• 대한수학회논문집
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• 제33권2호
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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.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제17권1호
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• pp.1-27
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• 2010

Wu and Zhao [17] provided a semilocal convergence analysis for a Newton-type method on a Banach space setting following the ideas of Frontini and Sormani [9]-[11]. In this study first: we point out inconsistencies between the hypotheses of Theorem 1 and the two examples given in [17], and then, we provide the proof in affine invariant form for this result. Then, we also establish new convergence results with the following advantages over the ones in [17]: weaker hypotheses, and finer error estimates on the distances involved. A numerical example is also provided to show that our results apply in cases other fail [17].

• Rapid Communication in Photoscience
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• 제1권1호
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• pp.1-9
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• 2012

Luminescent lanthanide complexes have been overviewed for advanced photonics applications. Lanthanide(III) ions ($Ln^{3+}$) were encapsulated by the luminescent ligands such as metalloporphyrins, naphthalenes, anthracene, push-pull diketone derivatives and boron dipyrromethene(bodipy). The energy levels of the luminescent ligands were tailored to maintain the effective energy transfer process from luminescent ligands to $Ln^{3+}$ ions for getting a higher optical amplification gain. Also, key parameters for emission enhancement and efficient energy transfer pathways for the sensitization of $Ln^{3+}$ ions by luminescent ligands were investigated. Furthermore, to enhance the optophysical properties of novel luminescent $Ln^{3+}$ complexes, aryl ether-functionalized dendrons as photon antennas have been incorporated into luminescent $Ln^{3+}$ complexes, yielding novel $Ln^{3+}$-cored dendrimer complex such as metalloporphyrins, naphthalenes, and anthracenes bearing the Fr$\acute{e}$chet aryl-ether dendrons, namely, ($Er^{3+}-[Gn-Pt-Por]_3$ (terpy), $Er^{3+}-[Gn-Naph]_3$(terpy) and $Er^{3+}-[Gn-An]_3$(terpy)). These complexs showed much stronger near-IR emission bands at 1530 nm, originated from the 4f-4f electronic transition of the first excited state ($^4I_{13/2}$) to the ground state ($^4I_{15/2}$) of the partially filled 4f shell. A significant decrease in the fluorescence of metalloporphyrins, naphthalenes and anthracene ligand were accompanied by a strong increase in the near IR emission of the $Ln^{3+}$ ions. The near IR emission intensities of $Ln^{3+}$ ions in the lanthanide(III)-encapsulated dendrimer complexes were dramatically enhanced with increasing the generation number (n) of dendrons, due to the site-isolation and the light-harvesting(LH) effects. Furthermore, it was first attempted to distinguish between the site-isolation and the light-harvesting effects in the present complexes. In this review, synthesis and photophysical studies of inert and stable luminescent $Ln^{3+}$ complexes will be dealt for the advanced photonics applications. Also, the review will include the exploratory investigation of the key parameters for emission enhancement and the effective energy transfer pathways from luminescent ligands to $Ln^{3+}$ ions with $Ln^{3+}$-chelated prototype complexes.