Abstract
We present some new conditions which assure that the Aitken-Steffensen method yields bilateral approximation for the solution of a nonlinear scalar equation. The auxiliary functions appearing in the method are constructed under the hypothesis that the nonlinear application is not differentiable on an interval containing the solution.
Author
Ion Păvăloiu
(Tiberiu Popoviciu Institute of Numerical Analysis)
Keywords
nonlinear equations in R; Aitken-Steffensen method; monotone iterations; bilateral approximations.
PDF-LaTeX file (on the journal website).
Cite this paper as:
I. Păvăloiu, Aitken-Steffensen-type methods for nonsmooth functions (II), Rev. Anal. Numér. Théor. Approx., 31 (2002) no. 2, pp. 191-196. https://doi.org/10.33993/jnaat312-724
About this paper
Publisher Name
Print ISSN
1222-9024
Online ISSN
2457-8126
References
[1] Balazs, M., A bilateral approximating method for finding the real roots of real equations , Rev. Anal. Num ́er. Th ́eor. Approx., 21, no. 2, pp. 111–117, 1992.
[2] Casulli, V. and Trigiante, D., The convergence order for iterative multipoint procedures, Calcolo, 13, no. 1, pp. 25–44, 1977.
[3] Cobzas ̧, S., Mathematical Analysis, Presa Universitar a Clujean a, Cluj-Napoca, 1997 (in Romanian).
[4] Ostrowski, A. M., Solution of Equations and Systems of Equations, Academic Press, New York, 1960.
[5] Pavaloiu, I., On the monotonicity of the sequences of approximations obtained by Steffensens’s method, Mathematica (Cluj), 35 (58), no. 1, pp. 71–76, 1993.
[6] Pavaloiu, I., Bilateral approximations for the solutions of scalar equations, Rev. Anal. Numer. Theor. Approx., 23 , no. 1, pp. 95–100, 1994.
[7] Pavaloiu, I., Approximation of the roots of equations by Aitken-Steffensen-type monotonic sequences, Calcolo, 32, no. 1–2, pp. 69–82, 1995.
[8] Pavaloiu, I., Aitken-Steffensen-type methods for nonsmooth functions (I), Rev. Anal.Numer. Theor. Approx., 31, no. 1, pp. 111–116, 2002.
[9] Traub, F. J., Iterative Methods for the Solution of Equations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964.
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Aitken-Steffensen-type methods for nonsmooth functions (II)∗
Aitken-Steffensen-type methods
Aitken-Steffensen-type methods
for nonsmooth functions
(II)∗
(Date: January 16, 2002.)
Abstract.
We present some new conditions which assure that the Aitken-Steffensen method yields bilateral approximation for the solution of a nonlinear scalar equation. The auxiliary functions appearing in the method are constructed under the hypothesis that the nonlinear application is not differentiable on an interval containing the solution.
∗This work
has been supported by the Romanian Academy under grant GAR
45/2002.
†“T. Popoviciu” Institute of Numerical Analysis, P.O. Box 68–1, 3400
Cluj-Napoca, Romania (pavaloiu@ictp.acad.ro).
{amssubject}
65H05.
{keyword}
Aitken-Steffensen iterations.
1. Introduction
In this note we continue the study of the convergence of the Aitken-Steffensen-type iterations
| (1) |
for solving
| (2) |
where
Under supplementary assumptions we shall show, as in [7], that (2) generates three monotone sequences, converging to the solution of (1).
Regarding the monotonicity and convexity of we shall consider the notions introduced in [3]. We shall also use Theorem 1 and Lemma 2 from [8].
For defining the functions and , we shall consider such that and defining then:
| (4) |
| (5) |
Regarding we shall make the following assumptions.
-
i.
-
ii.
is increasing on
-
iii.
is convex on and continuous at and ;
-
iv.
if is the solution of (1), then is differentiable at .
Remarks.
1∘ Hypothesiis iii. ensures the continuity of on , and therefore the existence of . Hypothesis ii. ensures the uniqueness of .
2∘ From hypotheses ii, iii and [8, Lm. 1.1], it follows that for any one obtains
| (6) |
i.e., is increasing and is decreasing on . ∎
Let be such that
-
a)
-
b)
2. The convergence of the Aitken-Steffensen-type iterations
We shall study in the following the convergence of the sequences to the solution . We obtain the following result.
Theorem 1.
Proof.
By and it follows , while by and the fact that is increasing it follows , i.e. .
Now, since , is decreasing one gets , i.e. the following relations hold:
| (8) |
Let since implies .
Repeating this reason, the induction shows that (7) holds for . This attracts in turn that the sequences and are increasing, i.e., statements j and jj.
We show next that is decreasing. Indeed, by for we get since is decreasing. Inequalities , , show that .
Let us notice that the sequences , , are monotone and bounded, so they converge. Let , and . We show that
We prove first that . Assume the contrary, , e.g. . Obviously, and Let be a positive number. Then there exists such that for .
This implies that
so is not the exact upper bound of the elements of the sequence . Hence, clearly, ,
i.e., . Since , it follows that
since is the unique solution of equation ∎
The above relations show that we have a control of the error at each iteration step, justified by
Analogously,
where we have denoted and by Lemma 2 from [8] This relation, together with the decreasing of lead to i.e.,
References
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