Journal of Applied Mathematics

  • J. Appl. Math.
  • Volume 2014, Special Issue (2014), Article ID 737305, 7 pages.

A New Biparametric Family of Two-Point Optimal Fourth-Order Multiple-Root Finders

Young Ik Kim and Young Hee Geum

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Abstract

We construct a biparametric family of fourth-order iterative methods to compute multiple roots of nonlinear equations. This method is verified to be optimally convergent. Various nonlinear equations confirm our proposed method with order of convergence of four and show that the computed asymptotic error constant agrees with the theoretical one.

Article information

Source
J. Appl. Math., Volume 2014, Special Issue (2014), Article ID 737305, 7 pages.

Dates
First available in Project Euclid: 27 February 2015

Permanent link to this document
https://projecteuclid.org/euclid.jam/1425050560

Digital Object Identifier
doi:10.1155/2014/737305

Mathematical Reviews number (MathSciNet)
MR3263578

Citation

Kim, Young Ik; Geum, Young Hee. A New Biparametric Family of Two-Point Optimal Fourth-Order Multiple-Root Finders. J. Appl. Math. 2014, Special Issue (2014), Article ID 737305, 7 pages. doi:10.1155/2014/737305. https://projecteuclid.org/euclid.jam/1425050560


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References

  • L. B. Rall, “Convergence of the Newton process to multiple solutions,” Numerische Mathematik, vol. 9, pp. 23–37, 1966.
  • H. T. Kung and J. F. Traub, “Optimal order of one-point and multipoint iteration,” Journal of the Association for Computing Machinery, vol. 21, pp. 643–651, 1974.
  • P. Jarratt, “Some efficient fourth order multipoint methods for solving equations,” BIT, vol. 9, pp. 119–124, 1969.
  • C. Dong, “A family of multipoint iterative functions for finding multiple roots of equations,” International Journal of Computer Mathematics, vol. 21, pp. 363–367, 1987.
  • Y. H. Geum and Y. I. Kim, “Cubic convergence of parameter-controlled Newton-secant method for multiple zeros,” Journal of Computational and Applied Mathematics, vol. 233, no. 4, pp. 931–937, 2009.
  • Y. I. Kim and Y. H. Geum, “A two-parameter family of fourth-order iterative methods with optimal convergence for multiple zeros,” Journal of Applied Mathematics, vol. 2013, Article ID 369067, 7 pages, 2013.
  • A. N. Johnson and B. Neta, “High-order nonlinear solver for multiple roots,” Computers & Mathematics with Applications, vol. 55, no. 9, pp. 2012–2017, 2008.
  • V. Kanwar, S. Bhatia, and M. Kansal, “New optimal class of higher-order methods for multiple roots, permitting ${f}^{'}\left({x}_{n}\right)=0$,” Applied Mathematics and Computation, vol. 222, pp. 564–574, 2013.
  • X. Li, C. Mu, J. Ma, and L. Hou, “Fifth-order iterative method for finding multiple roots of nonlinear equations,” Numerical Algorithms, vol. 57, no. 3, pp. 389–398, 2011.
  • B. Neta, “Extension of Murakami's high-order non-linear solver to multiple roots,” International Journal of Computer Mathematics, vol. 87, no. 5, pp. 1023–1031, 2010.
  • M. Petkovic, L. Petkovic, and J. Dzunic, “Accelerating generators of iterative methods for finding multiple roots of nonlinear equations,” Computers & Mathematics with Applications, vol. 59, no. 8, pp. 2784–2793, 2010.
  • J. R. Sharma and R. Sharma, “Modified Jarratt method for computing multiple roots,” Applied Mathematics and Computation, vol. 217, no. 2, pp. 878–881, 2010.
  • J. R. Sharma and R. Sharma, “Modified Chebyshev-Halley type method and its variants for computing multiple roots,” Numerical Algorithms, vol. 61, no. 4, pp. 567–578, 2012.
  • L. Shengguo, L. Xiangke, and C. Lizhi, “A new fourth-order iterative method for finding multiple roots of nonlinear equations,” Applied Mathematics and Computation, vol. 215, no. 3, pp. 1288–1292, 2009.
  • S. G. Li, L. Z. Cheng, and B. Neta, “Some fourth-order nonlinear solvers with closed formulae for multiple roots,” Computers & Mathematics with Applications, vol. 59, no. 1, pp. 126–135, 2010.
  • X. Zhou, X. Chen, and Y. Song, “Constructing higher-order methods for obtaining the multiple roots of nonlinear equations,” Journal of Computational and Applied Mathematics, vol. 235, no. 14, pp. 4199–4206, 2011.
  • J. F. Traub, Iterative Methods for the Solution of Equations, Prentice-Hall, New Jersey, NJ, USA, 1964.
  • B. Neta, Numerical Methods for the Solution of Equations, Net-A-Sof, Monterey, Calif, USA, 1983.
  • S. Wolfram, The Mathematica Book, Cambridge University Press, Cambridge, UK, 4th edition, 1999. \endinput