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Two new classes of optimal Jarratt-type fourth-order methods. (English) Zbl 1239.65030
Summary: We investigate the construction of some two-step without memory iterative classes of methods for finding simple roots of nonlinear scalar equations. The classes are built through the approach of weight functions and these obtained classes reach the optimal order four using one function and two first derivative evaluations per full cycle. This shows that our classes can be considered as Jarratt-type schemes. The accuracy of the classes is tested on a number of numerical examples. And eventually, it is observed that our contributions take less number of iterations than the compared existing methods of the same type to find more accurate approximate solutions of the nonlinear equations.
65H05Single nonlinear equations (numerical methods)
[1]Khattri, S. K.: Optimal eighth order iterative methods, Math. comput. Sci. (2011)
[2]Sargolzaei, P.; Soleymani, F.: Accurate fourteenth-order methods for solving nonlinear equations, Numer. algorithms (2011)
[3]Soleymani, F.; Sharifi, M.; Mousavi, B. S.: An improvement of Ostrowski’s and King’s techniques with optimal convergence order eight, J. optim. Theory appl. (2011)
[4]Chun, C.; Neta, B.: Certain improvements of Newton’s method with fourth-order convergence, Appl. math. Comput. 215, 821-828 (2009) · Zbl 1192.65049 · doi:10.1016/j.amc.2009.06.007
[5]Weerakoon, S.; Fernando, G. I.: A variant of Newton’s method with accelerated third-order convergence, Appl. math. Lett. 17, 87-93 (2000) · Zbl 0973.65037 · doi:10.1016/S0893-9659(00)00100-2
[6]Frontini, M.; Sormani, E.: Some variants of Newton’s method with third-order convergence, Appl. math. Comput. 140, 419-426 (2003) · Zbl 1037.65051 · doi:10.1016/S0096-3003(02)00238-2
[7]Homeier, H. H. H.: On Newton-type methods with cubic convergence, J. comput. Appl. math. 176, 425-432 (2005) · Zbl 1063.65037 · doi:10.1016/j.cam.2004.07.027
[8]Jarratt, P.: Some efficient fourth order multipoint methods for solving equations, Bit 9, 119-124 (1969) · Zbl 0188.22101 · doi:10.1007/BF01933248
[9]Khattri, S. K.; Abbasbandy, S.: Optimal fourth order family of iterative methods, Mat. vesnik 63, 67-72 (2011)
[10]Kung, H. T.; Traub, J. F.: Optimal order of one-point and multipoint iteration, J. assoc. Comput. Mach 21, 634-651 (1974) · Zbl 0289.65023 · doi:10.1145/321850.321860
[11]Soleymani, F.; Vanani, S. Karimi; Khan, M.; Sharifi, M.: Some modifications of King’s family with optimal eighth order of convergence, Math. comput. Modelling (2011)
[12]Soleymani, F.; Vanani, S. Karimi: Optimal Steffensen-type methods with eighth order of convergence, Comput. math. Appl. (2011)
[13]Soleymani, F.: A novel and precise sixth-order method for solving nonlinear equations, Int. J. Math. models methods appl. Sci. 5, 730-737 (2011)