Abstract
The aim of this paper is to obtain a unified treatment of some iterative methods. We obtain some conditions for which a given equation admits a unique solution in a certain ball. The main obtained result refers to the convergence of the modified chord method.
Authors
Ion Păvăloiu
(Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy)
Keywords
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Cite this paper as:
I. Păvăloiu, A unified treatment of the modified Newton and chord methods, Carpathian J. Math. 25 (2009) no. 2, pp. 192-196.
About this paper
Publisher Name
Universitatea Tehnica din Cluj-Napoca
JSTOR permalink
Print ISSN
1584-2851
Online ISSN
1843-4401
Google Scholar Profile
References
[1] Berinde, V., Iterative Approximation of Fixed Point, Lecture Notes in Mathematics, Springer, 2007
[2] Krasnoselskij, M. A., Vainikko, G. M., Zabreiko, P. P., Rutitki, Ia. B. and Otetenko, BIA, Priblijenie resenie operatornıh uravnenii, Iydatelstva Nauka, Glavnaia Redactia Fizico-Mathematicescoi Literaturı, Moskva, 1969
[3] Pavaloiu, I., La resolution des equations par interpolation, Mathematica 23 (46) (1981), No. 1, 61-72
[4] Rus, A. I., Petrusel, A. and Petrusel, Gabriela, Fixed Point Theory. Cluj University Press, 2008
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A unified treatment of the modified Newton and chord methods
A unified treatment of the modified Newton and chord methods
Abstract
The aim of this paper is to obtain a unified treatment of some iterative methods. We obtain some conditions for which a given equation admits a unique solution in a certain ball. The main obtained result refers to the convergence of the modified chord method.
1 Introduction
Let be a Banach space and . Given , we consider the ball with , and we assume that .
In this paper we shall study certain aspects of the problems regarding the existence, uniqueness, and the approximation of the solution of an operatorial equation of the form
| (1) |
where
Let be a linear and continuous operator. For the approximation of the solution of equation (1) we shall consider the sequence given by (2)
| (2) |
In the following we shall obtain conditions under which equation (1) has a unique solution in the ball and, moreover, the sequence generated by (2) is convergent and
We shall apply the obtained results to the unified study of the convergence of some sequences of the form (2), namely generated by the modified Newton method, and the modified chord method. For the modified Newton method we shall retrieve in this way a known result, which for the modified chord method the result is new.
2 Existence, uniqueness and approximation
Regarding the existence and uniqueness of solution of equation (1) in the ball and the convergence of sequence (2), the following result holds:
Theorem 1
If and verify the following assumptions:
Proof. Consider the operator given by
| (7) |
In the following we shall prove that the image of the ball by remains in and moreover, is a contraction on this ball.
In this sense we notice that may be written as
| (8) |
By (8) and (3) for and by we get
| (9) |
for all By (9) it follows that for all given by (7) belongs to
Consider now By (7) we get:
whence, taking into account (3) it follows
i.e., is a contraction on Consequently, has a unique fixed point Since is a linear invertible operator by (3) it follows that equation (1) has the unique solution
Properties and are immediate consequences of the fact that is a Banach space and is a contraction on the invariant ball (see [1], [4]).Denoting then it can easily seen that
| (10) |
These relations are obtained from (8) for and
From (10), if we replace and we obtain lower and upper error bounds at the -th step:
| (11) |
The last relations offer a posteriori error bounds.
3 The modified chord method
We consider now the divided differences of at some given points in . Let denote the set of linear operators from to
Definition [3]. The linear operator is called first order divided difference of at the points if
- a)
-
- b)
-
if is Fréchet differentiable at then
We shall use in the following the usual notation In order to approximate the solution of equation (1) we consider the sequence given by
| (12) |
We obtain the following result.
Theorem 2
If and verify the conditions
4 The modified Newton method
Assume that is Fréchet differentiable at all points Let and Assume that is invertible.
In order to approximate we shall consider the sequence generated by the modified Newton method.
| (18) |
Concerning the convergence of this sequence the following result holds:
Theorem 3
. [2]. If , and verify
- i
-
- ii
-
is Fréchet differentiable at all points and is a linear continuous operator;
- iii
-
the operator is invertible
- iv
-
there exists , such that for all one has.
(19) - v
-
one has
then the following relations hold:
- j
-
equation (1) has a unique solution
- jj
-
the sequence given by (18) is convergent and
- jjj
-
the following estimations hold true:
(20) (21) (22) (23)
Let be an operator differentiable at all points Then it is known (see, e.g.,[2]) that for all one has
| (24) |
Let now take given by
| (25) |
then
| (26) |
If we take into account (25) and (26), by (24) we get
which shows that (3) is verified for Theorem 3 is a consequence of Theorem 1, and therefore properties –are true.
References
- [1] Berinde, Vasile, Iterative Approximation of Fixed Point, Lecture Notes in Mathematics, Springer (2007).
- [2] Krasnoselskij, M.A., Vainikko, G.M., Zabreiko, P.P., Rutiţki, Ia. B., Oteţenko, BIA, Priblijenie resenie operatornîh uravnenii. Iydatelstva Nauka, Glavnaia Redactia Fizico-Mathematicescoi Literaturî, Moskva, 1969.
- [3] Păvăloiu, Ion, La résolution des equations par intérpolation, Mathematica, 23(46), 1, (1981), pp. 61-72.
- [4] Rus, A. Ioan, Petruşel, Adrian, Petruşel Gabriela. Fixed Point Theorhy. Cluj University Press. (2008).