Journal of the Chungcheong Mathematical Society (충청수학회지)

The Chungcheong Mathematical Society (충청수학회)
권호 표지
DOI QR코드

DOI QR Code

ON CONSTRUCTING A HIGHER-ORDER EXTENSION OF DOUBLE NEWTON'S METHOD USING A SIMPLE BIVARIATE POLYNOMIAL WEIGHT FUNCTION

  • Received : 2015.07.20
  • Accepted : 2015.08.05
  • Published : 2015.08.15
Download PDF
⟨ Previous Next ⟩

Abstract

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.

Keywords

References

  1. C. Chun, Some fourth-order iterative methods for solving nonlinear equation, Appl. Math. Comput. 195 (2008), 454-459. https://doi.org/10.1016/j.amc.2007.04.105
  2. S. D. Conte and C. de Boor, Elementary Numerical Analysis, McGraw-Hill Inc., 1980.
  3. M. Frontini and E. Sormani, Some variant of Newtons method with third-order convergence, Appl. Math. Comput. 140 (2003), 419-426. https://doi.org/10.1016/S0096-3003(02)00238-2
  4. R. R. Goldberg, Methods of Real Analysis, 2nd ed., John Wiley and Sons Ltd., 1976.
  5. P. Jarratt, Some fourth-order multipoint iterative methods for solving equations, Math. Comput. 20 (1966), no. 95, 434-437. https://doi.org/10.1090/S0025-5718-66-99924-8
  6. A. B. Kasturiarachi, Leap-frogging Newtons method, Int. J. Math. Educ. Sci. Technol. 37 (2002), 521-527.
  7. R. King, A family of fourth-order methods for nonlinear equations, SIAM J. Numer. Anal. 10 (1973), no. 5, 876-879. https://doi.org/10.1137/0710072
  8. H. T. Kung and J. F. Traub, Optimal order of one-point and multipoint iteration, J. of the Association for Computing Machinery 21 (1974), 643-651. https://doi.org/10.1145/321850.321860
  9. A. Ostrowski, Solution of equations and systems of equations, Academic press, New York, 1960.
  10. A. Y. Ozban Some New Variants of Newtons Method with Accelerated Third-order Convergence, Appl. Math. Lett. 17 (2004), 677-682. https://doi.org/10.1016/S0893-9659(04)90104-8
  11. J. F. Traub,Iterative Methods for the Solution of Equations, Chelsea Publishing Company, 1982.
  12. S. Wolfram, The Mathematica Book, 5th ed., Wolfram Media, 2003.