We suggest and analyze a family of multistep iterative methods for solving nonlinear equations using the decomposition technique mainly due to Rafiq et al.
[13]
.
1. INTRODUCTION
In recent years, much attention has been given to develop several iterative methods for solving nonlinear equations (see for example
[1
,
6

12
,
14]
). 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]
.
In
[11]
, Noor developed twostep and threestep iterative methods by using the Adomian decomposition technique and by combining the wellknown Newton method with other onestep and twostep methods.
In
[1
,
11

12]
, the authors have used the higher order derivatives which is a drawback. To overcome this drawback, following the lines of
[11]
, we suggest and analyze a family of multistep iterative methods which do not involve the highorder derivatives of the function for solving nonlinear equations using the decomposition technique mainly due to Rafiq et al.
[13]
. 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 sufficiently close to
α
. We can rewrite (1) as a coupled system using the Taylor series.
We can rewrite (3) in the following form as
We can rewrite (4) in the following equivalent form as
where
and
In order to prove the multistep iterative methods, He
[8]
and Lao
[10]
have considered the case with the definition that
and Noor and Noor
[12]
have considered the case
which is actually
and is the stronger one. We rectify this error and also remove such kind of conditions. For this purpose, we substituting (3) into (7) to obtain
with
We now construct a new family of iterative methods by using the following decomposition method mainly due to Rafiq et al.
[13]
. 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 (5) having the series form
The nonlinear operator
N
can be decomposed as
where
A_{i}
are the functions which are known as the new Adomian polynomials depending on
x
_{0}
,
x
_{1}
, ⋯ ; given by a formula
First few new Adomian polynomials are as follows
Substituting (12) and (13) into (5), we obtain
It follows from (6), (12) and (17), that
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
.
From (10) and (17), we have
Again using (12), (15), (16), (17) and (18) , 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
From (11), (17)(21), we conclude that
Using this, we can suggest and analyze the following twostep iterative method for solving (1).
Algorithm 3. (AP)
For a given
x
_{0}
, compute the approximate solution
x
_{n+1}
by the iterative scheme
PredictorStep
CorrectorStep
Algorithm 3 is called the twostep iterative method for solving (1).
Again using (11) and (16), we have
From (11), (18) – (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
If
f'
(
y_{n}
) = 0 then Algorithm 3 reduces to the following method
Now using finite difference approximation, we obtain
Combining (22) and (24), we suggest the following new iterative method for solving (1) as follows
Also we can suggest and analyze the following iterative method for solving (1).
Algorithm 5. (A)
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 sufficiently differentiable function
f
:
I
⊆ ℝ → ℝ
for an open interval I
.
If
x
_{0}
is sufficiently close to β
,
then the twostep iterative method defined by Algorithm 3 has the fourthorder 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 (27) and (28), we have
From (18) and (29), we get
Now expanding
f
(
y_{n}
) about
β
, we have
From (28) and (31), we have
Now expanding
f'
(
y_{n}
) about
β
and using (32), we have
From (31) and (34), we get
From (29) and (35), we have
From (22), (32) and (35), one obtains
Hence it is proved. ☐
Theorem 2.
Let β
∈
I be a simple zero of a sufficiently differentiable function
f
:
I
⊆ ℝ → ℝ
for an open interval I
.
If
x
_{0}
is sufficiently close to β
,
then the iterative method defined by Algorithm 4 has the fourth order convergence
.
Proof.
From (28) and (33), we have
From (35) and (37), we get
From (23), (32), (35) and (38), one obtains
Hence it is proved. ☐
Theorem 3.
Let β
∈
I be a simple zero of a sufficiently differentiable function
f
:
I
⊆ ℝ → ℝ
for an open interval I
.
If
x
_{0}
is sufficiently close to β
,
then iterative method defined by Algorithm 5 has the third order convergence
.
Proof
. From (27) and (31), we have
From (32) and (40), we get
From (26), (30), (32) and (41), one obtains
Hence it is proved. ☐
4. NUMERICAL EXAMPLES
We provide some examples to illustrate the efficiency of the new developed iterative methods. Put
ϵ
= 10
^{−15}
.
The following stopping criteria is used for the computer programs

(1) xn+1−xn <ϵ,

(2) f(xn+1) <ϵ.
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+ 3cosx+ 5,

F7(x) =ex2+7x−30− 1.
Also 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}
, the value
f
(
x
_{0}
) and
δ
. We compare the Newton method (NM), the Double Newton method (DNM), the method of Noor (NR)
[12]
and the method (AP) , introduced in the Algorithm 3 (see
Table 1
).
Now we compare the Newton method (NM), the Double Newton method (DNM), the method of Weerakoon and Fernando
[14]
, the method of Frontini and Sormani
[7]
, the method of Homeier
[9]
, the method of Noor (NR)
[12]
and the method (A), introduced in the Algorithm 5 (see
Table 2
).
5. CONCLUSION
We have suggested a family of twostep iterative methods for solving nonlinear equations by using a new decomposition technique mainly due to Rafiq et al.
[13]
. It is important to note that the implementation of these methods does not require the computation of higher order derivatives compared to most other methods of the same order.
2010 Mathematics Subject Classification. 26A18, 39B12.
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