A biparametric family of four-step sixteenth-order root-finding methods with the optimal efficiency index. (English) Zbl 1225.65051
A biparametric family of four-step multipoint iterative methods to solve nonlinear equations is given. Convergence of sixteenth order is established, and the efficiency index is found to be the optimal one \(16^{1/5}\approx 1.741101\). Numerical experiments compare the new method with existing methods.
Reviewer: János Karátson (Budapest)
MSC:
| 65H05 | Numerical computation of solutions to single equations |
Keywords:
optimal order; biparametric family; asymptotic error constant; efficiency index; four-step multipoint iterative methods; nonlinear equations; convergence of sixteenth order; numerical experimentsSoftware:
MathematicaReferences:
| [1] | Ahlfors, L. V., Complex Analysis (1979), McGraw-Hill Book, Inc. · Zbl 0395.30001 |
| [2] | Bi, W.; Ren, H.; Wu, Q., Three-step iterative methods with eighth-order convergence for solving nonlinear equations, Journal of Computational and Applied Mathematics, 225, 105-112 (2009) · Zbl 1161.65039 |
| [3] | Bi, W.; Wu, Q.; Ren, H., A new family of eighth-order iterative methods for solving nonlinear equations, Applied Mathematics and Computation, 214, 236-245 (2009) · Zbl 1173.65030 |
| [4] | Chun, C.; Neta, B., Some modification of Newtons method by the method of undetermined coefficients, Computers and Mathematics with Applications, 56, 2528-2538 (2008) · Zbl 1165.65343 |
| [5] | Geum, Y. H.; Kim, Y. I., A multi-parameter family of three-step eighth-order iterative methods locating a simple root, Applied Mathematics and Computation, 215, 3375-3382 (2010) · Zbl 1183.65049 |
| [6] | Kung, H. T.; Traub, J. F., Optimal order of one-point and multipoint iteration, Journal of the Association for Computing Machinery, 21, 643-651 (1974) · Zbl 0289.65023 |
| [7] | Liu, L.; Wang, X., Eighth-order methods with high efficiency index for solving nonlinear equations, Applied Mathematics and Computation, 215, 3449-3454 (2010) · Zbl 1183.65051 |
| [8] | Neta, B., On a family of multipoint methods for non-linear equations, International Journal of Computer Mathematics, 9, 353-361 (1981) · Zbl 0466.65027 |
| [9] | Traub, J. F., Iterative Methods for the Solution of Equations (1982), Chelsea Publishing Company · Zbl 0472.65040 |
| [10] | Wang, X.; Liu, L., Modified Ostrowski’s method with eighth-order convergence and high efficiency index, Applied Mathematics Letters, 23, 549-554 (2010) · Zbl 1191.65050 |
| [11] | Wang, X.; Liu, L., Two new families of sixth-order methods for solving non-linear equations, Applied Mathematics and Computation, 213, 73-78 (2009) · Zbl 1170.65032 |
| [12] | Wolfram, Stephen, The Mathematica Book (2003), Wolfram Media · Zbl 0924.65002 |
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