Some new variants of Cauchy’s methods for solving nonlinear equations. (English) Zbl 1268.65067

Liu, Tianbao; Li, Hengyan

J. Appl. Math. 2012, Article ID 927450, 13 p. (2012).

Summary: We present and analyze some variants of Cauchy’s methods free from second derivative for obtaining simple roots of nonlinear equations. The convergence analysis of the methods is discussed. It is established that the methods have convergence order three. Per iteration the new methods require two function and one first derivative evaluations. Numerical examples show that the new methods are comparable with the well-known existing methods and give better numerical results in many aspects.

MSC:

65H05

Numerical computation of solutions to single equations

Keywords:

Cauchy’s method; simple roots of nonlinear equations; convergence order three; numerical examples

References:

[1]	A. M. Ostrowski, Solution of equations in Euclidean and Banach spaces, Academic Press, New York, NY, USA, 1973. · Zbl 0304.65002
[2]	J. F. Traub, Iterative Methods for the Solution of Equations, Chelsea, New York, NY, USA, 1977. · Zbl 0383.68041
[3]	J. Kou, “Some variants of Cauchy’s method with accelerated fourth-order convergence,” Journal of Computational and Applied Mathematics, vol. 213, no. 1, pp. 71-78, 2008. · Zbl 1135.65024
[4]	J. Kou, “Fourth-order variants of Cauchy’s method for solving non-linear equations,” Applied Mathematics and Computation, vol. 192, no. 1, pp. 113-119, 2007. · Zbl 1193.65066
[5]	I. K. Argyros, D. Chen, and Q. Qian, “The Jarratt method in Banach space setting,” Journal of Computational and Applied Mathematics, vol. 51, no. 1, pp. 103-106, 1994. · Zbl 0809.65054
[6]	C. Chun, “Some variants of Chebyshev-Halley methods free from second derivative,” Applied Mathematics and Computation, vol. 191, no. 1, pp. 193-198, 2007. · Zbl 1193.65053
[7]	C. Chun, “Some second-derivative-free variants of Chebyshev-Halley methods,” Applied Mathematics and Computation, vol. 191, no. 2, pp. 410-414, 2007. · Zbl 1193.65054
[8]	S. K. Khattri and T. Log, “Constructing third-order derivative-free iterative methods,” International Journal of Computer Mathematics, vol. 88, no. 7, pp. 1509-1518, 2011. · Zbl 1214.65023
[9]	C. Chun, M. Y. Lee, B. Neta, and J. D, “On optimal fourth-order iterative methods free from second derivative and their dynamics,” Applied Mathematics and Computation, vol. 218, no. 11, pp. 6427-6438, 2012. · Zbl 1277.65031
[10]	J. Kou, “Fourth-order variants of Cauchy’s method for solving non-linear equations,” Applied Mathematics and Computation, vol. 192, no. 1, pp. 113-119, 2007. · Zbl 1193.65066
[11]	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
[12]	S. Weerakoon and T. G. I. Fernando, “A variant of Newton’s method with accelerated third-order convergence,” Applied Mathematics Letters, vol. 13, no. 8, pp. 87-93, 2000. · Zbl 0973.65037
[13]	J. R. Sharma, “A composite third order Newton-Steffensen method for solving nonlinear equations,” Applied Mathematics and Computation, vol. 169, no. 1, pp. 242-246, 2005. · Zbl 1084.65054
[14]	A. M. Ostrowski, Solutions of Equations and System of Equations, Academic Press, New York, NY, USA, 1960. · Zbl 0115.11201
[15]	S. K. Khattri, M. A. Noor, and E. Al-Said, “Unifying fourth-order family of iterative methods,” Applied Mathematics Letters, vol. 24, no. 8, pp. 1295-1300, 2011. · Zbl 1225.65053