On Constructing a Cubically Convergent Numerical Method for Multiple Roots

Authors: Young Hee Geum

Abstract:

We propose the numerical method defined by

xn+1 = xn − λ[f(xn − μh(xn))/]f'(xn) , n ∈ N,

and determine the control parameter λ and μ to converge cubically. In addition, we derive the asymptotic error constant. Applying this proposed scheme to various test functions, numerical results show a good agreement with the theory analyzed in this paper and are proven using Mathematica with its high-precision computability.

Keywords: Asymptotic error constant, iterative method , multiple root, root-finding.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1336548

References:

[1] R. G. Bartle, The Elements of Real Analysis, 2nd ed., JohnWiley & 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, Vol. 66, No. 217(1997), pp.345-361.
[5] Y. H. Geum, The asymptotic error constant of leap-frogging Newtons method locating a simple real zero, Mathematics of Computation, Vol. 66, No. 217(1997), pp.345-361.
[6] A. Bathi Kasturiarachi, Leap-frogging Newton’s Method, INT. J. MATH. EDUC. SCI. TECHNOL., Vol. 33, No. 4(2002), pp.521-527.
[7] L. D. Petkovic, M. S. Petkovic and D. Zivkovic, Hansen-Patrick’s Family Is of Laguerre’s Type, Novi Sad J. Math., Vol. 33, No. 1(2003), pp.109-115.
[8] Kenneth A. Ross, Elementary Analysis, Springer-Verlag New York Inc., 1980.
[9] J. Stoer and R. Bulirsh, Introduction to Numerical Analysis, pp.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.