Optimal Eighth Order Iterative Methods
Tóm tắt
We develop an eighth order family of methods, consisting of three steps and three parameters, for solving nonlinear equations. Per iteration the methods require four evaluations (three function evaluations and one evaluation of the first derivative). Convergence analysis shows that the family is eighth-order convergent which is also substantiated through the numerical work. Computational results ascertain that family of methods are efficient and demonstrate equal or better performance as compared with other well known methods.
Tài liệu tham khảo
Argyros I.K.: Computational theory of iterative methods, series. In: Chui, C.K., Wuytack, L. (eds) Studies in Computational Mathematics, vol. 15, Elsevier, New York (2007)
Khattri S.K., Log T.: Constructing third-order derivative-free iterative methods. Int. J. Comput. Math. 88, 1509–1518 (2011)
Khattri, S.K., Log, T.: Derivative free algorithm for solving nonlinear equations. Computing 1–11 (2010)
Khattri S.K., Agarwal R.: Derivative-free optimal iterative methods. Comput. Methods Appl. Math. 10, 368–375 (2010)
Neta B.: On a family of multipoint methods for non-linear equations. Int. J. Comput. Math. 9, 353–361 (1981)
Khattri S.K., Agarwal R.: Derivative-free optimal iterative methods. Comput. Methods Appl. Math. 10, 368–375 (2010)
Ostrowski A.M.: Solutions of Equations and System Equations. Academic Press, New York (1960)
Traub J.F.: Iterative Methods for the Solution of Equations. Chelsea Publishing Company, New York (1977)
Ren H., Wu Q., Bi W.: New variants of Jarratt’s method with sixth-order convergence. Numer. Algorithms 52, 585–603 (2009)
Wang X., Kou J., Li Y.: A variant of Jarratt method with sixth-order convergence. Appl. Math. Comput. 204, 14–19 (2008)
Sharma J.R., Guha R.K.: A family of modified Ostrowski methods with accelerated sixth order convergence. Appl. Math. Comput. 190, 111–115 (2007)
Chun C., Ham Y.: Some sixth-order variants of Ostrowski root-finding methods. Appl. Math. Comput. 193, 389–394 (2003)
Neta B.: A sixth-order family of methods for nonlinear equations. Int. J. Comput. Math. 7, 157–161 (1979)
Geuma Y.H., Kim Y.I.: A multi-parameter family of three-step eighth-order iterative methods locating a simple root. Appl. Math. Comput. 215, 3375–3382 (2010)
Bi W., Ren H., Wu Q.: Three-step iterative methods with eighth-order convergence for solving nonlinear equations. J. Comput. Appl. Math. 225, 105–112 (2009)
Bi W., Wu Q., Ren H.: A new family of eighth-order iterative methods for solving nonlinear equations. Appl. Math. Comput. 214, 236–254 (2009)
Li X., Mu C., Ma J., Wang C.: Sixteenth order method for nonlinear equations. Appl. Math. Comput. 215, 3769–4054 (2009)
King R.F.: A family of fourth order methods for nonlinear equations. SIAM J. Numer. Anal. 10, 876–879 (1973)
Argyros I.K., Chen D., Qian Q.: The Jarrat method in Banach space setting. J. Comput. Appl. Math. 51, 103–106 (1994)
Kung H.T., Traub J.F.: Optimal order of one-point and multipoint iteration. J. Assoc. Comput. Math. 21, 634–651 (1974)
Chun C.: Construction of Newton-like iteration methods for solving nonlinear equations. Numer. Math. 104(3), 297–315 (2006)
ARPREC, C++/Fortran-90 arbitrary precision package. http://crd.lbl.gov/~dhbailey/mpdist/