충청수학회지 (Journal of the Chungcheong Mathematical Society)

  • 제23권4호
  • /
  • Pages.845-852
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  • 2010
  • /
  • 1226-3524(pISSN)
  • /
  • 2383-6245(eISSN)
충청수학회 (The Chungcheong Mathematical Society)
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A THIRD-ORDER VARIANT OF NEWTON-SECANT METHOD FINDING A MULTIPLE ZERO

  • Kim, Young Ik (Department of Applied Mathematics Dankook University) ;
  • Lee, Sang Deok (Department of Applied Mathematics Dankook University)
  • 투고 : 2010.10.20
  • 심사 : 2010.11.09
  • 발행 : 2010.12.30
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초록

A nonlinear algebraic equation f(x) = 0 is considered to find a root with integer multiplicity $m{\geq}1$. A variant of Newton-secant method for a multiple root is proposed below: for n = 0, 1, $2{\cdots}$ $$x_{n+1}=x_n-\frac{f(x_n)^2}{f^{\prime}(x_n)\{f(x_n)-{\lambda}f(x_n-\frac{f(x_n)}{f^{\prime}(x_n)})\}$$, $$\lambda=\{_{1,\;if\;m=1.}^{(\frac{m}{m-1})^{m-1},\;if\;m{\geq}2$$ It is shown that the method has third-order convergence and its asymptotic error constant is expressed in terms of m. Numerical examples successfully verified the proposed scheme with high-precision Mathematica programming.

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참고문헌

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