A family of fifth-order iterative methods for finding multiple roots of nonlinear equations
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 24 (2021) no. 2, pp. 213-227.

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We present a family of fifth order iterative methods for finding multiple roots of nonlinear equations. Numerical examples are considered to check the validity of theoretical results. The results show that the new methods are competitive to other methods for multiple roots. Basins of attraction for new methods and some existing methods are drawn to observe the dynamics in the complex plane.
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J. R. Sharma; H. Arora. A family of fifth-order iterative methods for finding multiple roots of nonlinear equations. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 24 (2021) no. 2, pp. 213-227. http://geodesic.mathdoc.fr/item/SJVM_2021_24_2_a6/

[1] Schroder E., “Über unendlich viele Algorithmen zur Auflosung der Gleichungen”, Math. Ann., 2 (1870), 317–365 | DOI | MR

[2] Traub J. F., Iterative Methods for the Solution of Equations, Prentice-Hall, Englewood Cliffs, 1964 | MR | Zbl

[3] Biazar J., Ghanbari B., “A new third-order family of nonlinear solvers for multiple roots”, Comput. Math. Appl., 59:10 (2010), 3315–3319 | DOI | MR | Zbl

[4] Chun C., Neta B., “A third-order modification of Newton's method for multiple roots”, Appl. Math. Comput., 211:2 (2009), 474–479 | MR | Zbl

[5] Chun C., Bae H. J., Neta B., “New families of nonlinear third-order solvers for finding multiple roots”, Comput. Math. Appl., 57:9 (2009), 1574–1582 | DOI | MR | Zbl

[6] Hansen E., Patrick M., “A family of root finding methods”, Numer. Math., 27 (1976), 257–269 | DOI | MR

[7] Neta B., “New third order nonlinear solvers for multiple roots”, Appl. Math. Comput., 202 (2008), 162–170 | MR | Zbl

[8] Osada N., “An optimal multiple root-finding method of order three”, J. Comput. Appl. Math., 51:1 (1994), 131–133 | DOI | MR | Zbl

[9] Sharma R., Bahl A., “General family of third order methods for multiple roots of nonlinear equations and basin attractors for various methods”, Advances in Numerical Analysis, 2014 (2014), 963878 | MR | Zbl

[10] Sharma J. R., Sharma R., “Modified Chebyshev-Halley type method and its variants for computing multiple roots”, Numer. Algor., 61 (2012), 567–578 | DOI | MR | Zbl

[11] Dong C., “A basic theorem of constructing an iterative formula of the higher order for computing multiple roots of an equation”, Math. Numer. Sinica, 11 (1982), 445–450 | MR

[12] Homeier H. H. H., “On Newton-type methods for multiple roots with cubic convergence”, J. Comput. Appl. Math., 231:1 (2009), 249–254 | DOI | MR | Zbl

[13] Kumar S., Kanwar V., Singh S., “On some modified families of multipoint iterative methods for multiple roots of nonlinear equations”, Appl. Math. Comput., 218 (2012), 7382–7394 | MR | Zbl

[14] Li S., Li H., Cheng L., “Some second-derivative-free variants of Halley's method for multiple roots”, Appl. Math. Comput., 215:6 (2009), 2192–2198 | MR | Zbl

[15] Neta B., “New third order nonlinear solvers for multiple roots”, Appl. Math. Comput., 202 (2008), 162–170 | MR | Zbl

[16] Victory H. D., Neta B., “A higher order method for multiple zeros of nonlinear functions”, Inter. J. Comput. Math., 12:3–4 (1983), 329–335 | DOI | MR | Zbl

[17] Zhou X., Chen X., Song Y., “Families of third and fourth order methods for multiple roots of nonlinear equations”, Appl. Math. Comput., 219 (2013), 6030–6038 | MR | Zbl

[18] Dong C., “A family of multipoint iterative functions for finding multiple roots of equations”, Inter. J. Comput. Math., 21:3–4 (1987), 363–367 | DOI | Zbl

[19] Li S.G., Cheng L. Z., Neta B., “Some fourth-order nonlinear solvers with closed formulae for multiple roots”, Comput. Math. Appl., 59:1 (2010), 126–135 | DOI | MR | Zbl

[20] Li S.G., Liao X., Cheng L. Z., “A new fourth-order iterative method for finding multiple roots of nonlinear equations”, Appl. Math. Comput., 215 (2009), 1288–1292 | MR | Zbl

[21] Neta B., “Extension of Murakami's high-order non-linear solver to multiple roots”, Inter. J. Comput. Math., 87:5 (2010), 1023–1031 | DOI | MR | Zbl

[22] Neta B., Johnson A. N., “High-order nonlinear solver for multiple roots”, Comput. Math. Appl., 55:9 (2008), 2012–2017 | DOI | MR | Zbl

[23] Ostrowski A. M., Solution of Equations and Systems of Equations, Academic Press, New York, 1966 | MR | Zbl

[24] Wolfram S., The Mathematica Book, 5-th ed., Wolfram Media, 2003 | MR

[25] Jay L. O., “A note on Q-order of convergence”, BIT Numerical Mathematics, 41 (2001), 422–429 | DOI | MR | Zbl

[26] Scott M., Neta B., Chun C., “Basin attractors for various methods”, Appl. Math. Comput., 218 (2011), 2584–2599 | MR | Zbl

[27] Magrenan A. A., “A new tool to study real dynamics: the convergence plane”, Appl. Math. Comput., 248 (2014), 215–224 | MR | Zbl