We study the local convergence of a Aitken-Steffensen type method for approximating the solutions of nonlinear scalar equations. We show that under some usual assumptions on the nonlinear function (involving monotony and convexity), the considered method generates bilateral approximations for the solution. Therefore, one obtains an evaluation of the error at each iteration step.
Ion Păvăloiu
(Tiberiu Popoviciu Institute of Numerical Analysis)
I. Păvăloiu, Monotone sequences for approximating the solutions of equations, Bul. Ştiinţ. Univ. Baia Mare Ser. B Mat.-Inf., 15 (1999) nos. 1-2, pp. 103-110.
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Monotone sequences for approximating the solutions of equations
Bul. Ştiinţ. Univ. Baia Mare, Ser. B,
Matematică-Informatică, Vol. XV (1999), No. 1–2, 103–110
Dedicated to Professor Ion PĂVĂLOIU on his 60th anniversary
Monotone sequences for approximating the solutions of equations
Ion Păvăloiu
1. Introduction
We shall consider in the following the Aitken-Steffensen-like methods and some
conditions under which they generate bilateral sequences for the approximation
of the solutions of the scalar equations.
Let be an interval of the
real axis and consider the equation
where . Let moreover,
| (2) |
|
|
|
|
|
|
|
|
with be other two equations.
We shall assume that if is a root of
(1), then it also satisfies both equations from
(2).
The Aitken-Seffensen method consists in the construction of the sequences
generated by the
following iterative process:
| (3) |
|
|
|
where denotes the first order divided difference of
at the points and
The second order divided differences of will be denoted by
In this paper we shall show that in the study of the convergence of the
sequences generated by (3), an important role is played by the
hypothesis
of convexity on the function We bring some completions and
specifications to the results obtained in [5]–[7].
Concerning the convexity and the monotonicity of the functions we shall
consider the following definitions (see, for example, [3, pp. 288–299 and
p. 327]).
Definition 1.1.
The function is called increasing
(nondecreasing, decreasing, resp. nonincreasing) on the interval if for
all it follows that resp. .
Definition 1.2.
The function is called convex
(nonconcave, concave, resp. nonconvex) if for all it follows that
Some of the usual properties of the convex functions will be used in the
following, and we remind them without proof (see, e.g. [3, pp. 288-299]).
Denote the slope of the function at The following
results hold:
Proposition 1.1.
Let be an arbitrary
function and
-
(1)
if is convex on then is increasing on
-
(2)
If is nonconcave on then is nondecreasing on
Proposition 1.2.
If is nonconcave, then
admits the left derivative and the
right derivative at any point
Moreover, the functions and are nondecreasing on and for all
Proposition 1.3.
If is a convex function on
then
-
(1)
the function is continuous at any point
-
(2)
the function satisfies the Lipschitz condition on any compact
interval contained by
-
(3)
the function is derivable on excepting a subset of at most countable.
Proposition 1.4.
Let .
The following statements are equivalent:
-
(1)
the function is convex on
-
(2)
for any there exists the left derivative of
at which is finite and is
increasing as a function on
-
(3)
for any there exists the right derivative
of at which is finite and is
increasing as a function on
Taking into account the properties expressed in Propositions
1.1–1.4, we are interested in the present
note to simplify the hypotheses requested in [5]–[7]. As we shall see, the convexity properties of the function from
equation (1) play an essential role in the construction of the
functions and from (2).
2. The monotonicity of the sequences generated by the Aitken-Steffensen
method
We shall consider the following hypotheses concerning the functions
and :
-
(a)
the function is convex on
-
(b)
the functions and are continuous on
-
(c)
the function is increasing on
-
(d)
the function is decreasing on
-
(e)
equation (1) has a unique solution
-
(f)
for any it follows that
Concerning the convergence of the sequences
and , the
following result holds.
Theorem 2.1.
If the functions satisfy conditions (a)–(f)
and, moreover,
-
i
the function is increasing on
-
ii
there exists such that and
then the sequences generated by (3) with the
initial approximation considered above, have the following properties:
-
j
the sequences and are increasing and bounded;
-
jj
the sequence is decreasing and bounded;
-
jjj
-
jv
the following relations hold:
|
|
|
|
|
|
Proof.
Since is increasing on and
is the unique solution of on it follows that
By c) and f), for we get Now, for one obtains
when and when By c) and it
follows i.e.
Since one
gets By d) and it results i.e.
By
and it results Hypothesis also implies whence, by (3), one obtains
It can be easily verified that the following identities hold for all :
| (4) |
|
|
|
| (5) |
|
|
|
Since it
follows and using (4) one obtains Now, if in (5) we set
and we take into account (3) we get
|
|
|
But is a convex function, so and consequently
Summarizing, we have obtained the following relations
|
|
|
It‘ remains to prove that satisfies hypothesis and the above
reasoning may be repeated.
Since is decreasing, is increasing and the
following inequalities are true:
From i.e.
which shows
that
Consider now with and If in the above reasoning we take
we obtain
| (6) |
|
|
|
and so the affirmations and of the theorem are
proved. In order to prove we denote and and we shall prove that Indeed, by (6) and (b) we get
|
|
|
i.e. and so Since is convex on Proposition
1.3 assures that is continuous in and by
(3), passing to limit it follows i.e.
The inequality implies
Finally, and since and, at the same time, (4) implies
we obtain
∎
Analogous results hold in the case when is decreasing and convex, or
increasing, resp. decreasing and concave (see [7]).
3. The Steffensen method
This method is obtained from (3) for for
all For the sake of simplicity we shall denote in this section
So, the Steffensen method reads as
| (7) |
|
|
|
We observe that the hypotheses (b), (c) and (f) from the previous section are
automatically satisfied for the function we have considered here.
Concerning the functions and it remains here to make the following assumptions:
-
(a1)
the function is convex on ;
-
(b
the function is decreasing and
continuous on ;
-
(c
equations (1) and have each a unique solution which is the same.
We obtain the following consequences concerning the convergence of the method
(7):
Corollary 3.1.
If the functions and obey and,
moreover, is increasing on , there exists and the point in (7) may be chosen
such that and then
the sequences and verify the following properties:
-
j
the sequence is increasing
and bounded;
-
jj
the sequence is decreasing and bounded;
-
jjj
-
jv
-
v
We shall assume in the following that the function from equation
(1) has the form In
this case (7) becomes
| (8) |
|
|
|
Concerning the convergence of these iterates we obtain from Corollary
3.1 the following result
Corollary 3.2.
If is increasing and concave on and equation
has a unique solution there exists and the
initial approximation is chosen such that
with then the sequences and generated
by (8) verify the conclusions of Corollary 3.1.
Proof.
Since is decreasing on it follows that for any we have
and so i.e. for all which implies that is increasing. On
the other hand, for all we have that and since is concave we obtain that is
convex. One can see that the hypotheses of Corollary 3.1 are satisfied.
∎
4. Applications
In this section we shall show that the functions (resp.) from
the auxiliary equations (2) (resp. ) may be
determined in different ways, under convexity and monotonicity assumptions on
the function from (1), such that the essential hypotheses of
Theorem 2.1, resp. Corollaries 3.1 and
3.2 are automatically satisfied.
We shall assume that is increasing and convex on , i.e. for all
we have Let Choose
|
|
|
(the existence of the lateral derivatives and is assumed by Proposition
1.4). Obviously,
and since we have assumed that
is increasing on From the assumption of convexity on it follows that
is continuous on and hence and
are both continuous on therefore
satisfying hypothesis (b). On the other hand, for all we have
|
|
|
and since is convex we get that i.e. (in other
words, is an increasing function on ).
A similar reasoning leads to the conclusion that is a decreasing
function on
Resuming, one can see that hypotheses (c) and (d) are both satisfied. The
function is assumed to be increasing and so hypothesis (e) is verified.
Hypothesis (f) is obviously satisfied from relation
and from the fact that
We choose now in (3) and we assume that in which case the functions
and satisfy in an obvious manner the hypotheses of Theorem
2.1.
References
-
[1]
M. Balázs,
††margin:
available soon,
refresh and click here
A bilateral approximating method for
finding the real roots of real equations, Rev. Anal. Numér. Théor.
Approx., 21 (1992) 2, pp. 111–117.
-
[2]
V. Casulli, D. Trigiante, The convergence
order for iterative multipoint procedures, Calcolo, 13 (1977) 1, pp. 25–44.
-
[3]
S. Cobzaş, Mathematical Analysis, Presa
Universitară Clujeană, Cluj-Napoca (1977) (in Romanian).
-
[4]
A. M. Ostrowski, Solution of Equations and
Systems of Equations, 1960, Academic Press, New York and London.
-
[5]
††margin:
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I. Păvăloiu,
On the monotonicity of the
sequences of approximations obtained by Steffensen’s method, Mathematica
(Cluj), (35) (58) (1993) 1, pp. 71–76.
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[6]
††margin:
clickable
I. Păvăloiu,
Bilateral approximations for
the solutions of scalar equations,
Rev. Anal. Numér. Théor. Approx.,
23 (1994) no. 1, pp. 95–100.
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[7]
††margin:
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I. Păvăloiu,
Approximation of the roots
of equations by Aitken-Steffensen-type mono- tonic sequences, Calcolo, vol. 32 (1995)
nos. 1–2, pp. 69–82.
-
[8]
F. J. Traub, Iterative Methods for the Solution of
Equations, Prentice-Hall, Inc., Englewood Cliffs, N. J. 1964.
North University of Baia Mare
Department of Mathematics and Computer Science
Victoriei 76, 4800 Baia Mare
