article

A family of root finding methods

Authors:

Eldon Hansen,

Merrell PatrickAuthors Info & Claims

Numerische Mathematik, Volume 27, Issue 3

Pages 257 - 269

https://doi.org/10.1007/BF01396176

Published: 01 September 1976 Publication History

Abstract

A one parameter family of iteration functions for finding roots is derived. The family includes the Laguerre, Halley, Ostrowski and Euler methods and, as a limiting case, Newton's method. All the methods of the family are cubically convergent for a simple root (except Newton's which is quadratically convergent). The superior behavior of Laguerre's method, when starting from a pointz for which |z| is large, is explained. It is shown that other methods of the family are superior if |z| is not large. It is also shown that a continuum of methods for the family exhibit global and monotonic convergence to roots of polynomials (and certain other functions) if all the roots are real.

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Sharma HBehl RKansal MRamos H(2024)A robust iterative family for multiple roots of nonlinear equationsJournal of Computational and Applied Mathematics10.1016/j.cam.2024.115795444:COnline publication date: 1-Jul-2024
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Herceg ĐPetković I(2022)Computer visualization and dynamic study of new families of root-solversJournal of Computational and Applied Mathematics10.1016/j.cam.2021.113775401:COnline publication date: 1-Feb-2022
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Sugiura HHasegawa T(2018)On the global convergence of Schröder’s iterative formulae for real roots of algebraic equationsJournal of Computational and Applied Mathematics10.1016/j.cam.2018.05.041344:C(313-322)Online publication date: 15-Dec-2018
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A family of root finding methods
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Numerische Mathematik Volume 27, Issue 3

September 1976

99 pages

ISSN:0029-599X

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Springer-Verlag

Berlin, Heidelberg

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Published: 01 September 1976

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Sharma HBehl RKansal MRamos H(2024)A robust iterative family for multiple roots of nonlinear equationsJournal of Computational and Applied Mathematics10.1016/j.cam.2024.115795444:COnline publication date: 1-Jul-2024
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Herceg ĐPetković I(2022)Computer visualization and dynamic study of new families of root-solversJournal of Computational and Applied Mathematics10.1016/j.cam.2021.113775401:COnline publication date: 1-Feb-2022
https://dl.acm.org/doi/10.1016/j.cam.2021.113775
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https://dl.acm.org/doi/10.1016/j.cam.2018.05.041
Chun CNeta B(2018)Comparative study of methods of various orders for finding repeated roots of nonlinear equationsJournal of Computational and Applied Mathematics10.1016/j.cam.2018.02.009340:C(11-42)Online publication date: 1-Oct-2018
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Farmer MLoizou GMaybank S(2017)Iteration Functions re-visitedJournal of Computational and Applied Mathematics10.1016/j.cam.2016.08.021311:C(484-496)Online publication date: 1-Feb-2017
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Kansal MKanwar VBhatia S(2016)Efficient derivative-free variants of Hansen-Patrick's family with memory for solving nonlinear equationsNumerical Algorithms10.1007/s11075-016-0127-673:4(1017-1036)Online publication date: 1-Dec-2016
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Derivative-free family of higher order root finding methods

On a family of Weierstrass-type root-finding methods with accelerated convergence

On the Order of Convergence of a Determinantal Family of Root-Finding Methods

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