1. Introduction
Finding the roots of nonlinear equations is one of the most important problems in numerical analysis. In this study, we use iterative methods to find a multiple root
x
⋆
of multiplicity
m
(
m
>
1
)
; that is,
f
(
j
)
(
x
⋆
)
=
0
,
j
=
0,1
,
…
,
m

1
, and
f
(
m
)
(
x
⋆
)
≠
0
, of a nonlinear equation
f
(
x
)
=
0
.
It is known that the modified Newton method for multiple roots is given by
(1)
x
n
+
1
=
x
n

m
f
(
x
n
)
f
′
(
x
n
)
,
which converges quadratically [1].
There exists a cubically convergent method for multiple roots, presented by Hansen and Patrick [2]. Consider
(2)
x
n
+
1
=
x
n

(
f
(
x
n
)
)
×
(
m
+
1
2
m
f
′
(
x
n
)

f
(
x
n
)
f
′′
(
x
n
)
2
f
′
(
x
n
)
)

1
,
which is an extension of the classical Halley method of the third order.
Another cubically convergent method for multiple roots is proposed by Traub [3]. Consider
(3)
x
n
+
1
=
x
n

m
(
3

m
)
2
f
(
x
n
)
f
′
(
x
n
)

m
2
2
f
(
x
n
)
2
f
′′
(
x
n
)
f
′
(
x
n
)
3
,
which is an extension of the wellknown Chebyshev method of the third order.
In recent years, a lot of methods for multiple roots have been presented and analyzed, which require the knowledge of the multiplicity
m
; see [4–24] and references therein.
Based on King's fourthorder method (for simple roots) [25], Dong [4] has developed two thirdorder methods for multiple roots, requiring two evaluations of the function and one of its first derivative. Consider
(4)
y
n
=
x
n

m
f
(
x
n
)
f
′
(
x
n
)
,
x
n
+
1
=
y
n

m
(
1

1
m
)
(
1

m
)
f
(
y
n
)
f
′
(
x
n
)
,
(5)
y
n
=
x
n

f
(
x
n
)
f
′
(
x
n
)
,
x
n
+
1
=
y
n

f
(
y
n
)
(
(
m

1
)
/
m
)
m

1
f
(
x
n
)

f
(
y
n
)
f
(
x
n
)
f
′
(
x
n
)
.
Using the same information, Victory Jr. and Neta [5] have developed a third method. Consider
(6)
y
n
=
x
n

f
(
x
n
)
f
′
(
x
n
)
,
x
n
+
1
=
y
n

f
(
y
n
)
f
′
(
x
n
)
f
(
x
n
)
+
A
f
(
y
n
)
f
(
x
n
)
+
B
f
(
y
n
)
,
where
(7)
A
=
(
m
m

1
)
2
m

(
m
m

1
)
m
+
1
,
B
=

(
m
/
(
m

1
)
)
m
(
m

2
)
(
m

1
)
+
1
(
m

1
)
2
.
Neta [9] has developed another thirdorder method requiring the same information
(8)
y
n
=
x
n

α
f
(
x
n
)
f
′
(
x
n
)
,
x
n
+
1
=
y
n

(
β

α
)
f
(
x
n
)
+
γ
f
(
y
n
)
f
′
(
x
n
)
,
where
(9)
α
=
1
2
m
(
m
+
3
)
m
+
1
,
β
=
m
3
+
4
m
2
+
9
m
+
2
(
m
+
3
)
2
,
γ
=
2
m
+
1
(
m
2

1
)
(
m
+
3
)
2
(
(
m

1
)
/
(
m
+
1
)
)
m
.
Based on Halley’s method, Li et al. [15] have proposed a family of third methods using the same information. Consider
(10)
y
n
=
x
n

α
f
(
x
n
)
f
′
(
x
n
)
,
x
n
+
1
=
x
n

m
α
2
μ
m
f
(
x
n
)
(
m

α
+
α
2
)
μ
m
f
(
x
n
)

(
m

α
)
f
(
y
n
)
f
(
x
n
)
f
′
(
x
n
)
,
where
α
is a real parameter and
α
≠
0
,
m
, and
μ
=
(
m

α
)
/
m
.
Note also that, based on Traub’s method [2], Homeier [16] has suggested a family of third methods using the same information. Consider
(11)
y
n
=
x
n

α
f
(
x
n
)
f
′
(
x
n
)
,
x
n
+
1
=
x
n

β
f
(
x
n
)
f
′
(
x
n
)

f
(
y
n
)
γ
f
′
(
x
n
)
,
where
α
≠
0
,
m
is a real parameter,
β
=
(
m
/
α
2
)
(
α
2
+
α

m
)
, and
γ
=
(
1
/
m
)
(
1

α
/
m
)
m
(
α
2
/
(
m

α
)
)
.
In this paper, we propose two new families of thirdorder methods for multiple roots; each of the methods requires twofunction and onederivative evaluation per iteration, respectively. The presented methods are obtained by investigating the following two iteration schemes:
(12)
(
I
)
{
y
n
=
x
n

α
f
(
x
n
)
f
′
(
x
n
)
,
x
n
+
1
=
y
n

a
f
(
x
n
)
+
b
f
(
y
n
)
c
f
(
x
n
)
+
d
f
(
y
n
)
f
(
y
n
)
f
′
(
x
n
)
,
(13)
(
II
)
{
y
n
=
x
n

α
f
(
x
n
)
f
′
(
x
n
)
,
x
n
+
1
=
y
n

a
f
(
x
n
)
+
b
f
(
y
n
)
c
f
(
x
n
)
+
d
f
(
y
n
)
f
(
x
n
)
f
′
(
x
n
)
,
where
α
,
a
,
b
,
c
, and
d
are parameters to be determined. By specially choosing the parameters in (12) and (13), we get two new families of thirdorder methods, which include methods (4)–(6), (8), (10), and (11). In fact, the mild conditions to assure the cubic convergence of (I)type iteration (12) or (II)type iteration (13) are given. Divided differences are adopted successfully in developing our methods, which will be useful in developing more new methods. Finally, we use some numerical examples to compare the presented methods with the modified Newton method and some known thirdorder methods.
2. Preliminaries
We need the definitions of divided differences and their properties.
Definition 1 (see [<xref reftype="bibr" rid="B26">26</xref>]).
The divided differences
f
[
a
0
,
a
1
,
…
,
a
k
]
on
k
+
1
distinct points
a
0
,
a
1
,
…
,
a
k
of a function
f
(
x
)
are defined by
(14)
v
v
v
v
v
v
v
v
v
f
[
a
0
]
=
f
(
a
0
)
,
v
v
v
v
v
v
f
[
a
0
,
a
1
]
=
f
[
a
0
]

f
[
a
1
]
a
0

a
1
,
vvvvvvvvvvvvv
⋮
f
[
a
0
,
a
1
,
…
,
a
k
]
=
f
[
a
0
,
a
1
,
…
,
a
k

1
]

f
[
a
1
,
a
2
,
…
,
a
k
]
a
0

a
k
.
If the function
f
is sufficiently differentiable, then its divided differences
f
[
a
0
,
a
1
,
…
,
a
k
]
can be defined if some of the arguments
a
i
coincide. For instance, if
f
(
x
)
has a derivative of the
k
th order at
a
0
, then it makes sense to define
(15)
f
[
a
0
,
a
0
,
…
,
a
0
︸
k
+
1
]
=
f
(
k
)
(
a
0
)
k
!
.
Lemma 2 (see [<xref reftype="bibr" rid="B26">26</xref>]).
The divided differences
f
[
a
0
,
a
1
,
…
,
a
k
]
are symmetric functions of their arguments; that is, they are invariant to permutations of the
a
0
,
a
1
,
…
,
a
k
.
Lemma 3 (see [<xref reftype="bibr" rid="B26">26</xref>]).
If the function
f
has
(
k
+
1
)
st
derivative, then, for every argument
x
, the following interpolation formula holds:
(16)
f
(
x
)
=
f
[
a
0
]
+
∑
i
=
1
k
f
[
a
0
,
a
1
,
…
,
a
i
]
∏
j
=
0
i

1
(
x

a
j
)
+
f
[
a
0
,
a
1
,
…
,
a
k
,
x
]
∏
i
=
0
k
(
x

a
i
)
.
Lemma 4.
If the function
f
has a derivative of the
(
m
+
1
)
th order, and
x
⋆
is a multiple root of multiplicity
m
, then, for every argument
x
, the following formulae hold:
(17)
f
(
x
)
=
f
[
x
⋆
,
x
⋆
,
…
,
x
⋆
︸
m
,
x
]
(
x

x
⋆
)
m
,
(18)
f
′
(
x
)
=
f
[
x
⋆
,
x
⋆
,
…
,
x
⋆
︸
m
,
x
,
x
]
(
x

x
⋆
)
m
+
m
f
[
x
⋆
,
x
⋆
,
…
,
x
⋆
︸
m
,
x
]
(
x

x
⋆
)
m

1
.
Proof.
Applying Lemma 3 to the case of the multiple zero
x
⋆
of multiplicity
m
and using (15), we get (17). Differentiating both sides of (17) gives (18).
3. Development of New Families of ThirdOrder Methods
We would like to find the five parameters
α
,
a
,
b
,
c
, and
d
in Itype iteration (12) and IItype iteration (13) so as to maximize its order of convergence to a root
x
⋆
of multiplicity
m
, respectively. Let
e
n
,
d
n
be the errors at the
n
th step; that is,
(19)
e
n
=
x
n

x
⋆
,
d
n
=
y
n

x
⋆
.
Define functions
g
(
x
)
and
h
(
x
)
as follows:
(20)
g
(
x
)
=
f
[
x
⋆
,
x
⋆
,
…
,
x
⋆
︸
,
m
x
]
,
h
(
x
)
=
f
[
x
⋆
,
x
⋆
,
…
,
x
⋆
︸
m
,
x
,
x
]
.
Write
(21)
g
n
=
g
(
x
n
)
,
h
n
=
h
(
x
n
)
,
g
⋆
=
g
(
x
⋆
)
,
h
⋆
=
h
(
x
⋆
)
.
In view of (17) and (18), we get the following:
(22)
f
(
x
n
)
=
g
n
e
n
m
,
(23)
f
′
(
x
n
)
=
h
n
e
n
m
+
m
g
n
e
n
m

1
,
(24)
f
(
y
n
)
=
g
(
y
n
)
d
n
m
.
Using the definitions of divided differences, we get the following:
(25)
g
(
y
n
)
=
g
(
y
n
)

g
(
x
n
)
+
g
(
x
n
)
=
f
[
x
⋆
,
x
⋆
,
…
,
x
⋆
︸
m
,
y
n
,
x
n
]
(
d
n

e
n
)
+
g
n
=
(
f
[
x
⋆
,
x
⋆
,
…
,
x
⋆
︸
m
,
y
n
,
x
n
]

h
(
x
n
)
+
h
(
x
n
)
)
×
(
d
n

e
n
)
+
g
n
=
p
n
(
d
n

e
n
)
2
+
h
n
(
d
n

e
n
)
+
g
n
,
where
(26)
p
n
=
f
[
x
⋆
,
x
⋆
,
…
,
x
⋆
︸
m
,
y
n
,
x
n
,
x
n
]
.
In view of (22), (23), (12), and (13), we get in turn
(27)
f
(
x
n
)
f
′
(
x
n
)
=
g
n
e
n
h
n
e
n
+
m
g
n
,
(28)
d
n
=
e
n

α
g
n
e
n
h
n
e
n
+
m
g
n
=
h
n
e
n
+
(
m

α
)
g
n
h
n
e
n
+
m
g
n
e
n
,
(29)
d
n

e
n
=

α
g
n
e
n
h
n
e
n
+
m
g
n
.
Substituting (29) into (25) yields
(30)
g
(
y
n
)
=
p
n
(

α
g
n
e
n
h
n
e
n
+
m
g
n
)
2
+
h
n
(

α
g
n
e
n
h
n
e
n
+
m
g
n
)
+
g
n
=
g
n
(
h
n
e
n
+
m
g
n
)
2
(
[
α
2
p
n
g
n
+
(
1

α
)
h
n
2
]
e
n
2
v
v
v
v
v
v
v
v
v
v
+
(
2

α
)
m
h
n
g
n
e
n
+
m
2
g
n
2
)
.
Substituting (30) and (28) into (24) leads to
(31)
f
(
y
n
)
=
g
n
[
h
n
e
n
+
(
m

α
)
g
n
]
m
(
h
n
e
n
+
m
g
n
)
m
+
2
×
(
[
α
2
p
n
g
n
+
(
1

α
)
h
n
2
]
e
n
2
+
(
2

α
)
m
h
n
g
n
e
n
+
m
2
g
n
2
)
e
n
m
.
Write
(32)
A
n
=
h
n
e
n
+
m
g
n
,
B
n
=
h
n
e
n
+
(
m

α
)
g
n
,
C
n
=
[
α
2
p
n
g
n
+
(
1

α
)
h
n
2
]
e
n
2
+
(
2

α
)
m
h
n
g
n
e
n
+
m
2
g
n
2
.
Using (23), (24), and (32), we get
(33)
f
′
(
x
n
)
=
A
n
e
n
m

1
,
f
(
y
n
)
=
g
n
B
n
m
C
n
A
n
m
+
2
e
n
m
.
Then, we can get the error equations as follows:
(34)
e
n
+
1
=
B
n
A
n
e
n

a
g
n
e
n
m
+
b
(
g
n
B
n
m
C
n
/
A
n
m
+
2
)
e
n
m
c
g
n
e
n
m
+
d
(
g
n
B
n
m
C
n
/
A
n
m
+
2
)
e
n
m
×
(
g
n
B
n
m
C
n
/
A
n
m
+
2
)
e
n
m
A
n
e
n
m

1
=

(
[
(
a
A
n
m
+
2
+
b
B
n
m
C
n
)
g
n
B
n
m

1
C
n

(
c
A
n
m
+
2
+
d
B
n
m
C
n
)
A
n
m
+
2
]
B
n
e
n
)
×
(
(
c
A
n
m
+
2
+
d
B
n
m
C
n
)
A
n
m
+
3
)

1
,
for Itype iteration (12), and
(35)
e
n
+
1
=
B
n
A
n
e
n

a
g
n
e
n
m
+
b
(
g
n
B
n
m
C
n
/
A
n
m
+
2
)
e
n
m
c
g
n
e
n
m
+
d
(
g
n
B
n
m
C
n
/
A
n
m
+
2
)
e
n
m
×
g
n
e
n
m
A
n
e
n
m

1
=

(
[
(
a
A
n
m
+
2
+
b
B
n
m
C
n
)
g
n

(
c
A
n
m
+
2
+
d
B
n
m
C
n
)
B
n
]
e
n
)
×
(
(
c
A
n
m
+
2
+
d
B
n
m
C
n
)
A
n
)

1
,
for IItype iteration (13).
In view of (34) and (35), the order of convergence for Itype or IItype iteration will arrive at three provided that
(36)
Φ
n
(
c
A
n
m
+
2
+
d
B
n
m
C
n
)
A
n
m
+
3
=
O
(
e
n
2
)
,
or
(37)
Ψ
n
(
c
A
n
m
+
2
+
d
B
n
m
C
n
)
A
n
=
O
(
e
n
2
)
,
holds true, respectively. Here
(38)
Φ
n
=
[
(
a
A
n
m
+
2
+
b
B
n
m
C
n
)
g
n
B
n
m

1
C
n

(
c
A
n
m
+
2
+
d
B
n
m
C
n
)
A
n
m
+
2
]
B
n
,
Ψ
n
=
(
a
A
n
m
+
2
+
b
B
n
m
C
n
)
g
n

(
c
A
n
m
+
2
+
d
B
n
m
C
n
)
B
n
.
Write
(39)
λ
=
m

α
.
In view of (32), we can get, as
n
→
∞
,
(40)
A
n
⟶
m
g
⋆
=
m
f
[
x
⋆
,
x
⋆
,
…
,
x
⋆
︸
m
+
1
]
=
m
f
(
m
)
(
x
⋆
)
m
!
≠
0
,
B
n
⟶
λ
g
⋆
,
C
n
⟶
(
m
g
⋆
)
2
,
and then
(41)
(
c
A
n
m
+
2
+
d
B
n
m
C
n
)
A
n
m
+
3
⟶
(
c
m
m
+
d
λ
m
)
m
m
+
5
g
⋆
2
m
+
5
,
v
v
v
v
v
v
v
v
v
v
v
v
v
(
n
⟶
∞
)
,
(
c
A
n
m
+
2
+
d
B
n
m
C
n
)
A
n
⟶
(
c
m
m
+
d
λ
m
)
m
3
g
⋆
m
+
3
,
v
v
v
v
v
v
v
v
v
v
(
n
⟶
∞
)
,
which show that, in order to assure the denominators in (36) and (37) are not equal to zero, we demand naturally that
(42)
c
m
m
+
d
λ
m
≠
0
.
It is obvious that, under the condition (42), the error relations (36) and (37) are equivalent to
(43)
Φ
n
=
O
(
e
n
2
)
,
(44)
Ψ
n
=
O
(
e
n
2
)
,
respectively.
Next we will find conditions to assure (43) and (44). Note that the factor
B
n
of
Φ
n
plays an important role on the order of
Φ
n
. In fact, using the Taylor formula, we get from (32) the following:
(45)
B
n
=
λ
g
⋆
+
O
(
e
n
)
.
Then, in the case
λ
=
0
, to assure the relation (43) holds true, the following estimate is needed:
(46)
Δ
n
=
(
a
A
n
m
+
2
+
b
B
n
m
C
n
)
g
n
B
n
m

1
C
n

(
c
A
n
m
+
2
+
d
B
n
m
C
n
)
A
n
m
+
2
=
O
(
e
n
)
,
which demands
(47)
Δ
n
⟶

c
(
m
g
⋆
)
2
m
+
4
=
0
,
(
n
⟶
∞
)
;
that is,
c
=
0
. This is a contradiction to (42). Hence, in the case
λ
=
0
, the relation (43) cannot be satisfied; that is, we cannot choose parameters so that the order of convergence of (I)type iteration (12) arrives at three.
In what follows, we suppose
λ
≠
0
. In this case,
B
n
→
λ
g
⋆
≠
0
(
n
→
∞
)
, and then (43) is equivalent to
(48)
Δ
n
=
O
(
e
n
2
)
.
In view of (32), and by a straight computation, we can get
(49)
Δ
n
=
u
n
+
v
n
e
n
+
O
(
e
n
2
)
,
where
(50)
u
n
=
(
m
m
λ
m

1
a
+
λ
2
m

1
b

m
2
m
c

m
m
λ
m
d
)
m
4
g
n
2
m
+
4
,
(51)
v
n
=
(
m
m
λ
m

2
[
λ
2
+
4
λ
+
m
(
m

1
)
]
a
+
λ
2
m

2
[
2
λ
2
+
(
4

2
m
)
λ
+
m
(
2
m

1
)
]
b

(
2
m
+
4
)
m
2
m
c

m
m
λ
m

1
(
λ
2
+
4
λ
+
m
2
)
d
)
m
3
g
n
2
m
+
3
h
n
.
In view of (49) and in order to assure the relation (48) holds, we should choose parameters
λ
,
a
,
b
,
c
, and
d
such that
(52)
m
m
λ
m

1
a
+
λ
2
m

1
b

m
2
m
c

m
m
λ
m
d
=
0
,
m
m
λ
m

2
[
λ
2
+
4
λ
+
m
(
m

1
)
]
a
+
λ
2
m

2
[
2
λ
2
+
(
4

2
m
)
λ
+
m
(
2
m

1
)
]
b

(
2
m
+
4
)
m
2
m
c

m
m
λ
m

1
(
λ
2
+
4
λ
+
m
2
)
d
=
0
.
By a straight computation, we deduce that
(53)
c
=
m
m
+
1
λ
m

1
a
+
[
m

(
λ

m
)
2
]
λ
2
m

1
b
(
λ

m
)
2
m
2
m
,
(54)
d
=
[
(
λ

m
)
2

m
]
m
m
a
+
[
2
(
λ

m
)
2

m
]
λ
m
b
λ
(
λ

m
)
2
m
m
.
Substituting (53) and (54) into the left side of (42), we get
(55)
c
m
m
+
d
λ
m
=
m
m
+
1
λ
m

1
a
+
[
m

(
λ

m
)
2
]
λ
2
m

1
b
(
λ

m
)
2
m
m
+
[
(
λ

m
)
2

m
]
m
m
λ
m

1
a
+
[
2
(
λ

m
)
2

m
]
λ
2
m

1
b
(
λ

m
)
2
m
m
=
m
m
a
+
λ
m
b
m
m
λ
m

1
,
which shows the condition (42) is equivalent to
(56)
a
m
m
+
b
(
m

α
)
m
≠
0
.
We summarize our development of new methods done so far in the following theorem.
Theorem 5.
Let
x
⋆
∈
I
be a multiple root of multiplicity
m
(
m
>
1
)
of a sufficiently differentiable function
f
:
I
→
R
for an open interval
I
. If
x
0
is sufficiently close to
x
⋆
, then the methods defined by (I)type iteration (12) are cubically convergent for any parameters
α
,
a
,
b
,
c
, and
d
such that
α
≠
0
and (53), (54), and (56) hold.
Next, we turn to find the proper conditions to establish the relation (44). In view of (32), and by a straight computation, we can get
(57)
Ψ
n
=
r
n
+
s
n
e
n
+
O
(
e
n
2
)
,
where
(58)
r
n
=
(
m
m
a
+
λ
m
b

m
m
λ
c

λ
m
+
1
d
)
m
2
g
n
m
+
3
,
s
n
=
(
(
m
+
2
)
m
m
+
1
a
+
[
(
2

α
)
m
λ
m
+
m
3
λ
m

1
]
b

[
m
m
+
2
+
m
m
+
1
(
m
+
2
)
λ
]
c

[
(
2

α
)
m
λ
m
+
1
+
m
2
(
m
+
1
)
λ
m
]
d
)
g
n
m
+
2
h
n
.
In view of (57) and in order to assure the relation (44) holds, we should choose parameters
λ
,
a
,
b
,
c
, and
d
such that
(59)
m
m
a
+
λ
m
b

m
m
λ
c

λ
m
+
1
d
=
0
,
(
m
+
2
)
m
m
a
+
[
(
2

α
)
λ
+
m
2
]
λ
m

1
b

[
m
+
(
m
+
2
)
λ
]
m
m
c

[
(
2

α
)
λ
+
m
(
m
+
1
)
]
λ
m
d
=
0
.
By a straight computation, we deduce that
(60)
c
=
[
(
λ

m
)
2
+
m
]
m
m

1
a
+
λ
m
b
λ
(
λ

m
)
2
m
m

1
,
d
=

m
m
+
1
a
+
[
(
λ

m
)
2

m
]
λ
m
b
λ
m
+
1
(
λ

m
)
2
.
Substituting (60) into the left side of (42), we get
(61)
c
m
m
+
d
λ
m
=
m
m
a
+
λ
m
b
λ
,
which shows the condition (42) is also equivalent to the condition given by (56).
We can summarize the development of new methods involving (II)type iteration (13) done so far in the following theorem.
Theorem 6.
Let
x
⋆
∈
I
be a multiple root of multiplicity
m
(
m
>
1
)
of a sufficiently differentiable function
f
:
I
→
R
for an open interval
I
. If
x
0
is sufficiently close to
x
⋆
, then the methods defined by (II)type iteration (13) are cubically convergent for any parameters
α
,
a
,
b
,
c
, and
d
such that
α
≠
0
and (60) and (56) hold.
Choosing
α
=
m
,
a
=
1
, and
b
=
0
, we can deduce from (53), (54), and (56) that
(62)
c
=
(
m

m
)
m

1
m
m
,
(63)
d
=
0
,
(64)
a
m
m
+
b
(
m

α
)
m
=
m
m
≠
0
.
Using the parameters
α
,
a
,
b
,
c
, and
d
given above in (I)type iteration (12), we can obtain Dong's method (4), and its order of convergence arrives at three by Theorem 5.
Choosing
α
=
1
,
a
=
1
, and
b
=
0
, we can deduce from (53), (54), and (56) that
(65)
c
=
(
m

1
)
m

1
m
m

1
,
d
=

1
,
a
m
m
+
b
(
m

α
)
m
=
m
m
≠
0
.
Using the parameters
α
,
a
,
b
,
c
, and
d
given above in (I)type iteration (12), we can obtain Dong's method (5), and its order of convergence arrives at three by Theorem 5.
Choosing
α
=
1
,
a
=
1
, and
b
=
(
m
/
(
m

1
)
)
2
m

(
m
/
(
m

1
)
)
m
+
1
, we can deduce from (53), (54), and (56) that
(66)
c
=
1
,
(67)
d
=

(
m
/
(
m

1
)
)
m
(
m

2
)
(
m

1
)
+
1
(
m

1
)
2
,
(68)
a
m
m
+
b
(
m

α
)
m
=
m
m
[
1
+
(
m
m

1
)
m

m
m

1
]
≠
0
.
Using the parameters
α
,
a
,
b
,
c
, and
d
given above in (I)type iteration (12), we can obtain Victory and Neta's method (6), and its order of convergence arrives at three by Theorem 5.
Let
α
=
(
1
/
2
)
(
m
(
m
+
3
)
/
(
m
+
1
)
)
,
a
=
(
m
3
+
4
m
2
+
9
m
+
2
)
/
(
m
+
3
)
2
−
(
1
/
2
)
(
m
(
m
+
3
)
/
(
m
+
1
)
)
,
b
=
2
m
+
1
(
m
2

1
)
/
(
m
+
3
)
2
(
(
m

1
)
/
(
m
+
1
)
)
m
,
c
=
0
, and
d
=
1
. Using the definition of
λ
, we get
(69)
λ
=
m
(
m

1
)
2
(
m
+
1
)
.
We can verify that the parameters given above satisfy (52), and thus they also satisfy (53) and (54) (as
c
given in (53) and
d
given in (54) are solved from (52)). Furthermore, it is easy to verify that the condition (42) holds:
(70)
c
m
m
+
d
λ
m
=
(
m
(
m

1
)
2
(
m
+
1
)
)
m
≠
0
.
This means that the condition (56) is also true, since (56) is equivalent to (42). Using the parameters
α
,
a
,
b
,
c
, and
d
given in (I)type iteration (12), we can obtain Neta's method (8), and its order of convergence arrives at three by Theorem 5.
We can verify that the family of methods (11) given by Homeier [16] satisfies all conditions in Theorem 5. First, we can rewrite (11) as follows:
(71)
y
n
=
x
n

α
f
(
x
n
)
f
′
(
x
n
)
,
x
n
+
1
=
y
n

(
β

α
)
f
(
x
n
)
+
(
1
/
γ
)
f
(
y
n
)
f
(
y
n
)
f
(
y
n
)
f
′
(
x
n
)
,
where
α
≠
0
,
m
is a real parameter,
β
=
(
m
/
α
2
)
(
α
2
+
(
α

m
)
)
=
(
m
/
α
2
)
(
α
2

λ
)
, and
γ
=
(
1
/
m
)
(
1

α
/
m
)
m
(
α
2
/
(
m

α
)
)
=
(
λ
m

1
α
2
/
m
m
+
1
)
. Choosing
a
=
β

α
and
b
=
1
/
γ
, we can deduce from (53), (54), and (56) that
c
=
0
,
d
=
1
, and
(72)
c
m
m
+
d
λ
m
=
(
m

α
)
m
≠
0
.
This means that the condition (56) is also true. Using the parameters
α
,
a
,
b
,
c
, and
d
given in (I)type iteration (12), we can obtain Homeier's family of methods (11), which has cubic convergence by Theorem 5.
We can verify that the family of methods (10) given by Homeier [16] satisfies all conditions in Theorem 6. First, we can rewrite (10) as follows:
(73)
y
n
=
x
n

α
f
(
x
n
)
f
′
(
x
n
)
,
(74)
x
n
+
1
=
y
n

(
(
m
α
2

m
α
+
α
2

α
3
)
×
μ
m
f
(
x
n
)
+
α
(
m

α
)
f
(
y
n
)
(
m
α
2

m
α
+
α
2

α
3
)
)
×
(
(
m

α
+
α
2
)
μ
m
f
(
x
n
)

(
m

α
)
f
(
y
n
)
)

1
×
f
(
x
n
)
f
′
(
x
n
)
,
where
α
is a real parameter and
α
≠
0
,
m
, and
μ
=
(
m

α
)
/
m
.
Let
a
=
λ
m
+
1
α
(
α

1
)
/
m
m
,
b
=
α
λ
,
c
=
(
λ
+
α
2
)
λ
m
/
m
m
, and
d
=

λ
. We can verify that the parameters given above satisfy (60) and (56). Using the parameters
α
,
a
,
b
,
c
, and
d
given in (II)type iteration (13), we can obtain Shengguo et al.'s family of methods (10), which has cubic convergence by Theorem 6.
4. Some Concrete Methods
In this section, we give some concrete iterative forms of (I)type iteration (12) and (II)type iteration (13).
Method 1. Choosing
α
=
1
,
a
=
0
, and
b
=
1
, we obtain from (63) and (64) that
c
=
(
(
m

1
)
/
m
)
2
m
,
d
=
(
2

m
)
(
m

1
)
m

1
/
m
m
, and
a
m
m
+
b
(
m

α
)
m
=
(
m

1
)
m
≠
0
. Using these parameters in (12), we get a new method. Consider
(75)
y
n
=
x
n

f
(
x
n
)
f
′
(
x
n
)
,
x
n
+
1
=
y
n

(
f
(
y
n
)
)
×
(
(
2

m
)
(
m

1
)
m

1
m
m
f
(
y
n
)
(
m

1
m
)
2
m
f
(
x
n
)
vvvvvvvv
vvvv
+
(
2

m
)
(
m

1
)
m

1
m
m
f
(
y
n
)
)

1
×
f
(
y
n
)
f
′
(
x
n
)
,
which has cubic convergence by Theorem 5.
Method 2. Choosing
α
=
1
,
c
=
1
, and
d
=
0
, we can obtain from (61) and (62) that
a
=
(
2

m
)
m
m
/
(
m

1
)
m

1
,
b
=
m
2
m
/
(
m

1
)
2
m

2
, and
c
m
m
+
d
(
m

α
)
m
=
m
m
≠
0
. Using these parameters in (12), we get a new method. Consider
(76)
y
n
=
x
n

f
(
x
n
)
f
′
(
x
n
)
,
x
n
+
1
=
y
n

(
+
m
2
m
(
m

1
)
2
m

2
f
(
y
n
)
(
2

m
)
m
m
(
m

1
)
m

1
f
(
x
n
)
+
m
2
m
(
m

1
)
2
m

2
f
(
y
n
)
)
×
(
f
(
x
n
)
)

1
×
f
(
y
n
)
f
′
(
x
n
)
,
which has cubic convergence by Theorem 5.
Method 3. Choosing
α
=
1
,
b
=
1
, and
d
=
1
, we can obtain from (61) and (62) that
a
=
(
(
m

1
)
m

1
(
2

m
)

m
m
)
/
m
m
,
c
=
(
(
m

1
)
2
m

2

m
m
+
1
(
m

1
)
m

1
)
/
m
2
m
, and
c
m
m
+
d
(
m

α
)
m
=
(
m

1
)
m

1
[
(
m

1
)
m

1

m
m
]
/
m
m
≠
0
. Using these parameters in (12), we get a new method. Consider
(77)
y
n
=
x
n

f
(
x
n
)
f
′
(
x
n
)
,
x
n
+
1
=
y
n

(
(
m

1
)
m

1
(
2

m
)

m
m
m
m
f
(
x
n
)
+
f
(
y
n
)
)
×
(
(
m

1
)
2
m

2

m
m
+
1
(
m

1
)
m

1
m
2
m
×
f
(
x
n
)
+
f
(
y
n
)
(
m

1
)
2
m

2

m
m
+
1
(
m

1
)
m

1
m
2
m
)

1
×
f
(
y
n
)
f
′
(
x
n
)
,
which has cubic convergence by Theorem 5.
Method 4. Choosing
α
=
1
,
a
=
1
, and
b
=
0
, we can obtain from (70) and (71) that
c
=
(
m
+
1
)
/
(
m

1
)
,
d
=

(
m
/
(
m

1
)
)
m
+
1
, and
a
m
m
+
b
(
m

α
)
m
=
m
m
≠
0
. Using these parameters in (13), we get a new method. Consider
(78)
y
n
=
x
n

f
(
x
n
)
f
′
(
x
n
)
,
x
n
+
1
=
y
n

(
f
(
x
n
)
)
×
(
m
+
1
m

1
f
(
x
n
)

(
m
m

1
)
m
+
1
f
(
y
n
)
)

1
×
f
(
x
n
)
f
′
(
x
n
)
,
which has cubic convergence by Theorem 6.
Method 5. Choosing
α
=
1
,
a
=
1
, and
c
=
0
, we can obtain from (68) and (69) that
b
=

(
m
+
1
)
m
m

1
/
(
m

1
)
m
,
d
=

m
m

1
/
(
m

1
)
m
+
1
, and
c
m
m
+
d
(
m

α
)
m
=

m
m

1
/
(
m

1
)
≠
0
. Using these parameters in (13), we get a new method. Consider
(79)
y
n
=
x
n

f
(
x
n
)
f
′
(
x
n
)
,
x
n
+
1
=
y
n
+
(
f
(
x
n
)

(
m
+
1
)
m
m

1
(
m

1
)
m
f
(
y
n
)
)
×
(
m
m

1
(
m

1
)
m
+
1
f
(
y
n
)
)

1
×
f
(
x
n
)
f
′
(
x
n
)
,
which has cubic convergence by Theorem 6.
Method 6. Choosing
α
=
m
,
a
=
1
, and
b
=
0
, we can obtain from (70) and (71) that
c
=
2
/
(
m

m
)
,
d
=

m
m
/
(
m

m
)
(
m
+
1
)
, and
a
m
m
+
b
(
m

α
)
m
=
m
m
≠
0
. Using these parameters in (13), we get a new method. Consider
(80)
y
n
=
x
n

m
f
(
x
n
)
f
′
(
x
n
)
,
x
n
+
1
=
y
n

(
f
(
x
n
)
)
×
(
2
m

m
f
(
x
n
)

m
m
(
m

m
)
m
+
1
f
(
y
n
)
)

1
×
f
(
x
n
)
f
′
(
x
n
)
,
which has cubic convergence by Theorem 6.