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Some new bi-accelerator two-point methods for solving nonlinear equations

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Abstract

In this work, we extract some new and efficient two-point methods with memory from their corresponding optimal methods without memory, to estimate simple roots of a given nonlinear equation. Applying two accelerator parameters in each iteration, we try to increase the convergence order from four to seven without any new functional evaluation. To this end, firstly we modify three optimal methods without memory in such a way that we could generate methods with memory as efficient as possible. Then, convergence analysis is put forward. Finally, the applicability of the developed methods on some numerical examples is examined and illustrated by means of dynamical tools, both in smooth and in nonsmooth functions.

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References

  • Babajee DKR (2012) Several improvements of the 2-point third order midpoint iterative method using weight functions. Appl Math Comput 218:7958–7966

    Article  MathSciNet  MATH  Google Scholar 

  • Chicharro FI, Cordero A, Torregrosa JR (2013) Drawing dynamical and parameters planes of iterative families and methods. Sci World J. Article ID 780153, 11 pp

  • Chun C, Lee MY (2013) A new optimal eighth-order family of iterative methods for the solution of nonlinear equations. Appl Math Comput 223:506–519

    Article  MathSciNet  MATH  Google Scholar 

  • Cordero A, Hueso JL, Martínez E, Torregrosa JR (2010) New modifications of Potra–Ptàk’s method with optimal fourth and eighth orders of convergence. J Comput Appl Math 234:2969–2976

    Article  MathSciNet  MATH  Google Scholar 

  • Cordero A, Lotfi T, Bakhtiari P, Torregrosa JR (2014) An efficient two-parametric family with memory for nonlinear equations. Numer Algor. doi:10.1007/s11075-014-9846-8

  • Geum YH, Kim YI (2011) A uniparametric family of three-step eighth-order multipoint iterative methods for simple roots. Appl Math Lett 24:929–935

    Article  MathSciNet  MATH  Google Scholar 

  • Heydari M, Hosseini SH, Loghmani GB (2011) On two new families of iterative methods for solving nonlinear equations with optimal order. Appl Anal Discret Math 5:93–109

    Article  MathSciNet  MATH  Google Scholar 

  • Jay IO (2001) A note on Q-order of convergence. BIT Numer Math 41:422–429

    Article  MathSciNet  MATH  Google Scholar 

  • Khattri SK, Steihaug T (2013) Algorithm for forming derivative-free optimal methods. Numer Algor. doi:10.1007/s11075-013-9715-x

    MathSciNet  MATH  Google Scholar 

  • Kou J, Wang X, Li Y (2010) Some eighth-order root-finding three-step methods. Commun Nonlinear Sci Numer Simul 15:536–544

    Article  MathSciNet  MATH  Google Scholar 

  • Kung HT, Traub JF (1974) Optimal order of one-point and multipoint iteration. J Assoc Comput Math 21:634–651

  • Liu X, Wang X (2012) A convergence improvement factor and higher-order methods for solving nonlinear equations. Appl Math Comput 218:7871–7875

    Article  MathSciNet  MATH  Google Scholar 

  • Lotfi T, Tavakoli E (2014) On a new efficient Steffensen-like iterative class by applying a suitable self-accelerator parameter. Sci World J. Article ID 769758, 9 pp. doi:10.1155/2014/769758

  • Lotfi T, Soleymani F, Shateyi S, Assari P, Khaksar Haghani F (2014a) New mono- and biaccelerator iterative methods with memory for nonlinear equations. Abstr Appl Anal. Article ID 705674, 8 pp. doi:10.1155/2014/705674

  • Lotfi T, Soleymani F, Noori Z, Kiliman A, Khaksar Haghani F (2014b) Efficient iterative methods with and without memory possessing high efficiency indices. Discret Dyn Nat Soc. Article ID 912796, 9 pp. doi:10.1155/2014/912796

  • Magreñan AA (2014) A new tool to study real dynamics: the convergence plane. arXiv:1310.3986 [math.NA]

  • Ortega JM, Rheimbolt WC (1970) Iterative solution of nonlinear equations in several variables. Academic Press, New York

    Google Scholar 

  • Ostrowski AM (1966) Solutions of equations and systems of equations. Academic Press, New York-London

    MATH  Google Scholar 

  • Petković MS, Ilić S, Džunić J (2010) Derivative free two-point methods with and without memory for solving nonlinear equations. Appl Math Comput 217(5):1887–1895

    Article  MathSciNet  MATH  Google Scholar 

  • Petković MS, Neta B, Petković LD, Džunić J (2014) Multipoint methods for solving nonlinear equations: a survey. Appl Math Comput 226(2):635–660

  • Ren H, Wu Q, Bi W (2009) A class of two-step Steffensen type methods with fourth-order convergence. Appl Math Comput 209:206–210

    Article  MathSciNet  MATH  Google Scholar 

  • Soleymani F, Sharifi M, Mousavi S (2012) An improvement of Ostrowski’s and King’s techniques with optimal convergence order eight. J Optim Theory Appl 153:225–236

    Article  MathSciNet  MATH  Google Scholar 

  • Soleimani F, Soleymani F, Shateyi S (2013) Some iterative methods free from derivatives and their basins of attraction for nonlinear equations. Discret Dyn Nat Soc. Article ID 301718, 10 pp

  • Thukral R (2011) Eighth-order iterative methods without derivatives for solving nonlinear equation. ISRN Appl Math. Article ID 693787, 12 pp

  • Traub JF (1964) Iterative methods for the solution of equations. Prentice Hall, New York

    MATH  Google Scholar 

  • Zheng Q, Li J, Huang F (2011) An optimal Steffensen-type family for solving nonlinear equations. Appl Math Comput 217:9592–9597

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

The authors thank to the anonymous referees for their suggestions to improve the final version of the paper. The second author would like to thank Hamedan Brach of Islamic Azad University for partial financial support in this research.

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Authors and Affiliations

  1. Instituto de Matemática Multidisciplinar, Universitat Politècnica de València, Valencia, Spain

    Alicia Cordero & Juan R. Torregrosa

  2. Department of Mathematics, Hamedan Branch, Islamic Azad University, Hamedan, Iran

    Taher Lotfi, Paria Assari & Katayoun Mahdiani

Authors
  1. Alicia Cordero
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  2. Taher Lotfi
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  3. Juan R. Torregrosa
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  4. Paria Assari
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  5. Katayoun Mahdiani
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Corresponding author

Correspondence to Taher Lotfi.

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Communicated by Pascal Frey.

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Cordero, A., Lotfi, T., Torregrosa, J.R. et al. Some new bi-accelerator two-point methods for solving nonlinear equations. Comp. Appl. Math. 35, 251–267 (2016). https://doi.org/10.1007/s40314-014-0192-1

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  • DOI: https://doi.org/10.1007/s40314-014-0192-1

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