Geodesic
New family of iterative methods based on the Ermakov–Kalitkin scheme for solving nonlinear systems of equations
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 55 (2015) no. 12, pp. 1986-1998

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A new one-parameter family of iterative methods for solving nonlinear equations and systems is constructed. It is proved that their order of convergence is three for both equations and systems. An analysis of the dynamical behavior of the methods shows that they have a larger domain of convergence than previously known iterative schemes of the second to fourth orders. Numerical results suggest that the methods are also preferable in terms of their relative stability and the number of iteration steps. The methods are compared with previously known techniques as applied to a system of two nonlinear equations describing the dynamics of a passively gravitating mass in the Newtonian circular restricted four-body problem formulated on the basis of Lagrange’s triangular solutions to the threebody problem. 
@article{ZVMMF_2015_55_12_a1,
     author = {D. A. Budzko and A. Cordero and J. R. Torregrosa},
     title = {New family of iterative methods based on the {Ermakov{\textendash}Kalitkin} scheme for solving nonlinear systems of equations},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1986--1998},
     publisher = {mathdoc},
     volume = {55},
     number = {12},
     year = {2015},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2015_55_12_a1/}
}
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D. A. Budzko; A. Cordero; J. R. Torregrosa. New family of iterative methods based on the Ermakov–Kalitkin scheme for solving nonlinear systems of equations. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 55 (2015) no. 12, pp. 1986-1998. http://geodesic.mathdoc.fr/item/ZVMMF_2015_55_12_a1/