• Title, Summary, Keyword: high-order convergence

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A FOURTH-ORDER ACCURATE FINITE DIFFERENCE SCHEME FOR THE EXTENDED-FISHER-KOLMOGOROV EQUATION

  • Kadri, Tlili;Omrani, Khaled
    • Bulletin of the Korean Mathematical Society
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    • v.55 no.1
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    • pp.297-310
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    • 2018
  • In this paper, a nonlinear high-order difference scheme is proposed to solve the Extended-Fisher-Kolmogorov equation. The existence, uniqueness of difference solution and priori estimates are obtained. Furthermore, the convergence of the difference scheme is proved by utilizing the energy method to be of fourth-order in space and second-order in time in the discrete $L^{\infty}-norm$. Some numerical examples are given in order to validate the theoretical results.

STUDY OF OPTIMAL EIGHTH ORDER WEIGHTED-NEWTON METHODS IN BANACH SPACES

  • Argyros, Ioannis K.;Kumar, Deepak;Sharma, Janak Raj
    • 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.

ON CONSTRUCTING A HIGHER-ORDER EXTENSION OF DOUBLE NEWTON'S METHOD USING A SIMPLE BIVARIATE POLYNOMIAL WEIGHT FUNCTION

  • LEE, SEON YEONG;KIM, YOUNG IK
    • Journal of the Chungcheong Mathematical Society
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    • v.28 no.3
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    • pp.491-497
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    • 2015
  • In this paper, we have suggested an extended double Newton's method with sixth-order convergence by considering a control parameter ${\gamma}$ and a weight function H(s, u). We have determined forms of ${\gamma}$ and H(s, u) in order to induce the greatest order of convergence and established the main theorem utilizing related properties. The developed theory is ensured by numerical experiments with high-precision computation for a number of test functions.

ON THE ORDER AND RATE OF CONVERGENCE FOR PSEUDO-SECANT-NEWTON'S METHOD LOCATING A SIMPLE REAL ZERO

  • Kim, Young Ik
    • Journal of the Chungcheong Mathematical Society
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    • v.19 no.2
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    • pp.133-139
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    • 2006
  • By combining the classical Newton's method with the pseudo-secant method, pseudo-secant-Newton's method is constructed and its order and rate of convergence are investigated. Given a function $f:\mathbb{R}{\rightarrow}\mathbb{R}$ that has a simple real zero ${\alpha}$ and is sufficiently smooth in a small neighborhood of ${\alpha}$, the convergence behavior is analyzed near ${\alpha}$ for pseudo-secant-Newton's method. The order of convergence is shown to be cubic and the rate of convergence is proven to be $\(\frac{f^{{\prime}{\prime}}(\alpha)}{2f^{\prime}(\alpha)}\)^2$. Numerical experiments show the validity of the theory presented here and are confirmed via high-precision programming in Mathematica.

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A Fourth-Order Accurate Numerical Boundary Scheme for the Planar Dielectric Interface: a 2-D TM Case

  • Hwang, Kyu-Pyung
    • Journal of electromagnetic engineering and science
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    • v.11 no.1
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    • pp.11-15
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    • 2011
  • Preserving high-order accuracy in high-order FDTD solutions across dielectric interfaces is very important for practical time-domain electromagnetic simulations. This paper presents a fourth-order accurate numerical boundary scheme for the planar dielectric interface to be used in the fourth-order FDTD method proposed earlier by the author. The interface scheme for the two-dimensional (2-D) transverse magnetic (TM) polarization case is derived and validated by monitoring the $L_2$ norm errors in the numerical solutions of a partially-filled cavity demonstrating its fourth-order convergence and long-time numerical stability in the presence of the planar dielectric interface.

External Cavity Lasers Composed of Higher Order Gratings and SLDs Integrated on PLC Platform

  • Shin, Jang-Uk;Oh, Su-Hwan;Park, Yoon-Jung;Park, Sang-Ho;Han, Young-Tak;Sung, Hee-Kyung;Oh, Kwang-Ryong
    • ETRI Journal
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    • v.29 no.4
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    • pp.452-456
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    • 2007
  • Very compact 4-channel 200-GHz-spacing external cavity lasers (ECLs) were fabricated by hybrid integration of reflection gratings and superluminescent laser diodes on a planar lightwave circuit chip. The fifth-order gratings as reflection gratings were formed using a conventional contact-mask photo-lithography process to achieve low-cost fabrication. The lasing wavelength of the fabricated ECLs matched the ITU grid with an accuracy of ${\pm}0.1$ nm, and optical powers were more than 0.4 mW at the injection current of 80 mA for all channels. The ECLs showed single mode operations with more than 30 dB side lobe suppression.

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ON THE HIGH-ORDER CONVERGENCE OF THE k-FOLD PSEUDO-CAUCHY'S METHOD FOR A SIMPLE ROOT

  • Kim, Young Ik
    • Journal of the Chungcheong Mathematical Society
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    • v.21 no.1
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    • pp.107-116
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    • 2008
  • In this study the k-fold pseudo-Cauchy's method of order k+3 is proposed from the classical Cauchy's method defined by an iteration $x_{n+1}=x_n-{\frac{f^{\prime}(x_n)}{f^{{\prime}{\prime}}(x_n)}}{\cdot}(1-{\sqrt{1-2f(x_n)f^{{\prime}{\prime}}(x_n)/f^{\prime}(x_n)^2}})$. The convergence behavior of the asymptotic error constant is investigated near the corresponding simple zero. A root-finding algorithm with the k-fold pseudo-Cauchy's method is described and computational examples have successfully confirmed the current analysis.

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Multilevel acceleration of scattering-source iterations with application to electron transport

  • Drumm, Clif;Fan, Wesley
    • Nuclear Engineering and Technology
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    • v.49 no.6
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    • pp.1114-1124
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    • 2017
  • Acceleration/preconditioning strategies available in the SCEPTRE radiation transport code are described. A flexible transport synthetic acceleration (TSA) algorithm that uses a low-order discrete-ordinates ($S_N$) or spherical-harmonics ($P_N$) solve to accelerate convergence of a high-order $S_N$ source-iteration (SI) solve is described. Convergence of the low-order solves can be further accelerated by applying off-the-shelf incomplete-factorization or algebraic-multigrid methods. Also available is an algorithm that uses a generalized minimum residual (GMRES) iterative method rather than SI for convergence, using a parallel sweep-based solver to build up a Krylov subspace. TSA has been applied as a preconditioner to accelerate the convergence of the GMRES iterations. The methods are applied to several problems involving electron transport and problems with artificial cross sections with large scattering ratios. These methods were compared and evaluated by considering material discontinuities and scattering anisotropy. Observed accelerations obtained are highly problem dependent, but speedup factors around 10 have been observed in typical applications.

ASYMPTOTIC ERROR ANALYSIS OF k-FOLD PSEUDO-NEWTON'S METHOD LOCATING A SIMPLE ZERO

  • Kim, Young Ik
    • Journal of the Chungcheong Mathematical Society
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    • v.21 no.4
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    • pp.483-492
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    • 2008
  • The k-fold pseudo-Newton's method is proposed and its convergence behavior is investigated near a simple zero. The order of convergence is proven to be at least k + 2. The asymptotic error constant is explicitly given in terms of k and the corresponding simple zero. High-precison numerical results are successfully implemented via Mathematica and illustrated for various examples.

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Variable Length Optimum Convergence Factor Algorithm for Adaptive Filters (적응 필터를 위한 가변 길이 최적 수렴 인자 알고리듬)

  • Boo, In-Hyoung;Kang, Chul-Ho
    • The Journal of the Acoustical Society of Korea
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    • v.13 no.4
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    • pp.77-85
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    • 1994
  • In this study an adaptive algorithm with optimum convergence factor for steepest descent method is proposed, which controls automatically the filter order to take the appropriate level. So far, fixed order filters have been used when adaptive filter is employed according to the priori knowledge or experience in various adaptive signal processing applications. But, it is so difficult to know the filter order needed in real implementations that high order filters have to be performed. As a result, redundant calculations are increased in the case of high order filters. The proposed variable length optimum convergence factor (VLOCF) algorithm takes the appropriated filter order within the given one so that the redundant calculation is decreased to get the enhancement of convergence speed and smaller convergence error during the steady state. The proposed algorithm is evaluated to prove the validity by computer simulation for system Identification.

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