Steffensen type methods for approximating solutions of differential equations


The implicit methods for numerical solving of ODEs lead to nonlinear equations which are usually solved by the Newton method. We study the use of a Steffensen type method instead, and we give conditions under which this method provides bilateral approximations for the solution of these equations; this approach offers a more rigorous control of the errors. Moreover, the method can be applied even in the case when certain functions are not differentiable on the definition domain. The convergence order is the same as for Newton method.


Flavius Patrulescu
(Tiberiu Popoviciu Institute of Numerical Analysis,
Romanian Academy)


initial value problems; stiff equations; Steffensen method; Newton method; convergence order

Cite this paper as

F. Pătrulescu, Steffensen type methods for approximating solutions of differential equations, Studia Universitatis Babeş-Bolyai, seria Mathematica, vol. 56, no. 2 (2011), pp. 505-515.


About this paper

Publisher Name

Universitatea Babeş-Bolyai, Cluj-Napoca; Cluj University Press, Cluj-Napoca

Print ISSN


Online ISSN






Google Scholar


Paper in html format


[1] Agratini, I. Chiorean, Gh. Coman, R. Trimbitas, Numerical Analysis and Approximation Theory III, Presa Universitara Clujeana, Cluj-Napoca (2002) .
[2] C. Butcher, Numerical Methods for Ordinary Differential Equations, John Willey & Sons, Chichester (2003).
[3] Catinas, Methods of Newton and Newton-Krylov Type, Risoprint, Cluj-Napoca (2007).
[4] Crouzeix, A.L. Mignot, Analyse numerique des equations differentielles, Masson, Paris (1989).
[5] D. Lambert, Numerical Methods for Ordinary Differential Systems. The Initial Value Problem, John Wiley & Sons, Chichester (1990).
[6] Matheij, J. Molennar, Ordinary Differential Equations in Theory and Practice, SIAM, Philadelphia (2002).
[7] Pavaloiu, On the monotonicity of the sequences of approximations obtained by Steffensen’s method, Mathematica, 35, no. 1 (1993), 95-101.
[8] Pavaloiu,Bilateral approximation for the solution of scalar Equations, Rev. Anal. Numer. Theor. Approx.,23, no. 1 (1994), 95-101.
[9] Pavaloiu, Approximation of the roots of equations by Aitken-Steffensen-type monotonic sequences, Calcolo, 32 (1995), 69-82.
[10] Pavaloiu, Aitken-Steffensen type methods for nonsmooth functions(III), Rev. Anal. Numer. Theor. Approx., 32, no. 1 (2003), 73-77.


Related Posts