Abstract
In this paper we deal with iterative methods of interpolatory type, for solving nonlinear equations in Banach spaces. We show that the convergence order of the iterations may considerably grow if the nodes are properly controlled.
Author
Ion Păvăloiu
(Tiberiu Popoviciu Institute of Numerical Analysis)
Keywords
nonlinear equations in Banach spaces; interpolatory method.
PDF-LaTeX file (on the journal website).
Cite this paper as:
I. Păvăloiu, Accelerating the convergence of the iterative methods of interpolatory type, Rev. Anal. Numér. Théor. Approx., 34 (2005) no. 2, pp. 169-173. https://doi.org/10.33993/jnaat342-803
About this paper
Publisher Name
Print ISSN
1222-9024
Online ISSN
2457-8126
References
Argyros, I., Polynomial Operator Equations in Abstract Spaces and Applications, CRC Press, LLC, 1998.
Ostrowski, A.M., Solution of Equations and Systems of Equations, Academic Press, New York, 1960.
Păvăloiu, I., Interpolation dans des éspaces linéaires normèes et applications, Mathematica, 12 (35), no. 1, pp. 149-150, 1970.
Păvăloiu, I., Introduction to the Theory of Approximating the Solutions of Equations, Ed. Dacia, Cluj-Napoca, Romania, 1976 (in Romanian).
Păvăloiu, I., On the convergence order of some Aitken-Steffensen-type methods, Rev. Anal. Numér. Théor. Approx., 32, no. 2, pp. 193-202, 2003, https://ictp.acad.ro/jnaat/journal/article/view/2003-vol32-no2-art8
Păvăloiu, I., Local convergence of general Steffensen type methods, Rev. Anal. Numér. Théor. Approx., 33, no. 1, pp. 79-86, 2004, https://ictp.acad.ro/jnaat/journal/article/view/2004-vol33-no1-art10
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Accelerating the convergence of the iterative methods of interpolatory type
Accelerating the convergence of the iterative methods of interpolatory type
(Date: March 11, 2005.)
Abstract.
In this paper we deal with iterative methods of interpolatory type, for solving nonlinear equations in Banach spaces. We show that the convergence order of the iterations may considerably grow if the nodes are properly controlled.
Key words and phrases:
Nonlinear equations, iterative methods of interpolatory type.1991 Mathematics Subject Classification:
65H05.
∗This work has been supported by the Romanian Academy under grant GAR 14/2005.
†“T. Popoviciu” Institute of Numerical Analysis, P.O. Box 68-1, Cluj-Napoca, Romania,
e-mail: pavaloiu@ictp.acad.ro.
1. Introduction
Let be a Banach space, a subset, and a nonlinear mapping. Consider the equation
| (1) |
where is the zero vector of
Regarding we make the following assumptions:
-
a)
is a one to one mapping;
-
b)
equation (1) has a solution
-
c)
the operator is Fréchet differentiable on and where is the null linear operator.
In order to accelerate the convergence of the iterative methods of interpolatory type, we also consider an equation, equivalent to (1), of the form
| (2) |
where
We make the following assumptions regarding :
-
a′)
is times differentiable on the whole set for some
-
b′)
we have but where denotes the -linear operator. Also, for some
By hypotheses a) and b) we get the relation:
| (4) |
The relation above and (3) for attract that
| (5) |
whence we deduce an approximation for of the form:
| (6) |
The error for this approximation is bounded by
| (7) |
where the norm of is considered in the space of -linear operators.
In [3] it is shown that the convergence order of the iterative methods given by (6) cannot be greater than , even if the number of the interpolation nodes is arbitrarily increased. However, the convergence order can be increased if we use the auxilliary function considered above.
Let be an initial approximation to Denote and Replacing in (6) the interpolation nodes by we obtain for a first approximation, denoted by :
| (8) |
If is an approximation for then we obtain the next approximation in the following way. Denote and Analogously to (8) we get
| (9) |
with the error
| (10) |
In the following we shall analyze some particular instance of (9).
If we take in (9), we get
| (11) |
Taking into account the identities and we notice that we are lead to the Steffensen method:
| (12) |
If we take and recall that
| (13) | ||||
and we get
| (14) | ||||
a corrected Steffensen method.
2. Local convergence
From (10), using the finite growth formula (see, e.g., [5]), we get
| (15) |
where we have assumed that there exist such that
| (16) | ||||
| (17) |
From the hypotheses a′) and b′), using the Taylor formula we get
| (18) |
where and
We make the following notations:
| (21) | ||||
| (24) | ||||
| (25) |
Taking into account the above considerations, we obtain the following result.
Theorem 2.1.
If the hypotheses a)–c), a′), b′) hold and, moreover,
-
i.
-
ii.
then the elements of the sequence generated by (9) remain in the set and for all we have
| (28) |
Proof.
Let and so
which shows that
Relations (28) can be easily proved, and further, taking into consideration i., we obviously get ∎
Corollary 2.1.
In the case of the Steffensen method, we obtain the well known result (see, e.g., [6]) for and if and if
Corollary 2.2.
In the case of method (14) we get
We conclude that the iterative methods of interpolatory type may attain a substantially higher convergence order if the nodes are properly controlled.
References
- [1]
- [2] Argyros, I., Polynomial Operator Equations in Abstract Spaces and Applications, CRC Press, LLC, 1998.
- [3] Ostrowski, A.M., Solution of Equations and Systems of Equations, Academic Press, New York, 1960.
- [4] Păvăloiu, I., Interpolation dans des éspaces linéaires normèes et applications, Mathematica, 12 (35), no. 1, pp. 149–150, 1970.
- [5] Păvăloiu, I., Introduction to the Theory of Approximating the Solutions of Equations, Ed. Dacia, Cluj-Napoca, Romania, 1976 (in Romanian).
- [6] Păvăloiu, I., On the convergence order of some Aitken-Steffensen-type methods, Rev. Anal. Numér. Théor. Approx., 32, no. 2, pp. 193–202, 2003.
- [7] Păvăloiu, I., Local convergence of general Steffensen type methods, Rev. Anal. Numér. Théor. Approx., 33, no. 1, pp. 79–86, 2004.
- [8]