• Title/Summary/Keyword: higher order of convergence

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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.

AN EXTRAPOLATED HIGHER ORDER CHARACTERISTIC FINITE ELEMENT METHOD FOR NONLINEAR SOBOLEV EQUATIONS

  • Ohm, Mi Ray;Shin, Jun Yong
    • East Asian mathematical journal
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    • v.34 no.5
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    • pp.601-614
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    • 2018
  • In this paper, we introduce an extrapolated higher order characteristic finite element method to approximate solutions of nonlinear Sobolev equations with a convection term and we establish the higher order of convergence in the temporal and the spatial directions with respect to $L^2$ norm.

On Weak Convergence of Some Rescaled Transition Probabilities of a Higher Order Stationary Markov Chain

  • Yun, Seok-Hoon
    • Journal of the Korean Statistical Society
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    • v.25 no.3
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    • pp.313-336
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    • 1996
  • In this paper we consider weak convergence of some rescaled transi-tion probabilities of a real-valued, k-th order (k $\geq$ 1) stationary Markov chain. Under the assumption that the joint distribution of K + 1 consecutive variables belongs to the domain of attraction of a multivariate extreme value distribution, the paper gives a sufficient condition for the weak convergence and characterizes the limiting distribution via the multivariate extreme value distribution.

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AN EXTRAPOLATED HIGHER ORDER CHARACTERISTIC FINITE ELEMENT METHOD FOR SOBOLEV EQUATIONS

  • Ohm, Mi Ray;Shin, Jun Yong
    • East Asian mathematical journal
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    • v.33 no.5
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    • pp.511-525
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    • 2017
  • We introduce an extrapolated higher order characteristic finite element method to construct approximate solutions of a Sobolev equation with a convection term. The higher order of convergence in both the temporal direction and the spatial direction in $L^2$ normed space is established and some computational results to support our theoretical results are presented.

LOW REGULARITY SOLUTIONS TO HIGHER-ORDER HARTREE-FOCK EQUATIONS WITH UNIFORM BOUNDS

  • Changhun Yang
    • Journal of the Chungcheong Mathematical Society
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    • v.37 no.1
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    • pp.27-40
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    • 2024
  • In this paper, we consider the higher-order HartreeFock equations. The higher-order linear Schrödinger equation was introduced in [5] as the formal finite Taylor expansion of the pseudorelativistic linear Schrödinger equation. In [13], the authors established global-in-time Strichartz estimates for the linear higher-order equations which hold uniformly in the speed of light c ≥ 1 and as their applications they proved the convergence of higher-order Hartree-Fock equations to the corresponding pseudo-relativistic equation on arbitrary time interval as c goes to infinity when the Taylor expansion order is odd. To achieve this, they not only showed the existence of solutions in L2 space but also proved that the solutions stay bounded uniformly in c. We address the remaining question on the convergence of higherorder Hartree-Fock equations when the Taylor expansion order is even. The distinguished feature from the odd case is that the group velocity of phase function would be vanishing when the size of frequency is comparable to c. Owing to this property, the kinetic energy of solutions is not coercive and only weaker Strichartz estimates compared to the odd case were obtained in [13]. Thus, we only manage to establish the existence of local solutions in Hs space for s > $\frac{1}{3}$ on a finite time interval [-T, T], however, the time interval does not depend on c and the solutions are bounded uniformly in c. In addition, we provide the convergence result of higher-order Hartree-Fock equations to the pseudo-relativistic equation with the same convergence rate as the odd case, which holds on [-T, T].

GENERAL CONVERGENCE ANALYSIS OF THE LVCMS ALGORITHM (LVCMS 알고리즘에 대한 일반적인 수렴 특성 분석)

  • Nam, Seung-Hyon;Kim, Yong-Hoh
    • The Journal of Natural Sciences
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    • v.8 no.2
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    • pp.63-67
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    • 1996
  • Adaptive algorithms based on the higher order error criterion such as the LVCMS and the LMF show performance degradation if input signal contains additive noise with a heavier-tailed density. Conventional analysis often neglects higher order terms in the recursion and my not suit for prediction exact behavior of these higher order algorithms. This paper presents a new convergence analysis which contains all the higher order term in the recursion. The analysis shows that the higher order terms, which are often neglected, dose not affect the upper bound on the step size but the misadjustment. However, the effect decreases sharply proportional to the square of the step size.

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A HIGHER ORDER ITERATIVE ALGORITHM FOR MULTIVARIATE OPTIMIZATION PROBLEM

  • Chakraborty, Suvra Kanti;Panda, Geetanjali
    • Journal of applied mathematics & informatics
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    • v.32 no.5_6
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    • pp.747-760
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    • 2014
  • In this paper a higher order iterative algorithm is developed for an unconstrained multivariate optimization problem. Taylor expansion of matrix valued function is used to prove the cubic order convergence of the proposed algorithm. The methodology is supported with numerical and graphical illustration.

AN EXTRAPOLATED CRANK-NICOLSON CHARACTERISTIC FINITE ELEMENT METHOD FOR NONLINEAR SOBOLEV EQUATIONS

  • OHM, MI RAY;SHIN, JUN YONG
    • Journal of applied mathematics & informatics
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    • v.36 no.3_4
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    • pp.257-270
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    • 2018
  • An extrapolated Crank-Nicolson characteristic finite element method is introduced for approximate solutions of nonlinear Sobolev equations with a convection term. And we obtain the higher order of convergence for approximate solutions in the temporal and the spatial directions with respect to $L^2$ norm.

AN EXTRAPOLATED CRANK-NICOLSON CHARACTERISTIC FINITE ELEMENT METHOD FOR SOBOLEV EQUATIONS

  • Ohm, Mi Ray;Shin, Jun Yong
    • Bulletin of the Korean Mathematical Society
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    • v.54 no.4
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    • pp.1409-1419
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    • 2017
  • We introduce an extrapolated Crank-Nicolson characteristic finite element method to approximate solutions of a convection dominated Sobolev equation. We obtain the higher order of convergence in both the spatial direction and the temporal direction in $L^2$ normed space for the extrapolated Crank-Nicolson characteristic finite element method.