• Journal of the Chungcheong Mathematical Society
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• v.28 no.4
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• pp.513-523
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• 2015

We present a local convergence analysis of some Newton-like methods of R-order at least three in order to approximate a solution of a nonlinear equation in a Banach space. Our sufficient convergence conditions involve only hypotheses on the first and second $Fr{\acute{e}}chet$-derivative of the operator involved. These conditions are weaker that the corresponding ones given by Hernandez, Romero [10] and others [1], [4]-[9] requiring hypotheses up to the third $Fr{\acute{e}}chet$ derivative. Numerical examples are also provided in this study.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.15 no.4
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• pp.267-275
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• 2011

This paper proposes a new quickly convergent pattern search quasi-Newton algorithm that employs the multi-step version of the Symmetric Rank One (SRI). The new algorithm works on the factorizations of the inverse Hessian approximations to make available a sequence of convergent positive bases required by the pattern search process. The algorithm, in principle, resembles that developed in [1] with multi-step methods dominating the dervation and with numerical improvements incurred, as shown by the numerical results presented herein.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.5 no.2
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• pp.101-115
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• 2001

Recent theoretical analysis of numerical methods for solving nonlinear systems of equations is represented by alpha theory of Newton method developed Smale et al. The theory was extended to Secant method by providing convergence conditions by Yakoubsohn which the Secant method is treated as an operator defined for analytical functions. We use Secant methods as an iterative scheme with approximations, which results in new convergence conditions. We compare the two conditions and show that the new conditions represent the features of Secant method in a more precise way.

• Journal of applied mathematics & informatics
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• v.7 no.1
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• pp.29-40
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• 2000

We provide semilocal convergence theorems for Newton-like methods in Banach space using outer and generalized inverses. In contrast to earlier results we use hypotheses on the second instead of the first Frechet-derivative. This way our Newton-Kantorovich hypotheses differ from earlier ones. Our results can be used to solve undetermined systems, nonlinear least square problems and ill-posed nonlinear operator equations.

• Journal of applied mathematics & informatics
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• v.27 no.1_2
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• pp.335-341
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• 2009

In this paper, a system of nonsmooth equations reformulated from a nonlinear complementarity problem with nonsmooth data is studied. The formulas of some subdifferentials for related functions in this system of nonsmooth equations are developed. The present work can be applied to Newton methods for solving this kind of nonlinear complementarity problem.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.6 no.1
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• pp.91-107
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• 2002

This paper presents two new self-scaling variable-metric algorithms. The first is based on a known two-parameter family of rank-two updating formulae, the second employs an initial scaling of the estimated inverse Hessian which modifies the first self-scaling algorithm. The algorithms are compared with similar published algorithms, notably those due to Oren, Shanno and Phua, Biggs and with BFGS (the best known quasi-Newton method). The best of these new and published algorithms are also modified to employ inexact line searches with marginal effect. The new algorithms are superior, especially as the problem dimension increases.

• Journal of applied mathematics & informatics
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• v.27 no.5_6
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• pp.1195-1209
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• 2009

In this paper, we point out some errors in a recent paper by Cordero and Torregrosa [7]. We prove the convergence of the variants of Newton's method for solving the system of nonlinear equations using two different approaches. Several examples are given, which illustrate the cubic convergence of these methods and verify the theoretical results.

• East Asian mathematical journal
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• v.29 no.5
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• pp.511-519
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• 2013

There are various methods for evaluating a matrix square root, which is a solvent of the quadratic matrix equation $X^2-A=0$. We consider new iterative methods for solving matrix square roots of M-matrices. Particulary we show that the relaxed binomial iteration is more efficient than Newton-Schulz iteration in some cases. And we construct a formula to find relaxation coefficients through statistical experiments.

• Journal of the Society of Naval Architects of Korea
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• v.28 no.1
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• pp.182-188
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• 1991

The modified arc-length algorithms for the automatic incremental solution of nonlinear finite element equations proposed by Riks are presented, which comprise the cylindrical arc-length method and the normal arc-length method. These methods are developed to trace the nonlinear path of large displacement problems such as a pre and post bucking/collapse response of general structures. These methods are applied to analyse the nonlinear behavior of arch and shell problems in parallel with the standard and modified Newton-Raphson method.

• Journal of Advanced Marine Engineering and Technology
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• v.24 no.1
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• pp.18-24
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• 2000

As ship's propulsion shafting system has been complicated, many linear methods that have been used until now are not sufficient enough to produce proper solutions and these solutions are ofter unreasonable. So we need to solve nonlinear systems, and many methods for solving nonlinear vibration system have been developed. In this study, the propulsion shafting system was modeled with Duffing's nonlinear vibration system and multi-degree-of-freedom, and analyzed by using Quasi-Newton method. And for the purpose of confirming the reliability of the calculating results for nonlinear forced torsional vibration of the propulsion shafting system, the nonlinear calculated results were compared with the linear calculated ones for ship's propulsion shafting system. In the result, for analysis of the forced torsional vibration of the propulsion systems with nonlinear elements, the modified Newton's method is confirmed reasonable.