Variants of Steffensen-Secant method and applications. (English) Zbl 1200.65036

Let \(f:D \rightarrow\mathbb R\) be a sufficiently differentiable function with a simple root \(a \in D\), with \(D \subset\mathbb R\) an open set. The authors define a parametric variant of the Steffensen-secant method as follows. Let \(A_{n+1} = f(x_n)\) and \(B_{n+1} = f(x_n+\lambda_nA_{n+1})\). Also, let \(\bar{x_{n+1}} = x_n-\frac{\lambda_nA^2_{n+1}}{B_{n+1}-A_{n+1}}\), and \(C = f(\bar{x_{n+1}})\). Then
\[ x_{n+1} = x_n - \frac{\lambda_nA^3_{n+1}}{[B_{n+1}-A_{n+1}][A_{n+1}-C_{n+1}]}. \]
Note that this only requires three evaluations of the function at each step. The authors prove that one obtains at least cubic convergence.
For three judicious choices of \(\lambda_n\), the authors are able to prove \(\lim_{n\rightarrow \infty}\lambda_n = - 1/f'(a)\), and thus that the third-order asymptotic convergence constant is 0. This gives super cubic convergence for these choices of \(\lambda_n\).
The authors also present modifications of Steffensen-secant methods for multiple roots. With appropriate hypotheses, a modified parametric variant of the Steffensen-secant method is linearly convergent, and the modified variant is quadratically convergent.
In addition to proving the efficiency of their methods, the authors show by experimentation that their methods are nearly always faster and converge faster than previously proposed methods. They also apply their methods to the “multiple-shooting method” for solving boundary value problems.


65H05 Numerical computation of solutions to single equations
65L10 Numerical solution of boundary value problems involving ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
Full Text: DOI


[1] Ortega, J.M.; Rheinboldt, W.G., Iterative solution of nonlinear equations in several variables, (1970), Academic Press New York · Zbl 0241.65046
[2] Halley, E., Methodus nova, accurata and facillis inveniendi radices aequationum quarumcumque generaliter, sine praevia reductione, Philos. trans. roy. soc. London, 18, 136-148, (1694)
[3] Candela, V.; Marquina, A., Recurrence relations for rational cubic methods I: the Halley method, Computing, 44, 169-184, (1990) · Zbl 0701.65043
[4] Gutierrez, J.M.; Herandez, M.A., A family of chebyshev – halley type methods in Banach spaces, Bull. aust. math. soc., 55, 113-130, (1997) · Zbl 0893.47043
[5] Traub, J.F., Iterative methods for the solution of equations, (1964), Prentice-Hall Englewood Cliffs, New Jersey · Zbl 0121.11204
[6] Amat, S.; Busquier, S., A two-step steffensen’s method under modified convergence conditions, J. math. anal. appl., 324, 1084-1092, (2006) · Zbl 1103.65060
[7] Zheng, Q.; Wang, J.; Zhao, P.; Zhang, L., A Steffensen-like method and its higher-order variants, Appl. math. comput., 214, 10-16, (2009) · Zbl 1179.65052
[8] Zheng, Q.; Liu, Z.; Bai, R., Variations of Steffensen method with cubic convergence, J. comput. anal. appl., 9, 431-436, (2007) · Zbl 1126.65048
[9] Zheng, Q.; Wang, C.; Sun, G., A kind of Steffensen method and its third-order variant, J. comput. anal. appl., 11, 234-238, (2009) · Zbl 1221.65120
[10] Zhang, H.; Li, D.-S.; Liu, Y.-Z., A new method of secant-like for nonlinear equations, Commun. nonlinear sci. numer. simul., 14, 2923-2927, (2009) · Zbl 1221.65114
[11] Jain, P., Steffensen-type methods for solving nonlinear equations, Appl. math. comput., 194, 527-533, (2007) · Zbl 1193.65063
[12] Alarcón, V.; Amat, S.; Busquier, S.; López, D.J., A steffensen’s type method in Banach spaces with applications on boundary-value problems, J. comput. appl. math., 216, 243-250, (2008) · Zbl 1139.65040
[13] Hernández, M.A.; Rubio, M.J., The secant method and divided differences Hölder continuous, Appl. math. comput., 124, 137-149, (2001) · Zbl 1024.65043
[14] Babajee, D.K.R.; Dauhoo, M.Z., An analysis of the properties of the variants of newtons method with third-order convergence, Appl. math. comput., 183, 659-684, (2006) · Zbl 1123.65036
[15] Homeier, H.H.H., On Newton-type methods for multiple roots with cubic convergence, J. comput. appl. math., 231, 249-254, (2009) · Zbl 1168.65024
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.