Geodesic
Necessary and sufficient conditions for the convergence of two- and three-point Newton-type iterations
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 57 (2017) no. 7, pp. 1093-1102 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Necessary and sufficient conditions under which two- and three-point iterative methods have the order of convergence $p$ ($2\leqslant p\leqslant 8$) are formulated for the first time. These conditions can be effectively used to prove the convergence of iterative methods. In particular, the order of convergence of some known optimal methods is verified using the proposed sufficient convergence tests. The optimal set of parameters making it possible to increase the order of convergence is found. It is shown that the parameters of the known iterative methods with the optimal order of convergence have the same asymptotic behavior. The simplicity of choosing the parameters of the proposed methods is an advantage over the other known methods. 
@article{ZVMMF_2017_57_7_a1,
     author = {T. Zhanlav and V. Ulziibayar and O. Chuluunbaatar},
     title = {Necessary and sufficient conditions for the convergence of two- and three-point {Newton-type} iterations},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1093--1102},
     year = {2017},
     volume = {57},
     number = {7},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2017_57_7_a1/}
}
TY  - JOUR
AU  - T. Zhanlav
AU  - V. Ulziibayar
AU  - O. Chuluunbaatar
TI  - Necessary and sufficient conditions for the convergence of two- and three-point Newton-type iterations
JO  - Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
PY  - 2017
SP  - 1093
EP  - 1102
VL  - 57
IS  - 7
UR  - http://geodesic.mathdoc.fr/item/ZVMMF_2017_57_7_a1/
LA  - ru
ID  - ZVMMF_2017_57_7_a1
ER  - 
%0 Journal Article
%A T. Zhanlav
%A V. Ulziibayar
%A O. Chuluunbaatar
%T Necessary and sufficient conditions for the convergence of two- and three-point Newton-type iterations
%J Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
%D 2017
%P 1093-1102
%V 57
%N 7
%U http://geodesic.mathdoc.fr/item/ZVMMF_2017_57_7_a1/
%G ru
%F ZVMMF_2017_57_7_a1
T. Zhanlav; V. Ulziibayar; O. Chuluunbaatar. Necessary and sufficient conditions for the convergence of two- and three-point Newton-type iterations. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 57 (2017) no. 7, pp. 1093-1102. http://geodesic.mathdoc.fr/item/ZVMMF_2017_57_7_a1/

[1] Bi W., Ren H., Wu Q., “Three-step iterative methods with eighth-order convergence for solving nonlinear equations”, J. Comput. Appl. Math., 225 (2009), 105–112 | DOI | MR | Zbl

[2] Bi W., Wu Q., Ren H., “A new family of eighth-order iterative methods for solving nonlinear equations”, Appl. Math. Comput., 211 (2009), 236–245 | MR

[3] Cordero A., Hueso J. L., Martinez E., Torregrosa J. R., “New modifications of Potra-Ptak's method with optimal fourth and eighth orders of convergence”, J. Comput. Appl. Math., 234 (2010), 2969–2976 | DOI | MR | Zbl

[4] Cordero A., Torregrosa J. R., Vassileva M. P., “Three-step iterative methods with optimal eighth-order convergence”, J. Comput. Appl. Math., 235 (2011), 3189–3194 | DOI | MR | Zbl

[5] Thukral R., Petkovic M. S., “A family of the three-point methods of optimal order for solving nonlinear equations”, J. Comput. Appl. Math., 233 (2010), 2278–2284 | DOI | MR | Zbl

[6] Wang X., Liu L., “New eighth-order iterative methods for solving nonlinear equations”, J. Comput. Appl. Math., 234 (2010), 1611–1620 | DOI | MR | Zbl

[7] Wang X., Liu L., “Modified Ostrowski's method with eighth-order convergence and high efficient index”, Appl. Math. Lett., 23 (2010), 549–554 | DOI | MR | Zbl

[8] Sharma J. R., Sharma R., “A new family of modified Ostrowski's methods with accelerated eighth-order convergence”, Numer. Algor., 54 (2010), 445–458 | DOI | MR | Zbl

[9] Zhanlav T., Chuluunbaatar O., “Convergence of a continuous analog of Newton's method for solving nonlinear equations”, Numerical Methods and Programming, 10 (2009), 402–407

[10] Chun C., Neta B., “Comparison of several families of optimal eighth order methods”, Appl. Math. Comput., 274 (2010), 762–773 | MR

[11] Chun C., Neta B., “A analysis of a new family of eighth-order optimal methods”, Appl. Math. Comput., 245 (2014), 86–107 | DOI | MR | Zbl

[12] Ham Y., Chun C., Lee S., “Some higher-order modifications of Newton's method for solving nonlinear equations”, J. Comput. Appl. Math., 222 (2008), 477–486 | DOI | MR | Zbl

[13] Zhanlav T., Puzynin I. V., “The convergence of iteration based on a continuous analogy of Newton's method”, Comput. Math. and Math. Phys., 32 (1992), 729–737 | MR | Zbl

[14] King R., “A family of fourth order methods for nonlinear equations”, SIAM. Journal on Numerical Analysis, 10 (1973), 876–879 | DOI | MR | Zbl

[15] Chun C., “A family of composite fourth-order iterative methods for solving nonlinear equations”, Appl. Math. Comput., 187 (2007), 951–956 | MR | Zbl

[16] Maheshwari A. K., “A fourth-order iterative methods for solving nonlinear equations”, Appl. Math. Comput., 211 (2009), 383–391 | MR | Zbl

[17] Zhanlav T., Ulziibayar V., “Modified King's methods with optimal eighth-order convergence and high efficiency index”, Amer. J. Comput. Appl. Math., 6:5 (2016), 177–181

[18] Kou J., Li Y., Wang X., “Some variants of Ostrowski's method with seventh-order convergence”, J. Comput. Appl. Math., 209 (2007), 153–159 | DOI | MR | Zbl

[19] Chun C., Ham Y., “Some sixth-order variants of Ostrowski's methods with seventh-order convergence”, Appl. Math. Comput., 193 (2007), 389–394 | MR | Zbl

[20] Chun C., Neta B., “A new sixth-order scheme for nonlinear equations”, Appl. Math. Lett., 25 (2012), 185–189 | DOI | MR | Zbl

[21] Traub G. B., Iterative methods for solution of equations, Prentice-Hall, Englewood Cliffs, NJ, 1964 | MR | Zbl