On some iterative methods for solving nonlinear equations

Authors

  • Emil Cătinaş Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy

Abstract

Let \(h(x):=f(x)+g(x)=0\), with \(f\) differentiable and \(g\) continuous, where \(f,g,h:X \rightarrow X\), \(X\) a Banach space. 

The Newton method cannot be applied for solving \(h(x)=0\), and we propose an iterative method for solving the nonlinear equation, by combining the Newton method (for the differentiable part) with the chord/secant method (for the nondifferentiable part): \[x_{k+1} = \big(f^\prime(x_k)+[x_{k-1},x_k;g]\big)^{-1}(f(x_k)+g(x_k)).\]

We show that the r-convergence order of the method is the same as of the chord/secant method.

We provide some numerical examples and compare different methods for a nonlinear system in \(\mathbb{R}^2\).

 

[Editor note: for a series of papers dealing with the notion of convergence orders, see the convergence orders category]

Downloads

Download data is not yet available.

References

Goldner G., Balazs, M., On the Method of Chord and on Its Modification for Solving the Nonlinear Operatorial Equations, Studii şi cercetări matematice, Tomul 20, 1968, Editura Academiei.

Goldner, G., Balazs, M., Remarks on Divided Differences and Method of Chords, Revista de analiză numerică şi teoria aproximaţiei, 3,1 (1974), pp. 19-30.

Tetsuro, Z., A Note on a Posteriori Error Bound of Zabrejko and Nguen for Zincenko's Iteration, Numer. Funct. Anal. and Optimiz., 9 (9&10) (1987), pp. 987-994, https://doi.org/10.1080/01630568708816270

Tetsuro, Y., Ball Convergence. Theorem and Error Estimates for Certain Iterative Methods for Nonlinear Equations, Japan Journal of Applied Mathematics, 7, 1 (1990), pp. 131-143, https://doi.org/10.1007/bf03167895

Xiaojun, C., Te4tsuro, Y., Convergence Domains of Certain Iterative Methods for Solving Nonlinear Equations, Numer. Funct. Anal. and Optimiz., 10, (1&2), (1989), pp. 37-48, https://doi.org/10.1080/01630568908816289

Downloads

Published

1994-08-01

How to Cite

Cătinaş, E. (1994). On some iterative methods for solving nonlinear equations. Rev. Anal. Numér. Théor. Approx., 23(1), 47–53. Retrieved from https://ictp.acad.ro/jnaat/journal/article/view/1994-vol23-no1-art4

Issue

Section

Articles