# A New Optimal Eighth-Order Ostrowski-Type Family of Iterative Methods for Solving Nonlinear Equations

Authors: Taher Lotfi; Tahereh Eftekhari
Journal: Chinese Journal of Mathematics
 (2014)

---

## Abstract

Based on Ostrowski's method, a new family of eighth-order iterative methods for solving nonlinear equations by using weight function methods is presented. Per iteration the new methods require
three evaluations of the function and one evaluation of its first derivative. Therefore, this family of methods has the efficiency index which equals 1.682. Kung and Traub conjectured that a multipoint iteration without
memory based onn evaluations could achieve optimal convergence order
2
n
−
1. Thus, we provide a new class which agrees with the conjecture of Kung-Traub for n
=
4. Numerical comparisons are made to show the performance of the presented methods.


---

## Body

## 1. Introduction

In this paper, we consider iterative methods to find a simple rootα of a nonlinear equation f
(
x
)
=
0, where f
:
D
⊂
ℝ
→
ℝ is a scalar function on an open interval D. This problem is a prototype for many nonlinear numerical problems. Newton’s method is the most widely used algorithm for dealing with such problems, and it is defined by
(1)
x
n
+
1
=
x
n
-
f
(
x
n
)
f
′
(
x
n
)
,
which converges quadratically in some neighborhood of α (see [1, 2]).To improve the local order of convergence, many modified methods have been proposed in the open literature; see [3–17] and references therein. King [3] developed a one-parameter family of fourth-order methods, which is written as:
(2)
y
n
=
x
n
-
f
(
x
n
)
f
′
(
x
n
)
,
x
n
+
1
=
y
n
-
f
(
x
n
)
+
β
f
(
y
n
)
f
(
x
n
)
+
(
β
-
2
)
f
(
y
n
)
f
(
y
n
)
f
′
(
x
n
)
,
where β
∈
ℝ is a constant. In particular, the famous Ostrowski’s method [2] is a member of this family for the case β
=
0, and it can be written as
(3)
y
n
=
x
n
-
f
(
x
n
)
f
′
(
x
n
)
,
x
n
+
1
=
y
n
-
f
(
x
n
)
f
(
x
n
)
-
2
f
(
y
n
)
f
(
y
n
)
f
′
(
x
n
)
.
Kung and Traub [15] who conjectured that an iteration method without memory based on n evaluations of f or its derivatives could achieve optimal convergence order 2
n
-
1. Thus, the optimal order for a method with 3 functional evaluations per step would be 4. King’s method [3], Ostrowski’s method, and Jarrat’s method [16] are some of the optimal fourth-order methods, because they only perform three functional evaluations per step. Recently, based on Ostrowski’s or King’s methods, some higher-order multipoint methods have been proposed for solving nonlinear equations. Bi et al. developed a scheme of optimal order of convergence eight [17], estimating the first derivative of the function in the second and third steps and constructing a weight function as well in the following form:
(4)
y
n
=
x
n
-
f
(
x
n
)
f
′
(
x
n
)
,
z
n
=
y
n
-
2
f
(
x
n
)
-
f
(
y
n
)
2
f
(
x
n
)
-
5
f
(
y
n
)
f
(
y
n
)
f
′
(
x
n
)
,
x
n
+
1
=
z
n
-
f
(
x
n
)
+
(
γ
+
2
)
f
(
z
n
)
f
(
x
n
)
+
γ
f
(
z
n
)
×
f
(
z
n
)
f
[
z
n
,
y
n
]
+
f
[
z
n
,
x
n
,
x
n
]
(
z
n
-
y
n
)
,
where γ
∈
ℝ is constant. Liu and Wang in [18] presented the following family of optimal order eight:
(5)
y
n
=
x
n
-
f
(
x
n
)
f
′
(
x
n
)
,
z
n
=
y
n
-
f
(
x
n
)
f
(
x
n
)
-
2
f
(
y
n
)
f
(
y
n
)
f
′
(
x
n
)
,
x
n
+
1
=
z
n
-
[
(
f
(
x
n
)
-
f
(
y
n
)
f
(
x
n
)
-
2
f
(
y
n
)
)
2
+
f
(
z
n
)
f
(
y
n
)
-
μ
f
(
z
n
)
+
4
f
(
z
n
)
f
(
x
n
)
+
λ
f
(
z
n
)
]
f
(
z
n
)
f
′
(
x
n
)
,
where μ and λ are in ℝ. J. R. Sharma and R. Sharma in [12] produce optimal eighth-order method in the following form:
(6)
y
n
=
x
n
-
f
(
x
n
)
f
′
(
x
n
)
,
z
n
=
y
n
-
f
(
x
n
)
f
(
x
n
)
-
2
f
(
y
n
)
f
(
y
n
)
f
′
(
x
n
)
,
x
n
+
1
=
z
n
-
[
1
+
f
(
z
n
)
f
(
x
n
)
+
(
f
(
z
n
)
f
(
x
n
)
)
2
]
×
f
(
z
n
)
f
[
x
n
,
y
n
]
f
[
x
n
,
z
n
]
f
[
y
n
,
z
n
]
.In this paper, based on Ostrowski’s method, we present a new family of optimal eighth-order methods by using the method of weight functions and we apply a few weight functions to construct families of iterative methods with high convergence and high efficiency index. The convergence analysis is provided to establish their eighth order of convergence. In terms of computational cost, they require the evaluations of only three functions and one first-order derivative per iteration. This gives 1.682 as efficiency index of the presented methods. The new methods are comparable with Bi et al.’s method, Liu and Wang’s method, and Sharma’s method. The efficacy of the methods is tested on a number of numerical examples.We use the symbols→,  O, and ~ according to the following conventions [1]. If lim
⁡
x
n
→
∞
g
(
x
n
)
=
C, we write g
(
x
n
)
→
C or g
→
C. If lim
⁡
x
→
a
g
(
x
)
=
C, we write g
(
x
)
→
C or g
→
C. If f
/
g
→
C, where C is a nonzero constant, we write f
=
O
(
g
) or f
~
C
g. Let f
(
x
) be a function defined on an interval I, where I is the smallest interval containing k
+
1 distinct nodes x
1
,
x
2
,
…
,
x
k. The divided difference f
[
x
0
,
x
1
,
…
,
x
k
] with the kth order is defined as follows: f
[
x
0
]
=
f
(
x
0
)
(7)
f
[
x
0
]
=
f
[
x
1
]
-
f
[
x
0
]
x
1
-
x
0
,
…
,
f
[
x
0
,
x
1
,
…
,
x
k
]
=
f
[
x
1
,
x
2
,
…
,
x
k
]
-
f
[
x
0
,
x
1
,
…
x
k
-
1
]
x
k
-
x
0
.
Moreover, we recall the definition of efficiency index (EI) as E
=
p
1
/
n, where p is the order of convergence and n is the total number of function evaluations per iteration.


## 2. The Methods and Analysis of Convergence

In order to construct new methods, we consider an iteration scheme of the form(8)
y
n
=
x
n
-
f
(
x
n
)
f
′
(
x
n
)
,
z
n
=
y
n
-
f
(
x
n
)
f
(
x
n
)
-
2
f
(
y
n
)
f
(
y
n
)
f
′
(
x
n
)
,
x
n
+
1
=
z
n
-
f
(
z
n
)
f
′
(
z
n
)
.
This scheme includes three evaluations of the function and two evaluations of its first derivative. Therefore, this scheme has efficiency index equal to 1.516. To improve the efficiency index, we approximate f
′
(
z
n
) by the divided difference [12]
(9)
f
′
(
z
n
)
≈
f
[
x
n
,
z
n
]
f
[
y
n
,
z
n
]
f
[
x
n
,
y
n
]
.
Now, we present a new family of optimal eighth-order Ostrowski-type iterative methods by using the method of weight functions as follows:
(10)
y
n
=
x
n
-
f
(
x
n
)
f
′
(
x
n
)
,
z
n
=
y
n
-
f
(
x
n
)
f
(
x
n
)
-
2
f
(
y
n
)
f
(
y
n
)
f
′
(
x
n
)
,
x
n
+
1
=
z
n
-
{
K
(
t
1
)
×
L
(
t
2
)
×
P
(
t
3
)
}
×
f
(
z
n
)
f
[
x
n
,
y
n
]
f
[
x
n
,
z
n
]
f
[
y
n
,
z
n
]
,
where K
(
t
1
)
,
L
(
t
2
), and P
(
t
3
) are three real-valued weight functions when
(11)
t
1
=
f
(
z
)
f
(
x
)
,
t
2
=
f
(
y
)
f
(
x
)
,
t
3
=
f
(
z
)
f
(
y
)
,
without the index n, should be chosen such that the order of convergence arrives at the optimal level eight. If x
0 is an approximation to the zero α of f, then the corresponding iterative method is defined by
(12)
x
k
+
1
=
g
(
x
k
)
,
k
=
0,1
,
…
.
Regarding (10), let us define the errors
(13)
e
=
x
-
α
,
e
y
=
y
-
α
,
e
z
=
z
-
α
,
e
1
=
g
(
x
)
-
α
.
We will use Taylor’s expansion about the zero α to express f
(
x
)
,
f
(
y
), and f
(
z
) as series in e
,
e
y, and e
z, respectively. Then, according to (11), we represent t
1
,
t
2, and t
3 as Taylor’s polynomials in e.Assume thatx is sufficiently close to the zero α of f then t
1
,
t
2 and t
3 are close enough to 0. Let us represent real functions K
,
L, and P appearing in (10) by Taylor’s series about 0,
(14)
K
(
t
1
)
=
K
(
0
)
+
K
′
(
0
)
t
1
+
⋯
,
L
(
t
2
)
=
L
(
0
)
+
L
′
(
0
)
t
2
+
L
′′
(
0
)
2
t
2
2
+
L
′′′
(
0
)
6
t
2
3
+
L
(
4
)
(
0
)
24
t
2
4
+
⋯
,
P
(
t
3
)
=
P
(
0
)
+
P
′
(
0
)
t
3
+
P
′′
(
0
)
2
t
3
2
+
⋯
.
Symbolic computations reported here to find candidates for K
,
L, and P have been carried out in aMathematica  8.0 environment. We will find the coefficients K
(
0
)
,
K
′
(
0
)
,
…
,
P
′′
(
0
) of the developments (14) using a simple program inMathematica  8.0 and an interactive approach explained by the comments C1–C5. First, let us introduce the following abbreviations used in this program (see Algorithm 1):
(15)
c
k
=
f
(
k
)
(
α
)
k
!
f
′
(
α
)
,
e
=
x
-
α
,
e
1
=
g
(
x
)
-
α
.
f
[
e
]
=
f
(
x
)
,
f
[
e
y
]
=
f
(
y
)
,
f
[
e
z
]
=
f
(
z
)
,
f
′
[
e
]
=
f
′
(
x
)
,
f
1
a
=
f
′
(
α
)
,
K
0
=
K
(
0
)
,
K
1
=
K
′
(
0
)
,
K
2
=
K
′′
(
0
)
,
K
3
=
K
′′′
(
0
)
,
K
4
=
K
(
4
)
(
0
)
,
L
0
=
L
(
0
)
,
L
1
=
L
′
(
0
)
,
L
2
=
L
′′
(
0
)
,
L
3
=
L
′′′
(
0
)
,
L
4
=
L
(
4
)
(
0
)
,
P
0
=
P
(
0
)
,
P
1
=
P
′
(
0
)
,
P
2
=
P
′′
(
0
)
,
P
3
=
P
′′′
(
0
)
,
P
4
=
P
(
4
)
(
0
)
.Algorithm 1:
Program (written in Mathematica 8.0).
f
[
e
-
]
=
f
1
a
(
e
1
+
c
2
e
2
+
c
3
e
3
+
c
4
e
4
+
c
5
e
5
+
c
6
e
6
+
c
7
e
7
+
c
8
e
8
+
c
9
e
9
);
e
y
=
e
-
S
e
r
i
e
s
[
f
[
e
]
f
′
[
e
]
,
{
e
,
0,8
}
];
e
z
=
e
y
-
S
e
r
i
e
s
[
f
[
e
]
f
[
e
]
-
2
f
[
e
y
]
*
f
[
e
y
]
f
′
[
e
]
,
{
e
,
0,8
}
];
f
[
x
-
,
y
-
]
≔
f
[
x
]
-
f
[
y
]
x
-
y;
t
1
≔
f
[
e
z
]
f
[
e
]; t
2
≔
f
[
e
y
]
f
[
e
]; t
3
≔
f
[
e
z
]
f
[
e
y
];
K
=
K
0
+
K
1
*
t
1
+
1
2
*
K
2
*
t
1
2
+
1
6
*
K
3
*
t
1
3
+
1
24
*
K
4
*
t
1
4;
L
=
L
0
+
L
1
*
t
2
+
1
2
*
L
2
*
t
2
2
+
1
6
*
L
3
*
t
2
3
+
1
24
*
L
4
*
t
2
4;
P
=
P
0
+
P
1
*
t
3
+
1
2
*
P
2
*
t
3
2
+
1
6
*
P
3
*
t
3
3
+
1
24
*
P
4
*
t
3
4;
e
1
=
e
z
-
S
e
r
i
e
s
[
K
*
L
*
P
f
[
e
z
]
f
[
e
,
e
y
]
f
[
e
,
e
z
]
f
[
e
y
,
e
z
]
,
{
e
,
0,8
}
]
/
/
F
u
l
l
S
i
m
p
l
i
f
y
a
4
=
C
o
e
f
f
i
c
i
e
n
t
[
e
1
,
e
4
]
/
/
F
u
l
l
S
i
m
p
l
i
f
y
C
1: O
u
t
[
a
4
]
=
-
(
-
1
+
K
0
L
0
P
0
)
c
2
(
c
2
2
-
c
3
)
K
0
=
1; L
0
=
1; P
0
=
1; (* Vanish  coefficient  of  e
4  *)
a
5
=
C
o
e
f
f
i
c
i
e
n
t
[
e
1
,
e
5
]
/
/
F
u
l
l
S
i
m
p
l
i
f
y
C2: O
u
t
[
a
5
]
=
(
-
4
+
K
0
(
4
L
0
-
L
1
)
P
0
)
c
2
4
+
(
8
+
K
0
(
-
8
L
0
+
L
1
)
P
0
)
c
2
2
c
3
+
2
(
-
1
+
K
0
L
0
P
0
)
c
3
2
+
2
(
-
1
+
K
0
L
0
P
0
)
c
2
c
4
L
1
=
0; (* Vanish  coefficient  of  e
5  *)
a
6
=
C
o
e
f
f
i
c
i
e
n
t
[
e
1
,
e
6
]
/
/
F
u
l
l
S
i
m
p
l
i
f
y
C3: O
u
t
[
a
6
]
=
-
1
2
(
-
20
+
K
0
(
20
L
0
-
14
L
1
+
L
2
)
P
0
+
2
K
0
L
0
P
1
)
c
2
5
+
1
2
(
-
60
+
K
0
(
60
L
0
-
26
L
1
+
L
2
)
P
0
+
4
K
0
L
0
P
1
)
c
2
3
c
3
+
(
18
+
K
0
(
4
L
1
P
0
-
L
0
(
18
P
0
+
P
1
)
)
)
c
2
c
3
2
+
2
(
6
+
K
0
(
-
6
L
0
+
L
1
)
P
0
)
c
2
2
c
4
+
7
(
-
1
+
K
0
L
0
P
0
)
c
3
c
4
L
2
=
0; P
1
=
0; (* Vanish  coefficient  of  e
6  *)
a
7
=
C
o
e
f
f
i
c
i
e
n
t
[
e
1
,
e
7
]
/
/
F
u
l
l
S
i
m
p
l
i
f
y
C4: O
u
t
[
a
7
]
=
(
L
3
P
0
6
-
20
-
K
1
L
0
P
0
+
K
0
(
L
3
P
0
6
21
L
0
P
0
-
30
L
1
P
0
+
5
L
2
P
0
-
L
3
P
0
6
+
6
L
0
P
1
-
L
1
P
1
)
)
c
2
6
+
1
6
(
480
+
12
K
1
L
0
P
0
+
K
0
(
(
-
492
L
0
+
480
L
1
-
54
L
2
+
L
3
)
P
0
+
12
(
-
9
L
0
+
L
1
)
P
1
)
)
c
2
4
c
3
+
(
-
80
-
K
1
L
0
P
0
+
K
0
(
81
L
0
P
0
-
50
L
1
P
0
+
3
L
2
P
0
+
14
L
0
P
1
-
L
1
P
1
)
)
c
2
2
c
3
2
+
2
(
6
+
K
0
(
2
L
1
P
0
-
L
0
(
6
P
0
+
P
1
)
)
)
c
3
3
+
(
-
40
+
K
0
(
40
L
0
-
21
L
1
+
L
2
)
P
0
+
4
K
0
L
0
P
1
)
c
2
3
c
4
+
2
(
26
+
K
0
(
7
L
1
P
0
-
2
L
0
(
13
P
0
+
P
1
)
)
)
c
2
c
3
c
4
+
6
(
-
1
+
K
0
L
0
P
0
)
c
4
2
K
1
=
1; L
3
=
0; (* Vanish  coefficient  of  e
7  *)
e
1
/
/
F
u
l
l
S
i
m
p
l
i
f
y
C5: O
u
t
[
e
1
]
=
-
1
24
(
c
2
(
c
2
2
-
c
3
)
(
(
L
4
+
12
(
-
6
+
P
2
)
)
c
2
4
-
24
(
-
4
+
P
2
)
c
2
2
c
3
+
12
P
2
c
3
2
-
24
c
2
c
4
)
)
e
8
+
O
[
e
]
9
|
K
1
|
<
∞; |
L
3
|
<
∞;C
o
m
m
e
n
t
C
1: from the expression of the error e
1
=
g
(
x
)
-
α, we observe that e
1 is of the form
(16)
e
1
=
g
(
x
)
-
α
=
a
4
e
4
+
a
5
e
5
+
a
6
e
6
+
a
7
e
7
+
a
8
e
8
+
O
(
e
9
)
.
The iterative method x
k
+
1
=
g
(
x
k
) will have the order of convergence equal to eight if we determine the coefficients of the developments appearing in (14) in such a way that a
4
,
a
5
,
a
6, and a
7 (in (16)) vanish. We find these coefficients equalling shaded expressions in boxed formulas to 0. First, fromOut[a4], we have
(17)
-
1
+
K
(
0
)
L
(
0
)
P
(
0
)
=
0
.
Without loss of generality, we can take K
(
0
)
=
L
(
0
)
=
P
(
0
)
=
1 and hence, a
4
=
0. In what follows, a
5, a
6, a
7, and a
8 are calculated using the already found coefficients.C
o
m
m
e
n
t
C
2: fromOut[a5], we see that the choice L
′
(
0
)
=
0 gives a
5
=
0.C
o
m
m
e
n
t
C
3: we obtain a
6
=
0 choosing L
′′
(
0
)
=
P
′
(
0
)
=
0.C
o
m
m
e
n
t
C
4: a
7 vanishes if we choose simultaneously K
′
(
0
)
=
1
,
L
′′′
(
0
)
=
0.C
o
m
m
e
n
t
C
5: substituting the quantities K
(
0
)
,
K
′
(
0
)
,
…
,
P
′′
(
0
) in the expression of e
1, found in the described interactive procedure, we obtain
(18)
e
1
=
g
(
x
)
-
α
=
-
1
24
(
-
24
(
-
4
+
P
′′
(
0
)
)
c
2
2
c
3
+
12
P
′′
(
0
)
c
3
2
-
24
c
2
c
4
)
c
2
(
c
2
2
-
c
3
)
×
(
(
L
(
4
)
(
0
)
+
12
(
-
6
+
P
′′
(
0
)
)
)
c
2
4
-
24
(
-
4
+
P
′′
(
0
)
)
c
2
2
c
3
+
12
P
′′
(
0
)
c
3
2
-
24
c
2
c
4
)
)
×
e
n
8
+
O
(
e
n
9
)
,
where |
L
′′′
(
0
)
|
<
∞, |
P
′′
(
0
)
|
<
∞.According to the above analysis, we have proved the following theorem.Theorem 1.
Assume thatf
∈
C
5
(
D
). Suppose α
∈
D, f
(
α
)
=
0 and f
′
(
α
)
≠
0. If the initial point x
0 is sufficiently close to α, then the sequence x
n generated by any method of the family (10) has eighth order of convergence to α if K
,
L, and P are any functions with
(19)
K
(
0
)
=
K
′
(
0
)
=
1
,
L
(
0
)
=
1
,
L
′
(
0
)
=
L
′′
(
0
)
=
L
′′′
(
0
)
=
0
,
|
L
(
4
)
(
0
)
|
<
∞
,
P
(
0
)
=
1
,
P
′
(
0
)
=
0
,
|
P
′′
(
0
)
|
<
∞
.Remark 2.
Any method of the family (10) uses four evaluations per iteration and has eighth-order convergence and satisfies the conditions (19), which accords with the conjecture of Kung-Traub [15] that a multipoint iteration without memory based on n evaluations achieves optimal convergence order 2
n
-
1 for the case n
=
4.


## 3. The Concrete Iterative Methods

In what follows, we give some concrete iterative forms of scheme (10).Method 1.
The functionsK
(
t
)
,
L
(
t
), and P
(
t
) defined by
(20)
K
(
t
)
=
sin
(
t
)
+
cos
⁡
(
t
)
,
L
(
t
)
=
t
4
e
t
+
1
,
P
(
t
)
=
e
t
2
satisfy the conditions of Theorem 1. A new family of two-parameter eighth-order methods is obtained
(21)
y
n
=
x
n
-
f
(
x
n
)
f
′
(
x
n
)
,
z
n
=
y
n
-
f
(
x
n
)
f
(
x
n
)
-
2
f
(
y
n
)
f
(
y
n
)
f
′
(
x
n
)
,
x
n
+
1
=
z
n
-
(
sin
(
f
(
z
n
)
f
(
x
n
)
)
+
cos
⁡
(
f
(
z
n
)
f
(
x
n
)
)
)
×
(
1
+
f
4
(
y
n
)
f
4
(
x
n
)
e
(
f
(
y
n
)
/
f
(
x
n
)
)
)
e
(
f
2
(
z
n
)
/
f
2
(
y
n
)
)
×
f
(
z
n
)
f
[
x
n
,
y
n
]
f
[
x
n
,
z
n
]
f
[
y
n
,
z
n
]
,
and the error equation becomes
(22)
e
n
+
1
=
c
2
(
c
2
2
-
c
3
)
(
c
2
4
-
2
c
2
2
c
3
-
c
3
2
+
c
2
c
4
)
e
n
8
+
O
(
e
n
9
)
.Method 2.
The functionsK
(
t
)
,
L
(
t
), and P
(
t
) defined by
(23)
K
(
t
)
=
e
t
-
1
+
cos
⁡
(
t
)
,
L
(
t
)
=
e
t
4
,
P
(
t
)
=
1
-
t
+
sin
(
t
)
satisfy the conditions of Theorem 1. A new family of two-parameter eighth-order methods is obtained
(24)
y
n
=
x
n
-
f
(
x
n
)
f
′
(
x
n
)
,
z
n
=
y
n
-
f
(
x
n
)
f
(
x
n
)
-
2
f
(
y
n
)
f
(
y
n
)
f
′
(
x
n
)
,
x
n
+
1
=
z
n
-
(
e
(
f
(
z
n
)
/
f
(
x
n
)
)
-
1
+
cos
⁡
(
f
(
z
n
)
f
(
x
n
)
)
)
×
(
1
-
f
(
z
n
)
f
(
y
n
)
+
sin
(
f
(
z
n
)
f
(
y
n
)
)
)
e
f
4
(
y
n
)
/
f
4
(
x
n
)
×
f
(
z
n
)
f
[
x
n
,
y
n
]
f
[
x
n
,
z
n
]
f
[
y
n
,
z
n
]
,
and the error equation becomes
(25)
e
n
+
1
=
c
2
2
(
c
2
2
-
c
3
)
(
2
c
2
3
-
4
c
2
c
3
+
c
4
)
e
n
8
+
O
(
e
n
9
)
.Method 3.
The functionsK
(
t
)
,
L
(
t
), and P
(
t
) defined by
(26)
K
(
t
)
=
1
+
sin
(
t
)
,
L
(
t
)
=
1
+
t
4
cos
⁡
(
t
)
,
P
(
t
)
=
cos
⁡
(
t
)
satisfy the conditions of Theorem 1. A new family of two-parameter eighth-order methods is obtained
(27)
y
n
=
x
n
-
f
(
x
n
)
f
′
(
x
n
)
,
z
n
=
y
n
-
f
(
x
n
)
f
(
x
n
)
-
2
f
(
y
n
)
f
(
y
n
)
f
′
(
x
n
)
,
x
n
+
1
=
z
n
-
(
1
+
sin
(
f
(
z
n
)
f
(
x
n
)
)
)
×
(
1
+
f
4
(
y
n
)
f
4
(
x
n
)
cos
⁡
(
f
(
y
n
)
f
(
x
n
)
)
)
×
cos
⁡
(
f
(
z
n
)
f
(
y
n
)
)
f
(
z
n
)
f
[
x
n
,
y
n
]
f
[
x
n
,
z
n
]
f
[
y
n
,
z
n
]
,
and the error equation becomes
(28)
e
n
+
1
=
1
2
c
2
(
c
2
2
-
c
3
)
(
5
c
2
4
-
10
c
2
2
c
3
+
c
3
2
+
2
c
2
c
4
)
e
n
8
+
O
(
e
n
9
)
.
The families (21), (24), and (27) achieve eighth-order convergence. Per iteration the presented methods require three evaluations of the function and one evaluation of its first derivative. If we assume that all the evaluations have the same cost as function one, we have that the family of methods (10) has the efficiency index of 8
1
/
4
=
1.682 which is better than 2
1
/
2
=
1.414 of Newton’s method, 4
1
/
3
=
1.587 of King’s methods, 6
1
/
4
=
1.565 of some methods with sixth-order convergence, and 7
1
/
4
=
1.627 of some variants of Ostrowski’s method with seventh-order convergence [14].


## 4. Numerical Results

Now, Method1 (21), Method 2 (24), and Method 3 (27) are employed to solve some nonlinear equations and compared with Bi et al.’s method (BM), (4), (with γ
=
1), Liu and Wang’s method (LWM), (5), (with μ
=
5 and λ
=
-
7), and Sharma’s method (ShM), (6). The test functions of f
(
x
) are listed in Table 1.Table 1
The examples considered in this study.
Test functions
Zeros
f
1
(
x
)
=
x
2
-
e
x
-
3
x
+
2
α
≈
0.25753028543986079
f
2
(
x
)
=
x
e
x
2
-
sin
2
(
x
)
+
3
cos
⁡
(
x
)
+
5
α
≈
-
1.207647827130919
f
3
(
x
)
=
sin
(
x
)
e
x
+
Ln
(
x
2
+
1
)
α
≈
0
f
4
(
x
)
=
x
5
+
x
4
+
4
x
2
-
15
α
≈
1.3474280989683053
f
5
(
x
)
=
10
x
e
-
x
2
-
1
α
≈
1.6796306104284499
f
6
(
x
)
=
cos
⁡
(
x
)
-
x
α
≈
0.73908513321516064
f
7
(
x
)
=
e
-
x
2
+
x
+
2
-
1
α
≈
-
1.000000000000000
f
8
(
x
)
=
Ln
(
x
2
+
x
+
2
)
-
x
+
1
α
≈
4.1525907367571583Numerical computations reported here have been carried out in aMathematica  8.0 environment. Table 2 shows the difference of the root α and the approximation x
n to α, where α is the exact root computed with 800 significant digits (D
i
g
i
t
s
:
=
800) and x
n is calculated by using the same total number of function evaluations (TNFE) for all methods. The absolute values of the function (
|
f
(
x
n
)
|
) and the computational order of convergence (COC) are also shown in Table 2. Here, COC is defined by [19]
(29)
ρ
≈
Ln
|
(
x
n
+
1
-
α
)
(
x
n
-
α
)
|
Ln
|
(
x
n
-
α
)
(
x
n
-
1
-
α
)
|
.Table 2
Comparison of various iterative methods under the same total number of function evaluations (TNFE = 12).
(21)
(24)
(27)
(BM)
(LWM)
(ShM)
f
1
,
x
0
=
0
|
x
n
-
α
|
0.23
e
-
643
0.18
e
-
654
0.42
e
-
662
0.46
e
-
650
0.24
e
-
613
0.90
e
-
656
|
f
(
x
n
)
|
0.85
e
-
643
0.68
e
-
654
0.16
e
-
661
0.17
e
-
649
0.91
e
-
613
0.34
e
-
655
COC
8.000000
8.000000
8.000000
8.000000
8.000000
8.000000
f
2
,
x
0
=
-
1
|
x
n
-
α
|
0.24
e
-
307
0.39
e
-
293
0.26
e
-
297
0.48
e
-
217
0.14
e
-
340
0.65
e
-
430
|
f
(
x
n
)
|
0.49
e
-
306
0.79
e
-
292
0.53
e
-
296
0.97
e
-
216
0.28
e
-
339
0.13
e
-
428
COC
8.000001
8.000002
8.000002
7.999999
8.000000
8.000000
f
3
,
x
0
=
2
|
x
n
-
α
|
0.32
e
-
83
0.610
e
-
92
0.74
e
-
77
0.10
e
-
107
0.79
e
-
52
0.14
e
-
96
|
f
(
x
n
)
|
0.32
e
-
83
0.610
e
-
92
0.74
e
-
77
0.10
e
-
107
0.79
e
-
52
0.14
e
-
96
COC
7.986895
7.993900
7.984913
7.965999
7.899280
7.991216
f
4
,
x
0
=
1.6
|
x
n
-
α
|
0.12
e
-
368
0.11
e
-
359
0.21
e
-
361
0.66
e
-
303
0.97
e
-
307
0.51
e
-
341
|
f
(
x
n
)
|
0.45
e
-
367
0.41
e
-
358
0.78
e
-
360
0.24
e
-
301
0.36
e
-
305
0.19
e
-
339
COC
7.999999
7.999997
8.000018
7.999999
7.999999
8.000000
f
5
,
x
0
=
1.8
|
x
n
-
α
|
0.15
e
-
472
0.13
e
-
471
0.10
e
-
485
0.26
e
-
446
0.47
e
-
399
0.17
e
-
444
|
f
(
x
n
)
|
0.42
e
-
472
0.36
e
-
471
0.28
e
-
485
0.72
e
-
446
0.13
e
-
398
0.46
e
-
444
COC
8.000000
8.000000
8.000000
8.000000
8.000000
8.000000
f
6
,
x
0
=
1
|
x
n
-
α
|
0.11
e
-
632
0.33
e
-
572
0.99
e
-
558
0.98
e
-
664
0.49
e
-
581
0.71
e
-
570
|
f
(
x
n
)
|
0.18
e
-
632
0.56
e
-
572
0.17
e
-
557
0.16
e
-
663
0.81
e
-
581
0.12
e
-
569
COC
8.000000
8.000000
8.000000
8.000000
8.000000
8.000000
f
7
,
x
0
=
-
0.5
|
x
n
-
α
|
0.27
e
-
235
0.17
e
-
198
0.84
e
-
192
0.31
e
-
221
0.13
e
-
221
0.35
e
-
193
|
f
(
x
n
)
|
0.80
e
-
235
0.52
e
-
198
0.25
e
-
191
0.94
e
-
221
0.39
e
-
221
0.11
e
-
192
COC
8.015338
8.000253
8.000189
8.000058
8.000030
7.999924
f
8
,
x
0
=
3.2
|
x
n
-
α
|
0.31
e
-
518
0.10
e
-
485
0.12
e
-
476
0.19
e
-
618
0.58
e
-
487
0.26
e
-
479
|
f
(
x
n
)
|
0.19
e
-
518
0.62
e
-
486
0.69
e
-
477
0.11
e
-
618
0.35
e
-
487
0.16
e
-
479
COC
8.000000
8.000000
8.000000
8.000000
8.000000
8.000000


## 5. Conclusions

We have obtained a new family of variants of Ostrowski’s method. The convergence order of these methods is eight, which consist of three evaluations of the function and one evaluation of the first derivative per iteration, so they have an efficiency index equal to8
1
/
4
=
1.682. Therefore, the family of methods agrees with the conjecture of Kung-Traub for n
=
4. Numerical examples also show that the numerical results of our new methods, in equal iterations, improve the results of other existing three-step methods with eighth order convergence.



---

*Source: 369713-2014-03-13.xml*