Abstract
We are concerned with the approximation of the solution of scalar equations by Halley-Aitken method. We obtain local convergence result.
Authors
Ion Păvăloiu
(Tiberiu Popoviciu Institute of Numerical Analysis)
Keywords
nonlinear equations in R; iterative methods; Halley-Aitken method.
Cite this paper as:
I. Păvăloiu, A Halley-Aitken-type method for approximating the solutions of scalar equations, Bul. Ştiinţ. Univ. Baia Mare, 17 (2001) nos. 1-2, pp. 85-92.
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A Halley-Aitken type method for approximating the solutions of scalar equations
A Halley-Aitken type method for approximating the solutions of scalar equations
1 Introduction
Let where , and suppose that has the first order derivative, which is positive: . Consider the function
| (1.1) |
In [2] there is shown that the Halley method for solving
| (1.2) |
is in fact the Newton method for solving (1.1). This method consists therefore in generating the sequence by
| (1.3) |
The first and second order derivatives of are given by
| (1.4) |
and
| (1.5) |
These relations imply
| (1.6) |
and
| (1.7) |
where denotes the solution of (1.2). As shown in [1], equality (1.7) characterizes the Halley method, in the sense that ensures its convergence order The authors of [4], analyzing an algorithm of Heron for approximating , give a general algorithm which can be used for approximating the cubic root of any real positive number.
In [7] it is shown that the algorithm from [4] is nothing else than the chord method applied to equation where , with . In this case the equation has the form , when ,
It is clear that between the Heron algorithm and the Halley method there exists a connection, in the sense that the transformed equation to which we apply the Newton or the chord method is the same. In [7] and [10], the authors study the convergence and error bounds for the Steffensen and Aitken-Steffensen methods applied to (1.1). In this note we shall study a variant of the Aitken-Steffensen method, which differs from those presented in [10] and [11].
We shall consider other two equations, equivalent to (1.2), having the form
| (1.8) |
and
| (1.9) |
where will be convenably chosen.
We shall study the sequence given by
| (1.10) |
We shall consider the following assumptions on and :
i.
ii. equation (1.2) has a solution
iii.
iv. obeys where denotes the first order divided difference of on and :
v. obeys
2 The local convergence and error bounds
We shall use the following identities:
| (2.1) | ||||
and also the Newton identity
| (2.2) |
where is the second order divided difference of on
We notice that equality (1.6) and hypothesis i. ensure the existence of such that .
The following theorem holds:
Theorem 2.1
Let be such that
.
If the functions and the initial approximation
satisfy:
a) can be
chosen such that and
b) the hypotheses i-v an
satisfied.
Then the following properties are true:
j. for all we have
jj. there exists , , which does not depend on , such that
jjj. if is close enough to to obey then the sequences and converge to their common limit .
Proof. We shall analyses two cases.
I. Then i.e., . Denote and so , i.e. . Now we show that From it follows i.e. Next, we show that , where is obtained from (1.10) for . Since , and , we get that (we know that ) and so satisfies . We have used the fact that implies Now we show that . This inequality follows from , and from (2.1) for . we have shown that
| (2.3) |
and
| (2.4) |
II. . Similarly to the above reason, we get that
| (2.5) |
and
| (2.6) |
Denoting by the open interval determined by and , then obviously relations (2.3) - (2.6) may be sinthetized as
It can be easily seen that if we denote by the open interval determined by and then get
and
where is obtained from (1.10) for .
Let be the open interval determined by and for some . Then repeating the above reason, we may show that
| (2.7) |
and
where is determined by and . From the above reason and from (2.7) it follows j., which yields an error bound bound for each iteration step.
The mean value formulae for the divided differences lead us to
| (2.9) |
and
From i. and using the Lagrange formula it follows
| (2.10) |
The property jj. follows easily by denoting and taking into account iv, v, and the fact that
Property jjj is an immediate consequence of j and jj.
3 Determining the functions and
we shall present a modality of choosing and in order to obey the assumptions of Theorem 2.1
Suppose that is strictly convex on , i.e. . This assumption, together with , lead, by (1.4) to . Relation (1.4) again and and imply the existence of , such that These hypotheses ensure the existence of an interval for which . Since it follows that is increasing on .
Taking
and
with and and assuming that , then the functions and defined above obey hypotheses iv and v of Theorem 2.1. For in Theorem 2.1, hypothesis is automatically verified, but the assumption must be kept.
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