Improved Chebyshev–Halley family of methods with seventh and eighth order of convergence for simple roots

Improved Chebyshev–Halley family of methods with seventh and eighth order of convergence for... The principle aim of this manuscript is to introduce a new highly efficient improved modification of Chebyshev–Halley family with increasing order of convergence. The idea is based on the schemes proposed by Li et al. (Appl Math Comput 235:221–225, 2014) and Sharma (Appl Math Comput 256:119–124, 2015). The proposed scheme has at least seventh-order of convergence and also posses an optimal eighth-order of convergence for a particular value of the disposable parameter. Each member of the proposed scheme is free from second-order derivative and requires only four functional evaluations (viz. three evaluations of function f and one of first-order derivative $$f'$$ f ′ ) per full iteration. A variety of concrete numerical examples demonstrates that our proposed scheme performs better than the existing schemes proposed by Li et al. (Appl Math Comput 235:221–225, 2014) and Sharma (Appl Math Comput 256:119–124, 2015). Dynamical study of our methods also confirms the above conclusions to a great extent. Moreover, the local convergence of these methods is given using hypotheses only on the first derivative and Lipschitz constants. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png SeMA Journal Springer Journals

Improved Chebyshev–Halley family of methods with seventh and eighth order of convergence for simple roots

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Publisher
Springer Journals
Copyright
Copyright © 2017 by Sociedad Española de Matemática Aplicada
Subject
Mathematics; Mathematics, general; Applications of Mathematics
ISSN
2254-3902
eISSN
2281-7875
D.O.I.
10.1007/s40324-016-0106-9
Publisher site
See Article on Publisher Site

Abstract

The principle aim of this manuscript is to introduce a new highly efficient improved modification of Chebyshev–Halley family with increasing order of convergence. The idea is based on the schemes proposed by Li et al. (Appl Math Comput 235:221–225, 2014) and Sharma (Appl Math Comput 256:119–124, 2015). The proposed scheme has at least seventh-order of convergence and also posses an optimal eighth-order of convergence for a particular value of the disposable parameter. Each member of the proposed scheme is free from second-order derivative and requires only four functional evaluations (viz. three evaluations of function f and one of first-order derivative $$f'$$ f ′ ) per full iteration. A variety of concrete numerical examples demonstrates that our proposed scheme performs better than the existing schemes proposed by Li et al. (Appl Math Comput 235:221–225, 2014) and Sharma (Appl Math Comput 256:119–124, 2015). Dynamical study of our methods also confirms the above conclusions to a great extent. Moreover, the local convergence of these methods is given using hypotheses only on the first derivative and Lipschitz constants.

Journal

SeMA JournalSpringer Journals

Published: Jan 16, 2017

References

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