Some real-life applications of a newly constructed derivative free iterative scheme. (English) Zbl 1416.65136
Summary: In this study, we present a new higher-order scheme without memory for simple zeros which has two major advantages. The first one is that each member of our scheme is derivative free and the second one is that the present scheme is capable of producing many new optimal family of eighth-order methods from every 4-order optimal derivative free scheme (available in the literature) whose first substep employs a Steffensen or a Steffensen-like method. In addition, the theoretical and computational properties of the present scheme are fully investigated along with the main theorem, which demonstrates the convergence order and asymptotic error constant. Moreover, the effectiveness of our scheme is tested on several real-life problems like Van der Waal’s, fractional transformation in a chemical reactor, chemical engineering, adiabatic flame temperature, etc. In comparison with the existing robust techniques, the iterative methods in the new family perform better in the considered test examples. The study of dynamics on the proposed iterative methods also confirms this fact via basins of attraction applied to a number of test functions.
MSC:
| 65H05 | Numerical computation of solutions to single equations |
References:
| [1] | Griewank, A.; Walther, A.; ; Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation: Philadelphia, PA, USA 2008; . · Zbl 1159.65026 |
| [2] | Behl, R.; Maroju, P.; Motsa, S.S.; A family of second derivative free fourth order continuation method for solving nonlinear equations; J. Comut. Appl. Math.: 2017; Volume 318 ,38-46. · Zbl 1357.65056 |
| [3] | Cordero, A.; Torregrosa, J.R.; A class of Steffensen type methods with optimal order of convergence; Appl. Math. Comput.: 2011; Volume 217 ,7653-7659. · Zbl 1216.65055 |
| [4] | Cordero, A.; Hueso, J.L.; Martínez, E.; Torregrosa, J.R.; Steffensen type methods for solving nonlinear equations; Appl. Math. Comput.: 2012; Volume 236 ,3058-3064. · Zbl 1237.65049 |
| [5] | Kansal, M.; Kanwar, V.; Bhatia, S.; An optimal eighth-order derivative-free family of Potra-Pták’s method; Algorithms: 2015; Volume 8 ,309-320. · Zbl 1461.65078 |
| [6] | Khattri, S.K.; Steihaug, T.; Algorithm for forming derivative-free optimal methods; Numer. Algor.: 2014; Volume 65 ,809-824. · Zbl 1304.65143 |
| [7] | Kung, H.T.; Traub, J.F.; Optimal order of one-point and multi-point iteration; J. ACM: 1974; Volume 21 ,643-651. · Zbl 0289.65023 |
| [8] | Liu, Z.; Zheng, Q.; Zhao, P.; A variant of Steffensen’s method of fourth-order convergence and its applications; Appl. Math. Comput.: 2010; Volume 216 ,1978-1983. · Zbl 1208.65064 |
| [9] | Matthies, G.; Salimi, M.; Sharifi, S.; Varona, J.L.; An optimal eighth-order iterative method with its dynamics; Jpn. J. Ind. Appl. Math.: 2016; Volume 33 ,751-766. · Zbl 1365.65145 |
| [10] | Ren, H.; Wu, Q.; Bi, W.; A class of two-step Steffensen type methods with fourth-order convergence; Appl. Math. Comput.: 2009; Volume 209 ,206-210. · Zbl 1166.65338 |
| [11] | Salimi, M.; Lotfi, T.; Sharifi, S.; Siegmund, S.; Optimal Newton-Secant like methods without memory for solving nonlinear equations with its dynamics; Int. J. Comput. Math.: 2017; Volume 94 ,1759-1777. · Zbl 1391.65135 |
| [12] | Salimi, M.; Nik Long, N.M.A.; Sharifi, S.; Pansera, B.A.; A multi-point iterative method for solving nonlinear equations with optimal order of convergence; Jpn. J. Ind. Appl. Math.: 2018; Volume 35 ,497-509. · Zbl 1406.65034 |
| [13] | Sharifi, S.; Salimi, M.; Siegmund, S.; Lotfi, T.; A new class of optimal four-point methods with convergence order 16 for solving nonlinear equations; Math. Comput. Simul.: 2016; Volume 119 ,69-90. · Zbl 1540.65155 |
| [14] | Soleymani, F.; Vanani, S.K.; Optimal Steffensen-type methods with eighth order of convergence; Comput. Math. Appl.: 2011; Volume 62 ,4619-4626. · Zbl 1236.65056 |
| [15] | Traub, J.F.; ; Iterative Methods for the Solution of Equations: Upper Saddle River, NJ, USA 1964; . · Zbl 0121.11204 |
| [16] | Thukral, R.; Eighth-order iterative methods without derivatives for solving nonlinear equations; Int. Sch. Res. Net. Appl. Math.: 2011; Volume 2011 ,693787. · Zbl 1478.65038 |
| [17] | Zheng, Q.; Li, J.; Huang, F.; An optimal Steffensen-type family for solving nonlinear equations; Appl. Math. Comput.: 2011; Volume 217 ,9592-9597. · Zbl 1227.65044 |
| [18] | Zheng, Q.; Zhao, P.; Huang, F.; A family of fourth-order Steffensen-type methods with the applications on solving nonlinear ODEs; Appl. Math. Comput.: 2011; Volume 217 ,8196-8203. · Zbl 1223.65034 |
| [19] | Sharma, J.R.; Guhaa, R.K.; Gupta, P.; Improved King’s methods with optimal order of convergence based on rational approximations; Appl. Math. Lett.: 2013; Volume 26 ,473-480. · Zbl 1261.65047 |
| [20] | Jarratt, P.; Nudds, D.; The use of rational functions in the iterative solution of equations on a digital computer; Comput. J.: 1965; Volume 8 ,62-65. · Zbl 0296.65020 |
| [21] | Cordero, A.; Torregrosa, J.R.; Variants of Newton’s method using fifth-order quadrature formulas; Appl. Math. Comput.: 2007; Volume 190 ,686-698. · Zbl 1122.65350 |
| [22] | Shacham, M.; Numerical solution of constrained nonlinear algebraic equations; Int. J. Numer. Method Eng.: 1986; Volume 23 ,1455-1481. · Zbl 0597.65044 |
| [23] | Balaji, G.V.; Seader, J.D.; Application of interval Newton’s method to chemical engineering problems; Reliab. Comput.: 1995; Volume 1 ,215-223. · Zbl 0838.65058 |
| [24] | Shacham, M.; An improved memory method for the solution of a nonlinear equation; Chem. Eng. Sci.: 1989; Volume 44 ,1495-1501. |
| [25] | Shacham, M.; Kehat, E.; Converging interval methods for the iterative solution of nonlinear equations; Chem. Eng. Sci.: 1973; Volume 28 ,2187-2193. |
| [26] | Ezquerro, J.A.; Hernández, M.A.; An optimization of Chebyshev’s method; J. Complex.: 2009; Volume 25 ,343-361. · Zbl 1183.65058 |
| [27] | Ferrara, M.; Sharifi, S.; Salimi, M.; Computing multiple zeros by using a parameter in Newton-Secant method; SeMA J.: 2017; Volume 74 ,361-369. · Zbl 1380.65089 |
| [28] | Stewart, B.D.; Attractor Basins of Various Root-Finding Methods; Master’s Thesis: Monterey, CA, USA 2001; . |
| [29] | Varona, J.L.; Graphic and numerical comparison between iterative methods; Math. Intell.: 2002; Volume 24 ,37-46. · Zbl 1003.65046 |
| [30] | Hernández-Paricio, L.J.; Marañón-Grandes, M.; Rivas-Rodríguez, M.T.; Plotting basins of end points of rational maps with Sage; Tbil. Math. J.: 2012; Volume 5 ,71-99. · Zbl 1280.37001 |
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