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An Improvement of Ostrowski’s and King’s Techniques with Optimal Convergence Order Eight

Author

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  • Fazlollah Soleymani

    (Islamic Azad University, Zahedan Branch)

  • Mahdi Sharifi

    (Islamic Azad University, Zahedan Branch)

  • Bibi Somayeh Mousavi

    (Islamic Azad University, Zahedan Branch)

Abstract

In this paper, we first establish a new class of three-point methods based on the two-point optimal method of Ostrowski. Analysis of convergence shows that any method of our class arrives at eighth order of convergence by using three evaluations of the function and one evaluation of the first derivative per iteration. Thus, this order agrees with the conjecture of Kung and Traub (J. ACM 643–651, 1974) for constructing multipoint optimal iterations without memory. We second present another optimal eighth-order class based on the King’s fourth-order family and the first attained class. To support the underlying theory developed in this work, we examine some methods of the proposed classes by comparison with some of the existing optimal eighth-order methods in literature. Numerical experience suggests that the new classes would be valuable alternatives for solving nonlinear equations.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:153:y:2012:i:1:d:10.1007_s10957-011-9929-9
    DOI: 10.1007/s10957-011-9929-9
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    References listed on IDEAS

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    1. 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.
    2. 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.
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    Cited by:

    1. F. Soleymani, 2012. "Optimized Steffensen-Type Methods with Eighth-Order Convergence and High Efficiency Index," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2012, pages 1-18, September.
    2. 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.
    3. Xiaofeng Wang, 2022. "A Novel n -Point Newton-Type Root-Finding Method of High Computational Efficiency," Mathematics, MDPI, vol. 10(7), pages 1-22, April.

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    1. 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.
    2. 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.

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