• 충청수학회지
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• 제21권4호
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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.

• East Asian mathematical journal
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• 제24권3호
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• pp.295-304
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• 2008

We approximate a locally unique solution of a nonlinear operator equation containing a non-differentiable operator in a Banach space setting using Newton's method. Sufficient conditions for the semilocal convergence of Newton's method in this case have been given by several authors using mainly increasing sequences [1]-[6]. Here, we use center as well as Lipschitz conditions and decreasing majorizing sequences to obtain new sufficient convergence conditions weaker than before in many interesting cases. Numerical examples where our results apply to solve equations but earlier ones cannot [2], [5], [6] are also provided in this study.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제15권2호
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• pp.111-120
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• 2008

A semi local convergence analysis is provided for Newton's method in a Banach space setting. The operators involved are only locally Holderian. We make use of a point-based approximation and center-Holderian hypotheses. This approach can be used to approximate solutions of equations involving nonsmooth operators.

• 호남수학학술지
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• 제30권2호
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• pp.399-409
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• 2008

We consider the most generalized quadratic matrix equation, Q(X) = $A_7XA_6XA_5+A_4XA_3+A_2XA_1+A_0=0$, where X is m ${\times}$ n, $A_7$, $A_4$ and $A_2$ are p ${\times}$ m, $A_6$ is n ${\times}$ m, $A_5$, $A_3$ and $A_l$ are n ${\times}$ q and $A_0$ is p ${\times}$ q matrices with complex elements. The convergence of Newton's method for solving some different types of quadratic matrix equations are considered and we show that the elementwise minimal positive solvents can be found by Newton's method with the zero starting matrices. We finally give numerical results.

• East Asian mathematical journal
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• 제32권1호
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• pp.13-25
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• 2016

We consider the nonlinear matrix equation $X^p+AX^qB+CXD+E=0$, where p and q are positive integers, A, B and E are $n{\times}n$ nonnegative matrices, C and D are arbitrary $n{\times}n$ real matrices. A sufficient condition for the existence of the elementwise minimal nonnegative solution is derived. The monotone convergence of Newton's method for solving the equation is considered. Several numerical examples to show the efficiency of the proposed Newton's method are presented.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• 제14권2호
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• pp.113-124
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• 2010

We consider matrix polynomial which has the form $P_1(X)=A_oX^m+A_1X^{m-1}+...+A_m=0$ where X and $A_i$ are $n{\times}n$ matrices with real elements. In this paper, we propose an iterative method for the symmetric and generalized centro-symmetric solution to the Newton step for solving the equation $P_1(X)$. Then we show that a symmetric and generalized centro-symmetric solvent of the matrix polynomial can be obtained by our Newton's method. Finally, we give some numerical experiments that confirm the theoretical results.

• 전기학회논문지
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• 제66권7호
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• pp.1053-1058
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• 2017

In this paper, the design of iron loss minimization of 600W was performed by using Quasi-Newton method. Stator shoe, the width of stator teeth and yoke, and the length of d-axis flux path were selected as design parameters, and the output characteristics according to each design variable were considered. The objective function was set to minimize iron loss. Using the Quasi-Newton method, the variables converged to the target value while changing simultaneously and multiple times. As the algorithm advanced optimization, the correlation with the behavior of each variable was compared and analyzed.

• 호남수학학술지
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• 제35권3호
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• pp.417-433
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• 2013

We consider the iterative solution of a quadratic matrix equation with special coefficient matrices which arises in the quasibirth and death problem. In this paper, we show that the elementwise minimal positive solvent of the quadratic matrix equations can be obtained using Newton's method if there exists a positive solvent and the convergence rate of the Newton iteration is quadratic if the Fr$\acute{e}$chet derivative at the elementwise minimal positive solvent is nonsingular. Although the Fr$\acute{e}$chet derivative is singular, the convergence rate is at least linear. Numerical experiments of the convergence rate are given.

• 대한수학회지
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• 제57권5호
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• pp.1119-1133
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• 2020

In this paper, we derive an upper bound for the distance from a point in the immediate basin of a root of a complex polynomial to the root itself. We establish that if z is a point in the immediate basin of a root α of a polynomial p of degree d ≥ 12, then ${\mid}z-{\alpha}{\mid}{\leq}{\frac{3}{\sqrt{d}}\(6{\sqrt{310}}/35\)^d{\mid}N_p(z)-z{\mid}$, where N_p is the Newton map induced by p. This bound leads to a new bound of the expected total number of iterations of Newton's method required to reach all roots of every polynomial p within a given precision, where p is normalized so that its roots are in the unit disk.

• 한국수학교육학회지시리즈E:수학교육논문집
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• 제23권4호
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• pp.1079-1092
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• 2009

본 연구는 무한급수와 멱급수의 발생 배경과 발달 과정의 인식론적 토대가 되었던 뉴턴의 이항정리(binomial theorem)의 개념을 살펴보고, 그 발달 과정에서 얻어진 제곱근의 근삿값 구하는 방법, 뉴턴의 역유율법을 이용한 정적분 구하는 방법, 그리고 메르카토어 급수와 그레고리 급수의 발견 과정을 알아보고자 한다. 이 과정을 통하여 뉴턴의 이항정리가 가지는 수학사의 교수법적 논의를 제시하고자 한다.