• Title/Summary/Keyword: k-fold pseudo-secant-Newton's method

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ON THE ORDER AND RATE OF CONVERGENCE FOR PSEUDO-SECANT-NEWTON'S METHOD LOCATING A SIMPLE REAL ZERO

  • Kim, Young Ik
    • Journal of the Chungcheong Mathematical Society
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    • v.19 no.2
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    • pp.133-139
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    • 2006
  • By combining the classical Newton's method with the pseudo-secant method, pseudo-secant-Newton's method is constructed and its order and rate of convergence are investigated. Given a function $f:\mathbb{R}{\rightarrow}\mathbb{R}$ that has a simple real zero ${\alpha}$ and is sufficiently smooth in a small neighborhood of ${\alpha}$, the convergence behavior is analyzed near ${\alpha}$ for pseudo-secant-Newton's method. The order of convergence is shown to be cubic and the rate of convergence is proven to be $\(\frac{f^{{\prime}{\prime}}(\alpha)}{2f^{\prime}(\alpha)}\)^2$. Numerical experiments show the validity of the theory presented here and are confirmed via high-precision programming in Mathematica.

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