Abstract
(soon)
Authors
Ion Păvăloiu
(Tiberiu Popoviciu Institute of Numerical Analysis)
Keywords
(soon)
Scanned paper: on the journal website.
Cite this paper as:
I. Păvăloiu, A convergency theorem concerning the chord method, Rev. Anal. Numér. Théor. Approx., 22 (1993) no. 1, pp. 83-85.
About this paper
Publisher Name
Article on the journal website
Print ISSN
1222-9024
Online ISSN
2457-8126
References
[1] Argyros, K.I., The Secant Method and Point of Nonlinear Operators Mh. Math. 106 (1988), 85–94.
[2] Pavaloiu, I., Introduction in the theory of approximation of equations solutions, Dacia Ed., Cluj-Napoca 1976, (in Romanian).
[3] Pavaloiu, I., Remarks on the secant method for the solution of nonlinear operational equations, Research Seminars. Seminar on Mathematical analysis, Preprint 7, 127–132 (1991).
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A convergency theorem concerning the chord method
A convergency theorem concerning
A convergency theorem concerning
the chord method
Let be a Banach space, and let be a mapping to solve the equation:
| (1) |
the chord method is well known, consisting of approximating the solution of (1) by elements of the sequence generated by the following relations:
| (2) |
where stands for the divided difference of on It is clear that to generate the elements of the sequence by means of (2) we must ensure ourselves that at every iteration step the linear mapping is invertible. The mathematical literature dealing with the convergency of the chord method contains results which state by hypothesis that the mapping admits a bounded inverse for every where is a subset of
In this note we intend to establish convergency conditions for the method (2), supposing the existence of the inverse mapping only for the divided difference
Let be a real number, and write
Theorem.
If the mapping the real number and the element fulfil the conditions:
-
(i)
the mapping admits a bounded inverse mapping, and
-
(ii)
the bilinear mapping (the second order divided difference of on ) is bounded for every that is,
-
(iii)
-
(iv)
where
and -
(v)
then the following properties hold:
-
(j)
for every
-
(jj)
the mapping admits bounded inverse for every
-
(jjj)
equation (1) has at least one solution
-
(jv)
the sequence is convergent, and
-
(v)
Proof.
We shall firstly show that for every the following inequality holds:
| (3) |
Taking into account hypothesis (ii) and the definition of the second order divided difference [2], it results:
From the above inequality and hypothesis (i) there follows (3).
Using Banach’s lemma on inverse mapping continuousness, it results from (3) that there exists and:
Suppose now that the following properties hold:
-
(a)
-
(b)
and prove that they hold for too.
Indeed, to prove that we estimate the difference:
To prove (b) for we use Newton’s identity:
| (4) |
and the obvious identity:
| (5) |
Applying (4) and taking into account (2) and (5), we deduce:
and writing we obtain:
that is, the property (b) holds for too.
From these relations, for every we deduce:
| (6) | ||||
from which, taking into account the fact that there follows that the sequence is fundamental.
It is obvious that ∎
Remark.
References
- [1] Argyros, K.I., The Secant Method and Point of Nonlinear Operators Mh. Math. 106 (1988), 85–94.
- [2] ††margin: clickable Păvăloiu, I., Introduction in the theory of approximation of equations solutions, Dacia Ed., Cluj-Napoca 1976, (in Romanian).
- [3] ††margin: clickable Păvăloiu, I., Remarks on the secant method for the solution of nonlinear operational equations, Research Seminars. Seminar on Mathematical analysis, Preprint 7, 127–132 (1991).
Received 1.III.1992 Institutul de Calcul
Academia Română, Filiala Cluj-Napoca
C.P.68, Cluj-Napoca
Romania