In this paper, we suggest and analyze a family of multistep iterative methods which do not involve the highorder differentials of the function for solving nonlinear equations using a different type of decomposition (mainly due to Noor and Noor
[15]
). We also discuss the convergence of the new proposed methods. Several numerical examples are given to illustrate the efficiency and the performance of the new iterative method. Our results can be considered as an improvement and refinement of the previous results.
1. INTRODUCTION
In recent years, much attention has been given to develop several iterative methods for solving nonlinear equations (see for example
[1
,
6

16]
). These methods can be classified as onestep and twostep methods.
Abbasbandy
[1]
and Chun
[6]
have proposed and studied several onestep and twostep iterative methods with higher order convergence by using the decomposition technique of Adomian
[2

5]
.
In their methods, they have used the higher order differential derivatives which is a serious drawback. To overcome this drawback, Noor and Noor
[14

15]
developed twostep and three step iterative methods by combining the wellknown Newton method with other onestep and twostep methods.
Following the lines of
[14

15]
, we suggest and analyze a family of multistep iterative methods which do not involve the highorder differentials of the function for solving nonlinear equations using a different type of decomposition (mainly due to Noor and Noor
[15]
). We also discuss the convergence of the new proposed methods. Several numerical examples are given to illustrate the efficiency and the performance of the new iterative method. Our results can be considered as an improvement and refinement of the previous results.
2. ITERATIVE METHODS
Consider the nonlinear equation
We assume that α is a simple root of (1) and β is an initial guess suffciently close to α. We can rewrite (1) as a coupled system using the Taylor series
We can rewrite (3) in the following form
where
and
In order to prove the multistep iterative methods, He
[9]
and Lao
[11]
have considered the case with the definition that
and Noor and Noor
[15]
have considered the case
For the derivation of multistep iterative methods for solving nonlinear equations, the condition (9) introduced by Noor and Noor
[15]
which is actually
is the stronger one. We rectify this error and also remove such kind of conditions. For this purpose, we substitute (3) into (7) to obtain
We now construct a sequence of higher order iterative methods by using the following decomposition method which is mainly due to Noor and Noor
[15]
. This decomposition of the nonlinear operator
N
(
x
) is quite different than that of Adomian decomposition. The main idea of this technique is to look for a solution of (4) having the series form
The nonlinear operator
N
can be decomposed as
Combining (4), (12) and (13), we have
Thus we have the following iterative scheme
Then
and
It follows from (6), (11) and (15), that
and
From (14), (16) and (17) we have
This allows us to suggest the following onestep iterative method for solving (1).
Algorithm 1
For a given
x
_{0}
, compute the approximate solution
x
_{n+1}
by the iterative scheme
which is known as
“Newton’s Method”
and it has the
second order convergence
. Again using (15), (17)(19), we conclude that
Using this relation, we can suggest the following twostep iterative methods for solving (1).
Algorithm 2
For a given
x
_{0}
, compute the approximate solution
x
_{n+1}
by the iterative scheme
PredictorStep
CorrectorStep
This Algorithm is commonly known as
“DoubleNewton Method”
with the
third order convergence
.
Again using (11) and (15), we can calculate
From (11), (15)(20), we conclude that
Using this, we can suggest and analyze the following twostep iterative method for solving (1).
Algorithm 3 (AA)
For a given
x
_{0}
, compute the approximate solution
x
_{n+1}
by the iterative scheme PredictorStep
PredictorStep
CorrectorStep
Again using (11) and (15), we have
From (11), (17)(20), we have
Using this, we can suggest and analyze the following iterative method for solving (1).
Algorithm 4
For a given
x
_{0}
, compute the approximate solution
x
_{n+1}
by the iterative scheme
PredictorStep
CorrectorStep
3. CONVERGENCE ANALYSIS
Theorem 1.
Let β ∈ I be a simple zero of a sufficiently differentiable function
for an open interval I
.
If x_{0} is sufficiently close to β, then the threestep iterative method defined by Algorithm 3 has the secondorder convergence.
Proof. Let β
∈
I
be a simple zero of f. Since f is sufficiently differentiable function, by expanding
f
(
x_{n}
) and
f
'(
x_{n}
) about
β
, we get
where
k
= 1, 2, 3, ··· and
e_{n}
=
x_{n }

β
Now from (28) and (29), we have
From (18) and (30), we have
Now expanding
f
(
y_{n}
) about
β
and using (31), we get
Now from (29) and (32), we have
Now again expanding
f
(
y_{n}
+
z_{n}
) about
β
and using (31) and (33), we have
From (29) and (34), we get
From (23), (33) and (35), one obtains
Hence it is proved. □
Theorem 2.
Let β ∈ I be a simple zero of a sufficiently differentiable function
for an open interval I
.
If
x
_{0}
is sufficiently close to
β
,
then the fourstep iterative method defined by Algorithm 4 has the secondorder convergence
.
Proof
. From (26) and (35), we have
Now again expanding
f
(
y_{n}
+
z_{n}
+
w_{n}
) about
β
and using (23), (25) and (37), we get
From (29) and (38), we have
From (27), (31), (33), (37) and (39), one obtains
Hence it is proved. □
4. NUMERICAL EXAMPLES
We present some examples to illustrate the efficiency of the new developed threestep iterative methods. We compare the Newton method (NM), the method (NR1)
[14]
, the method (NR2)
[15]
and the method (AA). Put ∊ = 10
^{−15}
.
The following stopping criteria is used for computer programs

(1) xn+1−xn < ∊,

(2) f(xn+1) < ∊.
As for the convergence criteria, it was required that the distance of two consecutive approximations δ for the zero was less than 10
^{−15}
. Also displayed are the number of iterations (IT) to approximate the zero, the approximate zero
x
_{0}
and the value
f
(
x
_{0}
) and
δ
(see
Table 1
).
The examples are the same as in Chun
[6]
:

F1(x) = sin2x−x2+ 1,

F2(x) =x2−ex− 3x+ 2,

F3(x) = cosx−x,

F4(x) = (x− 1)3− 1,

F5(x) =x3− 10,

F6(x) =x·ex2− sin2x+ 3 cosx+ 5,

F7(x) =ex2+7x−30− 1.
5. CONCLUSIONS
We have suggested a family of onestep, twostep, threestep and fourstep iterative methods for solving nonlinear equations. It is important to note that the implementation of these multistep methods does not require the computation of higher order derivatives compared to most other methods of the same order.
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