• 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.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.14 no.2
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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.

• Journal of the Korean Mathematical Society
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• v.57 no.5
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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.

• Journal for History of Mathematics
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• v.15 no.3
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• pp.25-34
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• 2002

This paper explains methods of numerical analysis which appear on Cardano's Ars Magna and Newton's Principia. Cardano's method is secant method, but its actual al]plication is severely limited by technical difficulties. Newton's method is what nowadays called Newton-Raphson's method. But mysteriously, Newton's explanation had been forgotten for two hundred years, until Adams rediscovered it. Newton had even explained finding the root using the second degree Taylor's polynomial, which shows Newton's greatness.

• Journal of the Korean Mathematical Society
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• v.50 no.4
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• pp.755-770
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• 2013

One of the interesting nonlinear matrix equations is the quadratic matrix equation defined by $$Q(X)=AX^2+BX+C=0$$, where X is a $n{\times}n$ unknown real matrix, and A, B and C are $n{\times}n$ given matrices with real elements. Another one is the matrix polynomial $$P(X)=A_0X^m+A_1X^{m-1}+{\cdots}+A_m=0,\;X,\;A_i{\in}\mathbb{R}^{n{\times}n}$$. Newton's method is used to find the symmetric and bisymmetric solvents of the nonlinear matrix equations Q(X) and P(X). The method does not depend on the singularity of the Fr$\acute{e}$chet derivative. Finally, we give some numerical examples.

• Journal of the Chungcheong Mathematical Society
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• v.26 no.2
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• pp.275-285
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• 2013

In this paper a nonlinear matrix equation is considered which has the form $$P(X)=A_0X^m+A_1X^{m-1}+{\cdots}+A_{m-1}X+A_m=0$$ where X is an $n{\times}n$ unknown real matrix and $A_m$, $A_{m-1}$, ${\cdots}$, $A_0$ are $n{\times}n$ matrices with real elements. Newtons method is applied to find the skew-symmetric solvent of the matrix polynomial P(X). We also suggest an algorithm which converges the skew-symmetric solvent even if the Fr$\acute{e}$echet derivative of P(X) is singular.

• 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.

• Journal of the Korean Society of Surveying, Geodesy, Photogrammetry and Cartography
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• v.17 no.3
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• pp.257-264
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• 1999

The lens distortions were corrected directly using the high-order polynomial which was offered in camera calibration data for the forward transformation and the root of Newton-Raphson's $2\times{2}$ nonlinear system for the backward transformation. The 0.04~0.08 pixels increase in accuracy was indicated through the use of direct correction of lens distortions instead of least square methods of commercial software. The least square adjustment method of high-order polynomial requires many control points which has a same weight. But this suggested method which is unnecessary to determine control points was developed and applied. The algorithm showed improved efficacy.

• Journal of Information Technology Services
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• v.19 no.1
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• pp.145-158
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• 2020

With the development and widespread application of online shopping, the number of online consumers has increased. With one click of a mouse, people can buy anything they want without going out and have it sent right to the doors. As consumers benefit from online shopping, people are becoming more concerned about protecting their privacy. In the group buying scenario described in our paper, online shopping was regarded as intra-group communication. To protect the sensitive information of consumers, the polynomial-based encryption key sharing method (Piao et al., 2013; Piao and Kim, 2018) can be applied to online shopping communication. In this paper, we analyze security problems by using a polynomial-based scheme in the following ways : First, in Kamal's attack, they said it does not provide perfect forward and backward secrecy when the members leave or join the group because the secret key can be broken in polynomial time. Second, for simultaneous equations, the leaving node will compute the new secret key if it can be confirmed that the updated new polynomial is recomputed. Third, using Newton's method, attackers can successively find better approximations to the roots of a function. Fourth, the Berlekamp Algorithm can factor polynomials over finite fields and solve the root of the polynomial. Fifth, for a brute-force attack, if the key size is small, brute force can be used to find the root of the polynomial, we need to make a key with appropriately large size to prevent brute force attacks. According to these analyses, we finally recommend the use of a relatively reasonable hash-based mechanism that solves all of the possible security problems and is the most suitable mechanism for our application. The study of adequate and suitable protective methods of consumer security will have academic significance and provide the practical implications.

• Journal of applied mathematics & informatics
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• v.10 no.1_2
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• pp.63-73
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• 2002

The governing equation locating component centers in the degree-n bifurcation set is a polynomial with a very high degree and its root-finding lacks numerical accuracy. The equation is transformed to have its degree reduced by a factor(n-1). Newton's method applied to the transformed equation improves the accuracy with properly chosen initial values. The numerical implementation is done with Maple V using a large number of computational precision digits. Many cases are studied for 2 $\leq$ n $\leq$ 25 and show a remarkably improved computation.