis. Convergence rates of the proposed solutions are illustrated and reference results established for different values of the governing parameters.

Integral transform solution of eigenvalue problems

AbstractThe generalized integral transform technique is used to reduce eigenvalue problems described by partial differential equations to algebraic ones, that can be solved by existing codes for matrix eigensystem analysis. The method is illustrated for the operators that correspond to heat and mass diffusion, but can be employed in different fields. Three examples demonstrate the potential of the method.

Acceleration of Convergence and Spectrum Transformation of Implicit Finite Difference Operators Associated with Navier-Stokes Equations

Eigensystem analysis techniques are applied to finite difference formulations of Euler and Navier-Stokes equations in two dimensions. Spectrums of the resulting implicit difference operators are computed. The convergence and stability properties of the iterative methods are studied by making into account, the effect of grid geometry, time-step, numerical dissipation,, viscosity, boundary conditions, and the physics of the underlying flow. The largest eigenvalues are computed by using the Frechet derivative of the operators and Arnoldi's method. The accuracy of Arnoldi's method is tested by comparing the dominant eigenvalues with the rate of convergence of the iterative method. Based on the pattern of eigenvalues distributions for various flow configurations, the feasability of applying existing convergence-acceleration techniques like eigenvalue annihilation and relaxation are discussed. Finally a shifting of the implicit operators in question is devised. The idea of shifting is based on the power method of linear algebra and is very simple to implement. The procedure of shifting the spectrum is applied to ARC2D, a flow code developed and being used at NASA Ames Research Center. When compared to eigenvalue annihilation, the shifting method clearly establishes its superiority. For the ARC2D code, an efficiency of 20 to 33% has been achieved by this method.

Some exact algebraic expressions for the reliability of multipath switching networks

Abstract Some optical-fiber telecommunication systems have multipath switching designs that do not lend themselves to reliability evaluation using conventional methods for series and parallel components. Efficient and exact numerical procedures for reliability evaluation of such systems are described in previous publications of the authors but no exact algebraic results have been reported. Exact algebraic expressions for reliabilities of multipath systems are generally intractable, except possibly by means of computerized symbol manipulation. In this article, exact algebraic expressions for the reliabilities of several simple multipath systems are derived. The results are useful in their own right but also for the theoretical insight that they provide. In spite of the difficulty posed by their initial derivation, once discovered, the expressions have a surprising simplicity. The derivation is based on an eigensystem analysis of the Markov transition matrices involved in the reliability evaluation.

A magnetic model for the demonstration of phase transitions and excitations in molecular crystals

A magnetic model for the demonstration of structural properties and the lattice dynamics of molecular crystals is presented. It is used to visualize an orientational order–disorder phase transition. The orientational order of the low-temperature phase observed on the model is also obtained from a mean field calculation. A simple theory for harmonic excitations of the model is presented, which gives the dispersion relation for an infinite crystal. The frequencies of the observed resonances of the finite model are compared with the results of a numerical eigensystem analysis and with the dispersion relation. Within the limitations given by the construction of the model the results are in rather good agreement.

Analytical structures for eigensystem study of power flow oscillations in large power systems

The authors argue that since eigensystem analysis of large-order power systems is based on numerical methods, each engineering case study is uniquely discrete and lacks the analytical continuity to link it to other case studies so that universally applicable conclusions can be drawn. They provide some analytical structures to validate some of the engineering intuitions which have been formed from extensive inspection of numerical eigenvalues and eigenvectors. A correlation is found between the frequency of oscillation and the mode shape of power system load flow. It is concluded that there is an analytical basis for associating the low-oscillating-frequency modes with light or unstable damping. >

Computer-aided analysis of the convergence to steady state of discrete approximations to the euler equations

The behaviour of a centered finite volume scheme for the isoenergetic Euler equations in two space dimensions is studied by numerical differentiation and approximate eigensystem analysis. The entire semidiscrete approximation including boundary conditions is formulated as a large system of ODES, which are linearized by numerically approximating the Frechet derivative. An approximate eigensystem procedure that only needs the Frechet derivative is used to extract the least damped eigenmodes. The overall method has been applied to the case of transonic flow past an airfoil and has revealed that the most persistent transient modes are highly structured and are associated with eigenvalues of small modulus. Furthermore, they appear to be centered around the shock region, the stagnation region and the trailing edge/wake region of the airfoil. The beneficial effect of local time-step scaling and artificial dissipation is also demonstrated by the method.