Ion Păvăloiu
(Tiberiu Popoviciu Institute of Numerical Analysis)
I. Păvăloiu, Bilateral approximations for the solutions of scalar equations, Rev. Anal. Numér. Théor. Approx., 23(1994) no. 1, pp. 95-100.
[1] Balazs, M., A Bilateral Approximating Method for Finding the Real Roots of Real Equations., Revue d’analyse numerique et de theorie de l’approximation, 21, 2 (1992), pp. 111-117.
[2] Păvăloiu Ion, On the Monotonicity of the Sequences of Approximations Obtained by Steffensen’s Method., Mathematica, 35 (58), 1(1993), pp. 71-76.
[3] Păvăloiu Ion, Solving Equations by Interpolation. Ed. DACIA (1981), (in Romanian).
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Bilateral approximations for the Solutions of Scalar Equations
Bilateral approximations for the Solutions
of Scalar Equations
Ion Păvăloiu
(Cluj-Napoca)
1. Introduction
Let be an interval on the real axis. Consider
the equation:
with .
In paper [1], to solve equation (1), the author has
considered the sequences and generated by means of Steffensen’s
method for the case when is of the form:
where , and he has studied the conditions under
which the two above sequences are monotonous (one increasing, the other
decreasing), both converging to the solution of equation
(1).
In paper [2] the same problem has been studied, considering
Steffensen’s method for a more general case, that is, when and do not
satisfy equality (2), but it is supposed that equation (1)
is equivalent to the equation:
Paper [2] points out the advantages of Steffensen’s method in
the mentioned case ( and fulfill the above condition, hence
(2) does not hold).
As known, Steffensen’s method, studied in [1] and [2]), consists in generating the sequences and
through:
| (4) |
|
|
|
where stands for the first
order divided differences of on the points and , [3].
In the present note we shall study the problem of [1] and [2] for the Aitken-Steffensen method. For this purpose, consider the
following three
equations:
| (5) |
|
|
|
where .
Assuming that equations (5) are equivalent, in order to approximate
the root of equation (1) we shall consider the sequences
and
generated by the Aitken-Steffensen method, namely:
| (6) |
|
|
|
It is well known that the convergence order of Steffensen’s method for
sequence (4) is 2 if the functions and verify equality
(2).
In the case of the more general studied in [2], the convergence
order is if the sequence
generated by , has the
convergence order
The convergence order of the method (6) is
if the sequence and generated by , have the convergence
orders and , respectively.
From this viewpoint the results of [2] and those of this paper
can present certain advantages; more concretely, given the function , the
functions and respectively, may be chosen in infinitely
various ways. These will be classified at the end of this note.
We shall adopt the notation and with , for the first and second order divided
differences of the function , respectively. We shall also use in proofs the
following obvious identities:
| (7) |
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| (8) |
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where . As to the notions of monotonicity and convexity of the
function on the interval , we shall adopt the following definitions:
Definition 1.1.
The function is increasing
(nondecreasing,decreasing, nonincreasing) on if for every the
relation ( respectively) holds.
Definition 1.2.
The function is convex
(nonconcave, concave, nonconvex) on if for every the relation
(, respectively) holds.
2. Monotonicity of the sequences Generated by the Aitken-Steffensen
Method
In the sequel we shall suppose that the function fulfill the
following conditions:
-
(a)
the functions are continuous;
-
(b)
the function is increasing on
-
(c)
the equation has only one root
;
-
(d)
the function is decreasing on ;
-
(e)
the equations (5) are equivalent on .
As to the problem stated in Section 1, some theorems are verified, as follows:
Theorem 2.1.
If the functions fulfil the conditions
(a)–(e) and, in addition,
-
(i1).
is increasing and convex on
-
(ii1).
there exists for which and ,
then the sequences and have
the properties:
-
(j1).
the sequence and are increasing and convergent;
-
(jj1).
the sequence is decreasing and convergent;
-
(jjj1).
limx, where is
the root of equation (1).
Proof.
Since equations (5) are equivalent, and is the unique root
for equation it results that is
the common unique root of equations (5).
Since is increasing and , it follows that
. Observe now that from the fact that is the unique
root of is increasing, and it results that
for every . As , it results that , that is, . The function is decreasing, hence
, namely
Since , it follows that inequality which, together with
, and taking into account (6) for ,
leads to the inequality . From identity
(7) for and from the fact that it results that
, therefore .
Substituting in (8), and taking into
account (6) for , we get the identity:
|
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|
With this, and taking into account the convexity of and the above proved
results, we obtain , from which it results
hence .
In this way the following relations were proved:
|
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|
Since and is increasing, it follows that from which there results
because we assumed that is decreasing.
Let now be arbitrary element of the sequence generated by
(6) for which and . From it results
that . Repeating (for ) the above
procedure (corresponding to ), we obtain:
| (9) |
|
|
|
relations which prove the monotonicity of the two sequences. These relations
also prove that both sequences are bounded.
Now we show that these sequences have a common limit, , where .
Write and suppose that .
From the continuousness of and , and from the definition of
, we deduce:
| (10) |
|
|
|
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But, by virtue of (9), , hence
and , and, taking into account (10), it results
, namely , therefore . In other words, the
following inequalities hold:
from which, taking into account the monotonicity of , we get:
|
|
|
hence
But, since is increasing and is decreasing,
there results ,
from which we deduce , which, together
with , leads to , and
this one, together with , implies , which
contradicts the hypothesis .
Therefore ; because , we have .
Passing at limit in (6), and considering the continuousness of the
functions it results that is the root for
equation (1).
With this, Theorem 2.1 is completely proved.
The following theorems can be proved in a similar manner:
Theorem 2.2.
If the functions fulfil the conditions
(a)–(e) and, in addition:
-
(i2)
is increasing and concave on ;
-
(ii2)
there exists for which and ,
then the sequences have the properties:
-
(j2)
the sequences and are decreasing and convergent;
-
(jj2)
the sequence is increasing and convergent;
-
(jjj2)
, where is
the root of equation (1).
Theorem 2.3.
If the functions fulfil the conditions
(a)–(e) and, in addition,
-
(i3)
is decreasing and convex in
-
(ii3)
there exists for which and ,
then the sequences have the properties:
-
(j3)
the sequences and are decreasing and convergent;
-
(jj3)
the sequence is increasing and convergent;
-
(jjj3)
, where is
the root of equation (1).
Theorem 2.4.
If the functions fulfil the condition
(a)–(e) and, in addition,
-
(i4)
is decreasing and concave;
-
(ii
there exists for which and ,
then the sequences have the properties:
-
(j4)
the sequences and are increasing and convergent;
-
(jj4)
the sequence is decreasing and convergent;
-
(jjj4)
, where is
the root of equation (1).
3. Numerical example
Consider the equation
for . According to the above remark, we construct
the functions for , obtaining
|
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It is easy to see that, putting the functions and
fulfill the conditions of Theorem 9 on the interval
The sequence generated by relation (6) for this case can be stopped
at the step , because of the fact that as results from the table below:
References
-
[1]
Balázs, M.,
††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, 2 (1992), pp.111-117.
-
[2]
††margin:
clickable
Păvăloiu, I., On the monotonicity of the
sequences of approximations obtained by Steffensen’s method, Mathematica,
35, (58), 1 (1993), pp. 71-76.
-
[3]
Păvăloiu, I.,
††margin:
available soon,
refresh and click here
Solving Equations by
Interpolation. Ed. Dacia (1981) (in Romanian).
