Geodesic
On derivative free cubic convergence iterative methods for solving nonlinear equations
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 51 (2011) no. 4, pp. 555-561 
M. Dehghan; M. Hajarian. On derivative free cubic convergence iterative methods for solving nonlinear equations. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 51 (2011) no. 4, pp. 555-561. http://geodesic.mathdoc.fr/item/ZVMMF_2011_51_4_a1/
@article{ZVMMF_2011_51_4_a1,
     author = {M. Dehghan and M. Hajarian},
     title = {On derivative free cubic convergence iterative methods for solving nonlinear equations},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {555--561},
     year = {2011},
     volume = {51},
     number = {4},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2011_51_4_a1/}
}
TY  - JOUR
AU  - M. Dehghan
AU  - M. Hajarian
TI  - On derivative free cubic convergence iterative methods for solving nonlinear equations
JO  - Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
PY  - 2011
SP  - 555
EP  - 561
VL  - 51
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/ZVMMF_2011_51_4_a1/
LA  - en
ID  - ZVMMF_2011_51_4_a1
ER  - 
%0 Journal Article
%A M. Dehghan
%A M. Hajarian
%T On derivative free cubic convergence iterative methods for solving nonlinear equations
%J Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
%D 2011
%P 555-561
%V 51
%N 4
%U http://geodesic.mathdoc.fr/item/ZVMMF_2011_51_4_a1/
%G en
%F ZVMMF_2011_51_4_a1

Voir la notice de l'article provenant de la source Math-Net.Ru

Finding the zeros of a nonlinear equation is a classical problem of numerical analysis which has various applications in many science and engineering. In this problem we seek methods that lead to approximate solutions. Sometimes the applications of the iterative methods depended on derivatives are restricted in Physics, chemistry and engineering. In this paper, we propose two iterative formulas without derivatives. These methods are based on the central-difference and forward-difference approximations to derivatives. The convergence analysis shows that the methods are cubically and quadratically convergent respectively. The best property of these schemes are that they are derivative free. Several numerical examples are given to illustrate the efficiency and performance of the proposed methods.

[1] Alefeld G., “On the convergence of Halley's method”, Amer. Math. Monthly, 88 (1981), 530–538 | DOI | MR

[2] Chen D., “On the convergence of a class of generalized Steffensen's iterative procedures and error analysis”, Internat. J. Comput. Math., 31 (1989), 195–203 | DOI | Zbl

[3] Conte S. D., de Boor C., Elementary numerical analysis, an algorithmic approach, McGraw-Hill, 1981 | MR

[4] Dehghan M., Hajarian M., “New iterative method for solving non-linear equations with fourth-order convergence”, Internat. J. Comput. Math., 87 (2010), 834–839 | DOI | MR | Zbl

[5] Kasturiarachi A. B., “Leap frogging Newton's method”, Int. J. Math. Edu. Sci. Thecnol., 33 (2002), 521–527 | DOI | MR

[6] Kincaid D., Cheney W., Numerical analysis, Sec. Ed., 1996 | MR

[7] Orteg J. M., Rheinboldt W. C., Iterative solution of nonlinear equations in several variabels, Acad. Press, 1975 | MR

[8] Özban A. Y., “Some new variants of Newton's method”, Appl. Math. Letts, 17 (2004), 677–682 | DOI | MR

[9] Stoer J., Bulirsch R., Introduction to numerical analysis, Th. Ed., Springer, Inc., New-York, 2002 | MR

[10] Weerakoon S., Fernando T. G. I., “A variant of Newton's method with accelerated third–orderconvergence”, Appl. Math. Letts, 13 (2000), 87–93 | DOI | MR | Zbl

[11] Dehghan M., Hajarian M., “On some cubic convergence iterative formulas without derivatives for solving nonlinear equations”, Comuns Numer. Meth. Engng | DOI

[12] Dehghan M., Hajarian M., “Some derivative free quadratic and cubic convergence iterative formulas”, Comput. Appl. Math., 29:1 (2010), 19–30 | DOI | MR | Zbl