Abstract
In this note we study the convergence of a generalized Aitken type method for approximating the solutions of nonlinear equations in R. We obtain conditions which assure monotone convergence of the generated sequences. We also obtain a posteriori estimations for the errors.
Author
Keywords
nonlinear equations in R; Aitken method
Cite this paper as:
I. Păvăloiu, E. Cătinaş, On an Aitken type method, Rev. Anal. Numér. Théor. Approx., 36 (2007) no. 2, pp. 173-176. https://doi.org/10.33993/jnaat362-865
PDF-LaTeX file (on the journal website).
About this paper
Publisher Name
Print ISSN
1222-9024
Online ISSN
2457-8126
MR
1222-9024
Online ISSN
2457-8126
Google Scholar citations
[1] M.A. Ostrowski, Solution of Equations and Systems of Equations, Academic Press, New York and London, 1980.
[2] I. Pavaloiu, Solutions Equations by Interpolation, Dacia, Cluj-Napoca, 1981 (in Romanian).
[3] I. Pavaloiu, Bilateral approximation of solution of equations by order three Steffensen-type methods , Studia Univ. “Babes-Bolyai”, Mathematica, LI (2006) no. 3, pp. 105–114.
[4] I. Pavaloiu and N. Pop, Interpolation and Applications, Risoprint, Cluj-Napoca, 2005 (in Romanian).
[5] I. Pavaloiu, Approximation of the roots of equations by Aitken-Steffensen-type monotonic sequences , Calcolo, 32 (1995) nos. 1–2, pp. 69–82.
[6] I. Pavaloiu and E. Catinas, On a Steffensen type method, Proceedings of the Ninth International Symposion on Symbolic and Numeric Algoritms for Scientific Computing (SYNASC 2007), September 26–29, 2007, Timisoara, Romania, IEEE Computer Society, pp. 369–375.
[7] B.A. Turowicz, Sur les derivees d’ordre superieur d’une fonction inverse , Ann. Polon. Math., 8 (1960), pp. 265–269.
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ON AN AITKEN TYPE METHOD∗
Table 1. Numerical results
ON AN AITKEN TYPE METHOD∗
(Date: August 14, 2006.)
Abstract.
In this note we study the convergence of a generalized Aitken type method for approximating the solutions of nonlinear equations in . We obtain conditions which assure monotone convergence of the generated sequences. We also obtain a posteriori estimations for the errors.
Key words and phrases:
Nonlinear equations, Aitken method.1991 Mathematics Subject Classification:
65H05.
∗This work was supported by MEdC under grant 2CEEX06-11-96/19.09.2006.
†“T. Popoviciu” Institute of Numerical Analysis, P.O. Box 68-1,
Cluj-Napoca, Romania, e-mail: {pavaloiu,ecatinas}@ictp.acad.ro.
1. Introduction
Consider the equation
| (1) |
where
Consider also the following two equations, both equivalent to (1):
| (2) |
In order to approximate a root of (1), we consider the sequence generated by the following relations:
| (3) | ||||
where denotes the first order divided difference of on and
Relation (3) suggests an Aitken type method. If we assume that and for all denoting it is known that there exists and Moreover, the following relation holds
| (4) |
In [4] was studied the convergence of a Steffensen type method, analogous to (3). In order to obtain a control of the error at each iteration step, we have studied in [7] the cases when the generalized Steffensen type method leads to monotone approximations. Under some reasonable conditions of monotonicity and convexity (concavity) on in [7] were obtained monotone sequences which yield bilateral approximations to the solution of (1). A condition such that the Steffensen type method studied in [7] to lead to monotone approximations is that the sign of the expression
| (5) |
to be negative:
In this note we shall show that if one may construct functions and and on may choose the initial approximations such that, under some reasonable hypotheses on the sequence (3) converges monotonically.
In an analogous fashion as in [7], we can easily show that for a given then exists where is the smallest interval containing the points such that
| (6) |
2. The convergence of the Aitken method
We consider the following hypotheses on the functions and on the initial approximation
The following result holds:
Theorem 1.
If functions and the initial approximation verify hypotheses i.–viii., then the sequences generated by (3) have the following properties:
-
j.
the sequence is increasing;
-
jj.
the sequences are decreasing;
-
jjj.
-
jv.
the following relations hold:
Proof.
Let be an approximation of the solution such that From ii. and iii. it follows By v. we have Taking into account the fact that and using hypothesis iii. and iv., from (3) it follows that Since , by iii., i. and (6) we get
By v. and it follows and By v. it also follows that and These relations imply j. and jj.
In order to prove jjj., let For from (3) we obtain and hence Obviously, from the continuity of and it follows that and
Relations jv. are obvious, and allow us to evaluate the a posteriori error at each iteration step. ∎
3. Construction of functions and
The basic conditions on and on v. and vii.
We shall consider a function given by
| (7) |
where .
If then is decreasing. This attracts
i.e.,
| (8) |
But since and we obtain
or
| (9) |
In order to obtain we need that which leads to which in turn holds if is sufficiently close to
Let be two numbers such that
then functions given by
verify hypotheses v. and vii.
4. Numerical example
Consider the equation
with
Obviously, for all and If in (7) we take and then and verify v. and vii.
We have:
Obviously, and Function is given by
In the table below we have obtained the following results.
| 1 | 0.5235987755982988 | 1.127824791583588 | 1.248669994780646 | 1.208452031970579e+0 |
| 2 | 1.027717814817341 | 1.030632925047758 | 1.031215947093841 | 5.830220460833369e3 |
| 3 | 1.029866528928396 | 1.029866529462959 | 1.029866529569871 | 1.069125232788792e9 |
| 4 | 1.029866529322259 | 1.029866529322259 | 1.029866529322259 | 0 |
The numerical results confirm that inequalities
hold.
References
- [1]
- [2] Ostrowski, M.A., Solution of Equations and Systems of Equations, Academic Press, New York and London, 1980.
- [3] Păvăloiu, I., Solutions Equations by Interpolation, Dacia, Cluj-Napoca, 1981 (in Romanian).
- [4] Păvăloiu, I., Bilateral approximation of solution of equations by order three Steffensen-Type methods, Studia Univ. “Babeş-Bolyai”, Mathematica, LI (2006) no. 3, pp. 105–114.
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- [6] Păvăloiu, I., Approximation of the roots of equations by Aitken-Steffensen-type monotonic sequences, Calcolo, 32 (1995) nos. 1–2, pp. 69–82.
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