An optimal fourth order iterative method for solving nonlinear equations and its dynamics. (English) Zbl 1337.30009
Summary: We present a new fourth order method for finding simple roots of a nonlinear equation \(f(x) = 0\). In terms of computational cost, per iteration the method uses one evaluation of the function and two evaluations of its first derivative. Therefore, the method has optimal order with efficiency index 1.587 which is better than efficiency index 1.414 of Newton method and the same with Jarratt method and King’s family. Numerical examples are given to support that the method thus obtained is competitive with other similar robust methods. The conjugacy maps and extraneous fixed points of the presented method and other existing fourth order methods are discussed, and their basins of attraction are also given to demonstrate their dynamical behavior in the complex plane.
MSC:
| 30C15 | Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral) |
| 65A99 | Tables in numerical analysis |
Software:
MathematicaReferences:
| [1] | Ortega, J. M.; Rheinboldt, W. C., Iterative Solution of Nonlinear Equations in Several Variables (1970), New York, NY, USA: Academic Press, New York, NY, USA · Zbl 0241.65046 |
| [2] | Chapra, S. C.; Canale, R. P., Numerical Methods for Engineers (1988), New York, NY, USA: McGraw-Hill, New York, NY, USA |
| [3] | Traub, J. F., Iterative Methods for the Solution of Equations (1977), New York, NY, USA: Chelsea Publishing Company, New York, NY, USA · Zbl 0121.11204 |
| [4] | Petković, M. S.; Neta, B.; Petković, L. D.; Dzunić, J., Multipoint Methods for Solving Nonlinear Equations (2013), Academic Press · Zbl 1286.65060 |
| [5] | Ostrowski, A. M., Solution of Equations and Systems of Equations (1960), New York, NY, USA: Academic Press, New York, NY, USA · Zbl 0115.11201 |
| [6] | Neta, B., Numerical Methods for the Solution of Equations (1983), Monterey, Calif, USA: Net-A-Sof, Monterey, Calif, USA · Zbl 0514.65029 |
| [7] | Weerakoon, S.; Fernando, T. G. I., A variant of Newton’s method with accelerated third order convergence, Applied Mathematics Letters, 13, 8, 87-93 (2000) · Zbl 0973.65037 |
| [8] | Halley, E., A new and general method of finding the roots of equations, Philosophical Transactions of the Royal Society of London, 18, 136-138 (1694) |
| [9] | Hansen, E.; Patrick, M., A family of root finding methods, Numerische Mathematik, 27, 3, 257-269 (1977) · Zbl 0361.65041 · doi:10.1007/bf01396176 |
| [10] | King, R. F., A family of fourth order methods for nonlinear equations, SIAM Journal on Numerical Analysis, 10, 5, 876-879 (1973) · Zbl 0266.65040 · doi:10.1137/0710072 |
| [11] | Chun, C., A family of composite fourth-order iterative methods for solving nonlinear equations, Applied Mathematics and Computation, 187, 2, 951-956 (2007) · Zbl 1116.65054 · doi:10.1016/j.amc.2006.09.009 |
| [12] | Chun, C.; Ham, Y., A one-parameter fourth-order family of iterative methods for nonlinear equations, Applied Mathematics and Computation, 189, 1, 610-614 (2007) · Zbl 1122.65330 · doi:10.1016/j.amc.2006.11.113 |
| [13] | Chun, C.; Lee, M. Y.; Neta, B.; Džunić, J., On optimal fourth-order iterative methods free from second derivative and their dynamics, Applied Mathematics and Computation, 218, 11, 6427-6438 (2012) · Zbl 1277.65031 · doi:10.1016/j.amc.2011.12.013 |
| [14] | Cordero, A.; Hueso, J. L.; Martínez, E.; Torregrosa, J. R., New modifications of Potra-Pták’s method with optimal fourth and eighth orders of convergence, Journal of Computational and Applied Mathematics, 234, 10, 2969-2976 (2010) · Zbl 1191.65048 · doi:10.1016/j.cam.2010.04.009 |
| [15] | Kou, J.; Wang, X.; Li, Y., Some eighth-order root-finding three-step methods, Communications in Nonlinear Science and Numerical Simulation, 15, 3, 536-544 (2010) · Zbl 1221.65115 · doi:10.1016/j.cnsns.2009.04.013 |
| [16] | Kou, J.; Li, Y.; Wang, X., A composite fourth-order iterative method for solving non-linear equations, Applied Mathematics and Computation, 184, 2, 471-475 (2007) · Zbl 1114.65045 · doi:10.1016/j.amc.2006.05.181 |
| [17] | Jarratt, P., Some fourth order multipoint methods for solving equations, Mathematics of Computation, 20, 434-437 (1966) · Zbl 0229.65049 |
| [18] | Curry, J. H.; Garnett, L.; Sullivan, D., On the iteration of a rational function: computer experiments with Newton’s method, Communications in Mathematical Physics, 91, 2, 267-277 (1983) · Zbl 0524.65032 · doi:10.1007/bf01211162 |
| [19] | Vrscay, E. R., Julia sets and mandelbrot-like sets associated with higher order Schröder rational iteration functions: a computer assisted study, Mathematics of Computation, 46, 173, 151-169 (1986) · Zbl 0597.58013 · doi:10.2307/2008220 |
| [20] | Vrscay, E. R.; Gilbert, W. J., Extraneous fixed points, basin boundaries and chaotic dynamics for Schröder and König rational iteration functions, Numerische Mathematik, 52, 1, 1-16 (1987) · Zbl 0612.30025 · doi:10.1007/bf01401018 |
| [21] | Amat, S.; Busquier, S.; Plaza, S., Review of some iterative root-finding methods from a dynamical point of view, Scientia, Series A, 10, 3-35 (2004) · Zbl 1137.37316 |
| [22] | Amat, S.; Busquier, S.; Plaza, S., Dynamics of the king and jarratt iterations, Aequationes Mathematicae, 69, 3, 212-223 (2005) · Zbl 1068.30019 · doi:10.1007/s00010-004-2733-y |
| [23] | Blanchard, P., The dynamics of Newton’s method, Proceedings of Symposia in Applied Mathematics, 49, 139-154 (1994) · Zbl 0853.58086 |
| [24] | Chicharro, F.; Cordero, A.; Gutiérrez, J. M.; Torregrosa, J. R., Complex dynamics of derivative-free methods for nonlinear equations, Applied Mathematics and Computation, 219, 12, 7023-7035 (2013) · Zbl 1286.65059 · doi:10.1016/j.amc.2012.12.075 |
| [25] | Neta, B.; Chun, C.; Scott, M., Basins of attraction for optimal eighth order methods to find simple roots of nonlinear equations, Applied Mathematics and Computation, 227, 567-592 (2014) · Zbl 1364.65110 · doi:10.1016/j.amc.2013.11.017 |
| [26] | Babajee, D. K. R.; Cordero, A.; Soleymani, F.; Torregrosa, J. R., On improved three-step schemes with high efficiency index and their dynamics, Numerical Algorithms, 65, 1, 153-169 (2014) · Zbl 1300.65028 · doi:10.1007/s11075-013-9699-6 |
| [27] | Gautschi, W., Numerical Analysis: An Introduction (1997), Boston, Mass, USA: Birkhaüser, Boston, Mass, USA · Zbl 0877.65001 |
| [28] | Wolfram, S., The Mathematica Book (2003), Wolfram Media |
| [29] | Varona, J. L., Graphic and numerical comparison between iterative methods, The Mathematical Intelligencer, 24, 1, 37-46 (2002) · Zbl 1003.65046 |
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