I. Păvăloiu, On the monotonicity of the sequences of approximations obtained by Steffensen’s method, Mathematica, 35(58) (1993) no. 1, pp. 71-76.
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Mathematica – Revue d’analyse numérique et de théorie de l’approximation.
Mathematica
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References
[1] Balazs, M., A bilateral approximating method for finding the real roots of real equations. Rev. Anal. Numer. Theor. Approx., 21, no. 2, 111–117, (1992).
[2] Pavaloiu, I., Sur la methode de Steffensen pour la resolution des equations operationnelles non lineaires, Revue Roumaine des Mathematiques Pures et Appliquees, XIII, 6, 149–158 (1968).
[3] Pavaloiu, I., Rezolvarea ecuatiilor prin interpolare, Ed. Dacia, Cluj-Napoca (1981).
[4] Ul’m, S., Ob obobschenie metoda Steffensen dlya reshenia nelineinyh operatornyh uravnenii, Jurnal Vicisl, mat. i. mat-fiz. 4, 6, (1964).
On the monotonicity of the sequences of approximations obtained by
Steffensen’s method
Ion Păvăloiu
In a recent paper [1], M. Bálazs studied the conditions in
which the sequence generated by Steffensen’s
method is monotonic and converges to the solution of equation
(1)
where is a given function, and
is an interval of the real axis. Paper [1] considers the simple
case of the sequence generated by the recurrence relation
(2)
in which is given by and is the first order
divided difference of the function
We shall show further down that the properties resulting from Theorem
1 hold for more general Steffensen-type methods, while for the method
(2), if hypothesis (ii) is replaced by:
(ii1)
there exists for which and then hypothesis (iii) can be dropped,
and the conclusions of the theorem remain valid putting
Consider an arbitrary function whose fixed
points coincide with the real roots of equation (1), and reciprocally.
The following theorem holds:
Theorem 2.
If the functions and are continuous on and if the following conditions:
(i2)
the function is strictly decreasing on
(ii2)
the function is strictly increasing and concave on
(iii2)
there exists such that and
(iv2)
the equations and are equivalent, are fulfilled then equation (1) has a
unique solution
and the following properties hold:
(j2)
the sequence is
decreasing and convergent;
(jj2)
the sequence is increasing and convergent;
(jjj2)
where is the unique solution of equation (1), therefore fixed point of
in the interval
hence the equation has unique solution in the interval
i.e. has a unique fixed
point in this interval. Since the fixed points of coincide with
the roots of equation (1) and reciprocally, there follows that
is unique solution for equation (1) within the same
interval, and Now we shall show
that First show that
It is easy to verify that the terms of the sequence provided by the relations
coincide with those of the sequence
generated by (2). In other words the equalities
hold; for it follows that:
from which it results that
From Lagrange’s interpolating polynomial we get
and, since taking
into account (2), it follows that that
is, Since the function is increasing, it results that
, i.e. Let us
show that We have that is,
Repeating the above argument, putting , and
supposing that the hypotheses of Theorem 2 are fulfilled for
we obtain:
It results subsequently that the sequences
and fulfil the properties
(j2) and (jj2) of Theorem 2 and, in addition, are bounded.
Write
and we obtain
and Suppose that therefore
. But
since this shows that
At limit (for ), equalities (2) yield where the continuity of was also taken into account.
∎
Remark 1.
If we put in Theorem 2, since is decreasing, it follows that is increasing; since is convex, it
follows that for every hence
that is, is
concave. From it follows i.e. In this case, Theorem
1 in which hypotheses (ii) and (iii) are replaced by (ii2) is a
consequence of Theorem 2.
The results of Theorem 2 are graphically illustrated in Figure 1.
Figure 1.
In what follows we shall present, without proof. other cases in which
properties of monotonicity analogous to those given by Theorem 2 hold.∎
Theorem 3.
If the functions and are continuous on and if the following conditions are fulfilled:
(i3)
the function is strictly decreasing on
(ii3)
the function is strictly increasing and convex on
(iii3)
there exists for which and
(iv3)
the equations and are equivalent, then the sequence generated by (2) is increasing and convergent, the sequence
is decreasing and
convergent, and is the solution of
equation (1).
If the functions and are continuous on and if the following conditions are fulfilled:
(i4)
the function is strictly decreasing on
(ii4)
the function is strictly decreasing and convex on
(iii4)
there exists for which and
(iv4)
the equations and are equivalent, then the sequence generated by (2) is decreasing and convergent, the sequence
is increasing and convergent, and
The results of this theorem are illustrated by Figure 3.
Theorem 5.
If the functions and are continuous on and if the following conditions are fulfilled:
(i5)
the functions is strictly decreasing on
(ii5)
the function is strictly decreasing and concave on
(iii5)
there exists such that and
(iv5)
the equations and are equivalent then the sequence
generated by (2) is increasing and convergent, the sequence is decreasing and convergent, and
Remark 2.
The fact that the functions and from the above
theorems are related only by the equivalence of equations and offers large possibilities to choose these
functions (i.e. to choose when is known, and conversely).
It is clear that if keeps the same monotonicity and convexity on then
we can moot the question of determining a real number such that
be a decreasing function.
Under certain conditions can be determined, as it results from the
following example:
If is strictly increasing and strictly convex on
and if is differentiable, then is also derivable, and
for every
Then we can put and we have for every hence is
decreasing. It is clear that the equations and
have the same roots. If and
then it is
obvious that and Theorem 3 can be applied for ∎
Numerical example.
Consider the equation:
and the function given by the relation:
Since and it follows that
is decreasing on and is increasing and convex. One
shows by direct calculation that and
hence fulfils the hypotheses of Theorem
3. The table below lists the results of the calculations for
0
-2.000000000000000000
-1.249045772398254430
-0.750954227601745574
1
-1.414047729532868260
-1.400933154002817630
-0.013114575530050630
2
-1.404227441155695550
-1.404222310683232820
-0.000005130472462734
3
-1.404223602392559510
-1.404223602391771120
-0.000000000000788388
4
-1.404223602391969620
-1.404223602391969620
-0.000000000000000000
Table 1.
The numerical results agree with the conclusions of Theorem 3; as one
can see, after four iteration steps a solution approximation with 18 decimals
is obtained (obviously, if truncation and rounding errors are neglected).∎
