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Efficient derivative-free numerical methods for solving systems of nonlinear equations

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Abstract

We present iterative methods of convergence order four and six for solving systems of nonlinear equations. The algorithms are free from any derivative evaluations per full iteration. First-order divided difference operator for functions of several variables and direct computation by Taylor’s expansion are used to prove the local convergence order. Computational efficiency is discussed and the comparison between efficiencies of proposed techniques with existing derivative-free techniques is performed. It is shown that the new methods are especially efficient in solving large systems. Numerical tests are performed on some problems of different nature, which confirm robust and efficient convergence behavior of the proposed methods. Moreover, theoretical results are also verified in the examples.

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References

  • Cordero A, Torregrosa JR (2006) Variants of Newton’s method for functions of several variables. Appl Math Comput 183:199–208

    Article  MathSciNet  MATH  Google Scholar 

  • Cordero A, Torregrosa JR (2007) Variants of Newton’s method using fifth-order quadrature formulas. Appl Math Comput 190:686–698

    Article  MathSciNet  MATH  Google Scholar 

  • Cordero A, Martínez E, Torregrosa JR (2009) Iterative methods of order four and five for systems of nonlinear equations. J Comput Appl Math 231:541–551

    Article  MathSciNet  MATH  Google Scholar 

  • Cordero A, Hueso JL, Martínez E, Torregrosa JR (2010) A modified Newton–Jarratt’s composition. Numer Algorithms 55:87–99

    Article  MathSciNet  MATH  Google Scholar 

  • Cordero A, Hueso JL, Martínez E, Torregrosa JR (2012) Increasing the convergence order of an iterative method for nonlinear systems. Appl Math Lett 25:2369–2374

    Article  MathSciNet  MATH  Google Scholar 

  • Darvishi MT, Barati A (2007) A third-order Newton-type method to solve systems of nonlinear equations. Appl Math Comput 187:630–635

    Article  MathSciNet  MATH  Google Scholar 

  • Darvishi MT, Barati A (2007) A fourth-order method from quadrature formulae to solve systems of nonlinear equations. Appl Math Comput 188:257–261

    Article  MathSciNet  MATH  Google Scholar 

  • Frontini M, Sormani E (2004) Third-order methods from quadrature formulae for solving systems of nonlinear equations. Appl Math Comput 149:771–782

    Article  MathSciNet  MATH  Google Scholar 

  • Fousse L, Hanrot G, Lefèvre V, Pélissier P, Zimmermann P (2007) MPFR: a multiple-precision binary floating-point library with correct rounding. ACM Trans Math Softw 33(2) 15. Art. 13

  • Grau-Sánchez M, Grau À, Noguera M (2011) On the computational efficiency index and some iterative methods for solving systems of nonlinear equations. J Comput Appl Math 236:1259–1266

    Article  MathSciNet  MATH  Google Scholar 

  • Grau-Sánchez M, Grau À, Noguera M (2011) Ostrowski type methods for solving systems of nonlinear equations. Appl Math Comput 218:2377–2385

    Article  MathSciNet  MATH  Google Scholar 

  • Grau-Sánchez M, Noguera M (2012) A technique to choose the most efficient method between secant method and some variants. Appl Math Comput 218:6415–6426

    Article  MathSciNet  MATH  Google Scholar 

  • Grau-Sánchez M, Noguera M, Amat S (2013) On the approximation of derivatives using divided difference operators preserving the local convergence order of iterative methods. J Comput Appl Math 237:363–372

    Article  MathSciNet  MATH  Google Scholar 

  • Homeier HHH (2004) A modified Newton method with cubic convergence: the multivariate case. J Comput Appl Math 169:161–169

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • Liu Z, Zheng Q, Zhao P (2010) A variant of Steffensen’s method of fourth-order convergence and its applications. Appl Math Comput 216:1978–1983

    Article  MathSciNet  MATH  Google Scholar 

  • Noor MA, Waseem M (2009) Some iterative methods for solving a system of nonlinear equations. Comput Math Appl 57:101–106

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  • Ostrowski AM (1960) Solution of equations and system of equations. Academic Press, New York

    MATH  Google Scholar 

  • Petković MS (1989) Iterative methods for simultaneous inclusion of polynomial zeros. Springer, Berlin

    MATH  Google Scholar 

  • Petković MS, Neta B, Petković LD, Džunić J (2013) Multipoint methods for solving nonlinear equations. Elsevier, Boston

    MATH  Google Scholar 

  • Potra F-A, Pták V (1984) Nondiscrete induction and iterarive processes. Pitman Publishing, Boston

    MATH  Google Scholar 

  • 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 

  • Sharma JR, Arora H (2014) Efficient Jarratt-like methods for solving systems of nonlinear equations. Calcolo 51:193–210. doi:10.1007/s10092-013-0097-1

  • Sharma JR, Guha RK, Sharma R (2013) An efficient fourth order weighted-Newton method for systems of nonlinear equations. Numer Algorithms 62:307–323

    Article  MathSciNet  MATH  Google Scholar 

  • Steffensen JF (1933) Remarks on iteration. Skand Aktuar Tidskr 16:64–72

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  • Wang X, Zhang T (2013) A family of Steffensen type methods with seventh-order convergence. Numer Algorithms 62:229–444

    MathSciNet  MATH  Google Scholar 

  • Wolfram S (2003) The mathematica book, 5th edn. Wolfram Media, Champaign

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Sant Longowal Institute of Engineering and Technology, Longowal, Sangrur, 148106, India

    Janak Raj Sharma & Himani Arora

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  1. Janak Raj Sharma
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  2. Himani Arora
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Correspondence to Janak Raj Sharma.

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Communicated by Pascal Frey.

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Sharma, J.R., Arora , H. Efficient derivative-free numerical methods for solving systems of nonlinear equations. Comp. Appl. Math. 35, 269–284 (2016). https://doi.org/10.1007/s40314-014-0193-0

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  • DOI: https://doi.org/10.1007/s40314-014-0193-0

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