Path-Following Method to Determine the Field of Values of a Matrix with High Accuracy

We describe a novel and efficient algorithm for calculating the field of values boundary, $\partial\textrm{W}(\cdot)$, of an arbitrary complex square matrix: the boundary is described by a system of ordinary differential equations which are solved using Runge--Kutta (Dormand--Prince) numerical integration to obtain control points with derivatives then finally Hermite interpolation is applied to produce a dense output. The algorithm computes $\partial\textrm{W}(\cdot)$ both efficiently and with low error. Formal error bounds are proven for specific classes of matrix. Furthermore, we summarise the existing state of the art and make comparisons with the new algorithm. Finally, numerical experiments are performed to quantify the cost-error trade-off between the new algorithm and existing algorithms

Theory of single- and few-mode lightguides

The thesis is divided into three main parts. After introducing the thesis with some background material in Chapter 1, Part I (Chs.2-3) sets the scene by examining how lightguide studies tie in with other physical theories, and studying some one-dimensional examples. In Part II (Chs. 4-6) we examine certain waveguiding effects for specific fiber profile shapes, in particular, higher mode "splitting" due to polarization and noncircularity. In Part III (Chs.7-8) we investigate some techniques for analyzing fibers of arbitrary profile shape. We conclude with Chapter 9 which further develops the themes of Parts II and III. A more detailed outline is as follows. In Chapter 2 we provide a simple and unified formalism for the analogy between fiber-optics and mechanics. It is based on the scalar theory of light for an optical fiber with longitudinally-independent refractive index, and the mechanics of a particle in a time-independent potential. Illustrative examples are given. We also mention the conceptual analogies that arise when we include polarization and longitudinal- or time-dependent variations. This should prove useful both in teaching and as a bridge between research areas. In Chapter 3 we examine planar lightguides which are of interest in integrated optics and as simple models for fiber effects considered later in the thesis. In addition to some scalar solutions for specific profiles, we examine methods for general profiles and for "exact" numerical solutions recommend use of an adaptation of the Sammut-Pask shooting-extrapolation method developed originally for fibers. We also consider a Green function method and the planar lightguide form of the Gaussian approximation. Then we consider the inclusion of polarization, and a one-dimensional model of a visual photoreceptor. In Chapter 4 we examine the theory of few-mode polarization effects on circularly symmetric fibers. General equations are found which determine arbitrary order corrections to the scalar wave equation. The higher-order corrections are of particular interest when there is a degeneracy at lower orders, and are also required to increase the accuracy for large numerical aperture fibers. The TEom and TMom modes are the ones for which an excitation dependent polarization splitting occurs. These modes have corrections to all orders determined by knowledge of the scalar solutions: we obtain expressions suitable for computer algebraic evaluation. Explicit polarization splittings are found for the infinite-parabolic and clad power-law profiles. In particular we find that in contrast to polarization splittings for the fundamental modes, those between the TEQm and TMQm modes are highly profile dependent. In Chapter 5 we consider the application of few-mode polarization effect studies to absorption in visual photoreceptors. In particular, we investigate the direction, wavelength dependence and magnitude of the polarization dependence of absorption in photoreceptors, assuming as our model, bound mode theory of uniformly absorbing dielectric waveguides. This also has application to few-mode optical fibers. First, a physical understanding is given in terms of simple concepts from plane-wave theory. Then, in undertaking a modal analysis, we find (in the spirit of the Gaussian approximation) that the infinite-parabolic profile provides a simple qualitative understanding of trends. Quantitative numerical results are given for the step profile. In Chapter 6 we provide an understanding of the order in which modes of noncircular lightguides are cutoff, discuss their eigenvalues, and note some interesting degeneracies. In Chapter 7 we develop a comparison method which identifies a lightguide with properties that are well known and very similar to the one under investigation. It has a very simple mathematical basis which allows a trivial derivation of the moment method for circular crosssection fibers including W-fibers, and an extension to non-circular lightguides. We first use the scalar approximation, and then extend to the full vector theory which accounts for polarization. Our concern is with fundamental mode propagation constants and higher mode cutoffs. In Chapter 8 we firstly extend the moment method to symmetric slab lightguide, thus providing a simple description of both the fundamental mode and the second mode cutoff point. Secondly, we further develop the theory of equivalent step index fibers. The equivalent step method rests on the concept that properties of the fundamental mode are not very sensitive to refractive index profile details. We show how this idea may be incorporated into the original variational scheme of Snyder and Sammut in order to greatly simplify the calculations and to provide wavelength independent equivalent step parameters. The new approach uses the effective waveguide parameter and moments of the profile shape function. In Chapter 9 we firstly show that highly effective single-mode single polarization ( SMSP) fibers can be made with comparitively small values of birefringence, provided that the profile heights Δₓ,Δy differ and the fiber is bent. The effect is enhanced in anisotropic W-fibers. Secondly, „e consider a transformation of the scalar wave equation to an integral equation using the Green function for a step reference profile, give results for circular profiles as examples, and examine 1 generalization to noncircular lightguides

Internal Camera Calibration Using Rotation and Geometric Shapes

This paper describes a simple method for internal camera calibration for computer vision. This method is based on tracking image features through a sequence of images while the camera undergoes pure rotation. The location of the features relative to the camera or to each other need not be known and therefore this method can be used both for laboratory calibration and for self calibration in autonomous robots working in unstructured environments. A second method of calibration is also presented. This method uses simple geometric objects such as spheres and straight lines to The camera parameters. Calibration is performed using both methods and the results compared

Vibration attenuation by mass redistribution

A nontraditional approach for active structural vibration attenuation was proposed using mass redistribution. The focus was on pendulum structures where the objective was to examine the effectiveness of mass reconfiguration along or within a structure to attenuate its vibrational energy. The mechanics associated with a translating mass along a rotating structure give rise to a Coriolis inertia force which either opposes or increases angular oscillations, thereby producing positive or negative damping, respectively. A strategy of cycling the mass to maximize attenuation and minimize amplification required the mass be moved at twice the frequency of the structural vibrations and be properly coordinated with the angular oscillations. The desired coordination involved moving the mass away from the pivot as the pendulum nears its vertical position and moving the mass towards the pivot when the pendulum nears its maximum angular excursion. System mass reconfiguration was analyzed by studying various mass displacement profiles including sinusoidal, piece-wise constant velocity and modified proportional and derivative action patterns. These strategies were optimized for various time intervals to maximize the rate of energy attenuation or minimize the final energy state. For small amplitude oscillations with sinusoidal mass motion, the dynamic behavior was modeled by Mathieu-Hill equations to explain the beating phenomenon that occurred when the frequency of the mass motion remained constant. Several control systems were designed to generate aforementioned mass reconfiguration profiles. The methodologies included human operator, modified proportional and derivative action, knowledge or rule based and artificial neural network controllers. The human operator system improved with experience and was the most effective. Other systems depended on the chosen parameterization or the implementation of self-adjusting parameters. Several unique tools were developed during the course of this research, as referenced herein

Conference on the Programming Environment for Development of Numerical Software

Systematic approaches to numerical software development and testing are presented

Comparative methods of computing maximum likelihood estimates for non-linear econometric systems

This research is mainly concerned with numerical optimisation techniques applied to general non-linear econometric simultaneous equations systems. The method of estimation used is maximum likelihood. An estimation program which applies gradient-type procedures, specifically the Berndt-Hall-Hall-Hausman and Gill-Murray' Pitfield methods, is developed. This program allows the estimation of a general small-to-medium size model which is non-linear in parameters, variables or both. In the course of program development, a general differentiation program is written which will differentiate a set of econometric equations and thus provide the analytical gradients for the optimisation procedures. A comparative study has been made of the relative efficiency of the two methods by running a set of simulated non-linear models and also using a small macro- economic model of the British Economy specified by David F. Hendry. To improve the efficiency of the estimation program in terms of computing time, the Berndt-Hall-Hall-Hausman method was implemented on the ICL Distributed Array Processor (DAP)’ which employs parallel computations. The DAP runs show that for a model with a large sample size, the DAP is approximately 30 times faster than the conventional computer CDC 7600, but that for the present algorithm, the latter is a more efficient alternative for small sample sizes

Third International Conference on Inverse Design Concepts and Optimization in Engineering Sciences (ICIDES-3)

Papers from the Third International Conference on Inverse Design Concepts and Optimization in Engineering Sciences (ICIDES) are presented. The papers discuss current research in the general field of inverse, semi-inverse, and direct design and optimization in engineering sciences. The rapid growth of this relatively new field is due to the availability of faster and larger computing machines