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Higher-Order Method for the Solution of a Nonlinear Scalar Equation

Author

Listed:
  • A. Germani

    (University of L’Aquila
    Istituto di Analisi dei Sistemi ed Informatica A. Ruberti, CNR)

  • C. Manes

    (University of L’Aquila
    Istituto di Analisi dei Sistemi ed Informatica A. Ruberti, CNR)

  • P. Palumbo

    (Istituto di Analisi dei Sistemi ed Informatica A. Ruberti, CNR)

  • M. Sciandrone

    (Istituto di Analisi dei Sistemi ed Informatica A. Ruberti, CNR)

Abstract

A new iterative method to find the root of a nonlinear scalar function f is proposed. The method is based on a suitable Taylor polynomial model of order n around the current point x k and involves at each iteration the solution of a linear system of dimension n. It is shown that the coefficient matrix of the linear system is nonsingular if and only if the first derivative of f at x k is not null. Moreover, it is proved that the method is locally convergent with order of convergence at least n + 1. Finally, an easily implementable scheme is provided and some numerical results are reported.

Suggested Citation

  • A. Germani & C. Manes & P. Palumbo & M. Sciandrone, 2006. "Higher-Order Method for the Solution of a Nonlinear Scalar Equation," Journal of Optimization Theory and Applications, Springer, vol. 131(3), pages 347-364, December.
    • Handle: RePEc:spr:joptap:v:131:y:2006:i:3:d:10.1007_s10957-006-9154-0
    DOI: 10.1007/s10957-006-9154-0
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    Citations

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    Cited by:

    1. Min Chen & Tsu-Shuan Chang, 2011. "On the Higher-Order Method for the Solution of a Nonlinear Scalar Equation," Journal of Optimization Theory and Applications, Springer, vol. 149(3), pages 647-664, June.
    2. Fazlollah Soleymani & Mahdi Sharifi & Bibi Somayeh Mousavi, 2012. "An Improvement of Ostrowski’s and King’s Techniques with Optimal Convergence Order Eight," Journal of Optimization Theory and Applications, Springer, vol. 153(1), pages 225-236, April.
    3. Young Hee Geum & Young Ik Kim, 2014. "An Optimal Family of Fast 16th-Order Derivative-Free Multipoint Simple-Root Finders for Nonlinear Equations," Journal of Optimization Theory and Applications, Springer, vol. 160(2), pages 608-622, February.

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