This paper is the result of a study of the power method to find dominant eigenvalues of square matrices. It introduces ideas basic to the study and shows the development of the power method for the most well-behaved matrices possible, and it explores exactly which other types of matrices yield to the power method. The paper also discusses a type of matrix typically considered impossible for the power method, along with a modification of the power method which works for this type of matrix. It gives an overview of common extensions of the power method. The appendices contain BASIC versions of the power method and its modification.

Waveguides in Euclidian space are piecewise path connected subsets of R^n that can be written as the union of a compact domain with boundary and their cylindrical ends. The compact and non-compact parts share a common boundary. This boundary is assumed to be Lipschitz, piecewise smooth and piecewise path connected. The ends can be thought of as the cartesian product of the boundary with the positive real half-line. A notable feature of Euclidian waveguides is that the scattering matrix admits a meromorphic continuation to a certain Riemann surface with a countably infinite number of leaves [2], which we will describe in detail and deal with. In order to construct this meromorphic continuation, one usually first constructs a meromorphic continuation of the resolvent for the Laplace operator. In order to do this, we will use a well known glueing construction (see for example [5]), which we adapt to waveguides. The construction makes use of the meromorphic Fredholm theorem and the fact that the resolvent for the Neumann Laplace operator on the ends of the waveguide can be easily computed as an integral kernel. The resolvent can then be used to construct generalised eigenfunctions and, from them, the scattering matrix. Being in possession of the scattering matrix allows us to calculate resonances; poles of the scattering matrix. We are able to do this using a combination of numerical contour integration and Newton s method.

Thesis (Sc.D.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 1980. / Includes bibliographical references. / by Seshadri Ramaswamy. / Thesis (Sc.D.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 1980.

The electron is known to possess an anomalous magnetic moment, which interacts with the gradient of the electric field. This makes it necessary to compute its effects on the energy spectrum. Even though the Coulomb Dirac equation can be solved in closed form, this is no longer possible when the anomalous magnetic moment is included. In fact the interaction due to this term is so strong that it changes the domain of the Hamiltonian. From a differential equation point of view, the anomalous magnetic moment term is strongly singular near the origin. As usual, one has to resort to perturbation theory. This, however, only makes sense if the eigenvalues are stable. To prove stability is therefore a challenge one has to face before actually computing the energy shifts. The first stability results in this line were shown by Behncke for angular momenta κ ≥ 3, because the eigenfunctions of the unperturbed Hamiltonian decay fast enough near the origin. He achieved this by decoupling the system and then using the techniques available for second order differential equations. Later, Kalf and Schmidt extended Behncke’s results basing their analysis on the Prüfer angle technique and a comparison result for first order differential equations. The Prüfer angle method is particularly useful because it shows a better stability and because it obeys a first order differential equation. Nonetheless, Kalf and Schmidt had to exclude some coupling constants for κ > 0. This I believe is an artefact of their method. In this study, I make increasing use of asymptotic integration, a method which is rather well adapted to perturbation theory and is known to give stability results to any level of accuracy. Together with the Prüfer angle technique, this lead to a more general stability result and even allows for an energy shifts estimate. Hamiltonians traditionally treated in physics to describe the spin-orbit effect are not self adjoint i.e. they are not proper observables in quantum mechanics. Nonetheless, naive perturbation theory gives correct results regarding the spectrum. To solve this mystery, one has to study the nonrelativistic limit of the Dirac operator. In the second part of this study, I have not only given the higher order correction to the Dirac operator but also shown the effects of the spin-orbit term.

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In this thesis, we set out to provide an enhanced set of techniques for determining the eigenvalues of the Laplacian in polygonal domains. Currently, finite-element methods provide a numerical means by which we can approximate these eigenvalues with ease. However, we would like a more analytic method which may allow us to avoid a basic parameter sweep in finite-element software such as COMSOL to determine what could possibly be an "optimal" distribution of eigenvalues. The hope is that this would allow us to draw conclusions about the acoustic quality of a pentagonally-shaped room. First, we find the eigenvalues using a common finite-element method through COMSOL Multiphysics. We then examine another method which makes use of conformal maps and Schwarz-Christoffel transformations with the prospect that it might provide a more analytic understanding of the calculation of these eigenvalues and possibly allow for variation of certain parameters. This method, as far as we could find, had not yet been developed on the pentagon. We end up carrying this method through nearly all of the steps necessary in finding these eigenvalues. We find that the finite-element method is not only easier to use, but is also more efficient in terms of computing power. / Master of Science

In the first chapter of this dissertation, I explore how forecaster attention, or the degree to which new information is incorporated into forecasts, is reflected at the lower-dimensional factor representation of multivariate forecast data. When information is costly to acquire, forecasters may pay more attention to some sources of information and ignore others. How much attention they pay will determine the strength of the forecast correlation (factor) structure. Using a factor model representation, I show that a forecast made by a rationally inattentive agent will include an extra shrinkage and thresholding "attention matrix" relative to a full information benchmark, and propose an econometric procedure to estimate it. Differences in the degree of forecaster attentiveness can explain observed differences in empirical shrinkage in professional macroeconomic forecasts relative to a consensus benchmark. Forecasters share the same reduced-form model, but differ in their measured attention. Better-performing forecasters have higher measured attention (lower shrinkage) than their poorly-performing peers. Measured forecaster attention to multiple dimensions of the information space can largely be captured by a single scalar cost parameter. I propose a new class of information cost functions for the classic multivariate linear-quadratic Gaussian tracking problem called separable spectral cost functions. The proposed measure of attention and mapping from theoretical model of attention allocation to factor structure in the first chapter is valid for this set of cost functions. These functions are defined over the eigenvalues of prior and posterior variance matrices. Separable spectral cost functions both nest known cost functions and are consistent with the definition of Uniformly Posterior Separable cost functions, which have desirable theoretical properties. The third chapter is coauthored work with Professor Serena Ng. We estimate higher frequency values of monthly macroeconomic data using different factor based imputation methods. Monthly and weekly economic indicators are often taken to be the largest common factor estimated from high and low frequency data, either separately or jointly. To incorporate mixed frequency information without directly modeling them, we target a low frequency diffusion index that is already available, and treat high frequency values as missing. We impute these values using multiple factors estimated from the high frequency data. In the empirical examples considered, static matrix completion that does not account for serial correlation in the idiosyncratic errors yields imprecise estimates of the missing values irrespective of how the factors are estimated. Single equation and systems-based dynamic procedures that account for serial correlation yield imputed values that are closer to the observed low frequency ones. This is the case in the counterfactual exercise that imputes the monthly values of consumer sentiment series before 1978 when the data was released only on a quarterly basis. This is also the case for a weekly version of the CFNAI index of economic activity that is imputed using seasonally unadjusted data. The imputed series reveals episodes of increased variability of weekly economic information that are masked by the monthly data, notably around the 2014-15 collapse in oil prices.

A unified approach to structure and controller design optimization is examined. Difficult problems arise in a unified approach, namely, a high dimensioned design space, nonlinearity, complexity of constraints and many inequality constraints. As a candidate for overcoming the above problems, an optimization algorithm utilizing sequential linear programming and continuation methods is proposed. The second part of this dissertation examines various ideas associated with both theory and practical issues arising in optimizing for eigenvalue sensitivity and stability robustness with respect to parameter variations or unstructured uncertainties. In particular, the time domain approach to stability robustness is pursued. It is found that a recently proposed stability robustness criteria of Patel and Toda is related to well known concepts of numerical conditioning of the eigenvalue problem and may be derived concisely using eigenvalue conditioning concepts. In addition, we review the more direct and perhaps less rigorous approach of dealing with uncertainties, namely modal insensitivity theory. The mathematical conditions for achieving modal insensitivity and eigenvalue placement simultaneously are reviewed along with a discussion of the practical merit of these ideas. As an alternative, we derive a scalar measure of eigenvalue sensitivity which is a linearly predicted bound on weighted eigenvalue perturbation; we also introduce an algorithm for minimization of this index. Furthermore, the expressions for eigenvector derivatives are correctly derived for non-self-adjoint case. This latter contribution corrects errors present in at least two textbooks on the subject and serves to clear up confusion in the literature. Finally, we use examples to demonstrate the design algorithm proposed here and numerically examine various designs arising from corresponding cost functions, using a specific configuration (a flexible free-free beam with an attached rigid body.) The numerical results confirm the conservatism of the stability robustness bound for highly structured perturbations but nevertheless clearly supports the hypothesis that maximizing the robustness measure significantly increases the true robustness of a closed loop system. The numerical results also indicate that maximizing the stability robustness measure is better (more efficient computationally and produces more robust designs) than minimizing the eigenvalue sensitivities directly for improving true stability robustness with respect to perturbations in the closed loop system matrix. / Ph. D. / incomplete_metadata

In this thesis, we consider real matrix functions that depend on two parameters and study the problem of how to detect and approximate parameters' values where the singular values coalesce. We prove several results connecting the existence of coalescing points to the periodic structure of the smooth singular values decomposition computed around the boundary of a domain enclosing the points. This is further used to develop algorithms for the detection and approximation of coalescing points in planar regions. Finally, we present techniques for continuing curves of coalescing singular values of matrices depending on three parameters, and illustrate how these techniques can be used to locate coalescing singular values of complex-valued matrices depending on three parameters.

<p>This theses is dedicated to the study of generalized diagonal dominance and its<br />various beneflts. The starting point is the well known nonsingularity result of strictly diagonally dominant matrices, from which generalizations were formed in difierent directions. In theses, after a short overview of very well known results, special attention was turned to contemporary contributions, where overview of already published original material is given, together with new obtained results. Particulary, Ger&bull;sgorin-type localization theory for matrix pencils is developed, and application of the results in wireless sensor networks optimization problems is shown.</p> / <p><span class="fontstyle0">Ova teza je posvećena izučavanju generalizovane dijagonalne dominacije i njenih brojnih prednosti. Osnovu čini poznati rezultat o regularnosti strogo dijagonalnih matrica,<br />čija su uop&scaron;tenja formirana u brojnim pravcima. U tezi, nakon kratkog pregleda dobro poznatih rezultata, posebna pažnja je posvećena savremenim doprinosima, gde je dat i pregled već objavljenih autorovih rezultata, kao i detaljan tretman novih dobijenih rezultata. Posebno je razvijena teorija lokalizacije Ger&scaron;gorinovog tipa generalizovanih karakterističnih korena i pokazana je primena rezultata u problemima optimizacije bežičnih senzor mreža.</span></p>