New Ninth Order J-Halley Method for Solving Nonlinear Equations

doi:10.4236/am.2013.412233

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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: *[email protected], [email protected], sifat2003@gmail.com, [email protected]

Received October 22, 2013; revised November 22, 2013; accepted December 2, 2013

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





 

fy fx

yfx









Recently, Kou et al. [2] considered the following two-

step iteration scheme

Algorithm 2





 



nn n

yxJf fx















where





 

fy fx

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,where 0

nn n

nnn

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:



 

 



 

where 0

nnn

nn nnnn

nn nnn n

fxfx fx

yxfx n

xfxfxfx

fxy xfx





 















Algorithm 5 (ISHM [2]). For a given x0, compute the

approximate solution xnþ1 by the iterative schemes:



 

 







where 0

nnn

nn nnnn

nnn

nn nn

fxfx fx

yxfx n

xfxfxfx

fyfx fy

fx fx





 









 



On the basis of the above discussion a new iterative

technique is proposed below (named as FAJH):

Algorithm 6



 

nn n

yx fx

fx 

 

0

(2.1)







nn n

zxJf

  (2.2)



 

nn nn

fzf z

zfzL





  (2.3)

where

 

yfx







 (2.4)

and

 

zfx

Lzx





 (2.5)

3. Analysis of Convergence

In this section, we compute the convergence order of the

proposed method (FAJH).

Theorem: Let





be a simple zero of sufficiently

differentiable function for an open in-

terval :fIRR

If 0

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





 

nn n

yx fx

fx 0





 (3.1)



nn n

zxJf

  (3.2)



 

nn nn

fzf z

xzfz fzL





  (3.3)

where





 



fy fx

Jf fy fx









(3.4)

and





zfx

Lzx





 (3.5)

Let



be a simple zero of

. By Taylor’s expan-

sion, we have,











234

2345

67 8 91011

678910 ,

n nnnn

nnnn nn

x fecececece

cececececeO e













 

(3.6)











23 45

5678910

6789 10

12 3 4 5

678910

nnnn

nnnn nn

xf cececece

ce cece ceceOe













 

(3.7)

where



1,2,3,

and .













,

Using , and we have

(3.1) (3.6) (3.7 ),



223

232

12 44,

33 33

nnn n

ece cceOe





 



 (3.8)

by Taylor’s series, we have



223

232

17 378,

39 279

nnn n

fyfececc eOe

























(3.9)

and



223

334

23 24

241

1333

47.

327

fy fcec ce

cccceO e



















 















(3.10)

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F. AHMAD ET AL. 1711

Using , and we have (3.4)



3.7



(3.10),

 

22 3

223 234

nn n

fce cceccceOe









(3.11)

Using , , and (3.11 we have (3.2)



3.6 (3.7) ),



4223

2245

5322432

820

824

27 9

zcccce

ccccccceOe





 







  











(3.12)

by Taylor’s series, we have



4223

2245

5322432

820

824

27 9

fzcccc e

ccccccceOe











  









(3.13)

and



4223

2245

5322432

820

824

27 9

fzc c cce

ccccccceOe











  







(3.14)

From , and we have

(3.5)



3.7 (3.14),



23 4 5

32 4

32 23643

23 45

33613

nnn

Lc cecece

ccccccceO e

 





(3.15)

Using and (3.15 we get



(3.3), 3.7,(3.12),(3.14)),



2332

14234323232

62910

32 43

113

332

31 ,

254

cccccccc cc

cccceO e















 





(3.16)

implies



2332

1423 432 32

2462 910

32 3243

113

332

254

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





 as tolerance and also the following crite-

ria is used for estimating the zero:

1) 1,



















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

144

xx 1



221fx



 3

 





3exp sinln1fxxx



 3.237562984023









42exp 1fx x





−0.442854401002

541fxx 5



 1.631980805566





67301fpx x



  3





7exp 11fx



 1

825fx x



 2.690647448028









91exp



 1





10 11fx



 1

Table 2. Comparison of Methods for Example 1.

, 0.75fx

Numbers of iteration





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.

,2.fx9

Numbers of iteration





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.

, 2.9fx

Numbers of iteration





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.

,9fx

Numbers of iteration





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.

,0.fx

Table 7. Comparison of Methods for Example 6.

,2.fx8

Numbers of iteration





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.

,1.fx1

Numbers of iteration





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.

,2fx

Numbers of iteration





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.

Numbers of iteration





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

,1fx

Numbers of iteration





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

Table 11. Comparison of Methods for Example 10.

10 0

,1.fx

Numbers of iteration





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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