Cubic convergence of parameter-controlled Newton-secant method for multiple zeros

  • Authors:

  • Young Hee Geum;Young Ik Kim

  • Affiliations:

  • Faculty of Science, University of Malaya, 50603 Kuala Lumpur, Malaysia;Department of Applied Mathematics, Dankook University, Cheonan, 330-714, Republic of Korea

  • Venue:
  • Journal of Computational and Applied Mathematics
  • Year:
  • 2009

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Abstract

Let f:C-C have a multiple zero @a with integer multiplicity m=1 and be analytic in a sufficiently small neighborhood of @a. For parameter-controlled Newton-secant method defined by x"n"+"1=x"n-@lf(x"n)^2f^'(x"n)@?{f(x"n)-f(x"n-@mf(x"n)/f^'(x"n))},n=0,1,2,..., we investigate the maximal order of convergence and the theoretical asymptotic error constant by seeking the relationship between parameters @l and @m. For various test functions, the numerical method has shown a satisfactory result with high-precision Mathematica programming.