• The Pure and Applied Mathematics
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• v.16 no.4
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• pp.405-416
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• 2009

In this study we are concerned with the problem of approximating a locally unique solution of an equation in a Banach space setting using Newton's and modified Newton's methods. We provide weaker convergence conditions for both methods than before [5]-[7]. Then, we combine Newton's with the modified Newton's method to approximate locally unique solutions of operator equations. Finer error estimates, a larger convergence domain, and a more precise information on the location of the solution are obtained under the same or weaker hypotheses than before [5]-[7]. The results obtained here improve our earlier ones reported in [4]. Numerical examples are also provided.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.12 no.2
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• pp.55-68
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• 2008

One of well known and much studied nonlinear matrix equations is the matrix polynomial which has the form $P(X)=A_0X^m+A_1X^{m-1}+{\cdots}+A_m$, where $A_0$, $A_1$, ${\cdots}$, $A_m$ and X are $n{\times}n$ complex matrices. Newton's method was introduced a useful tool for solving the equation P(X)=0. Here, we suggest an improved approach to solve each Newton step and consider how to incorporate line searches into Newton's method for solving the matrix polynomial. Finally, we give some numerical experiment to show that line searches reduce the number of iterations for convergence.

• Communications of the Korean Mathematical Society
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• v.9 no.2
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• pp.491-502
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• 1994

Many problems arising in science and engineering require the numerical solution of a system of n nonlinear equations in n unknowns: (1) given F : $R^{n}$ $\rightarrow$ $R^{n}$ , find $x_{*}$ $\epsilon$ $R^{n}$ / such that F($x_{*}$) = 0. Nonlinear problems are generally solved by iteration. Davidson [3] and Broyden [1] introduced the methods which had led to a large amount of research and a class of algorithm. This work has been called by the quasi-Newton methods, secant updates, or modification methods. Newton's method is the classical method for the problem (1) and quasi-Newton methods have been proposed to circumvent computational disadvantages of Newton's method.(omitted)

• Journal of the Korean Mathematical Society
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• v.51 no.2
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• pp.251-266
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• 2014

In this paper, we use Newton's method to approximate a locally unique solution of an equation in Banach spaces and introduce recurrent functions to provide a weaker semilocal convergence analysis for Newton's method than before [1]-[13], in some interesting cases, provided that the Fr$\acute{e}$chet-derivative of the operator involved is p-H$\ddot{o}$lder continuous (p${\in}$(0, 1]). Numerical examples involving two boundary value problems are also provided.

• Journal of the Chungcheong Mathematical Society
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• v.25 no.2
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• pp.267-275
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• 2012

We present new results for the local convergence of Newton's method to a unique solution of an equation in a Banach space setting. Under a flexible gamma-type condition [12], [13], we extend the applicability of Newton's method by enlarging the radius and decreasing the ratio of convergence. The results can compare favorably to other ones using Newton-Kantorovich and Lipschitz conditions [3]-[7], [9]-[13]. Numerical examples are also provided.

• Journal of the Korean Mathematical Society
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• v.56 no.2
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• pp.311-327
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• 2019

The aim of this paper is to extend the applicability of Gauss-Newton method for solving underdetermined nonlinear least squares problems in cases not covered before. The novelty of the paper is the introduction of a restricted convergence domain. We find a more precise location where the Gauss-Newton iterates lie than in earlier studies. Consequently the Lipschitz constants are at least as small as the ones used before. This way and under the same computational cost, we extend the local as well the semilocal convergence of Gauss-Newton method. The new developmentes are obtained under the same computational cost as in earlier studies, since the new Lipschitz constants are special cases of the constants used before. Numerical examples further justify the theoretical results.

• East Asian mathematical journal
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• v.24 no.3
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• pp.289-293
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• 2008

We recently showed in [5] a semilocal convergence theorem that guarantees convergence of Newton's method to a locally unique solution of a nonlinear equation under hypotheses weaker than those of the Newton-Kantorovich theorem [7]. Here, we first weaken Miranda's theorem [1], [9], [10], which is a generalization of the intermediate value theorem. Then, we show that operators satisfying the weakened Newton-Kantorovich conditions satisfy those of the weakened Miranda’s theorem.

• Journal of applied mathematics & informatics
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• v.23 no.1_2
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• pp.361-367
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• 2007

In this paper we propose an implementable method for solving a nonsmooth convex optimization problem by combining Moreau-Yosida regularization, bundle and quasi-Newton ideas. The method we propose makes use of approximate subgradients of the objective function, which makes the method easier to implement. We also prove the convergence of the proposed method under some additional assumptions.

• Proceedings of the Korean Society of Tribologists and Lubrication Engineers Conference
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• 1999.11a
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• pp.323-330
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• 1999

The conventional analysis for the numerical computation of fluid film thickness with elastic deformation of contact region. is performed by Newton-Rephson method for its 18st convergence characteristics. However, both high load and relatively low sliding velocity frequently make it impossible for Newton-Rahpson method to get both converged and stable solutions. In particular, this method cannot provide converged Solution under the condition of high load above 1.0 GPa which frequently occurs in line contact of EHL problem. Multigird multi-level method for the solver of non-linear partial differential equation including solid deformation is preferred to Newton-Rshpson method for better convergence and stability and is applied to line contact EHL behavior in this study.

• Journal of applied mathematics & informatics
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• v.7 no.3
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• pp.765-772
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• 2000

In this study we are concerned with the problem of approximating a solution of a nonlinear equation in Banach space using Newton-like methods. Due to rounding errors the sequence of iterates generated on a computer differs from the sequence produced in theory. Using Lipschitz-type hypotheses on the second Frechet-derivative instead of the first one, we provide sufficient convergence conditions for the inexact Newton-like method that is actually generated on the computer. Moreover, we show that the ratio of convergence improves under our conditions. Furthermore, we provide a wider choice of initial guesses than before. Finally, a numerical example is provided to show that our results compare favorably with earlier ones.