• Communications of the Korean Mathematical Society
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• v.33 no.2
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• pp.677-693
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• 2018

In this work, we generalize a family of optimal eighth order weighted-Newton methods to Banach spaces and study its local convergence to approximate a locally-unique solution of a system of nonlinear equations. The convergence in this study is shown under hypotheses only on the first derivative. Our analysis avoids the usual Taylor expansions requiring higher order derivatives but uses generalized Lipschitz-type conditions only on the first derivative. Moreover, our new approach provides computable radius of convergence as well as error bounds on the distances involved and estimates on the uniqueness of the solution based on some functions appearing in these generalized conditions. Such estimates are not provided in the approaches using Taylor expansions of higher order derivatives which may not exist or may be very expensive or impossible to compute. The convergence order is computed using computational order of convergence or approximate computational order of convergence which do not require usage of higher derivatives. This technique can be applied to any iterative method using Taylor expansions involving high order derivatives. The study of the local convergence based on Lipschitz constants is important because it provides the degree of difficulty for choosing initial points. In this sense the applicability of the method is expanded. Finally, numerical examples are provided to verify the theoretical results and to show the convergence behavior.

• Journal of KIISE
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• v.43 no.1
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• pp.40-46
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• 2016

Digital road map generation is primarily based on artificial satellite photographing or in-site manual survey work. Therefore, these map generation procedures require a lot of time and a large budget to create and update road maps. Consequently, people have tried to develop automated map generation systems using GPS trajectory data sets obtained by public vehicles. A fundamental problem in this road generation procedure involves the extraction of representative trajectory such as main roads. Extracting a representative trajectory requires the base data set of piecewise line segments(GPS-trajectories), which have close starting and ending points. So, geometrically similar trajectories are selected for clustering before extracting one representative trajectory from among them. This paper proposes a new divide- and-conquer approach by partitioning the whole map region into regular grid sub-spaces. We then try to find similar trajectories by sweeping. Also, we applied the $Fr{\acute{e}}chet$ distance measure to compute the similarity between a pair of trajectories. We conducted experiments using a set of real GPS data with more than 500 vehicle trajectories obtained from Gangnam-gu, Seoul. The experiment shows that our grid partitioning approach is fast and stable and can be used in real applications for vehicle trajectory clustering.