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 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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},
     year = {2015},
     volume = {55},
     number = {12},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2015_55_12_a1/}
}
TY  - JOUR
AU  - D. A. Budzko
AU  - A. Cordero
AU  - J. R. Torregrosa
TI  - New family of iterative methods based on the Ermakov–Kalitkin scheme for solving nonlinear systems of equations
JO  - Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
PY  - 2015
SP  - 1986
EP  - 1998
VL  - 55
IS  - 12
UR  - http://geodesic.mathdoc.fr/item/ZVMMF_2015_55_12_a1/
LA  - ru
ID  - ZVMMF_2015_55_12_a1
ER  - 
%0 Journal Article
%A D. A. Budzko
%A A. Cordero
%A J. R. Torregrosa
%T New family of iterative methods based on the Ermakov–Kalitkin scheme for solving nonlinear systems of equations
%J Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
%D 2015
%P 1986-1998
%V 55
%N 12
%U http://geodesic.mathdoc.fr/item/ZVMMF_2015_55_12_a1/
%G ru
%F ZVMMF_2015_55_12_a1
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/

[1] Petković M., Neta V., Petković L., Dzunić J., Multipoint methods for solving nonlinear equations, Academic Press, New York, 2012

[2] Cordero A., Lotfi T., Mahdiani K., Torregrosa J. R., “Two optimal general classes of iterative methods with eighth-order”, Acta Applicandae Mathematicae, 2014 | DOI | MR

[3] Artidiello S., Cordero A., Torregrosa J. R., Vassileva M. P., “Optimal high-order methods for solving nonlinear equations”, J. Applied Math., 2014, 591638 | DOI | MR

[4] Popuzin V. V., Sumbatyan M. A., Tanyushin R. A., “Bystryi iteratsionnyi metod v zadache o vzaimodeistvii voln s sistemoi tonkikh ekranov”, Zh. vychisl. matem. i matem. fiz., 53:8 (2013), 1374–1386 | DOI | Zbl

[5] Ermakov V. V., Kalitkin N. N., “Optimalnyi shag i regulyarizatsiya metoda Nyutona”, Zh. vychisl. matem. i matem. fiz., 21:2 (1981), 491–497 | MR | Zbl

[6] Zhanlav T., Puzynin I. V., “O skhodimosti iteratsii na osnove nepreryvnogo analoga metoda Nyutona”, Zh. vychisl. matem. i matem. fiz., 32:6 (1992), 846–856 | MR | Zbl

[7] Madorskii V. M., Kvazinyutonovskie protsessy dlya resheniya nelineinykh uravnenii, BrGU, Brest, 2005

[8] Traub J. F., Iterative methods for the solution of equations, Prentice-Hall, Englewood Cliffs, 1964 | MR | Zbl

[9] Jarratt P., “Some fourth order multipoint iterative methods for solving equations”, Math. Comput., 20:95 (1966), 434–437 | DOI | Zbl

[10] Budzko D. A., Prokopenya A. N., “On the stability of equilibrium positions in the circular restricted four-body problem”, Lect. Notes in Computer Sci., 6885, 2011, 88–100 | DOI | Zbl

[11] Budzko D. A., Prokopenya A. N., “Stability of equilibrium positions in the spatial circular restricted four-body problem”, Lect. Notes in Computer Sci., 7442, 2012, 72–83 | DOI | Zbl

[12] Blanchard P., “The dynamics of Newton's method”, Proc. of Symposia in Applied Meth., 49 (1994), 139–154 | DOI | MR | Zbl

[13] Devaney R. L., “The Mandelbrot Set, the Farey Tree and the Fibonacci sequence”, Am. Math. Monthly, 106:4 (1999), 289–302 | DOI | MR | Zbl

[14] Chicharro F. I., Cordero A., Torregrosa J. R., “Drawing dynamical and parameters planes of iterative families and methods”, The Sci. World J., 2013, 780153

[15] Cordero A., Torregrosa J. R., “Variants of Newton's method using fifth-order quadrature formulas”, Appl. Math. Comput., 190 (2007), 686–698 | DOI | MR | Zbl

[16] Ortega J. M., Rheinboldt W. G., Iterative solutions of nonlinear equations in several variables, Academic Press, New-York, 1970 | MR

[17] Hermite C., “Sur la formule d'interpolation de Lagrange”, J. Reine Angew. Math., 84 (1878), 70–79 | DOI | MR

[18] Budzko D. A., Prokopenya A. N., “Symbolic-numerical analysis of equilibrium solutions in a restricted four-body problem”, Program. Comput. Software, 36:2 (2010), 68–74 | DOI | MR | Zbl

[19] Budzko D. A., Prokopenya A. N., “Symbolic-numerical methods for searching equilibrium states in a restricted four-body problem”, Program. Comput. Software, 39:2 (2013), 74–80 | DOI | MR | Zbl