On optimal fourth-order iterative methods free from second derivative and their dynamics. (English) Zbl 1277.65031

Summary: New fourth-order optimal root-finding methods for solving nonlinear equations are proposed. The classical Jarratt’s family of fourth-order methods is obtained as special case. We then present results which describe the conjugacy classes and dynamics of the presented optimal method for complex polynomials of degree two and three. The basins of attraction of existing optimal methods and our method are presented and compared to illustrate their performance.


65H05 Numerical computation of solutions to single equations
Full Text: DOI


[1] Traub, J. F., Iterative Methods for the Solution of Equations (1977), Chelsea publishing company: Chelsea publishing company New York · Zbl 0121.11204
[2] Gander, W., On Halley’s iteration method, Am. Math. Mon., 92, 2, 131-134 (1985) · Zbl 0574.65041
[3] Ostrowski, A. M., Solution of Equations in Euclidean and Banach Space (1973), Academic Press: Academic Press New York · Zbl 0304.65002
[4] Kung, H. T.; Traub, J. F., Optimal order of one-point and multipoint iteration, J. Assoc. Comput. Mach., 21, 643-651 (1974) · Zbl 0289.65023
[5] Kou, J.; Li, Y.; Wang, X., Fourth-order iterative methods free from second derivative, Appl. Math. Comput., 184, 880-885 (2007) · Zbl 1114.65046
[6] Sharma, J. R.; Goyal, R. K., Fourth-order derivative-free methods for solving non-linear equations, Int. J. Comput. Math., 83, 1, 101-106 (2006) · Zbl 1094.65048
[7] Gilbert, W. J., Generalizations of Newton’s method, Fractals, 9, 3, 251-262 (2001) · Zbl 1046.37027
[8] Drakopoulos, V., How is the dynamics of König iteration functions affected by their additional fixed points?, Fractals, 7, 3, 327-334 (1999) · Zbl 1020.37025
[9] Kneisl, K., Julia sets for the super-Newton method, Cauchy’s method and Halley’s method, Chaos, 11, 2, 359-370 (2001) · Zbl 1080.65532
[10] Plaza, S., Review of some iterative root-finding methods from a dynamical point of view, Scientia, 10, 3-35 (2004) · Zbl 1137.37316
[11] Scott, M.; Neta, B.; Chun, C., Basin attractors for various methods, Appl. Math. Comput., 218, 2584-2599 (2011) · Zbl 1478.65037
[13] Amat, S.; Busquier, S.; Plaza, S., Dynamics of the King and Jarratt iterations, Aequationes Math., 69, 212-223 (2005) · Zbl 1068.30019
[14] Milnor, J., Dynamics in One Complex Variable, (Annals of Mathematics Studies, vol. 160 (2006), Princeton Univ. Press: Princeton Univ. Press Princeton, NJ) · Zbl 1281.37001
[15] Jarratt, P., Some fourth-order multipoint iterative methods for solving equations, Math. Comput., 20, 434-437 (1966) · Zbl 0229.65049
[16] Beardon, A. F., Iteration of Rational Functions (1991), Springer-Verlag: Springer-Verlag New York · Zbl 0757.30034
[17] Barna, B., Über die Divergenzpunkte des Newtonsches Verfahrens zur Bestimmumg von Wurzeln algebraischen Gleichungen. II, Publ. Math. Debrecen, 4, 384-397 (1956) · Zbl 0073.10603
[18] Smale, S., On the efficiency of algorithms of analysis for solving equations, Bull. Am. Math. Soc., 13, 87-121 (1985) · Zbl 0592.65032
[19] Kalantari, B., Polynomial Root-Finding and Polynomiography (2009), World Scientific Publishing Co.: World Scientific Publishing Co. Singapore · Zbl 1218.37003
[20] King, R. F., A family of fourth order methods for nonlinear equations, SIAM J. Numer. Amal., 10, 876-879 (1973) · Zbl 0266.65040
[21] Kou, J.; Li, Y.; Wang, X., A composite fourth-order iterative method for solving non-linear equations, Appl. Math. Comput., 184, 2, 471-475 (2007) · Zbl 1114.65045
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.