## Another simple way of deriving several iterative functions to solve nonlinear equations.(English)Zbl 1268.65064

Summary: We present another simple way of deriving several iterative methods for solving nonlinear equations numerically. The presented approach of deriving these methods is based on exponentially fitted osculating straight line. These methods are the modifications of Newton’s method. Also, we obtain well-known methods as special cases, for example, Halley’s method, super-Halley method, Ostrowski’s square-root method, Chebyshev’s method, and so forth. Further, new classes of third-order multipoint iterative methods free from a second-order derivative are derived by semidiscrete modifications of cubically convergent iterative methods. Furthermore, a simple linear combination of two third-order multipoint iterative methods is used for designing new optimal methods of order four.

### MSC:

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

### References:

 [1] S. Amat, S. Busquier, and J. M. Gutiérrez, “Geometric constructions of iterative functions to solve nonlinear equations,” Journal of Computational and Applied Mathematics, vol. 157, no. 1, pp. 197-205, 2003. · Zbl 1024.65040 · doi:10.1016/S0377-0427(03)00420-5 [2] V. Kanwar, S. Singh, and S. Bakshi, “Simple geometric constructions of quadratically and cubically convergent iterative functions to solve nonlinear equations,” Numerical Algorithms, vol. 47, no. 1, pp. 95-107, 2008. · Zbl 1137.65030 · doi:10.1007/s11075-007-9149-4 [3] J. F. Traub, Iterative Methods for the Solution of Equations, Prentice-Hall Series in Automatic Computation, Prentice-Hall, Englewood Cliffs, NJ, USA, 1964. · Zbl 0121.11204 [4] J. M. Gutiérrez and M. A. Hernández, “An acceleration of Newton’s method: super-Halley method,” Applied Mathematics and Computation, vol. 117, no. 2-3, pp. 223-239, 2001. · Zbl 1023.65051 · doi:10.1016/S0096-3003(99)00175-7 [5] E. Hansen and M. Patrick, “A family of root finding methods,” Numerische Mathematik, vol. 27, no. 3, pp. 257-269, 1976/77. · Zbl 0361.65041 · doi:10.1007/BF01396176 [6] A. M. Ostrowski, Solutions of Equations and System of Equations, Academic Press, New York, NY, USA, 1960. · Zbl 0115.11201 [7] W. Werner, “Some improvements of classical iterative methods for the solution of nonlinear equations,” in Numerical Solution of Nonlinear Equations, vol. 878 of Lecture Notes in Mathematics, pp. 426-440, Springer, Berlin, Germany, 1981. · Zbl 0494.65033 · doi:10.1007/BFb0090691 [8] A. M. Ostrowski, Solution of Equations in Euclidean and Banach Spaces, Academic Press, London, UK, 1973. · Zbl 0304.65002 [9] L. W. Johnson and R. D. Roiess, Numerical Analysis, Addison-Wesely, Reading, Mass, USA, 1977. [10] V. Kanwar, R. Behl, and K. K. Sharma, “Simply constructed family of a Ostrowski’s method with optimal order of convergence,” Computers & Mathematics with Applications, vol. 62, no. 11, pp. 4021-4027, 2011. · Zbl 1236.65054 · doi:10.1016/j.camwa.2011.09.039 [11] R. F. King, “A family of fourth order methods for nonlinear equations,” SIAM Journal on Numerical Analysis, vol. 10, pp. 876-879, 1973. · Zbl 0266.65040 · doi:10.1137/0710072 [12] F. A. Potra and V. Pták, “Nondiscreate introduction and iterative processes,” in Research Notes in Mathematics, Pitman, Boston, Mass, USA, 1984. · Zbl 0549.41001 [13] H. T. Kung and J. F. Traub, “Optimal order of one-point and multipoint iteration,” Journal of the Association for Computing Machinery, vol. 21, pp. 643-651, 1974. · Zbl 0289.65023 · doi:10.1145/321850.321860 [14] P. Jarratt, “Some fourth-order multipoint methods for solving equations,” BIT, vol. 9, pp. 434-437, 1965. · Zbl 0229.65049 · doi:10.2307/2003602 [15] F. Soleymani, “Optimal fourth-order iterative methods free from derivatives,” Miskolc Mathematical Notes, vol. 12, no. 2, pp. 255-264, 2011. · Zbl 1265.65101 [16] M. Sharifi, D. K. R. Babajee, and F. Soleymani, “Finding the solution of nonlinear equations by a class of optimal methods,” Computers & Mathematics with Applications, vol. 63, no. 4, pp. 764-774, 2012. · Zbl 1247.65066 · doi:10.1016/j.camwa.2011.11.040 [17] F. Soleymani, S. K. Khattri, and S. Karimi Vanani, “Two new classes of optimal Jarratt-type fourth-order methods,” Applied Mathematics Letters, vol. 25, no. 5, pp. 847-853, 2012. · Zbl 1239.65030 · doi:10.1016/j.aml.2011.10.030 [18] Behzad Ghanbari, “A new general fourth-order family of methods for finding simple roots of nonlinear equations,” Journal of King Saud University-Science, vol. 23, pp. 395-398, 2011. [19] E. Schröder, “Über unendlichviele algorithm zur au osung der gleichungen,” Annals of Mathematics, vol. 2, pp. 317-365, 1870.
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.