A STUDY ON CONVERGENCE OF EXTENDED LEAP-FROGGING NEWTON'S METHOD LOCATING MULTIPLE ZEROS

  • 투고 : 2009.03.23
  • 심사 : 2009.05.21
  • 발행 : 2009.06.30

초록

Assuming that a given nonlinear function f : $\mathbf{R}{\rightarrow}\mathbf{R}$ has a zero $\alpha$with integer multiplicity $m{\geq}1$ and is sufficiently smooth in a small neighborhood of $\alpha$, we define extended leap-frogging Newton's method. We investigate the order of convergence and the asymptotic error constant of the proposed method as a function of multiplicity m. Numerical experiments for various test functions show a satisfactory agreement with the theory presented in this paper and are throughly verified via Mathematica programming with its high-precision computability.

키워드

참고문헌

  1. R. G. Bartle, The Elements of Real Analysis, 2nd ed., John Wiley & Sons., New York, 1976.
  2. Ward Cheney and David Kincaid, Numerical Mathematics and Computing, Brooks/Cole Publishing Company, Monterey, California 1980
  3. S. D. Conte and Carl de Boor, Elementary Numerical Analysis, McGraw-Hill Inc., 1980
  4. Qiang Du, Ming Jin, T. Y. Li and Z. Zeng, The Quasi-Laguerre Iteration, Mathematics of Computation, 66 (1997), no. 217, 345-361. https://doi.org/10.1090/S0025-5718-97-00786-2
  5. Y. H. Geum, The asymptotic error constant of leap-frogging Newtons method locating a simple real zero, Applied Mathematics of Computation 189 (2007), no. 217, 963-969. https://doi.org/10.1016/j.amc.2006.11.150
  6. A. Bathi Kasturiarachi, Leap-frogging Newton's Method, INT. J. MATH. EDUC. SCI. TECHNOL. 33 (2002), no. 4, 521-527. https://doi.org/10.1080/00207390210131786
  7. L. D. Petkovic, M. S. Petkovic and D. Zivkovic, Hansen-Patrick's Family Is of Laguerre's Type, Novi Sad J. Math. 33 (2003), no. 1, 109-115.
  8. Kenneth A. Ross, Elementary Analysis, Springer-Verlag New York Inc., 1980.
  9. J. Stoer and R. Bulirsh, Introduction to Numerical Analysis, 244-313, Springer-Verlag New York Inc., 1980.
  10. J. F. Traub, Iterative Methods for the Solution of Equations, Chelsea Publishing Company, 1982.
  11. Stephen Wolfram, The Mathematica Book, 4th ed., Cambridge University Press, 1999.