Applied Mathematics, 2013, 4, 1709-1713
Published Online December 2013 (http://www.scirp.org/journal/am)
http://dx.doi.org/10.4236/am.2013.412233
Open Access AM
New Ninth Order J-Halley Method for Solving Nonlinear
Equations
Farooq Ahmad1*, Sajjad Hussain2, Sifat Hussain2, Arif Rafiq3
1Punjab Higher Education Department, Principal, Govt. Degree College Darya Khan, Bhakkar, Pakistan
2Centre for Advanced Studies in Pure and Applied Mathematics, B. Z. Uni., Multan, Pakistan
3Department of Mathematics, COMSATS Institute of Information Technology,
Islamabad, Pakistan
Email: *farooqgujar@gmail.com, sajjad_h96@yahoo.com, sifat2003@gmail.com, arafiq@comsats.edu.pk
Received October 22, 2013; revised November 22, 2013; accepted December 2, 2013
Copyright © 2013 Farooq Ahmad et al. This is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
In the paper [1], authors have suggested and analyzed a predictor-corrector Halley method for solving nonlinear equa-
tions. In this paper, we modified this method by using the finite difference scheme, which had a quantic convergence.
We have compared this modified Halley method with some other iterative methods of ninth order, which shows that this
new proposed method is a robust one. Some examples are given to illustrate the efficiency and the performance of this
new method.
Keywords: Halley Method; Jarratt Method; Iterative Methods; Convergence Order; Numerical Examples
1. Introduction
In recent years, several iterative type methods have been
developed by using the Taylor series, decomposition and
quadrature formulae (see [1-14] and the references there-
in). Using the technique of updating the solution and
Taylor series expansion, Noor and Noor [1] have sug-
gested and analyzed a sixth-order predictor-corrector
iterative type Halley method for solving the nonlinear
equations. Also, Kou et al. [2-4] have also suggested a
class of fifth-order iterative methods. In the implementa-
tion of these methods, one has to evaluate the second
derivative of the function, which is a serious drawback of
these methods. To overcome these drawbacks, we mod-
ify the predictor-corrector Halley method by replacing
the second derivatives of the function by its finite dif-
ference scheme. We prove that the new modified predic-
tor-corrector method is of fifth-order convergence. We
also present the comparison of the new method with the
methods of Kou et al. [2-4] and Hu et al. [5]. In passing,
we would like to point out that the results presented by
Kou et al. [2-4] are incorrect. We also rectify this error.
Several examples are given to illustrate the efficiency
and robustness of the new proposed method.
2. Iterative Methods
The Jarratt’s fourth-order method [6] which improves the
order of convergence is defined by
Algorithm 1
where

 
3.
62
nn
nn
fy fx
Jf
f
yfx


Recently, Kou et al. [2] considered the following two-
step iteration scheme
Algorithm 2
 


1
,0
,
n
nn n
n
n
nn n
fx
yxJf fx
fx
fy
xy
fy


where

 
3.
62
nn
nn
fy fx
Jf
f
yfx


We now state some fifth-order iterative methods which
have been suggested by Noor and Noor [6] and Kou et al.
[2,3] using quite different techniques.
*Corresponding author.
F. AHMAD ET AL.
1710
Algorithm 3

 

 
2
12
2,where 0
2
2,
2
nn
nn n
nnn
nnn
nn
nnnn
fxf x
yx fx
fx fxfx
fxfyf x
xxfyfx fyfx
 








which is a two-step Halley method of fifth-order conver-
gent.
In a recent paper Kou et al. [2,3] have suggested fol-
lowing iterative methods.
Algorithm 4 (SHM [3]). For a given x0, compute the
approximate solution xnþ1 by the iterative schemes:


 
 


 
2
3
1
,
22
where 0
,
nnn
nn nnnn
n
n
nn nnn n
fxfx fx
yxfx n
f
xfxfxfx
fx
fy
xy
fxy xfx

 





Algorithm 5 (ISHM [2]). For a given x0, compute the
approximate solution xnþ1 by the iterative schemes:


 
 





2
3
13
,
22
where 0
.
2
nnn
nn nnnn
n
nnn
nn nn
fxfx fx
yxfx n
f
xfxfxfx
fx
fyfx fy
xy
fx fx

 


 

On the basis of the above discussion a new iterative
technique is proposed below (named as FAJH):
Algorithm 6

 
2,
3
n
nn n
n
fx
yx fx
fx
 
0
(2.1)

,
n
nn n
f
x
zxJf
f
x
 (2.2)

 
1
2,
2
nn
nn nn
fzf z
xz
f
zfzL
 (2.3)
where
 
 
3
62
n
nn
n
f
yfx
Jf
yfx


(2.4)
and
 
.
n
nn
n
f
zfx
Lzx

(2.5)
3. Analysis of Convergence
In this section, we compute the convergence order of the
proposed method (FAJH).
Theorem: Let
I
be a simple zero of sufficiently
differentiable function for an open in-
terval :fIRR
I
.
If 0
x
is close to ,
then the three-step al-
gorithm 6 has ni nt h or der of convergence.
Proof: The iterative technique is given by
 
2,
3
n
nn n
n
fx
yx fx
fx 0
(3.1)


,
n
nn n
fx
zxJf
fx
 (3.2)

 
1
2,
2
nn
nn nn
fzf z
xzfz fzL
 (3.3)
where

 
3
62
n
nn
fy fx
Jf fy fx


n
(3.4)
and

.
n
nn
n
f
zfx
Lzx

(3.5)
Let
be a simple zero of
f
. By Taylor’s expan-
sion, we have,

234
2345
67 8 91011
678910 ,
n nnnn
nnnn nn
5
n
f
x fecececece
cececececeO e

 
(3.6)

23
23 45
5678910
6789 10
12 3 4 5
678910
nnnn
nnnn nn
4
,
n
f
xf cececece
ce cece ceceOe


 
(3.7)
where



1,2,3,
!
and .
k
k
nn
f
ck
kf
ex





,
Using , and we have
(3.1) (3.6) (3.7 ),

223
232
12 44,
33 33
nnn n
4
n
y
ece cceOe

 

 (3.8)
by Taylor’s series, we have



223
232
17 378,
39 279
nnn n
fyfececc eOe
4
n




(3.9)
and



22
223
334
23 24
241
1333
84
47.
327
nn
nn
fy fcec ce
cccceO e






 


n
(3.10)
Open Access AM
F. AHMAD ET AL. 1711
Using , and we have (3.4)
3.7
(3.10),
 
22 3
223 234
26
22
9
nn n
4
.
n
J
fce cceccceOe




(3.11)
Using , , and (3.11 we have (3.2)

3.6 (3.7) ),

34
4223
2245
5322432
1
9
820
824
27 9
nn
nn
zcccce
ccccccceOe

 


  
6
,
(3.12)
by Taylor’s series, we have


34
4223
2245
5322432
1
9
820
824
27 9
nn
nn
fzcccc e
ccccccceOe





  
6
,
(3.13)
and


34
4223
2245
5322432
1
9
820
824
27 9
nn
nn
fzc c cce
ccccccceOe





  


6
.
(3.14)
From , and we have
(3.5)
3.7 (3.14),

23
23 4 5
32 4
32 23643
23 45
33613
nnn
nn
Lc cecece
ccccccceO e
 


5
.
(3.15)
Using and (3.15 we get

(3.3), 3.7,(3.12),(3.14)),

2332
14234323232
62910
32 43
113
3
332
31 ,
254
n
nn
24
x
cccccccc cc
cccceO e

 
(3.16)
implies

2332
1423 432 32
2462 910
32 3243
113
332
31
3.
254
n
nn
ecccccccc
cccccc eOe

 
Thus we observe that the new three-step method
(FAJH) has ninth order convergence.
4. Numerical Examples
In this section now we consider some numerical exam-
ples (see Table 1) to demonstrate the performance of the
newly developed iterative method. We compare classical
Newton method (NW), Kou et al. method (see, [2])
(VCM) and (VSHM), Noor et al. methods (see [1])
(NR1), (NR2) and also ninth order Zhongyong Hu et al.
(Z Hu) [5] with the new developed method (FAJH). All
the computations for above mentioned methods, are per-
formed using software Maple , precision digits
and
9128
15
10
as tolerance and also the following crite-
ria is used for estimating the zero:
1) 1,
nn
xx

2)
,
n
fx
3) Maximum numbers of iterations
500.
We used the following examples for comparison:
5. Conclusion
In Tables 2-11, we observe that our iterative method
Table 1. (Table of functions).
TABLE # 1 OF FUNCTIONS
Functions Roots
42
144
f
xx 1

23
221fx
 3
 
2
3exp sinln1fxxx
 3.237562984023
42exp 1fx x

0.442854401002
32
541fxx 5
 1.631980805566
2
67301fpx x
  3
7exp 11fx
 1
32
825fx x
 2.690647448028
91exp
f
xx
 1
10 11fx
1
Table 2. Comparison of Methods for Example 1.
10
, 0.75fx
Numbers of iteration

n
f
x
NW 10 7.1e-40 5.9e-21
VCM 33 0 1.9e-42
VSHM 8 1.0e-127 3.6e-25
NR1 5 1.8e-37 9.5e-20
NR2 11 3.4e-36 4.1e-19
Z Hu 6 1.5e-99 6.5e-20
FAJH 5 1.4e-103 6.4e-27
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F. AHMAD ET AL.
1712
Table 3. Comparison of Methods for Example 2.
20
,2.fx9
Numbers of iteration

n
f
x
NW 13 7.0e-44 1.6e-23
VCM DIVERGE --- ---
VSHM DIVERGE ---- ----
NR1 6 1.9e-31 8.8e-19
NR2 20 3.1e-29 3.5e-16
Z Hu 6 4.5e-65 1.5e-15
FAJH 5 4.2e-60 1.3e-16
Table 4. Comparison of Methods for Example 3.
30
, 2.9fx
Numbers of iteration

n
f
x
NW 7 1.1e-51 6.6e-27
VCM DIVERGE --- ----
VSHM 4 5.0e-127 1.9e-67
NR1 4 1.0e-9 1.2e-20
NR2 DIVERGE ---- ----
Z Hu 5 4.5e-65 1.5e-15
FAJH 4 5.0e-127 8.2e-34
Table 5. Comparison of Methods for Example 4.
40
,9fx
Numbers of iteration

n
f
x
NW 6 3.4e-29 5.5e-15
VCM 4 1.0e-127 4.8e-26
VSHM 4 7.0e-128 5.2e-73
NR1 4 3.8e-38 1.8e-19
NR2 45 9.6e-50 2.8e-25
Z Hu 4 4.5e-65 1.5e-15
FAJH 4 7.0e-129 1.2e-42
Table 6. Comparison of Methods for Example 5.
50
,0.fx
Table 7. Comparison of Methods for Example 6.
60
,2.fx8
Numbers of iteration

n
f
x
NW 17 8.2e-33 9.8e-18
VCM DIVERGE --- ----
VSHM DIVERGE 5.1e-37 1.0e-18
NR1 8 6.9e-52 2.8e-27
NR2 42 1.9e-33 4.7e-18
Z Hu 7 2.5e-65 5.5e-15
FAJH 6 1.0e-110 5.2e-27
Table 8. Comparison of Methods for Example 7.
70
,1.fx1
Numbers of iteration

n
f
x
NW 5 7.8e-42 3.9e-21
VCM 3 0 4.3e-39
VSHM 3 0 2.2e-42
NR1 3 2.4e-33 7.0e-17
NR2 4 4.9e-37 9.9e-19
Z Hu 4 4.5e-65 1.5e-15
FAJH 3 0 3.5e-23
Table 9. Comparison of Methods for Example 8.
80
,2fx
Numbers of iteration

n
f
x
NW 7 1.0e-37 1.3e-19
VCM 53 0 3.7e-29
VSHM 4 1.0e-126 2.8e-36
NR1 4 7.2e-38 1.0e-19
NR2 9 5.8e-51 3.1e-26
Z Hu 5 4.5e-65 1.5e-30
FAJH 4 1.0e-126 9.8e-33
Table 10. Comparison of Methods for Example 9.
9
Numbers of iteration

n
f
x
NW 7 6.1e-51 2.6e-26
VCM 6 1.0e-126 7.7e-43
VSHM 4 0 1.5e-67
NR1 4 5.6e-40 8.0e-21
NR2 14 1.6e-30 4.3e-18
Z Hu 6 4.5e-65 1.5e-15
FAJH 4 0 1.9e-39
90
,1fx
Numbers of iteration

n
f
x
NW 1 0 0
VCM DIVERGE --- ----
VSHM DIVERGE --- ----
NR1 1 0 0
NR2 DIVERGE --- ----
Z Hu 2 4.5e-65 1.5e-15
FAJH 1 0 0
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F. AHMAD ET AL.
Open Access AM
1713
5
Table 11. Comparison of Methods for Example 10.
10 0
,1.fx
Numbers of iteration

n
f
x
NW 7 2.9e-39 5.4e-20
VCM 4 4.6e-105 3.0e-18
VSHM 4 0 8.3e-41
NR1 3 1.2e-38 1.1e-19
NR2 DIVERGE --- ----
Z Hu 5 4.5e-6 7.5e-35
FAJH 4 0 6.5e-39
(FAJH) is comparable with all the methods cited in the
above mentioned tables and gives better results even than
ninth orders method of Hu et al. [5]. With the help of the
technique and idea of this paper, one can develop higher-
order multi-step iterative methods for solving nonlinear
equations, as well as a system of nonlinear equations.
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