Browsing by Subject "Difference"

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An analytical approach to computing step sizes for finite-difference derivatives
(2012-05) Mathur, Ravishankar; Ocampo, Cesar; Hull, David G.; Fowler, Wallace T.; Marchand, Belinda; Senent, Juan
Finite-difference methods for computing the derivative of a function with respect to an independent variable require knowledge of the perturbation step size for that variable. Although rules of thumb exist for determining the magnitude of the step size, their effectiveness diminishes for complicated functions or when numerically solving difficult optimization problems. This dissertation investigates the problem of determining the step size that minimizes the total error associated with finite-difference derivative approximations. The total error is defined as the sum of errors from numerical sources (roundoff error) and mathematical approximations (truncation error). Several finite-difference approximations are considered, and expressions are derived for the errors associated with each approximation. Analysis of these errors leads to an algorithm that determines the optimal perturbation step size that minimizes the total error. A benefit of this algorithm is that the computed optimal step size, when used with neighboring values of the independent variable, results in approximately the same magnitude of error in the derivative. This allows the same step size to be used for several successive iterations of the independent variable in an optimization loop. A range of independent variable values for which the optimal step size can safely remain constant is also computed. In addition to roundoff and truncation errors within the finite-difference method, numerical errors within the actual function implementation are also considered. It is shown that the optimal step size can be used to compute an upper bound for these condition errors, without any prior knowledge of the function implementation. Knowledge of a function's condition error is of great assistance during the debugging stages of simulation design. Although the fundamental analysis assumes a scalar function of a scalar independent variable, it is later extended to the general case of a vector function of a vector independent variable. Several numerical examples are shown, ranging from simple polynomial and trigonometric functions to complex trajectory optimization problems. In each example, the step size is computed using the algorithm developed herein, a rule-of-thumb method, and an alternative statistical algorithm, and the resulting finite-difference derivatives are compared to the true derivative where available.
From darbuka to dümbelek : the Turkish goblet-shaped drum and the construction of difference
(2012-05) Ragheb, Nicholas Joseph; Seeman, Sonia Tamar, 1958-; Moore, Robin
This report examines the linguistic, physical, and social construction of the goblet-shaped drum in recent Turkish history. Following recent trends in organological scholarship towards a more socially situated analysis of musical instruments, I examine the role of the goblet-shaped drum as a mediator within large complex webs of interaction and social significance. Drawing on theoretical frameworks for understanding the production of difference set forth by Ruth Solie, my analysis explores the modalities of language, visual and aural representation and positionality. In order to examine the specific process of “othering” that undergird these processes, I supplement this theoretical framework with the concept of “articulation” developed by Laclau and Hall. My analysis begins with a review of the popular and musicological literature on the drum and a discussion of the way in which different names such as darbuka and dümbelek are used to construct social groups defined by notions of gender, urbanity, and professionalism. I follow this with an examination of the workflows and techniques involved in the production of two types of goblet-shaped drums revealing how these processes shape the identity of the artisans and entrepreneurs involved in the production process. Finally, I discuss the representation of the instrument in Turkish television documentaries and the Internet revealing the role of the drum in the production of a Turkish modernist narrative. I argue that while these processes may appear isolated when viewed independently, an organological approach reveals important connections between these different mechanisms for the production of difference and issues such as the masculinization of the darbuka playing profession may be more effectively engaged through an understanding of these connections.