In this paper we present an approach to the nonlinear model reduction based on representing the nonlinear system with a piecewise-linear system and then reducing each of the pieces with a Krylov projection. However, rather than approximating the individual components to make a system with exponentially many different linear regions, we instead generate a small set of linearizations about the state trajectory which is the response to a 'training input'. Computational results and performance data are presented for a nonlinear circuit and a micromachined fixed-fixed beam example. These examples demonstrate that the macromodels obtained with the proposed reduction algorithm are significantly more accurate than models obtained with linear or the recently developed quadratic reduction techniques. Finally, it is shown tat the proposed technique is computationally inexpensive, and that the models can be constructed 'on-the-fly', to accelerate simulation of the system response. / Singapore-MIT Alliance (SMA)

Masters of Science / In a world where technology is advancing at a very rapid pace, global competition has significantly increased, and this is putting pressure on software companies to produce quality software. It has therefore become critically important to manage well the implementation of software engineering projects by employing effective methods that ensure the best product is produced. The most popular software project implementation methodologies are the Traditional methods and Agile methods. This research explored these two methodologies by comparing the strength and weakness of both approaches. The research was conducted using a constructionist epistemology with a critical inquiry using the grounded theory methodology, applying both quantitative and qualitative methods to the case studies. Findings were collected through participant observation using a designed questionnaire targeting a selected sample of the study population. This sample of the population consisted of Ecommerce organizations in the Western Cape province of South Africa to establish which of the Traditional or Agile methods would best lead to the successful implementation of Ecommerce software engineering projects. The research results showed that the Agile methodology was the preferred and recommended approach. Very few participants of the research supported the Traditional approach to still be considered and used for projects with well-known end goals. An Ecommerce website prototype for a local Cape Town business was constructed as following the Agile approach to measure and validate the findings of the research. The prototype was built successfully from conception to the final delivery product and on time confirming the Agile approach as best for Ecommerce software development. In conclusion, the Agile methodology is the choice approach based on reviewed literature, the research results, and the prototype construction. These results will help in critical decision making regarding an appropriate development methodology to follow for the Ecommerce industry in the Western Cape.