Research ArticleGeneral Family of Third Order Methods for Multiple Roots ofNonlinear Equations and Basin Attractors for Various Methods

Rajni Sharma1 and Ashu Bahl2

1 Department of Applied Sciences DAV Institute of Engineering and Technology Kabir Nagar Jalandhar 144008 India2Department of Mathematics DAV College Jalandhar 144008 India

Correspondence should be addressed to Rajni Sharma rajni davietyahoocom

Received 29 November 2013 Accepted 8 February 2014 Published 31 March 2014

Academic Editor Ting-Zhu Huang

Copyright copy 2014 R Sharma and A Bahl This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

A general scheme of third order convergence is described for finding multiple roots of nonlinear equations The proposed schemerequires one evaluation of 119891 1198911015840 and 11989110158401015840 each per iteration and contains several known one-point third order methods for findingmultiple roots as particular cases Numerical examples are included to confirm the theoretical results and demonstrate convergencebehavior of the proposed methods In the end we provide the basins of attraction for some methods to observe their dynamics inthe complex plane

1 Introduction

Solving nonlinear equations is one of the most importantproblems in numerical analysis [1 2] With the advancementof computers the problem of solving nonlinear equations bynumerical methods has gainedmore importance than beforeA large number of suchmethods have been derived for simpleroots However not many methods are known for multipleroot case In this paper we consider iterative methods to finda multiple root 120572 of multiplicity 119898 gt 1 that is 119891(119895)(120572) = 0119895 = 0 1 119898 minus 1 and 119891(119898)(120572) = 0 of a nonlinear equation119891(119909) = 0 where 119891(119909) is the continuously differentiable realor complex function The modified Newton method [3] is animportant and basic method for finding multiple roots

119909119896+1

= 119909119896minus 119898

119891 (119909119896)

1198911015840 (119909119896) (1)

which converges quadratically and requires the knowledge ofmultiplicity119898 of root 120572

In order to improve the order of convergence of (1)several third order methods with known multiplicity119898 havebeen proposed at the expense of an additional evaluation ofsecond derivative (see [4ndash14])

Inspired by the work in this direction here we present athird order scheme of one-point methods without memory

for multiple roots The proposed scheme involves one evalu-ation of 119891 1198911015840 and 11989110158401015840 each per step It is also shown that thewell known existing methods by Ostrowski [4] Hansen andPatrick [5] Traub [6] Neta [8] Osada [7] Chun et al [9 10]Biazar and Ghanbari [11] and J R Sharma and R Sharma[12 13] can be regarded as particular cases of the proposedfamily

The paper is organized in six sections In Section 2 thethird order family ofmethods formultiple roots requiring oneevaluation of 119891 1198911015840 and 11989110158401015840 each per step is proposed and itsconvergence behavior is discussed Some particular cases ofthe family are presented in Section 3 In Section 4 the newmethods are comparedwith the closest competitors in a seriesof numerical examples In Section 5 we investigate the basinsof attraction for some of the methods to provide the chaoticbehavior of such schemes Concluding remarks are given inSection 6

2 The Method and Its Convergence Analysis

Consider the following iterative scheme

119909119896+1

= 119909119896minus 119867(

119891 (119909119896) 11989110158401015840 (119909

119896)

1198911015840(119909119896)2

)119891 (119909119896)

1198911015840 (119909119896) (2)

Hindawi Publishing CorporationAdvances in Numerical AnalysisVolume 2014 Article ID 963878 8 pageshttpdxdoiorg1011552014963878

2 Advances in Numerical Analysis

where the function 119867(sdot) isin 1198622(R) In order to discussthe properties of the scheme defined by (2) we prove thefollowing theorem

Theorem 1 Let 120572 isin 119868 be a multiple root of multiplicity 119898of a sufficiently differentiable function 119891 119868 rarr R for anopen interval 119868 If 119909

0is sufficiently close to 120572 then the iterative

scheme defined by (2) has third order convergence provided119867(119906) = 119898 1198671015840(119906) = 11989822 and 11986710158401015840(119906) lt infin and for119906 = (119898 minus 1)119898 119890

119896= 119909119896minus 120572 one has

119890119896+1

= [1198984 + 31198983 minus 411986710158401015840 (119906)

2119898511986021minus1198602

119898] 1198903119896+ 119874 (1198904

119896) (3)

where 119860119894= 119898119891(119898+119894)(120572)(119898 + 119894)119891(119898)(120572) for 119894 isin N N being

the set of natural numbers

Proof Expand 119891(119909119896) 1198911015840(119909

119896) 11989110158401015840(119909

119896) in the Taylor series

about 120572 where 120572 is a multiple root of multiplicity 119898 Thususing the fact that 119891(119895)(120572) = 0 119895 = 0 1 119898 minus 1 and119891(119898)(120572) = 0 we have

119891 (119909119896) =

119891(119898) (120572) 119890119898119896

119898[1 + 119860

1119890119896+ 11986021198902119896+ 11986031198903119896+ 119874 (1198904

119896)]

1198911015840 (119909119896) =

119891(119898) (120572) 119890119898minus1119896

(119898 minus 1)[1 + 119861

1119890119896+ 11986121198902119896+ 11986131198903119896+ 119874 (1198904

119896)]

11989110158401015840 (119909119896) =

119891(119898) (120572) 119890119898minus2119896

(119898 minus 2)[1 + 119862

1119890119896+ 11986221198902119896+ 11986231198903119896+ 119874 (1198904

119896)]

(4)

where

119860119894=

119898119891(119898+119894) (120572)

(119898 + 119894)119891(119898) (120572) 119861

119894=

(119898 minus 1)119891(119898+119894) (120572)

(119898 + 119894 minus 1)119891(119898) (120572)

119862119894=

(119898 minus 2)119891(119898+119894) (120572)

(119898 + 119894 minus 2)119891(119898) (120572)

119894 = 1 2

(5)

From (4) we get

119891 (119909119896)

1198911015840 (119909119896)=

119890119896

119898minus11986011198902119896

1198982+

1

1198983[(119898 + 1)119860

2

1minus 2119898119860

2] 1198903119896

+ 119874 (1198904119896)

(6)

119891 (119909119896) 11989110158401015840 (119909

119896)

1198911015840(119909119896)2

=119898 minus 1

119898+21198601119890119896

1198982

minus3

1198983[(119898 + 1)119860

2

1minus 2119898119860

2] 1198902119896

+4

1198984[(119898 + 1)119860

3

1minus 119898 (3119898 + 4)119860

11198602

+311989821198603] 1198903119896+ 119874 (1198904

119896)

(7)

Let119891(119909119896)11989110158401015840(119909

119896)1198911015840(119909

119896)2 = 119906+V where 119906 = (119898minus1)119898Then

from (7) the remainder V = (119891(119909119896)11989110158401015840(119909

119896)1198911015840(119909

119896)2) minus 119906 is

infinitesimal with the same order of 119890119896

Thus we can use the Taylor series to expand119867(119891(119909

119896)11989110158401015840(119909

119896)1198911015840(119909

119896)2) = 119867(119906 + V) about 119906 and then

obtain

119867(119891 (119909119896) 11989110158401015840 (119909

119896)

1198911015840(119909119896)2

) = 119867 (119906) + 1198671015840 (119906) V +11986710158401015840 (119906) V2

2

+119867101584010158401015840 (119906) V3

3+ 119874 (1198904

119896)

(8)

By invocation of (2) (6) and (8) we get the error equation as

119890119896+1

= 1198701119890119896+ 11987021198902119896+ 11987031198903119896+ 119874 (1198904

119896) (9)

where

1198701= 1 minus

119867 (119906)

119898 119870

2=

119867 (119906)119898 minus 21198671015840 (119906)

11989831198601

1198703= minus

1

1198985[ ( (1198983 + 1198982)119867 (119906)

minus (31198982 + 5119898)1198671015840 (119906) + 211986710158401015840 (119906) )1198602

1

+ (611989821198671015840 (119906) minus 21198983119867(119906))1198602]

(10)

In order to achieve the third order convergence the coeffi-cients119870

1and119870

2must vanish Solving119870

1= 0 and119870

2= 0 we

obtain

119867(119906) = 119898 1198671015840 (119906) =1198982

2 (11)

Hence (9) becomes

119890119896+1

= [1198984 + 31198983 minus 411986710158401015840 (119906)

2119898511986021minus1198602

119898] 1198903119896+ 119874 (1198904

119896) (12)

This completes the proof of Theorem 1

Remark 2 From the error equation (9) we can find that theiterative method (2) contains the modified Newton method(1) as a second order method also

Remark 3 Obviously the number of function evaluations periteration required in the scheme defined by (2) is three Weconsider the definition of efficiency index [15] as 1199011119903 where119901 is the order of the method and 119903 is the number of functionevaluations per iteration required by themethodThe schemedefined by (2) has the efficiency index equal to 313 asymp 1442

which is much better than 212 asymp 1414 of the modifiedNewton method

3 Some Special Cases of Order Three

In this section we present some special cases of order threeof the presented scheme (2)

Advances in Numerical Analysis 3

Case 1 For the function119867 defined by119867(119905) = 119860119905+119861 1198671015840(119905) =119860

According to (11) solving the equations

119860119906 + 119861 = 119898 119860 =1198982

2 (13)

we get119860 = 11989822 119861 = 119898(3minus119898)2 With these values of119860 and119861 scheme (2) takes the form

119909119896+1

= 119909119896minus 119898[

119898

2

119891 (119909119896) 11989110158401015840 (119909

119896)

1198911015840(119909119896)2

+3 minus 119898

2]

119891 (119909119896)

1198911015840 (119909119896)

(14)

which is Chebyshevrsquos method for multiple roots proposed in[6 8]

Case 2 If119867(119905) = 119860 + (119861119905) then1198671015840(119905) = minus1198611199052According to (11) solving the equations

119860 +119861

119906= 119898 minus

119861

1199062=

1198982

2 (15)

we get 119860 = 119898(119898 + 1)2 119861 = minus(119898 minus 1)22 With these valuesscheme (2) leads to

119909119896+1

= 119909119896minus [

119898 (119898 + 1)

2minus(119898 minus 1)2

2

1198911015840(119909119896)2

119891 (119909119896) 11989110158401015840 (119909

119896)]

times119891 (119909119896)

1198911015840 (119909119896)

(16)

which is the third order Osada method for multiple roots [7]

Case 3 Taking119867(119905) = 1(119860119905+119861) then1198671015840(119905) = minus119860(119860119905+119861)2According to (11) solving the equations

1

119860119906 + 119861= 119898 minus

119860

(119860119906 + 119861)2=

1198982

2 (17)

we get 119860 = minus12 and 119861 = (119898 + 1)2119898 With these values (2)yields

119909119896+1

= 119909119896minus 119891 (119909

119896) times (((119898 + 1) (2119898)) 119891

1015840 (119909119896)

minus ((119891 (119909119896) 11989110158401015840 (119909

119896)) (21198911015840 (119909

119896))))minus1

(18)

which is Halleyrsquos method for multiple roots [5]

Case 4 Supposing that 119867(119905) = 1radic119860 + 119861119905 then 1198671015840(119905) =

minus1198612(119860 + 119861119905)32According to (11) solving the equations

1

radic119860 + 119861119906= 119898 minus

119861

2(119860 + 119861119906)32=

1198982

2 (19)

we get 119860 = 1119898 and 119861 = minus1119898 and hence (2) becomes

119909119896+1

= 119909119896minus[[

[

radic119898

radic1 minus (119891 (119909119896) 11989110158401015840 (119909

119896)) 1198911015840(119909

119896)2

]]

]

119891 (119909119896)

1198911015840 (119909119896)

(20)

which is Ostrowskirsquos square root iteration [4]

Case 5 If119867(119905) = (119860 + 119861119905)(119862 + 119863119905) we have1198671015840(119905) = (119861119862 minus119860119863)(119862 + 119863119905)2

According to (11) solving the equations

119860 + 119861119906

119862 + 119863119906= 119898

119861119862 minus 119860119863

(119862 + 119863119906)2=

1198982

2 (21)

we get

119860 =119862119898 (3 minus 119898) minus 119863(119898 minus 1)2

2 119861 =

119863119898 + (119862 + 119863)1198982

2

(22)

Subcase 1 The choice of 119862 = 2 and119863 = minus120573(120573 isin R) gives

119860 =2119898 (3 minus 119898) + 120573(119898 minus 1)2

2 119861 = minus

120573119898 + (120573 minus 2)1198982

2

(23)

With these values (2) corresponds to the Sharma-Sharmamethod [13]

119909119896+1

= 119909119896minus1

4

times [

[

120573(119898 minus 1)2 minus 2119898 (119898 minus 3)

+(120573 (119898 minus 1) minus 2119898)

2

(119891 (119909119896) 11989110158401015840 (119909

119896) 21198911015840(119909

119896)2

)

1 minus 120573 (119891 (119909119896) 11989110158401015840 (119909

119896) 21198911015840(119909

119896)2

)]

]

times119891 (119909119896)

1198911015840 (119909119896)

(24)

Subcase 2 For the choice of 119862 = (119898 minus 1)2 and119863 = 119898(3 minus 119898)we get119860 = 0 and 119861 = 21198982 These values exhibit the proposedscheme (2) as

119909119896+1

= 119909119896

minus21198982119891(119909

119896)2

11989110158401015840 (119909119896)

119898 (3 minus 119898)119891 (119909119896) 1198911015840 (119909

119896) 11989110158401015840 (119909

119896) + (119898 minus 1)21198911015840(119909

119896)3

(25)

which is the third order Chun-Neta method [9]

4 Advances in Numerical Analysis

Case 6 For

119867(119905) =(119860 + 119861119905 + 1198621199052)

(119863 + 119864119905 + 1198651199052)

1198671015840 (119905) =(119861119863 minus 119860119864) minus 2 (119860119865 minus 119862119863) 119905 + (119862119864 minus 119861119865) 1199052

(119863 + 119864119905 + 1198651199052)2

(26)

According to (11) solving the equations (119860 +119861119906+1198621199062)(119863+

119864119906 + 1198651199062) = 119898 and

(119861119863 minus 119860119864) minus 2 (119860119865 minus 119862119863) 119906 + (119862119864 minus 119861119865) 1199062

(119863 + 119864119906 + 1198651199062)2

=1198982

2

(27)

we get

119864 =119862(119898 minus 1)3 + 119861119898(1198982 minus 1) + 1198982 (119860 (119898 + 3) minus 4119863119898)

21198982 (119898 minus 1)

119865 = minus119862 (119898 minus 3) (119898 minus 1)2 + 119861119898(119898 minus 1)2 + 1198982 (119860 (119898 + 1))

2119898(119898 minus 1)2

+21198631198983

2119898(119898 minus 1)2

(28)

Subcase 1 Taking119863 = 0 and 119864 = 1 and solving (28) we get

119860 = minus(119898 minus 1)2 ((119865 + 1)1198982 + 119865119898 minus 2119862)

21198982

119861 =1198982 (119865119898 + 119898 + 2119865 + 1) minus 4119862119898 minus 3119865119898 + 4119862

2119898

(29)

With these values (2) corresponds to the third order familyof multiple roots developed by Biazar and Ghanbari [11]

119909119896+1

= 119909119896minus ( (119861119891 (119909

119896) 1198911015840(119909

119896)2

11989110158401015840 (119909119896)

+1198601198911015840(119909119896)4

+ 119862119891(119909119896)2

11989110158401015840(119909119896)2

)

times (1198911015840(119909119896)3

11989110158401015840 (119909119896)

+119865119891 (119909119896) 1198911015840 (119909

119896) 11989110158401015840(119909

119896)2

)minus1

)

(30)

where 119860 and 119861 are given in (29)

Subcase 2 Taking 119860 = 2(1198982 minus 3119898 + 2) 119861 = minus1198982 + 5119898 119862 =

31198982 and119863 = 21198982 minus 6119898 + 4 (28) gives 119864 = minus2(1198982 minus 4119898 + 1)and 119865 = 0 With these values (2) yields

119909119896+1

= 119909119896minus [1 +

119891 (119909k) 11989110158401015840 (119909119896)

21198911015840(119909119896)2

+ ((2119898 minus 1)2(119891 (119909

119896) 11989110158401015840 (119909

119896) 1198911015840(119909

119896)2

)2

)

times (2 ((1198982 minus 3119898 + 2) minus (1198982 minus 4119898 + 1)

times (119891(119909119896)11989110158401015840(119909

119896)1198911015840(119909

119896)2

)))minus1

]

times119891 (119909119896)

1198911015840 (119909119896)

(31)

which is the Sharma-Sharma method [12]

Subcase 3 With 120579 isin R taking

119860 = minus120579(119898 minus 1)2

2 119861 =

(2120579 minus 1)1198982 + (3 minus 2120579)119898

2

119862 =(1 minus 120579)1198982

2

(32)

and119863 = 0 (28) generates 119864 = 1 and 119865 = 0 So (2) leads to

119909119896+1

= 119909119896

minus [119898 [(2120579 minus 1)119898 + 3 minus 2120579]

2

minus120579(119898 minus 1)2

2

1198911015840(119909119896)2

119891 (119909119896) 11989110158401015840 (119909

119896)

+(1 minus 120579)1198982

2

119891 (119909119896) 11989110158401015840 (119909

119896)

1198911015840(119909119896)2

]119891 (119909119896)

1198911015840 (119909119896)

(33)

which is the one-parameter family of methods for multipleroots proposed by Chun et al [10]

Case 7 Further if 119867(119905) = 119860 + (119861119905) + (1198621199052) then 1198671015840(119905) =minus(1198611199052) minus (21198621199053)

According to (11) solving the equations

119860 +119861

119906+

119862

1199062= 119898 minus

119861

1199062minus2119862

1199063=

1198982

2 (34)

we get 119860 = ((119898 + 1198982)2) + (1198621198982(119898 minus 1)2) and 119861 = (minus(119898 minus

1)22) minus (2119862119898(119898 minus 1))

Advances in Numerical Analysis 5

Table 1 Test functions

119891(119909) 120572 119898

1198911(119909) = ((119909 minus 1)3 minus 1)6 20000000000000000000000000000 6

1198912(119909) = (ln(1199092 + 119909 + 2) minus 119909 + 1)4 41525907367571582749969890048 4

1198913(119909) = (ln2 (119909 minus 2) (119890119909minus3 minus 1) sin 120587119909

3) 30000000000000000000000000000 4

1198914(119909) = (1199092 minus 119890119909 minus 3119909 + 2)5 02575302854398607604553673049 5

1198915(119909) = (cos119909 minus 119909)3 073908513321516064165531208767 3

1198916(119909) = (radic119909 minus

1

119909minus 3)3 96335955628326951924063127092 3

1198917(119909) = (1199093 minus 10)8 21544346900318837217592935665 8

Table 2 Performance of the methods using 12 NFE

119891(119909) 1199090

|119890119896| = |119909

119896minus 120572|

MNM OM HM SSM (120573 = 32) CNM SBM (119862 = 12)

1198911

175 120 (minus34) 276 (minus22) 548 (minus34) 401 (minus41) 108 (minus42) 125 (minus22)3 155 (minus16) 275 (minus15) 493 (minus18) 236 (minus20) 678 (minus21) 237 (minus15)

1198912

4 453 (minus129) 446 (minus137) 146 (minus145) 117 (minus144) 111 (minus141) 175 (minus137)5 688 (minus86) 245 (minus84) 775 (minus91) 378 (minus90) 710 (minus88) 119 (minus84)

1198913

25 541 (minus67) 247 (minus76) 833 (minus57) 560 (minus63) 490 (minus58) 134 (minus74)325 783 (minus99) 232 (minus118) 139 (minus106) 261 (minus110) 260 (minus107) 235 (minus115)

1198914

minus2 155 (minus37) 130 (minus42) 145 (minus40) 152 (minus39) 147 (minus40) 252 (minus40)15 106 (minus57) 244 (minus50) 515 (minus50) 243 (minus40) 466 (minus38) 133 (minus49)

1198915

03 359 (minus59) 213 (minus44) 512 (minus64) 512 (minus64) 717 (minus53) 116 (minus47)17 325 (minus65) 535 (minus41) 225 (minus44) 225 (minus44) 184 (minus42) 166 (minus41)

1198916

9 513 (minus108) 251 (minus137) 150 (minus135) 150 (minus135) 115 (minus193) 943 (minus146)10 111 (minus61) 359 (minus158) 468 (minus157) 468 (minus157) 907 (minus214) 754 (minus167)

1198917

2 259 (minus72) 168 (minus78) 675 (minus99) 389 (minus89) 329 (minus92) 144 (minus78)3 279 (minus33) 319 (minus34) 297 (minus47) 907 (minus41) 873 (minus43) 292 (minus34)

Here 119886(minus119887) = 119886 times 10minus119887

Table 3 Computational order of convergence (120588)

119891(119909) 1199090

120588

MNM OM HM SSM (120573 = 32) CNM SBM (119862 = 12)

1198911

175 2 3 3 3 3 33 2 3 3 299 299 3

1198912

4 2 3 3 3 3 35 2 3 3 3 3 3

1198913

25 2 3 3 3 3 3325 2 3 3 3 3 3

1198914

minus2 2 3 3 3 3 315 2 3 3 3 3 3

1198915

03 2 3 3 3 3 317 2 299 299 299 299 3

1198916

9 2 3 3 3 3 310 2 3 3 3 3 3

1198917

2 2 3 3 3 3 33 2 299 299 299 299 3

6 Advances in Numerical Analysis

minus2 minus1 0 1 2

2

1

0

minus2

minus1

(a) The modified Newton method

minus2 minus1 0 1 2

2

1

0

minus2

minus1

(b) The Osada method

minus2 minus1 0 1 2

2

1

0

minus2

minus1

(c) Halleyrsquos method

minus2 minus1 0 1 2

2

1

0

minus2

minus1

(d) The Chun-Neta method

minus2 minus1 0 1 2

2

1

0

minus2

minus1

(e) The Sharma-Sharma method (120573 = 32)

minus2 minus1 0 1 2

2

1

0

minus2

minus1

(f) SBM (119862 = 12)

Figure 1 Basins of attraction for 119891(119911) = (1199112 minus 1)2 119911 isin 119863 for various methods of the family

Advances in Numerical Analysis 7

So we introduce a new one-parameter third ordermethod as

119909119896+1

= 119909119896minus[[

[

(119898 + 1198982

2+

1198621198982

(119898 minus 1)2)

+ (minus(119898 minus 1)2

2minus

2119862119898

119898 minus 1)

1198911015840(119909119896)2

119891 (119909119896) 11989110158401015840 (119909

119896)

+119862((1198911015840 (119909

119896))2

(119891 (119909119896) 11989110158401015840 (119909

119896)))

2

]]

]

119891 (119909119896)

1198911015840 (119909119896)

(35)

where 119862 isin R We call this method the Sharma-Bahl methoddenoted by SBM

4 Numerical Examples

In this section we employ the presented third order methodSBM (35) for 119862 = 12 to solve some nonlinear equationswhich not only illustrate the methods practically but alsoserve to check the validity of the theoretical results we havederived To check the theoretical order of convergence weobtain the computational order of convergence (120588) using theformula [16]

120588 asympln 1003816100381610038161003816(119909119896+1 minus 120572) (119909

119896minus 120572)

1003816100381610038161003816ln 1003816100381610038161003816(119909119896 minus 120572) (119909

119896minus1minus 120572)

1003816100381610038161003816 (36)

The performance is compared with the methods in ourfamilies including the special ones for some arbitrary chosenfunctions with roots of known multiplicity 119898 The methodscompared include the modified Newton method (MNM) (1)the Osada method (OM) (16) Halleyrsquos method (HM) (18)the Sharma-Sharma method (SSM) (24) for 120573 = 32 and theChun-Neta method (CNM) (25)

The test functions along with root 120572 correct up to 28decimal places and its multiplicity 119898 is displayed in Table 1Table 2 shows the values of initial approximation (119909

0) chosen

from both ends to the root and the values of the error |119890119896| =

|119909119896minus 120572| calculated by costing the same total number of

function evaluations (NFE) for each method Table 3 exhibitsthe computational order of convergence (120588) For numericalillustrations in Table 3 we use fixed stopping criterion 120598 =

05 times 10minus28 The NFE is counted as sum of the number ofevaluations of the function plus the number of evaluations ofthe derivativesWe decide to choose 12 NFE for eachmethodThat means that for MNM the error |119890

119896| is calculated at

the sixth iteration whereas for the remaining methods thisis calculated at the fourth iteration All computations areperformed by MATHEMATICA [17] using 600 significantdigits

It can also be observed from the numerical results ofTable 2 that all the presented methods are well behavedand that for most of the functions we tested the methodintroduced in this paper has similar performance comparedto the other methods as expected from methods of the same

order and same computational efficiency However Halleyrsquosmethod (18) and the Chun-Neta method (25) behave betterand these are special cases of our proposed scheme for linearrational function 119867(sdot) The results of Table 3 show that thecomputational order of convergence is in accordance withthe theoretical order of convergenceThe formulae have beenemployed on several other nonlinear equations and resultsare found at par with those presented here A reasonablyclose starting value is necessary for the method to convergeThe condition however practically applies to all iterativemethods for solving nonlinear equations

5 Finding the Basins

In this section we find the basins of attraction of complexroots for the a complex function We consider the simplestcase119891(119911) = (1199112minus1)2 which has roots 119911 = plusmn 1 ofmultiplicity 2Cayley [18] was the first who considered the Newton methodfor the roots of polynomial with iterations over the complexnumbers

We take the initial point as 1199110

isin 119863 where 119863 is arectangular region [minus25 25] times [minus25 25] sub C containingall the roots of 119891(119911) = 0 We consider the stopping criterionfor convergence as 10minus3 up to amaximumof 25 iterationsWetake a grid of 1024 times 1024 points in119863

To generate the pictures we use MATHEMATICA [17]We assign the light to dark colors based on the number ofiterations in which the considered initial point converges to aroot The root 119911 = 1 of 119891(119911) = 0 and the points in the regionconverging to this root are shaded in cyan color and magentacolor is used for the other root 119911 = minus1 in a similar way Ifwe have not obtained the desired tolerance in 25 iterationswe do not continue and we decide that the iterative methodstarting at 119911

0does not converge to any root and such points

are shaded in black color (see [19ndash22])The figures given here clearly show that the mod-

ified Newton method (Figure 1(a)) and Halleyrsquos method(Figure 1(c)) are the best followed by the Osada method(Figure 1(b)) the Chun-Neta method (Figure 1(d)) SBM(Figure 1(f)) and Sharma-Sharma method (Figure 1(e))From these pictures we can guess the behavior and suitabilityof any method depending upon the circumstances Thusconcluding we can say that the linear rational function givesthe best choice of weight function119867(sdot)

6 Conclusions

In this paper a general family of one-point third ordermethods for finding multiple roots of nonlinear equations ispresented This scheme contains some well known methodsand recently developed methods as its particular cases Thetheoretical results have been checked with some numericalexamples comparing ourmethodswith themodifiedNewtonmethod and some third order methodsThe presented basinsof attraction have also demonstrated the strictness of methodfor choice of starting point

8 Advances in Numerical Analysis

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors are grateful to the anonymous reviewers for theirvaluable suggestions and comments on this paper

References

[1] J M Ortega and W C Rheinboldt Iterative Solution of Nonlin-ear Equations in Several Variables Academic Press New YorkNY USA 1970

[2] S C Chapra and R P CanaleNumerical Methods for EngineersMcGraw-Hill Book Company New York NY USA 1988

[3] E Schroder ldquoUeber unendlich vieleAlgorithmen zurAuflosungder GleichungenrdquoMathematische Annalen vol 2 no 2 pp 317ndash365 1870

[4] AM Ostrowski Solution of Equations in Euclidean and BanachSpaces Academic Press New York NY USA 3rd edition 1973

[5] E Hansen and M Patrick ldquoA family of root finding methodsrdquoNumerische Mathematik vol 27 no 3 pp 257ndash269 1977

[6] J F Traub Iterative Methods for the Solution of EquationsPrentice-Hall Englewood Cliffs NJ USA 1964

[7] N Osada ldquoAn optimal multiple root-finding method of orderthreerdquo Journal of Computational and Applied Mathematics vol51 no 1 pp 131ndash133 1994

[8] B Neta ldquoNew third order nonlinear solvers for multiple rootsrdquoApplied Mathematics and Computation vol 202 no 1 pp 162ndash170 2008

[9] C Chun and B Neta ldquoA third-order modification of Newtonrsquosmethod for multiple rootsrdquoAppliedMathematics and Computa-tion vol 211 no 2 pp 474ndash479 2009

[10] C Chun H J Bae and B Neta ldquoNew families of nonlinearthird-order solvers for finding multiple rootsrdquo Computers andMathematics with Applications vol 57 no 9 pp 1574ndash15822009

[11] J Biazar and B Ghanbari ldquoA new third-order family of nonlin-ear solvers formultiple rootsrdquoComputers andMathematics withApplications vol 59 no 10 pp 3315ndash3319 2010

[12] J R Sharma and R Sharma ldquoNew third and fourth ordernonlinear solvers for computing multiple rootsrdquo Applied Math-ematics and Computation vol 217 no 23 pp 9756ndash9764 2011

[13] J R Sharma and R Sharma ldquoModified Chebyshev-Halleytype method and its variants for computing multiple rootsrdquoNumerical Algorithms vol 61 no 4 pp 567ndash578 2012

[14] S Kumar V Kanwar and S Singh ldquoOn some modified familiesof multipoint iterative methods for multiple roots of nonlinearequationsrdquo Applied Mathematics and Computation vol 218 no14 pp 7382ndash7394 2012

[15] W Gautschi Numerical Analysis An Introduction BirkhauserBoston Mass USA 1997

[16] S Weerakoon and T G I Fernando ldquoA variant of Newtonrsquosmethod with accelerated third-order convergencerdquo AppliedMathematics Letters vol 13 no 8 pp 87ndash93 2000

[17] S Wolfram The Mathematica Book Wolfram Media Cham-paign Ill USA 5th edition 2003

[18] A Cayley ldquoThe Newton-Fourier imaginary problemrdquo TheAmerican Journal of Mathematics vol 2 no 1 article 97 1879

[19] M L Sahari and I Djellit ldquoFractal Newton basinsrdquo DiscreteDynamics in Nature and Society vol 2006 Article ID 28756 16pages 2006

[20] J L Varona ldquoGraphic and numerical comparison betweeniterative methodsrdquo The Mathematical Intelligencer vol 24 no1 pp 37ndash46 2002

[21] M Scott B Neta and C Chun ldquoBasin attractors for variousmethodsrdquo Applied Mathematics and Computation vol 218 no6 pp 2584ndash2599 2011

[22] B Neta M Scott and C Chun ldquoBasin attractors for variousmethods for multiple rootsrdquo Applied Mathematics and Compu-tation vol 218 no 9 pp 5043ndash5066 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Advances in Numerical Analysis

where the function 119867(sdot) isin 1198622(R) In order to discussthe properties of the scheme defined by (2) we prove thefollowing theorem

Theorem 1 Let 120572 isin 119868 be a multiple root of multiplicity 119898of a sufficiently differentiable function 119891 119868 rarr R for anopen interval 119868 If 119909

0is sufficiently close to 120572 then the iterative

scheme defined by (2) has third order convergence provided119867(119906) = 119898 1198671015840(119906) = 11989822 and 11986710158401015840(119906) lt infin and for119906 = (119898 minus 1)119898 119890

119896= 119909119896minus 120572 one has

119890119896+1

= [1198984 + 31198983 minus 411986710158401015840 (119906)

2119898511986021minus1198602

119898] 1198903119896+ 119874 (1198904

119896) (3)

where 119860119894= 119898119891(119898+119894)(120572)(119898 + 119894)119891(119898)(120572) for 119894 isin N N being

the set of natural numbers

Proof Expand 119891(119909119896) 1198911015840(119909

119896) 11989110158401015840(119909

119896) in the Taylor series

about 120572 where 120572 is a multiple root of multiplicity 119898 Thususing the fact that 119891(119895)(120572) = 0 119895 = 0 1 119898 minus 1 and119891(119898)(120572) = 0 we have

119891 (119909119896) =

119891(119898) (120572) 119890119898119896

119898[1 + 119860

1119890119896+ 11986021198902119896+ 11986031198903119896+ 119874 (1198904

119896)]

1198911015840 (119909119896) =

119891(119898) (120572) 119890119898minus1119896

(119898 minus 1)[1 + 119861

1119890119896+ 11986121198902119896+ 11986131198903119896+ 119874 (1198904

119896)]

11989110158401015840 (119909119896) =

119891(119898) (120572) 119890119898minus2119896

(119898 minus 2)[1 + 119862

1119890119896+ 11986221198902119896+ 11986231198903119896+ 119874 (1198904

119896)]

(4)

where

119860119894=

119898119891(119898+119894) (120572)

(119898 + 119894)119891(119898) (120572) 119861

119894=

(119898 minus 1)119891(119898+119894) (120572)

(119898 + 119894 minus 1)119891(119898) (120572)

119862119894=

(119898 minus 2)119891(119898+119894) (120572)

(119898 + 119894 minus 2)119891(119898) (120572)

119894 = 1 2

(5)

From (4) we get

119891 (119909119896)

1198911015840 (119909119896)=

119890119896

119898minus11986011198902119896

1198982+

1

1198983[(119898 + 1)119860

2

1minus 2119898119860

2] 1198903119896

+ 119874 (1198904119896)

(6)

119891 (119909119896) 11989110158401015840 (119909

119896)

1198911015840(119909119896)2

=119898 minus 1

119898+21198601119890119896

1198982

minus3

1198983[(119898 + 1)119860

2

1minus 2119898119860

2] 1198902119896

+4

1198984[(119898 + 1)119860

3

1minus 119898 (3119898 + 4)119860

11198602

+311989821198603] 1198903119896+ 119874 (1198904

119896)

(7)

Let119891(119909119896)11989110158401015840(119909

119896)1198911015840(119909

119896)2 = 119906+V where 119906 = (119898minus1)119898Then

from (7) the remainder V = (119891(119909119896)11989110158401015840(119909

119896)1198911015840(119909

119896)2) minus 119906 is

infinitesimal with the same order of 119890119896

Thus we can use the Taylor series to expand119867(119891(119909

119896)11989110158401015840(119909

119896)1198911015840(119909

119896)2) = 119867(119906 + V) about 119906 and then

obtain

119867(119891 (119909119896) 11989110158401015840 (119909

119896)

1198911015840(119909119896)2

) = 119867 (119906) + 1198671015840 (119906) V +11986710158401015840 (119906) V2

2

+119867101584010158401015840 (119906) V3

3+ 119874 (1198904

119896)

(8)

By invocation of (2) (6) and (8) we get the error equation as

119890119896+1

= 1198701119890119896+ 11987021198902119896+ 11987031198903119896+ 119874 (1198904

119896) (9)

where

1198701= 1 minus

119867 (119906)

119898 119870

2=

119867 (119906)119898 minus 21198671015840 (119906)

11989831198601

1198703= minus

1

1198985[ ( (1198983 + 1198982)119867 (119906)

minus (31198982 + 5119898)1198671015840 (119906) + 211986710158401015840 (119906) )1198602

1

+ (611989821198671015840 (119906) minus 21198983119867(119906))1198602]

(10)

In order to achieve the third order convergence the coeffi-cients119870

1and119870

2must vanish Solving119870

1= 0 and119870

2= 0 we

obtain

119867(119906) = 119898 1198671015840 (119906) =1198982

2 (11)

Hence (9) becomes

119890119896+1

= [1198984 + 31198983 minus 411986710158401015840 (119906)

2119898511986021minus1198602

119898] 1198903119896+ 119874 (1198904

119896) (12)

This completes the proof of Theorem 1

Remark 2 From the error equation (9) we can find that theiterative method (2) contains the modified Newton method(1) as a second order method also

Remark 3 Obviously the number of function evaluations periteration required in the scheme defined by (2) is three Weconsider the definition of efficiency index [15] as 1199011119903 where119901 is the order of the method and 119903 is the number of functionevaluations per iteration required by themethodThe schemedefined by (2) has the efficiency index equal to 313 asymp 1442

which is much better than 212 asymp 1414 of the modifiedNewton method

3 Some Special Cases of Order Three

In this section we present some special cases of order threeof the presented scheme (2)

Advances in Numerical Analysis 3

Case 1 For the function119867 defined by119867(119905) = 119860119905+119861 1198671015840(119905) =119860

According to (11) solving the equations

119860119906 + 119861 = 119898 119860 =1198982

2 (13)

we get119860 = 11989822 119861 = 119898(3minus119898)2 With these values of119860 and119861 scheme (2) takes the form

119909119896+1

= 119909119896minus 119898[

119898

2

119891 (119909119896) 11989110158401015840 (119909

119896)

1198911015840(119909119896)2

+3 minus 119898

2]

119891 (119909119896)

1198911015840 (119909119896)

(14)

which is Chebyshevrsquos method for multiple roots proposed in[6 8]

Case 2 If119867(119905) = 119860 + (119861119905) then1198671015840(119905) = minus1198611199052According to (11) solving the equations

119860 +119861

119906= 119898 minus

119861

1199062=

1198982

2 (15)

we get 119860 = 119898(119898 + 1)2 119861 = minus(119898 minus 1)22 With these valuesscheme (2) leads to

119909119896+1

= 119909119896minus [

119898 (119898 + 1)

2minus(119898 minus 1)2

2

1198911015840(119909119896)2

119891 (119909119896) 11989110158401015840 (119909

119896)]

times119891 (119909119896)

1198911015840 (119909119896)

(16)

which is the third order Osada method for multiple roots [7]

Case 3 Taking119867(119905) = 1(119860119905+119861) then1198671015840(119905) = minus119860(119860119905+119861)2According to (11) solving the equations

1

119860119906 + 119861= 119898 minus

119860

(119860119906 + 119861)2=

1198982

2 (17)

we get 119860 = minus12 and 119861 = (119898 + 1)2119898 With these values (2)yields

119909119896+1

= 119909119896minus 119891 (119909

119896) times (((119898 + 1) (2119898)) 119891

1015840 (119909119896)

minus ((119891 (119909119896) 11989110158401015840 (119909

119896)) (21198911015840 (119909

119896))))minus1

(18)

which is Halleyrsquos method for multiple roots [5]

Case 4 Supposing that 119867(119905) = 1radic119860 + 119861119905 then 1198671015840(119905) =

minus1198612(119860 + 119861119905)32According to (11) solving the equations

1

radic119860 + 119861119906= 119898 minus

119861

2(119860 + 119861119906)32=

1198982

2 (19)

we get 119860 = 1119898 and 119861 = minus1119898 and hence (2) becomes

119909119896+1

= 119909119896minus[[

[

radic119898

radic1 minus (119891 (119909119896) 11989110158401015840 (119909

119896)) 1198911015840(119909

119896)2

]]

]

119891 (119909119896)

1198911015840 (119909119896)

(20)

which is Ostrowskirsquos square root iteration [4]

Case 5 If119867(119905) = (119860 + 119861119905)(119862 + 119863119905) we have1198671015840(119905) = (119861119862 minus119860119863)(119862 + 119863119905)2

According to (11) solving the equations

119860 + 119861119906

119862 + 119863119906= 119898

119861119862 minus 119860119863

(119862 + 119863119906)2=

1198982

2 (21)

we get

119860 =119862119898 (3 minus 119898) minus 119863(119898 minus 1)2

2 119861 =

119863119898 + (119862 + 119863)1198982

2

(22)

Subcase 1 The choice of 119862 = 2 and119863 = minus120573(120573 isin R) gives

119860 =2119898 (3 minus 119898) + 120573(119898 minus 1)2

2 119861 = minus

120573119898 + (120573 minus 2)1198982

2

(23)

With these values (2) corresponds to the Sharma-Sharmamethod [13]

119909119896+1

= 119909119896minus1

4

times [

[

120573(119898 minus 1)2 minus 2119898 (119898 minus 3)

+(120573 (119898 minus 1) minus 2119898)

2

(119891 (119909119896) 11989110158401015840 (119909

119896) 21198911015840(119909

119896)2

)

1 minus 120573 (119891 (119909119896) 11989110158401015840 (119909

119896) 21198911015840(119909

119896)2

)]

]

times119891 (119909119896)

1198911015840 (119909119896)

(24)

Subcase 2 For the choice of 119862 = (119898 minus 1)2 and119863 = 119898(3 minus 119898)we get119860 = 0 and 119861 = 21198982 These values exhibit the proposedscheme (2) as

119909119896+1

= 119909119896

minus21198982119891(119909

119896)2

11989110158401015840 (119909119896)

119898 (3 minus 119898)119891 (119909119896) 1198911015840 (119909

119896) 11989110158401015840 (119909

119896) + (119898 minus 1)21198911015840(119909

119896)3

(25)

which is the third order Chun-Neta method [9]

4 Advances in Numerical Analysis

Case 6 For

119867(119905) =(119860 + 119861119905 + 1198621199052)

(119863 + 119864119905 + 1198651199052)

1198671015840 (119905) =(119861119863 minus 119860119864) minus 2 (119860119865 minus 119862119863) 119905 + (119862119864 minus 119861119865) 1199052

(119863 + 119864119905 + 1198651199052)2

(26)

According to (11) solving the equations (119860 +119861119906+1198621199062)(119863+

119864119906 + 1198651199062) = 119898 and

(119861119863 minus 119860119864) minus 2 (119860119865 minus 119862119863) 119906 + (119862119864 minus 119861119865) 1199062

(119863 + 119864119906 + 1198651199062)2

=1198982

2

(27)

we get

119864 =119862(119898 minus 1)3 + 119861119898(1198982 minus 1) + 1198982 (119860 (119898 + 3) minus 4119863119898)

21198982 (119898 minus 1)

119865 = minus119862 (119898 minus 3) (119898 minus 1)2 + 119861119898(119898 minus 1)2 + 1198982 (119860 (119898 + 1))

2119898(119898 minus 1)2

+21198631198983

2119898(119898 minus 1)2

(28)

Subcase 1 Taking119863 = 0 and 119864 = 1 and solving (28) we get

119860 = minus(119898 minus 1)2 ((119865 + 1)1198982 + 119865119898 minus 2119862)

21198982

119861 =1198982 (119865119898 + 119898 + 2119865 + 1) minus 4119862119898 minus 3119865119898 + 4119862

2119898

(29)

With these values (2) corresponds to the third order familyof multiple roots developed by Biazar and Ghanbari [11]

119909119896+1

= 119909119896minus ( (119861119891 (119909

119896) 1198911015840(119909

119896)2

11989110158401015840 (119909119896)

+1198601198911015840(119909119896)4

+ 119862119891(119909119896)2

11989110158401015840(119909119896)2

)

times (1198911015840(119909119896)3

11989110158401015840 (119909119896)

+119865119891 (119909119896) 1198911015840 (119909

119896) 11989110158401015840(119909

119896)2

)minus1

)

(30)

where 119860 and 119861 are given in (29)

Subcase 2 Taking 119860 = 2(1198982 minus 3119898 + 2) 119861 = minus1198982 + 5119898 119862 =

31198982 and119863 = 21198982 minus 6119898 + 4 (28) gives 119864 = minus2(1198982 minus 4119898 + 1)and 119865 = 0 With these values (2) yields

119909119896+1

= 119909119896minus [1 +

119891 (119909k) 11989110158401015840 (119909119896)

21198911015840(119909119896)2

+ ((2119898 minus 1)2(119891 (119909

119896) 11989110158401015840 (119909

119896) 1198911015840(119909

119896)2

)2

)

times (2 ((1198982 minus 3119898 + 2) minus (1198982 minus 4119898 + 1)

times (119891(119909119896)11989110158401015840(119909

119896)1198911015840(119909

119896)2

)))minus1

]

times119891 (119909119896)

1198911015840 (119909119896)

(31)

which is the Sharma-Sharma method [12]

Subcase 3 With 120579 isin R taking

119860 = minus120579(119898 minus 1)2

2 119861 =

(2120579 minus 1)1198982 + (3 minus 2120579)119898

2

119862 =(1 minus 120579)1198982

2

(32)

and119863 = 0 (28) generates 119864 = 1 and 119865 = 0 So (2) leads to

119909119896+1

= 119909119896

minus [119898 [(2120579 minus 1)119898 + 3 minus 2120579]

2

minus120579(119898 minus 1)2

2

1198911015840(119909119896)2

119891 (119909119896) 11989110158401015840 (119909

119896)

+(1 minus 120579)1198982

2

119891 (119909119896) 11989110158401015840 (119909

119896)

1198911015840(119909119896)2

]119891 (119909119896)

1198911015840 (119909119896)

(33)

which is the one-parameter family of methods for multipleroots proposed by Chun et al [10]

Case 7 Further if 119867(119905) = 119860 + (119861119905) + (1198621199052) then 1198671015840(119905) =minus(1198611199052) minus (21198621199053)

According to (11) solving the equations

119860 +119861

119906+

119862

1199062= 119898 minus

119861

1199062minus2119862

1199063=

1198982

2 (34)

we get 119860 = ((119898 + 1198982)2) + (1198621198982(119898 minus 1)2) and 119861 = (minus(119898 minus

1)22) minus (2119862119898(119898 minus 1))

Advances in Numerical Analysis 5

Table 1 Test functions

119891(119909) 120572 119898

1198911(119909) = ((119909 minus 1)3 minus 1)6 20000000000000000000000000000 6

1198912(119909) = (ln(1199092 + 119909 + 2) minus 119909 + 1)4 41525907367571582749969890048 4

1198913(119909) = (ln2 (119909 minus 2) (119890119909minus3 minus 1) sin 120587119909

3) 30000000000000000000000000000 4

1198914(119909) = (1199092 minus 119890119909 minus 3119909 + 2)5 02575302854398607604553673049 5

1198915(119909) = (cos119909 minus 119909)3 073908513321516064165531208767 3

1198916(119909) = (radic119909 minus

1

119909minus 3)3 96335955628326951924063127092 3

1198917(119909) = (1199093 minus 10)8 21544346900318837217592935665 8

Table 2 Performance of the methods using 12 NFE

119891(119909) 1199090

|119890119896| = |119909

119896minus 120572|

MNM OM HM SSM (120573 = 32) CNM SBM (119862 = 12)

1198911

175 120 (minus34) 276 (minus22) 548 (minus34) 401 (minus41) 108 (minus42) 125 (minus22)3 155 (minus16) 275 (minus15) 493 (minus18) 236 (minus20) 678 (minus21) 237 (minus15)

1198912

4 453 (minus129) 446 (minus137) 146 (minus145) 117 (minus144) 111 (minus141) 175 (minus137)5 688 (minus86) 245 (minus84) 775 (minus91) 378 (minus90) 710 (minus88) 119 (minus84)

1198913

25 541 (minus67) 247 (minus76) 833 (minus57) 560 (minus63) 490 (minus58) 134 (minus74)325 783 (minus99) 232 (minus118) 139 (minus106) 261 (minus110) 260 (minus107) 235 (minus115)

1198914

minus2 155 (minus37) 130 (minus42) 145 (minus40) 152 (minus39) 147 (minus40) 252 (minus40)15 106 (minus57) 244 (minus50) 515 (minus50) 243 (minus40) 466 (minus38) 133 (minus49)

1198915

03 359 (minus59) 213 (minus44) 512 (minus64) 512 (minus64) 717 (minus53) 116 (minus47)17 325 (minus65) 535 (minus41) 225 (minus44) 225 (minus44) 184 (minus42) 166 (minus41)

1198916

9 513 (minus108) 251 (minus137) 150 (minus135) 150 (minus135) 115 (minus193) 943 (minus146)10 111 (minus61) 359 (minus158) 468 (minus157) 468 (minus157) 907 (minus214) 754 (minus167)

1198917

2 259 (minus72) 168 (minus78) 675 (minus99) 389 (minus89) 329 (minus92) 144 (minus78)3 279 (minus33) 319 (minus34) 297 (minus47) 907 (minus41) 873 (minus43) 292 (minus34)

Here 119886(minus119887) = 119886 times 10minus119887

Table 3 Computational order of convergence (120588)

119891(119909) 1199090

120588

MNM OM HM SSM (120573 = 32) CNM SBM (119862 = 12)

1198911

175 2 3 3 3 3 33 2 3 3 299 299 3

1198912

4 2 3 3 3 3 35 2 3 3 3 3 3

1198913

25 2 3 3 3 3 3325 2 3 3 3 3 3

1198914

minus2 2 3 3 3 3 315 2 3 3 3 3 3

1198915

03 2 3 3 3 3 317 2 299 299 299 299 3

1198916

9 2 3 3 3 3 310 2 3 3 3 3 3

1198917

2 2 3 3 3 3 33 2 299 299 299 299 3

6 Advances in Numerical Analysis

minus2 minus1 0 1 2

2

1

0

minus2

minus1

(a) The modified Newton method

minus2 minus1 0 1 2

2

1

0

minus2

minus1

(b) The Osada method

minus2 minus1 0 1 2

2

1

0

minus2

minus1

(c) Halleyrsquos method

minus2 minus1 0 1 2

2

1

0

minus2

minus1

(d) The Chun-Neta method

minus2 minus1 0 1 2

2

1

0

minus2

minus1

(e) The Sharma-Sharma method (120573 = 32)

minus2 minus1 0 1 2

2

1

0

minus2

minus1

(f) SBM (119862 = 12)

Figure 1 Basins of attraction for 119891(119911) = (1199112 minus 1)2 119911 isin 119863 for various methods of the family

Advances in Numerical Analysis 7

So we introduce a new one-parameter third ordermethod as

119909119896+1

= 119909119896minus[[

[

(119898 + 1198982

2+

1198621198982

(119898 minus 1)2)

+ (minus(119898 minus 1)2

2minus

2119862119898

119898 minus 1)

1198911015840(119909119896)2

119891 (119909119896) 11989110158401015840 (119909

119896)

+119862((1198911015840 (119909

119896))2

(119891 (119909119896) 11989110158401015840 (119909

119896)))

2

]]

]

119891 (119909119896)

1198911015840 (119909119896)

(35)

where 119862 isin R We call this method the Sharma-Bahl methoddenoted by SBM

4 Numerical Examples

In this section we employ the presented third order methodSBM (35) for 119862 = 12 to solve some nonlinear equationswhich not only illustrate the methods practically but alsoserve to check the validity of the theoretical results we havederived To check the theoretical order of convergence weobtain the computational order of convergence (120588) using theformula [16]

120588 asympln 1003816100381610038161003816(119909119896+1 minus 120572) (119909

119896minus 120572)

1003816100381610038161003816ln 1003816100381610038161003816(119909119896 minus 120572) (119909

119896minus1minus 120572)

1003816100381610038161003816 (36)

The performance is compared with the methods in ourfamilies including the special ones for some arbitrary chosenfunctions with roots of known multiplicity 119898 The methodscompared include the modified Newton method (MNM) (1)the Osada method (OM) (16) Halleyrsquos method (HM) (18)the Sharma-Sharma method (SSM) (24) for 120573 = 32 and theChun-Neta method (CNM) (25)

The test functions along with root 120572 correct up to 28decimal places and its multiplicity 119898 is displayed in Table 1Table 2 shows the values of initial approximation (119909

0) chosen

from both ends to the root and the values of the error |119890119896| =

|119909119896minus 120572| calculated by costing the same total number of

function evaluations (NFE) for each method Table 3 exhibitsthe computational order of convergence (120588) For numericalillustrations in Table 3 we use fixed stopping criterion 120598 =

05 times 10minus28 The NFE is counted as sum of the number ofevaluations of the function plus the number of evaluations ofthe derivativesWe decide to choose 12 NFE for eachmethodThat means that for MNM the error |119890

119896| is calculated at

the sixth iteration whereas for the remaining methods thisis calculated at the fourth iteration All computations areperformed by MATHEMATICA [17] using 600 significantdigits

It can also be observed from the numerical results ofTable 2 that all the presented methods are well behavedand that for most of the functions we tested the methodintroduced in this paper has similar performance comparedto the other methods as expected from methods of the same

order and same computational efficiency However Halleyrsquosmethod (18) and the Chun-Neta method (25) behave betterand these are special cases of our proposed scheme for linearrational function 119867(sdot) The results of Table 3 show that thecomputational order of convergence is in accordance withthe theoretical order of convergenceThe formulae have beenemployed on several other nonlinear equations and resultsare found at par with those presented here A reasonablyclose starting value is necessary for the method to convergeThe condition however practically applies to all iterativemethods for solving nonlinear equations

5 Finding the Basins

In this section we find the basins of attraction of complexroots for the a complex function We consider the simplestcase119891(119911) = (1199112minus1)2 which has roots 119911 = plusmn 1 ofmultiplicity 2Cayley [18] was the first who considered the Newton methodfor the roots of polynomial with iterations over the complexnumbers

We take the initial point as 1199110

isin 119863 where 119863 is arectangular region [minus25 25] times [minus25 25] sub C containingall the roots of 119891(119911) = 0 We consider the stopping criterionfor convergence as 10minus3 up to amaximumof 25 iterationsWetake a grid of 1024 times 1024 points in119863

To generate the pictures we use MATHEMATICA [17]We assign the light to dark colors based on the number ofiterations in which the considered initial point converges to aroot The root 119911 = 1 of 119891(119911) = 0 and the points in the regionconverging to this root are shaded in cyan color and magentacolor is used for the other root 119911 = minus1 in a similar way Ifwe have not obtained the desired tolerance in 25 iterationswe do not continue and we decide that the iterative methodstarting at 119911

0does not converge to any root and such points

are shaded in black color (see [19ndash22])The figures given here clearly show that the mod-

ified Newton method (Figure 1(a)) and Halleyrsquos method(Figure 1(c)) are the best followed by the Osada method(Figure 1(b)) the Chun-Neta method (Figure 1(d)) SBM(Figure 1(f)) and Sharma-Sharma method (Figure 1(e))From these pictures we can guess the behavior and suitabilityof any method depending upon the circumstances Thusconcluding we can say that the linear rational function givesthe best choice of weight function119867(sdot)

6 Conclusions

In this paper a general family of one-point third ordermethods for finding multiple roots of nonlinear equations ispresented This scheme contains some well known methodsand recently developed methods as its particular cases Thetheoretical results have been checked with some numericalexamples comparing ourmethodswith themodifiedNewtonmethod and some third order methodsThe presented basinsof attraction have also demonstrated the strictness of methodfor choice of starting point

8 Advances in Numerical Analysis

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors are grateful to the anonymous reviewers for theirvaluable suggestions and comments on this paper

References

[1] J M Ortega and W C Rheinboldt Iterative Solution of Nonlin-ear Equations in Several Variables Academic Press New YorkNY USA 1970

[2] S C Chapra and R P CanaleNumerical Methods for EngineersMcGraw-Hill Book Company New York NY USA 1988

[3] E Schroder ldquoUeber unendlich vieleAlgorithmen zurAuflosungder GleichungenrdquoMathematische Annalen vol 2 no 2 pp 317ndash365 1870

[4] AM Ostrowski Solution of Equations in Euclidean and BanachSpaces Academic Press New York NY USA 3rd edition 1973

[5] E Hansen and M Patrick ldquoA family of root finding methodsrdquoNumerische Mathematik vol 27 no 3 pp 257ndash269 1977

[6] J F Traub Iterative Methods for the Solution of EquationsPrentice-Hall Englewood Cliffs NJ USA 1964

[7] N Osada ldquoAn optimal multiple root-finding method of orderthreerdquo Journal of Computational and Applied Mathematics vol51 no 1 pp 131ndash133 1994

[8] B Neta ldquoNew third order nonlinear solvers for multiple rootsrdquoApplied Mathematics and Computation vol 202 no 1 pp 162ndash170 2008

[9] C Chun and B Neta ldquoA third-order modification of Newtonrsquosmethod for multiple rootsrdquoAppliedMathematics and Computa-tion vol 211 no 2 pp 474ndash479 2009

[10] C Chun H J Bae and B Neta ldquoNew families of nonlinearthird-order solvers for finding multiple rootsrdquo Computers andMathematics with Applications vol 57 no 9 pp 1574ndash15822009

[11] J Biazar and B Ghanbari ldquoA new third-order family of nonlin-ear solvers formultiple rootsrdquoComputers andMathematics withApplications vol 59 no 10 pp 3315ndash3319 2010

[12] J R Sharma and R Sharma ldquoNew third and fourth ordernonlinear solvers for computing multiple rootsrdquo Applied Math-ematics and Computation vol 217 no 23 pp 9756ndash9764 2011

[13] J R Sharma and R Sharma ldquoModified Chebyshev-Halleytype method and its variants for computing multiple rootsrdquoNumerical Algorithms vol 61 no 4 pp 567ndash578 2012

[14] S Kumar V Kanwar and S Singh ldquoOn some modified familiesof multipoint iterative methods for multiple roots of nonlinearequationsrdquo Applied Mathematics and Computation vol 218 no14 pp 7382ndash7394 2012

[15] W Gautschi Numerical Analysis An Introduction BirkhauserBoston Mass USA 1997

[16] S Weerakoon and T G I Fernando ldquoA variant of Newtonrsquosmethod with accelerated third-order convergencerdquo AppliedMathematics Letters vol 13 no 8 pp 87ndash93 2000

[17] S Wolfram The Mathematica Book Wolfram Media Cham-paign Ill USA 5th edition 2003

[18] A Cayley ldquoThe Newton-Fourier imaginary problemrdquo TheAmerican Journal of Mathematics vol 2 no 1 article 97 1879

[19] M L Sahari and I Djellit ldquoFractal Newton basinsrdquo DiscreteDynamics in Nature and Society vol 2006 Article ID 28756 16pages 2006

[20] J L Varona ldquoGraphic and numerical comparison betweeniterative methodsrdquo The Mathematical Intelligencer vol 24 no1 pp 37ndash46 2002

[21] M Scott B Neta and C Chun ldquoBasin attractors for variousmethodsrdquo Applied Mathematics and Computation vol 218 no6 pp 2584ndash2599 2011

[22] B Neta M Scott and C Chun ldquoBasin attractors for variousmethods for multiple rootsrdquo Applied Mathematics and Compu-tation vol 218 no 9 pp 5043ndash5066 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Numerical Analysis 3

Case 1 For the function119867 defined by119867(119905) = 119860119905+119861 1198671015840(119905) =119860

According to (11) solving the equations

119860119906 + 119861 = 119898 119860 =1198982

2 (13)

we get119860 = 11989822 119861 = 119898(3minus119898)2 With these values of119860 and119861 scheme (2) takes the form

119909119896+1

= 119909119896minus 119898[

119898

2

119891 (119909119896) 11989110158401015840 (119909

119896)

1198911015840(119909119896)2

+3 minus 119898

2]

119891 (119909119896)

1198911015840 (119909119896)

(14)

which is Chebyshevrsquos method for multiple roots proposed in[6 8]

Case 2 If119867(119905) = 119860 + (119861119905) then1198671015840(119905) = minus1198611199052According to (11) solving the equations

119860 +119861

119906= 119898 minus

119861

1199062=

1198982

2 (15)

we get 119860 = 119898(119898 + 1)2 119861 = minus(119898 minus 1)22 With these valuesscheme (2) leads to

119909119896+1

= 119909119896minus [

119898 (119898 + 1)

2minus(119898 minus 1)2

2

1198911015840(119909119896)2

119891 (119909119896) 11989110158401015840 (119909

119896)]

times119891 (119909119896)

1198911015840 (119909119896)

(16)

which is the third order Osada method for multiple roots [7]

Case 3 Taking119867(119905) = 1(119860119905+119861) then1198671015840(119905) = minus119860(119860119905+119861)2According to (11) solving the equations

1

119860119906 + 119861= 119898 minus

119860

(119860119906 + 119861)2=

1198982

2 (17)

we get 119860 = minus12 and 119861 = (119898 + 1)2119898 With these values (2)yields

119909119896+1

= 119909119896minus 119891 (119909

119896) times (((119898 + 1) (2119898)) 119891

1015840 (119909119896)

minus ((119891 (119909119896) 11989110158401015840 (119909

119896)) (21198911015840 (119909

119896))))minus1

(18)

which is Halleyrsquos method for multiple roots [5]

Case 4 Supposing that 119867(119905) = 1radic119860 + 119861119905 then 1198671015840(119905) =

minus1198612(119860 + 119861119905)32According to (11) solving the equations

1

radic119860 + 119861119906= 119898 minus

119861

2(119860 + 119861119906)32=

1198982

2 (19)

we get 119860 = 1119898 and 119861 = minus1119898 and hence (2) becomes

119909119896+1

= 119909119896minus[[

[

radic119898

radic1 minus (119891 (119909119896) 11989110158401015840 (119909

119896)) 1198911015840(119909

119896)2

]]

]

119891 (119909119896)

1198911015840 (119909119896)

(20)

which is Ostrowskirsquos square root iteration [4]

Case 5 If119867(119905) = (119860 + 119861119905)(119862 + 119863119905) we have1198671015840(119905) = (119861119862 minus119860119863)(119862 + 119863119905)2

According to (11) solving the equations

119860 + 119861119906

119862 + 119863119906= 119898

119861119862 minus 119860119863

(119862 + 119863119906)2=

1198982

2 (21)

we get

119860 =119862119898 (3 minus 119898) minus 119863(119898 minus 1)2

2 119861 =

119863119898 + (119862 + 119863)1198982

2

(22)

Subcase 1 The choice of 119862 = 2 and119863 = minus120573(120573 isin R) gives

119860 =2119898 (3 minus 119898) + 120573(119898 minus 1)2

2 119861 = minus

120573119898 + (120573 minus 2)1198982

2

(23)

With these values (2) corresponds to the Sharma-Sharmamethod [13]

119909119896+1

= 119909119896minus1

4

times [

[

120573(119898 minus 1)2 minus 2119898 (119898 minus 3)

+(120573 (119898 minus 1) minus 2119898)

2

(119891 (119909119896) 11989110158401015840 (119909

119896) 21198911015840(119909

119896)2

)

1 minus 120573 (119891 (119909119896) 11989110158401015840 (119909

119896) 21198911015840(119909

119896)2

)]

]

times119891 (119909119896)

1198911015840 (119909119896)

(24)

Subcase 2 For the choice of 119862 = (119898 minus 1)2 and119863 = 119898(3 minus 119898)we get119860 = 0 and 119861 = 21198982 These values exhibit the proposedscheme (2) as

119909119896+1

= 119909119896

minus21198982119891(119909

119896)2

11989110158401015840 (119909119896)

119898 (3 minus 119898)119891 (119909119896) 1198911015840 (119909

119896) 11989110158401015840 (119909

119896) + (119898 minus 1)21198911015840(119909

119896)3

(25)

which is the third order Chun-Neta method [9]

4 Advances in Numerical Analysis

Case 6 For

119867(119905) =(119860 + 119861119905 + 1198621199052)

(119863 + 119864119905 + 1198651199052)

1198671015840 (119905) =(119861119863 minus 119860119864) minus 2 (119860119865 minus 119862119863) 119905 + (119862119864 minus 119861119865) 1199052

(119863 + 119864119905 + 1198651199052)2

(26)

According to (11) solving the equations (119860 +119861119906+1198621199062)(119863+

119864119906 + 1198651199062) = 119898 and

(119861119863 minus 119860119864) minus 2 (119860119865 minus 119862119863) 119906 + (119862119864 minus 119861119865) 1199062

(119863 + 119864119906 + 1198651199062)2

=1198982

2

(27)

we get

119864 =119862(119898 minus 1)3 + 119861119898(1198982 minus 1) + 1198982 (119860 (119898 + 3) minus 4119863119898)

21198982 (119898 minus 1)

119865 = minus119862 (119898 minus 3) (119898 minus 1)2 + 119861119898(119898 minus 1)2 + 1198982 (119860 (119898 + 1))

2119898(119898 minus 1)2

+21198631198983

2119898(119898 minus 1)2

(28)

Subcase 1 Taking119863 = 0 and 119864 = 1 and solving (28) we get

119860 = minus(119898 minus 1)2 ((119865 + 1)1198982 + 119865119898 minus 2119862)

21198982

119861 =1198982 (119865119898 + 119898 + 2119865 + 1) minus 4119862119898 minus 3119865119898 + 4119862

2119898

(29)

With these values (2) corresponds to the third order familyof multiple roots developed by Biazar and Ghanbari [11]

119909119896+1

= 119909119896minus ( (119861119891 (119909

119896) 1198911015840(119909

119896)2

11989110158401015840 (119909119896)

+1198601198911015840(119909119896)4

+ 119862119891(119909119896)2

11989110158401015840(119909119896)2

)

times (1198911015840(119909119896)3

11989110158401015840 (119909119896)

+119865119891 (119909119896) 1198911015840 (119909

119896) 11989110158401015840(119909

119896)2

)minus1

)

(30)

where 119860 and 119861 are given in (29)

Subcase 2 Taking 119860 = 2(1198982 minus 3119898 + 2) 119861 = minus1198982 + 5119898 119862 =

31198982 and119863 = 21198982 minus 6119898 + 4 (28) gives 119864 = minus2(1198982 minus 4119898 + 1)and 119865 = 0 With these values (2) yields

119909119896+1

= 119909119896minus [1 +

119891 (119909k) 11989110158401015840 (119909119896)

21198911015840(119909119896)2

+ ((2119898 minus 1)2(119891 (119909

119896) 11989110158401015840 (119909

119896) 1198911015840(119909

119896)2

)2

)

times (2 ((1198982 minus 3119898 + 2) minus (1198982 minus 4119898 + 1)

times (119891(119909119896)11989110158401015840(119909

119896)1198911015840(119909

119896)2

)))minus1

]

times119891 (119909119896)

1198911015840 (119909119896)

(31)

which is the Sharma-Sharma method [12]

Subcase 3 With 120579 isin R taking

119860 = minus120579(119898 minus 1)2

2 119861 =

(2120579 minus 1)1198982 + (3 minus 2120579)119898

2

119862 =(1 minus 120579)1198982

2

(32)

and119863 = 0 (28) generates 119864 = 1 and 119865 = 0 So (2) leads to

119909119896+1

= 119909119896

minus [119898 [(2120579 minus 1)119898 + 3 minus 2120579]

2

minus120579(119898 minus 1)2

2

1198911015840(119909119896)2

119891 (119909119896) 11989110158401015840 (119909

119896)

+(1 minus 120579)1198982

2

119891 (119909119896) 11989110158401015840 (119909

119896)

1198911015840(119909119896)2

]119891 (119909119896)

1198911015840 (119909119896)

(33)

which is the one-parameter family of methods for multipleroots proposed by Chun et al [10]

Case 7 Further if 119867(119905) = 119860 + (119861119905) + (1198621199052) then 1198671015840(119905) =minus(1198611199052) minus (21198621199053)

According to (11) solving the equations

119860 +119861

119906+

119862

1199062= 119898 minus

119861

1199062minus2119862

1199063=

1198982

2 (34)

we get 119860 = ((119898 + 1198982)2) + (1198621198982(119898 minus 1)2) and 119861 = (minus(119898 minus

1)22) minus (2119862119898(119898 minus 1))

Advances in Numerical Analysis 5

Table 1 Test functions

119891(119909) 120572 119898

1198911(119909) = ((119909 minus 1)3 minus 1)6 20000000000000000000000000000 6

1198912(119909) = (ln(1199092 + 119909 + 2) minus 119909 + 1)4 41525907367571582749969890048 4

1198913(119909) = (ln2 (119909 minus 2) (119890119909minus3 minus 1) sin 120587119909

3) 30000000000000000000000000000 4

1198914(119909) = (1199092 minus 119890119909 minus 3119909 + 2)5 02575302854398607604553673049 5

1198915(119909) = (cos119909 minus 119909)3 073908513321516064165531208767 3

1198916(119909) = (radic119909 minus

1

119909minus 3)3 96335955628326951924063127092 3

1198917(119909) = (1199093 minus 10)8 21544346900318837217592935665 8

Table 2 Performance of the methods using 12 NFE

119891(119909) 1199090

|119890119896| = |119909

119896minus 120572|

MNM OM HM SSM (120573 = 32) CNM SBM (119862 = 12)

1198911

175 120 (minus34) 276 (minus22) 548 (minus34) 401 (minus41) 108 (minus42) 125 (minus22)3 155 (minus16) 275 (minus15) 493 (minus18) 236 (minus20) 678 (minus21) 237 (minus15)

1198912

4 453 (minus129) 446 (minus137) 146 (minus145) 117 (minus144) 111 (minus141) 175 (minus137)5 688 (minus86) 245 (minus84) 775 (minus91) 378 (minus90) 710 (minus88) 119 (minus84)

1198913

25 541 (minus67) 247 (minus76) 833 (minus57) 560 (minus63) 490 (minus58) 134 (minus74)325 783 (minus99) 232 (minus118) 139 (minus106) 261 (minus110) 260 (minus107) 235 (minus115)

1198914

minus2 155 (minus37) 130 (minus42) 145 (minus40) 152 (minus39) 147 (minus40) 252 (minus40)15 106 (minus57) 244 (minus50) 515 (minus50) 243 (minus40) 466 (minus38) 133 (minus49)

1198915

03 359 (minus59) 213 (minus44) 512 (minus64) 512 (minus64) 717 (minus53) 116 (minus47)17 325 (minus65) 535 (minus41) 225 (minus44) 225 (minus44) 184 (minus42) 166 (minus41)

1198916

9 513 (minus108) 251 (minus137) 150 (minus135) 150 (minus135) 115 (minus193) 943 (minus146)10 111 (minus61) 359 (minus158) 468 (minus157) 468 (minus157) 907 (minus214) 754 (minus167)

1198917

2 259 (minus72) 168 (minus78) 675 (minus99) 389 (minus89) 329 (minus92) 144 (minus78)3 279 (minus33) 319 (minus34) 297 (minus47) 907 (minus41) 873 (minus43) 292 (minus34)

Here 119886(minus119887) = 119886 times 10minus119887

Table 3 Computational order of convergence (120588)

119891(119909) 1199090

120588

MNM OM HM SSM (120573 = 32) CNM SBM (119862 = 12)

1198911

175 2 3 3 3 3 33 2 3 3 299 299 3

1198912

4 2 3 3 3 3 35 2 3 3 3 3 3

1198913

25 2 3 3 3 3 3325 2 3 3 3 3 3

1198914

minus2 2 3 3 3 3 315 2 3 3 3 3 3

1198915

03 2 3 3 3 3 317 2 299 299 299 299 3

1198916

9 2 3 3 3 3 310 2 3 3 3 3 3

1198917

2 2 3 3 3 3 33 2 299 299 299 299 3

6 Advances in Numerical Analysis

minus2 minus1 0 1 2

2

1

0

minus2

minus1

(a) The modified Newton method

minus2 minus1 0 1 2

2

1

0

minus2

minus1

(b) The Osada method

minus2 minus1 0 1 2

2

1

0

minus2

minus1

(c) Halleyrsquos method

minus2 minus1 0 1 2

2

1

0

minus2

minus1

(d) The Chun-Neta method

minus2 minus1 0 1 2

2

1

0

minus2

minus1

(e) The Sharma-Sharma method (120573 = 32)

minus2 minus1 0 1 2

2

1

0

minus2

minus1

(f) SBM (119862 = 12)

Figure 1 Basins of attraction for 119891(119911) = (1199112 minus 1)2 119911 isin 119863 for various methods of the family

Advances in Numerical Analysis 7

So we introduce a new one-parameter third ordermethod as

119909119896+1

= 119909119896minus[[

[

(119898 + 1198982

2+

1198621198982

(119898 minus 1)2)

+ (minus(119898 minus 1)2

2minus

2119862119898

119898 minus 1)

1198911015840(119909119896)2

119891 (119909119896) 11989110158401015840 (119909

119896)

+119862((1198911015840 (119909

119896))2

(119891 (119909119896) 11989110158401015840 (119909

119896)))

2

]]

]

119891 (119909119896)

1198911015840 (119909119896)

(35)

where 119862 isin R We call this method the Sharma-Bahl methoddenoted by SBM

4 Numerical Examples

In this section we employ the presented third order methodSBM (35) for 119862 = 12 to solve some nonlinear equationswhich not only illustrate the methods practically but alsoserve to check the validity of the theoretical results we havederived To check the theoretical order of convergence weobtain the computational order of convergence (120588) using theformula [16]

120588 asympln 1003816100381610038161003816(119909119896+1 minus 120572) (119909

119896minus 120572)

1003816100381610038161003816ln 1003816100381610038161003816(119909119896 minus 120572) (119909

119896minus1minus 120572)

1003816100381610038161003816 (36)

The performance is compared with the methods in ourfamilies including the special ones for some arbitrary chosenfunctions with roots of known multiplicity 119898 The methodscompared include the modified Newton method (MNM) (1)the Osada method (OM) (16) Halleyrsquos method (HM) (18)the Sharma-Sharma method (SSM) (24) for 120573 = 32 and theChun-Neta method (CNM) (25)

The test functions along with root 120572 correct up to 28decimal places and its multiplicity 119898 is displayed in Table 1Table 2 shows the values of initial approximation (119909

0) chosen

from both ends to the root and the values of the error |119890119896| =

|119909119896minus 120572| calculated by costing the same total number of

function evaluations (NFE) for each method Table 3 exhibitsthe computational order of convergence (120588) For numericalillustrations in Table 3 we use fixed stopping criterion 120598 =

05 times 10minus28 The NFE is counted as sum of the number ofevaluations of the function plus the number of evaluations ofthe derivativesWe decide to choose 12 NFE for eachmethodThat means that for MNM the error |119890

119896| is calculated at

the sixth iteration whereas for the remaining methods thisis calculated at the fourth iteration All computations areperformed by MATHEMATICA [17] using 600 significantdigits

It can also be observed from the numerical results ofTable 2 that all the presented methods are well behavedand that for most of the functions we tested the methodintroduced in this paper has similar performance comparedto the other methods as expected from methods of the same

order and same computational efficiency However Halleyrsquosmethod (18) and the Chun-Neta method (25) behave betterand these are special cases of our proposed scheme for linearrational function 119867(sdot) The results of Table 3 show that thecomputational order of convergence is in accordance withthe theoretical order of convergenceThe formulae have beenemployed on several other nonlinear equations and resultsare found at par with those presented here A reasonablyclose starting value is necessary for the method to convergeThe condition however practically applies to all iterativemethods for solving nonlinear equations

5 Finding the Basins

In this section we find the basins of attraction of complexroots for the a complex function We consider the simplestcase119891(119911) = (1199112minus1)2 which has roots 119911 = plusmn 1 ofmultiplicity 2Cayley [18] was the first who considered the Newton methodfor the roots of polynomial with iterations over the complexnumbers

We take the initial point as 1199110

isin 119863 where 119863 is arectangular region [minus25 25] times [minus25 25] sub C containingall the roots of 119891(119911) = 0 We consider the stopping criterionfor convergence as 10minus3 up to amaximumof 25 iterationsWetake a grid of 1024 times 1024 points in119863

To generate the pictures we use MATHEMATICA [17]We assign the light to dark colors based on the number ofiterations in which the considered initial point converges to aroot The root 119911 = 1 of 119891(119911) = 0 and the points in the regionconverging to this root are shaded in cyan color and magentacolor is used for the other root 119911 = minus1 in a similar way Ifwe have not obtained the desired tolerance in 25 iterationswe do not continue and we decide that the iterative methodstarting at 119911

0does not converge to any root and such points

are shaded in black color (see [19ndash22])The figures given here clearly show that the mod-

ified Newton method (Figure 1(a)) and Halleyrsquos method(Figure 1(c)) are the best followed by the Osada method(Figure 1(b)) the Chun-Neta method (Figure 1(d)) SBM(Figure 1(f)) and Sharma-Sharma method (Figure 1(e))From these pictures we can guess the behavior and suitabilityof any method depending upon the circumstances Thusconcluding we can say that the linear rational function givesthe best choice of weight function119867(sdot)

6 Conclusions

In this paper a general family of one-point third ordermethods for finding multiple roots of nonlinear equations ispresented This scheme contains some well known methodsand recently developed methods as its particular cases Thetheoretical results have been checked with some numericalexamples comparing ourmethodswith themodifiedNewtonmethod and some third order methodsThe presented basinsof attraction have also demonstrated the strictness of methodfor choice of starting point

8 Advances in Numerical Analysis

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors are grateful to the anonymous reviewers for theirvaluable suggestions and comments on this paper

References

[1] J M Ortega and W C Rheinboldt Iterative Solution of Nonlin-ear Equations in Several Variables Academic Press New YorkNY USA 1970

[2] S C Chapra and R P CanaleNumerical Methods for EngineersMcGraw-Hill Book Company New York NY USA 1988

[3] E Schroder ldquoUeber unendlich vieleAlgorithmen zurAuflosungder GleichungenrdquoMathematische Annalen vol 2 no 2 pp 317ndash365 1870

[4] AM Ostrowski Solution of Equations in Euclidean and BanachSpaces Academic Press New York NY USA 3rd edition 1973

[5] E Hansen and M Patrick ldquoA family of root finding methodsrdquoNumerische Mathematik vol 27 no 3 pp 257ndash269 1977

[6] J F Traub Iterative Methods for the Solution of EquationsPrentice-Hall Englewood Cliffs NJ USA 1964

[7] N Osada ldquoAn optimal multiple root-finding method of orderthreerdquo Journal of Computational and Applied Mathematics vol51 no 1 pp 131ndash133 1994

[8] B Neta ldquoNew third order nonlinear solvers for multiple rootsrdquoApplied Mathematics and Computation vol 202 no 1 pp 162ndash170 2008

[9] C Chun and B Neta ldquoA third-order modification of Newtonrsquosmethod for multiple rootsrdquoAppliedMathematics and Computa-tion vol 211 no 2 pp 474ndash479 2009

[10] C Chun H J Bae and B Neta ldquoNew families of nonlinearthird-order solvers for finding multiple rootsrdquo Computers andMathematics with Applications vol 57 no 9 pp 1574ndash15822009

[11] J Biazar and B Ghanbari ldquoA new third-order family of nonlin-ear solvers formultiple rootsrdquoComputers andMathematics withApplications vol 59 no 10 pp 3315ndash3319 2010

[12] J R Sharma and R Sharma ldquoNew third and fourth ordernonlinear solvers for computing multiple rootsrdquo Applied Math-ematics and Computation vol 217 no 23 pp 9756ndash9764 2011

[13] J R Sharma and R Sharma ldquoModified Chebyshev-Halleytype method and its variants for computing multiple rootsrdquoNumerical Algorithms vol 61 no 4 pp 567ndash578 2012

[14] S Kumar V Kanwar and S Singh ldquoOn some modified familiesof multipoint iterative methods for multiple roots of nonlinearequationsrdquo Applied Mathematics and Computation vol 218 no14 pp 7382ndash7394 2012

[15] W Gautschi Numerical Analysis An Introduction BirkhauserBoston Mass USA 1997

[16] S Weerakoon and T G I Fernando ldquoA variant of Newtonrsquosmethod with accelerated third-order convergencerdquo AppliedMathematics Letters vol 13 no 8 pp 87ndash93 2000

[17] S Wolfram The Mathematica Book Wolfram Media Cham-paign Ill USA 5th edition 2003

[18] A Cayley ldquoThe Newton-Fourier imaginary problemrdquo TheAmerican Journal of Mathematics vol 2 no 1 article 97 1879

[19] M L Sahari and I Djellit ldquoFractal Newton basinsrdquo DiscreteDynamics in Nature and Society vol 2006 Article ID 28756 16pages 2006

[20] J L Varona ldquoGraphic and numerical comparison betweeniterative methodsrdquo The Mathematical Intelligencer vol 24 no1 pp 37ndash46 2002

[21] M Scott B Neta and C Chun ldquoBasin attractors for variousmethodsrdquo Applied Mathematics and Computation vol 218 no6 pp 2584ndash2599 2011

[22] B Neta M Scott and C Chun ldquoBasin attractors for variousmethods for multiple rootsrdquo Applied Mathematics and Compu-tation vol 218 no 9 pp 5043ndash5066 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Advances in Numerical Analysis

Case 6 For

119867(119905) =(119860 + 119861119905 + 1198621199052)

(119863 + 119864119905 + 1198651199052)

1198671015840 (119905) =(119861119863 minus 119860119864) minus 2 (119860119865 minus 119862119863) 119905 + (119862119864 minus 119861119865) 1199052

(119863 + 119864119905 + 1198651199052)2

(26)

According to (11) solving the equations (119860 +119861119906+1198621199062)(119863+

119864119906 + 1198651199062) = 119898 and

(119861119863 minus 119860119864) minus 2 (119860119865 minus 119862119863) 119906 + (119862119864 minus 119861119865) 1199062

(119863 + 119864119906 + 1198651199062)2

=1198982

2

(27)

we get

119864 =119862(119898 minus 1)3 + 119861119898(1198982 minus 1) + 1198982 (119860 (119898 + 3) minus 4119863119898)

21198982 (119898 minus 1)

119865 = minus119862 (119898 minus 3) (119898 minus 1)2 + 119861119898(119898 minus 1)2 + 1198982 (119860 (119898 + 1))

2119898(119898 minus 1)2

+21198631198983

2119898(119898 minus 1)2

(28)

Subcase 1 Taking119863 = 0 and 119864 = 1 and solving (28) we get

119860 = minus(119898 minus 1)2 ((119865 + 1)1198982 + 119865119898 minus 2119862)

21198982

119861 =1198982 (119865119898 + 119898 + 2119865 + 1) minus 4119862119898 minus 3119865119898 + 4119862

2119898

(29)

With these values (2) corresponds to the third order familyof multiple roots developed by Biazar and Ghanbari [11]

119909119896+1

= 119909119896minus ( (119861119891 (119909

119896) 1198911015840(119909

119896)2

11989110158401015840 (119909119896)

+1198601198911015840(119909119896)4

+ 119862119891(119909119896)2

11989110158401015840(119909119896)2

)

times (1198911015840(119909119896)3

11989110158401015840 (119909119896)

+119865119891 (119909119896) 1198911015840 (119909

119896) 11989110158401015840(119909

119896)2

)minus1

)

(30)

where 119860 and 119861 are given in (29)

Subcase 2 Taking 119860 = 2(1198982 minus 3119898 + 2) 119861 = minus1198982 + 5119898 119862 =

31198982 and119863 = 21198982 minus 6119898 + 4 (28) gives 119864 = minus2(1198982 minus 4119898 + 1)and 119865 = 0 With these values (2) yields

119909119896+1

= 119909119896minus [1 +

119891 (119909k) 11989110158401015840 (119909119896)

21198911015840(119909119896)2

+ ((2119898 minus 1)2(119891 (119909

119896) 11989110158401015840 (119909

119896) 1198911015840(119909

119896)2

)2

)

times (2 ((1198982 minus 3119898 + 2) minus (1198982 minus 4119898 + 1)

times (119891(119909119896)11989110158401015840(119909

119896)1198911015840(119909

119896)2

)))minus1

]

times119891 (119909119896)

1198911015840 (119909119896)

(31)

which is the Sharma-Sharma method [12]

Subcase 3 With 120579 isin R taking

119860 = minus120579(119898 minus 1)2

2 119861 =

(2120579 minus 1)1198982 + (3 minus 2120579)119898

2

119862 =(1 minus 120579)1198982

2

(32)

and119863 = 0 (28) generates 119864 = 1 and 119865 = 0 So (2) leads to

119909119896+1

= 119909119896

minus [119898 [(2120579 minus 1)119898 + 3 minus 2120579]

2

minus120579(119898 minus 1)2

2

1198911015840(119909119896)2

119891 (119909119896) 11989110158401015840 (119909

119896)

+(1 minus 120579)1198982

2

119891 (119909119896) 11989110158401015840 (119909

119896)

1198911015840(119909119896)2

]119891 (119909119896)

1198911015840 (119909119896)

(33)

which is the one-parameter family of methods for multipleroots proposed by Chun et al [10]

Case 7 Further if 119867(119905) = 119860 + (119861119905) + (1198621199052) then 1198671015840(119905) =minus(1198611199052) minus (21198621199053)

According to (11) solving the equations

119860 +119861

119906+

119862

1199062= 119898 minus

119861

1199062minus2119862

1199063=

1198982

2 (34)

we get 119860 = ((119898 + 1198982)2) + (1198621198982(119898 minus 1)2) and 119861 = (minus(119898 minus

1)22) minus (2119862119898(119898 minus 1))

Advances in Numerical Analysis 5

Table 1 Test functions

119891(119909) 120572 119898

1198911(119909) = ((119909 minus 1)3 minus 1)6 20000000000000000000000000000 6

1198912(119909) = (ln(1199092 + 119909 + 2) minus 119909 + 1)4 41525907367571582749969890048 4

1198913(119909) = (ln2 (119909 minus 2) (119890119909minus3 minus 1) sin 120587119909

3) 30000000000000000000000000000 4

1198914(119909) = (1199092 minus 119890119909 minus 3119909 + 2)5 02575302854398607604553673049 5

1198915(119909) = (cos119909 minus 119909)3 073908513321516064165531208767 3

1198916(119909) = (radic119909 minus

1

119909minus 3)3 96335955628326951924063127092 3

1198917(119909) = (1199093 minus 10)8 21544346900318837217592935665 8

Table 2 Performance of the methods using 12 NFE

119891(119909) 1199090

|119890119896| = |119909

119896minus 120572|

MNM OM HM SSM (120573 = 32) CNM SBM (119862 = 12)

1198911

175 120 (minus34) 276 (minus22) 548 (minus34) 401 (minus41) 108 (minus42) 125 (minus22)3 155 (minus16) 275 (minus15) 493 (minus18) 236 (minus20) 678 (minus21) 237 (minus15)

1198912

4 453 (minus129) 446 (minus137) 146 (minus145) 117 (minus144) 111 (minus141) 175 (minus137)5 688 (minus86) 245 (minus84) 775 (minus91) 378 (minus90) 710 (minus88) 119 (minus84)

1198913

25 541 (minus67) 247 (minus76) 833 (minus57) 560 (minus63) 490 (minus58) 134 (minus74)325 783 (minus99) 232 (minus118) 139 (minus106) 261 (minus110) 260 (minus107) 235 (minus115)

1198914

minus2 155 (minus37) 130 (minus42) 145 (minus40) 152 (minus39) 147 (minus40) 252 (minus40)15 106 (minus57) 244 (minus50) 515 (minus50) 243 (minus40) 466 (minus38) 133 (minus49)

1198915

03 359 (minus59) 213 (minus44) 512 (minus64) 512 (minus64) 717 (minus53) 116 (minus47)17 325 (minus65) 535 (minus41) 225 (minus44) 225 (minus44) 184 (minus42) 166 (minus41)

1198916

9 513 (minus108) 251 (minus137) 150 (minus135) 150 (minus135) 115 (minus193) 943 (minus146)10 111 (minus61) 359 (minus158) 468 (minus157) 468 (minus157) 907 (minus214) 754 (minus167)

1198917

2 259 (minus72) 168 (minus78) 675 (minus99) 389 (minus89) 329 (minus92) 144 (minus78)3 279 (minus33) 319 (minus34) 297 (minus47) 907 (minus41) 873 (minus43) 292 (minus34)

Here 119886(minus119887) = 119886 times 10minus119887

Table 3 Computational order of convergence (120588)

119891(119909) 1199090

120588

MNM OM HM SSM (120573 = 32) CNM SBM (119862 = 12)

1198911

175 2 3 3 3 3 33 2 3 3 299 299 3

1198912

4 2 3 3 3 3 35 2 3 3 3 3 3

1198913

25 2 3 3 3 3 3325 2 3 3 3 3 3

1198914

minus2 2 3 3 3 3 315 2 3 3 3 3 3

1198915

03 2 3 3 3 3 317 2 299 299 299 299 3

1198916

9 2 3 3 3 3 310 2 3 3 3 3 3

1198917

2 2 3 3 3 3 33 2 299 299 299 299 3

6 Advances in Numerical Analysis

minus2 minus1 0 1 2

2

1

0

minus2

minus1

(a) The modified Newton method

minus2 minus1 0 1 2

2

1

0

minus2

minus1

(b) The Osada method

minus2 minus1 0 1 2

2

1

0

minus2

minus1

(c) Halleyrsquos method

minus2 minus1 0 1 2

2

1

0

minus2

minus1

(d) The Chun-Neta method

minus2 minus1 0 1 2

2

1

0

minus2

minus1

(e) The Sharma-Sharma method (120573 = 32)

minus2 minus1 0 1 2

2

1

0

minus2

minus1

(f) SBM (119862 = 12)

Figure 1 Basins of attraction for 119891(119911) = (1199112 minus 1)2 119911 isin 119863 for various methods of the family

Advances in Numerical Analysis 7

So we introduce a new one-parameter third ordermethod as

119909119896+1

= 119909119896minus[[

[

(119898 + 1198982

2+

1198621198982

(119898 minus 1)2)

+ (minus(119898 minus 1)2

2minus

2119862119898

119898 minus 1)

1198911015840(119909119896)2

119891 (119909119896) 11989110158401015840 (119909

119896)

+119862((1198911015840 (119909

119896))2

(119891 (119909119896) 11989110158401015840 (119909

119896)))

2

]]

]

119891 (119909119896)

1198911015840 (119909119896)

(35)

where 119862 isin R We call this method the Sharma-Bahl methoddenoted by SBM

4 Numerical Examples

In this section we employ the presented third order methodSBM (35) for 119862 = 12 to solve some nonlinear equationswhich not only illustrate the methods practically but alsoserve to check the validity of the theoretical results we havederived To check the theoretical order of convergence weobtain the computational order of convergence (120588) using theformula [16]

120588 asympln 1003816100381610038161003816(119909119896+1 minus 120572) (119909

119896minus 120572)

1003816100381610038161003816ln 1003816100381610038161003816(119909119896 minus 120572) (119909

119896minus1minus 120572)

1003816100381610038161003816 (36)

The performance is compared with the methods in ourfamilies including the special ones for some arbitrary chosenfunctions with roots of known multiplicity 119898 The methodscompared include the modified Newton method (MNM) (1)the Osada method (OM) (16) Halleyrsquos method (HM) (18)the Sharma-Sharma method (SSM) (24) for 120573 = 32 and theChun-Neta method (CNM) (25)

The test functions along with root 120572 correct up to 28decimal places and its multiplicity 119898 is displayed in Table 1Table 2 shows the values of initial approximation (119909

0) chosen

from both ends to the root and the values of the error |119890119896| =

|119909119896minus 120572| calculated by costing the same total number of

function evaluations (NFE) for each method Table 3 exhibitsthe computational order of convergence (120588) For numericalillustrations in Table 3 we use fixed stopping criterion 120598 =

05 times 10minus28 The NFE is counted as sum of the number ofevaluations of the function plus the number of evaluations ofthe derivativesWe decide to choose 12 NFE for eachmethodThat means that for MNM the error |119890

119896| is calculated at

the sixth iteration whereas for the remaining methods thisis calculated at the fourth iteration All computations areperformed by MATHEMATICA [17] using 600 significantdigits

It can also be observed from the numerical results ofTable 2 that all the presented methods are well behavedand that for most of the functions we tested the methodintroduced in this paper has similar performance comparedto the other methods as expected from methods of the same

order and same computational efficiency However Halleyrsquosmethod (18) and the Chun-Neta method (25) behave betterand these are special cases of our proposed scheme for linearrational function 119867(sdot) The results of Table 3 show that thecomputational order of convergence is in accordance withthe theoretical order of convergenceThe formulae have beenemployed on several other nonlinear equations and resultsare found at par with those presented here A reasonablyclose starting value is necessary for the method to convergeThe condition however practically applies to all iterativemethods for solving nonlinear equations

5 Finding the Basins

In this section we find the basins of attraction of complexroots for the a complex function We consider the simplestcase119891(119911) = (1199112minus1)2 which has roots 119911 = plusmn 1 ofmultiplicity 2Cayley [18] was the first who considered the Newton methodfor the roots of polynomial with iterations over the complexnumbers

We take the initial point as 1199110

isin 119863 where 119863 is arectangular region [minus25 25] times [minus25 25] sub C containingall the roots of 119891(119911) = 0 We consider the stopping criterionfor convergence as 10minus3 up to amaximumof 25 iterationsWetake a grid of 1024 times 1024 points in119863

To generate the pictures we use MATHEMATICA [17]We assign the light to dark colors based on the number ofiterations in which the considered initial point converges to aroot The root 119911 = 1 of 119891(119911) = 0 and the points in the regionconverging to this root are shaded in cyan color and magentacolor is used for the other root 119911 = minus1 in a similar way Ifwe have not obtained the desired tolerance in 25 iterationswe do not continue and we decide that the iterative methodstarting at 119911

0does not converge to any root and such points

are shaded in black color (see [19ndash22])The figures given here clearly show that the mod-

ified Newton method (Figure 1(a)) and Halleyrsquos method(Figure 1(c)) are the best followed by the Osada method(Figure 1(b)) the Chun-Neta method (Figure 1(d)) SBM(Figure 1(f)) and Sharma-Sharma method (Figure 1(e))From these pictures we can guess the behavior and suitabilityof any method depending upon the circumstances Thusconcluding we can say that the linear rational function givesthe best choice of weight function119867(sdot)

6 Conclusions

In this paper a general family of one-point third ordermethods for finding multiple roots of nonlinear equations ispresented This scheme contains some well known methodsand recently developed methods as its particular cases Thetheoretical results have been checked with some numericalexamples comparing ourmethodswith themodifiedNewtonmethod and some third order methodsThe presented basinsof attraction have also demonstrated the strictness of methodfor choice of starting point

8 Advances in Numerical Analysis

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors are grateful to the anonymous reviewers for theirvaluable suggestions and comments on this paper

References

[1] J M Ortega and W C Rheinboldt Iterative Solution of Nonlin-ear Equations in Several Variables Academic Press New YorkNY USA 1970

[2] S C Chapra and R P CanaleNumerical Methods for EngineersMcGraw-Hill Book Company New York NY USA 1988

[3] E Schroder ldquoUeber unendlich vieleAlgorithmen zurAuflosungder GleichungenrdquoMathematische Annalen vol 2 no 2 pp 317ndash365 1870

[4] AM Ostrowski Solution of Equations in Euclidean and BanachSpaces Academic Press New York NY USA 3rd edition 1973

[5] E Hansen and M Patrick ldquoA family of root finding methodsrdquoNumerische Mathematik vol 27 no 3 pp 257ndash269 1977

[6] J F Traub Iterative Methods for the Solution of EquationsPrentice-Hall Englewood Cliffs NJ USA 1964

[7] N Osada ldquoAn optimal multiple root-finding method of orderthreerdquo Journal of Computational and Applied Mathematics vol51 no 1 pp 131ndash133 1994

[8] B Neta ldquoNew third order nonlinear solvers for multiple rootsrdquoApplied Mathematics and Computation vol 202 no 1 pp 162ndash170 2008

[9] C Chun and B Neta ldquoA third-order modification of Newtonrsquosmethod for multiple rootsrdquoAppliedMathematics and Computa-tion vol 211 no 2 pp 474ndash479 2009

[10] C Chun H J Bae and B Neta ldquoNew families of nonlinearthird-order solvers for finding multiple rootsrdquo Computers andMathematics with Applications vol 57 no 9 pp 1574ndash15822009

[11] J Biazar and B Ghanbari ldquoA new third-order family of nonlin-ear solvers formultiple rootsrdquoComputers andMathematics withApplications vol 59 no 10 pp 3315ndash3319 2010

[12] J R Sharma and R Sharma ldquoNew third and fourth ordernonlinear solvers for computing multiple rootsrdquo Applied Math-ematics and Computation vol 217 no 23 pp 9756ndash9764 2011

[13] J R Sharma and R Sharma ldquoModified Chebyshev-Halleytype method and its variants for computing multiple rootsrdquoNumerical Algorithms vol 61 no 4 pp 567ndash578 2012

[14] S Kumar V Kanwar and S Singh ldquoOn some modified familiesof multipoint iterative methods for multiple roots of nonlinearequationsrdquo Applied Mathematics and Computation vol 218 no14 pp 7382ndash7394 2012

[15] W Gautschi Numerical Analysis An Introduction BirkhauserBoston Mass USA 1997

[16] S Weerakoon and T G I Fernando ldquoA variant of Newtonrsquosmethod with accelerated third-order convergencerdquo AppliedMathematics Letters vol 13 no 8 pp 87ndash93 2000

[17] S Wolfram The Mathematica Book Wolfram Media Cham-paign Ill USA 5th edition 2003

[18] A Cayley ldquoThe Newton-Fourier imaginary problemrdquo TheAmerican Journal of Mathematics vol 2 no 1 article 97 1879

[19] M L Sahari and I Djellit ldquoFractal Newton basinsrdquo DiscreteDynamics in Nature and Society vol 2006 Article ID 28756 16pages 2006

[20] J L Varona ldquoGraphic and numerical comparison betweeniterative methodsrdquo The Mathematical Intelligencer vol 24 no1 pp 37ndash46 2002

[21] M Scott B Neta and C Chun ldquoBasin attractors for variousmethodsrdquo Applied Mathematics and Computation vol 218 no6 pp 2584ndash2599 2011

[22] B Neta M Scott and C Chun ldquoBasin attractors for variousmethods for multiple rootsrdquo Applied Mathematics and Compu-tation vol 218 no 9 pp 5043ndash5066 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Numerical Analysis 5

Table 1 Test functions

119891(119909) 120572 119898

1198911(119909) = ((119909 minus 1)3 minus 1)6 20000000000000000000000000000 6

1198912(119909) = (ln(1199092 + 119909 + 2) minus 119909 + 1)4 41525907367571582749969890048 4

1198913(119909) = (ln2 (119909 minus 2) (119890119909minus3 minus 1) sin 120587119909

3) 30000000000000000000000000000 4

1198914(119909) = (1199092 minus 119890119909 minus 3119909 + 2)5 02575302854398607604553673049 5

1198915(119909) = (cos119909 minus 119909)3 073908513321516064165531208767 3

1198916(119909) = (radic119909 minus

1

119909minus 3)3 96335955628326951924063127092 3

1198917(119909) = (1199093 minus 10)8 21544346900318837217592935665 8

Table 2 Performance of the methods using 12 NFE

119891(119909) 1199090

|119890119896| = |119909

119896minus 120572|

MNM OM HM SSM (120573 = 32) CNM SBM (119862 = 12)

1198911

175 120 (minus34) 276 (minus22) 548 (minus34) 401 (minus41) 108 (minus42) 125 (minus22)3 155 (minus16) 275 (minus15) 493 (minus18) 236 (minus20) 678 (minus21) 237 (minus15)

1198912

4 453 (minus129) 446 (minus137) 146 (minus145) 117 (minus144) 111 (minus141) 175 (minus137)5 688 (minus86) 245 (minus84) 775 (minus91) 378 (minus90) 710 (minus88) 119 (minus84)

1198913

25 541 (minus67) 247 (minus76) 833 (minus57) 560 (minus63) 490 (minus58) 134 (minus74)325 783 (minus99) 232 (minus118) 139 (minus106) 261 (minus110) 260 (minus107) 235 (minus115)

1198914

minus2 155 (minus37) 130 (minus42) 145 (minus40) 152 (minus39) 147 (minus40) 252 (minus40)15 106 (minus57) 244 (minus50) 515 (minus50) 243 (minus40) 466 (minus38) 133 (minus49)

1198915

03 359 (minus59) 213 (minus44) 512 (minus64) 512 (minus64) 717 (minus53) 116 (minus47)17 325 (minus65) 535 (minus41) 225 (minus44) 225 (minus44) 184 (minus42) 166 (minus41)

1198916

9 513 (minus108) 251 (minus137) 150 (minus135) 150 (minus135) 115 (minus193) 943 (minus146)10 111 (minus61) 359 (minus158) 468 (minus157) 468 (minus157) 907 (minus214) 754 (minus167)

1198917

2 259 (minus72) 168 (minus78) 675 (minus99) 389 (minus89) 329 (minus92) 144 (minus78)3 279 (minus33) 319 (minus34) 297 (minus47) 907 (minus41) 873 (minus43) 292 (minus34)

Here 119886(minus119887) = 119886 times 10minus119887

Table 3 Computational order of convergence (120588)

119891(119909) 1199090

120588

MNM OM HM SSM (120573 = 32) CNM SBM (119862 = 12)

1198911

175 2 3 3 3 3 33 2 3 3 299 299 3

1198912

4 2 3 3 3 3 35 2 3 3 3 3 3

1198913

25 2 3 3 3 3 3325 2 3 3 3 3 3

1198914

minus2 2 3 3 3 3 315 2 3 3 3 3 3

1198915

03 2 3 3 3 3 317 2 299 299 299 299 3

1198916

9 2 3 3 3 3 310 2 3 3 3 3 3

1198917

2 2 3 3 3 3 33 2 299 299 299 299 3

6 Advances in Numerical Analysis

minus2 minus1 0 1 2

2

1

0

minus2

minus1

(a) The modified Newton method

minus2 minus1 0 1 2

2

1

0

minus2

minus1

(b) The Osada method

minus2 minus1 0 1 2

2

1

0

minus2

minus1

(c) Halleyrsquos method

minus2 minus1 0 1 2

2

1

0

minus2

minus1

(d) The Chun-Neta method

minus2 minus1 0 1 2

2

1

0

minus2

minus1

(e) The Sharma-Sharma method (120573 = 32)

minus2 minus1 0 1 2

2

1

0

minus2

minus1

(f) SBM (119862 = 12)

Figure 1 Basins of attraction for 119891(119911) = (1199112 minus 1)2 119911 isin 119863 for various methods of the family

Advances in Numerical Analysis 7

So we introduce a new one-parameter third ordermethod as

119909119896+1

= 119909119896minus[[

[

(119898 + 1198982

2+

1198621198982

(119898 minus 1)2)

+ (minus(119898 minus 1)2

2minus

2119862119898

119898 minus 1)

1198911015840(119909119896)2

119891 (119909119896) 11989110158401015840 (119909

119896)

+119862((1198911015840 (119909

119896))2

(119891 (119909119896) 11989110158401015840 (119909

119896)))

2

]]

]

119891 (119909119896)

1198911015840 (119909119896)

(35)

where 119862 isin R We call this method the Sharma-Bahl methoddenoted by SBM

4 Numerical Examples

In this section we employ the presented third order methodSBM (35) for 119862 = 12 to solve some nonlinear equationswhich not only illustrate the methods practically but alsoserve to check the validity of the theoretical results we havederived To check the theoretical order of convergence weobtain the computational order of convergence (120588) using theformula [16]

120588 asympln 1003816100381610038161003816(119909119896+1 minus 120572) (119909

119896minus 120572)

1003816100381610038161003816ln 1003816100381610038161003816(119909119896 minus 120572) (119909

119896minus1minus 120572)

1003816100381610038161003816 (36)

The performance is compared with the methods in ourfamilies including the special ones for some arbitrary chosenfunctions with roots of known multiplicity 119898 The methodscompared include the modified Newton method (MNM) (1)the Osada method (OM) (16) Halleyrsquos method (HM) (18)the Sharma-Sharma method (SSM) (24) for 120573 = 32 and theChun-Neta method (CNM) (25)

The test functions along with root 120572 correct up to 28decimal places and its multiplicity 119898 is displayed in Table 1Table 2 shows the values of initial approximation (119909

0) chosen

from both ends to the root and the values of the error |119890119896| =

|119909119896minus 120572| calculated by costing the same total number of

function evaluations (NFE) for each method Table 3 exhibitsthe computational order of convergence (120588) For numericalillustrations in Table 3 we use fixed stopping criterion 120598 =

05 times 10minus28 The NFE is counted as sum of the number ofevaluations of the function plus the number of evaluations ofthe derivativesWe decide to choose 12 NFE for eachmethodThat means that for MNM the error |119890

119896| is calculated at

the sixth iteration whereas for the remaining methods thisis calculated at the fourth iteration All computations areperformed by MATHEMATICA [17] using 600 significantdigits

It can also be observed from the numerical results ofTable 2 that all the presented methods are well behavedand that for most of the functions we tested the methodintroduced in this paper has similar performance comparedto the other methods as expected from methods of the same

order and same computational efficiency However Halleyrsquosmethod (18) and the Chun-Neta method (25) behave betterand these are special cases of our proposed scheme for linearrational function 119867(sdot) The results of Table 3 show that thecomputational order of convergence is in accordance withthe theoretical order of convergenceThe formulae have beenemployed on several other nonlinear equations and resultsare found at par with those presented here A reasonablyclose starting value is necessary for the method to convergeThe condition however practically applies to all iterativemethods for solving nonlinear equations

5 Finding the Basins

In this section we find the basins of attraction of complexroots for the a complex function We consider the simplestcase119891(119911) = (1199112minus1)2 which has roots 119911 = plusmn 1 ofmultiplicity 2Cayley [18] was the first who considered the Newton methodfor the roots of polynomial with iterations over the complexnumbers

We take the initial point as 1199110

isin 119863 where 119863 is arectangular region [minus25 25] times [minus25 25] sub C containingall the roots of 119891(119911) = 0 We consider the stopping criterionfor convergence as 10minus3 up to amaximumof 25 iterationsWetake a grid of 1024 times 1024 points in119863

To generate the pictures we use MATHEMATICA [17]We assign the light to dark colors based on the number ofiterations in which the considered initial point converges to aroot The root 119911 = 1 of 119891(119911) = 0 and the points in the regionconverging to this root are shaded in cyan color and magentacolor is used for the other root 119911 = minus1 in a similar way Ifwe have not obtained the desired tolerance in 25 iterationswe do not continue and we decide that the iterative methodstarting at 119911

0does not converge to any root and such points

are shaded in black color (see [19ndash22])The figures given here clearly show that the mod-

ified Newton method (Figure 1(a)) and Halleyrsquos method(Figure 1(c)) are the best followed by the Osada method(Figure 1(b)) the Chun-Neta method (Figure 1(d)) SBM(Figure 1(f)) and Sharma-Sharma method (Figure 1(e))From these pictures we can guess the behavior and suitabilityof any method depending upon the circumstances Thusconcluding we can say that the linear rational function givesthe best choice of weight function119867(sdot)

6 Conclusions

In this paper a general family of one-point third ordermethods for finding multiple roots of nonlinear equations ispresented This scheme contains some well known methodsand recently developed methods as its particular cases Thetheoretical results have been checked with some numericalexamples comparing ourmethodswith themodifiedNewtonmethod and some third order methodsThe presented basinsof attraction have also demonstrated the strictness of methodfor choice of starting point

8 Advances in Numerical Analysis

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors are grateful to the anonymous reviewers for theirvaluable suggestions and comments on this paper

References

[1] J M Ortega and W C Rheinboldt Iterative Solution of Nonlin-ear Equations in Several Variables Academic Press New YorkNY USA 1970

[2] S C Chapra and R P CanaleNumerical Methods for EngineersMcGraw-Hill Book Company New York NY USA 1988

[3] E Schroder ldquoUeber unendlich vieleAlgorithmen zurAuflosungder GleichungenrdquoMathematische Annalen vol 2 no 2 pp 317ndash365 1870

[4] AM Ostrowski Solution of Equations in Euclidean and BanachSpaces Academic Press New York NY USA 3rd edition 1973

[5] E Hansen and M Patrick ldquoA family of root finding methodsrdquoNumerische Mathematik vol 27 no 3 pp 257ndash269 1977

[6] J F Traub Iterative Methods for the Solution of EquationsPrentice-Hall Englewood Cliffs NJ USA 1964

[7] N Osada ldquoAn optimal multiple root-finding method of orderthreerdquo Journal of Computational and Applied Mathematics vol51 no 1 pp 131ndash133 1994

[8] B Neta ldquoNew third order nonlinear solvers for multiple rootsrdquoApplied Mathematics and Computation vol 202 no 1 pp 162ndash170 2008

[9] C Chun and B Neta ldquoA third-order modification of Newtonrsquosmethod for multiple rootsrdquoAppliedMathematics and Computa-tion vol 211 no 2 pp 474ndash479 2009

[10] C Chun H J Bae and B Neta ldquoNew families of nonlinearthird-order solvers for finding multiple rootsrdquo Computers andMathematics with Applications vol 57 no 9 pp 1574ndash15822009

[11] J Biazar and B Ghanbari ldquoA new third-order family of nonlin-ear solvers formultiple rootsrdquoComputers andMathematics withApplications vol 59 no 10 pp 3315ndash3319 2010

[12] J R Sharma and R Sharma ldquoNew third and fourth ordernonlinear solvers for computing multiple rootsrdquo Applied Math-ematics and Computation vol 217 no 23 pp 9756ndash9764 2011

[13] J R Sharma and R Sharma ldquoModified Chebyshev-Halleytype method and its variants for computing multiple rootsrdquoNumerical Algorithms vol 61 no 4 pp 567ndash578 2012

[14] S Kumar V Kanwar and S Singh ldquoOn some modified familiesof multipoint iterative methods for multiple roots of nonlinearequationsrdquo Applied Mathematics and Computation vol 218 no14 pp 7382ndash7394 2012

[15] W Gautschi Numerical Analysis An Introduction BirkhauserBoston Mass USA 1997

[16] S Weerakoon and T G I Fernando ldquoA variant of Newtonrsquosmethod with accelerated third-order convergencerdquo AppliedMathematics Letters vol 13 no 8 pp 87ndash93 2000

[17] S Wolfram The Mathematica Book Wolfram Media Cham-paign Ill USA 5th edition 2003

[18] A Cayley ldquoThe Newton-Fourier imaginary problemrdquo TheAmerican Journal of Mathematics vol 2 no 1 article 97 1879

[19] M L Sahari and I Djellit ldquoFractal Newton basinsrdquo DiscreteDynamics in Nature and Society vol 2006 Article ID 28756 16pages 2006

[20] J L Varona ldquoGraphic and numerical comparison betweeniterative methodsrdquo The Mathematical Intelligencer vol 24 no1 pp 37ndash46 2002

[21] M Scott B Neta and C Chun ldquoBasin attractors for variousmethodsrdquo Applied Mathematics and Computation vol 218 no6 pp 2584ndash2599 2011

[22] B Neta M Scott and C Chun ldquoBasin attractors for variousmethods for multiple rootsrdquo Applied Mathematics and Compu-tation vol 218 no 9 pp 5043ndash5066 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Advances in Numerical Analysis

minus2 minus1 0 1 2

2

1

0

minus2

minus1

(a) The modified Newton method

minus2 minus1 0 1 2

2

1

0

minus2

minus1

(b) The Osada method

minus2 minus1 0 1 2

2

1

0

minus2

minus1

(c) Halleyrsquos method

minus2 minus1 0 1 2

2

1

0

minus2

minus1

(d) The Chun-Neta method

minus2 minus1 0 1 2

2

1

0

minus2

minus1

(e) The Sharma-Sharma method (120573 = 32)

minus2 minus1 0 1 2

2

1

0

minus2

minus1

(f) SBM (119862 = 12)

Figure 1 Basins of attraction for 119891(119911) = (1199112 minus 1)2 119911 isin 119863 for various methods of the family

Advances in Numerical Analysis 7

So we introduce a new one-parameter third ordermethod as

119909119896+1

= 119909119896minus[[

[

(119898 + 1198982

2+

1198621198982

(119898 minus 1)2)

+ (minus(119898 minus 1)2

2minus

2119862119898

119898 minus 1)

1198911015840(119909119896)2

119891 (119909119896) 11989110158401015840 (119909

119896)

+119862((1198911015840 (119909

119896))2

(119891 (119909119896) 11989110158401015840 (119909

119896)))

2

]]

]

119891 (119909119896)

1198911015840 (119909119896)

(35)

where 119862 isin R We call this method the Sharma-Bahl methoddenoted by SBM

4 Numerical Examples

In this section we employ the presented third order methodSBM (35) for 119862 = 12 to solve some nonlinear equationswhich not only illustrate the methods practically but alsoserve to check the validity of the theoretical results we havederived To check the theoretical order of convergence weobtain the computational order of convergence (120588) using theformula [16]

120588 asympln 1003816100381610038161003816(119909119896+1 minus 120572) (119909

119896minus 120572)

1003816100381610038161003816ln 1003816100381610038161003816(119909119896 minus 120572) (119909

119896minus1minus 120572)

1003816100381610038161003816 (36)

The performance is compared with the methods in ourfamilies including the special ones for some arbitrary chosenfunctions with roots of known multiplicity 119898 The methodscompared include the modified Newton method (MNM) (1)the Osada method (OM) (16) Halleyrsquos method (HM) (18)the Sharma-Sharma method (SSM) (24) for 120573 = 32 and theChun-Neta method (CNM) (25)

The test functions along with root 120572 correct up to 28decimal places and its multiplicity 119898 is displayed in Table 1Table 2 shows the values of initial approximation (119909

0) chosen

from both ends to the root and the values of the error |119890119896| =

|119909119896minus 120572| calculated by costing the same total number of

function evaluations (NFE) for each method Table 3 exhibitsthe computational order of convergence (120588) For numericalillustrations in Table 3 we use fixed stopping criterion 120598 =

05 times 10minus28 The NFE is counted as sum of the number ofevaluations of the function plus the number of evaluations ofthe derivativesWe decide to choose 12 NFE for eachmethodThat means that for MNM the error |119890

119896| is calculated at

the sixth iteration whereas for the remaining methods thisis calculated at the fourth iteration All computations areperformed by MATHEMATICA [17] using 600 significantdigits

It can also be observed from the numerical results ofTable 2 that all the presented methods are well behavedand that for most of the functions we tested the methodintroduced in this paper has similar performance comparedto the other methods as expected from methods of the same

order and same computational efficiency However Halleyrsquosmethod (18) and the Chun-Neta method (25) behave betterand these are special cases of our proposed scheme for linearrational function 119867(sdot) The results of Table 3 show that thecomputational order of convergence is in accordance withthe theoretical order of convergenceThe formulae have beenemployed on several other nonlinear equations and resultsare found at par with those presented here A reasonablyclose starting value is necessary for the method to convergeThe condition however practically applies to all iterativemethods for solving nonlinear equations

5 Finding the Basins

In this section we find the basins of attraction of complexroots for the a complex function We consider the simplestcase119891(119911) = (1199112minus1)2 which has roots 119911 = plusmn 1 ofmultiplicity 2Cayley [18] was the first who considered the Newton methodfor the roots of polynomial with iterations over the complexnumbers

We take the initial point as 1199110

isin 119863 where 119863 is arectangular region [minus25 25] times [minus25 25] sub C containingall the roots of 119891(119911) = 0 We consider the stopping criterionfor convergence as 10minus3 up to amaximumof 25 iterationsWetake a grid of 1024 times 1024 points in119863

To generate the pictures we use MATHEMATICA [17]We assign the light to dark colors based on the number ofiterations in which the considered initial point converges to aroot The root 119911 = 1 of 119891(119911) = 0 and the points in the regionconverging to this root are shaded in cyan color and magentacolor is used for the other root 119911 = minus1 in a similar way Ifwe have not obtained the desired tolerance in 25 iterationswe do not continue and we decide that the iterative methodstarting at 119911

0does not converge to any root and such points

are shaded in black color (see [19ndash22])The figures given here clearly show that the mod-

ified Newton method (Figure 1(a)) and Halleyrsquos method(Figure 1(c)) are the best followed by the Osada method(Figure 1(b)) the Chun-Neta method (Figure 1(d)) SBM(Figure 1(f)) and Sharma-Sharma method (Figure 1(e))From these pictures we can guess the behavior and suitabilityof any method depending upon the circumstances Thusconcluding we can say that the linear rational function givesthe best choice of weight function119867(sdot)

6 Conclusions

In this paper a general family of one-point third ordermethods for finding multiple roots of nonlinear equations ispresented This scheme contains some well known methodsand recently developed methods as its particular cases Thetheoretical results have been checked with some numericalexamples comparing ourmethodswith themodifiedNewtonmethod and some third order methodsThe presented basinsof attraction have also demonstrated the strictness of methodfor choice of starting point

8 Advances in Numerical Analysis

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors are grateful to the anonymous reviewers for theirvaluable suggestions and comments on this paper

References

[1] J M Ortega and W C Rheinboldt Iterative Solution of Nonlin-ear Equations in Several Variables Academic Press New YorkNY USA 1970

[2] S C Chapra and R P CanaleNumerical Methods for EngineersMcGraw-Hill Book Company New York NY USA 1988

[3] E Schroder ldquoUeber unendlich vieleAlgorithmen zurAuflosungder GleichungenrdquoMathematische Annalen vol 2 no 2 pp 317ndash365 1870

[4] AM Ostrowski Solution of Equations in Euclidean and BanachSpaces Academic Press New York NY USA 3rd edition 1973

[5] E Hansen and M Patrick ldquoA family of root finding methodsrdquoNumerische Mathematik vol 27 no 3 pp 257ndash269 1977

[6] J F Traub Iterative Methods for the Solution of EquationsPrentice-Hall Englewood Cliffs NJ USA 1964

[7] N Osada ldquoAn optimal multiple root-finding method of orderthreerdquo Journal of Computational and Applied Mathematics vol51 no 1 pp 131ndash133 1994

[8] B Neta ldquoNew third order nonlinear solvers for multiple rootsrdquoApplied Mathematics and Computation vol 202 no 1 pp 162ndash170 2008

[9] C Chun and B Neta ldquoA third-order modification of Newtonrsquosmethod for multiple rootsrdquoAppliedMathematics and Computa-tion vol 211 no 2 pp 474ndash479 2009

[10] C Chun H J Bae and B Neta ldquoNew families of nonlinearthird-order solvers for finding multiple rootsrdquo Computers andMathematics with Applications vol 57 no 9 pp 1574ndash15822009

[11] J Biazar and B Ghanbari ldquoA new third-order family of nonlin-ear solvers formultiple rootsrdquoComputers andMathematics withApplications vol 59 no 10 pp 3315ndash3319 2010

[12] J R Sharma and R Sharma ldquoNew third and fourth ordernonlinear solvers for computing multiple rootsrdquo Applied Math-ematics and Computation vol 217 no 23 pp 9756ndash9764 2011

[13] J R Sharma and R Sharma ldquoModified Chebyshev-Halleytype method and its variants for computing multiple rootsrdquoNumerical Algorithms vol 61 no 4 pp 567ndash578 2012

[14] S Kumar V Kanwar and S Singh ldquoOn some modified familiesof multipoint iterative methods for multiple roots of nonlinearequationsrdquo Applied Mathematics and Computation vol 218 no14 pp 7382ndash7394 2012

[15] W Gautschi Numerical Analysis An Introduction BirkhauserBoston Mass USA 1997

[16] S Weerakoon and T G I Fernando ldquoA variant of Newtonrsquosmethod with accelerated third-order convergencerdquo AppliedMathematics Letters vol 13 no 8 pp 87ndash93 2000

[17] S Wolfram The Mathematica Book Wolfram Media Cham-paign Ill USA 5th edition 2003

[18] A Cayley ldquoThe Newton-Fourier imaginary problemrdquo TheAmerican Journal of Mathematics vol 2 no 1 article 97 1879

[19] M L Sahari and I Djellit ldquoFractal Newton basinsrdquo DiscreteDynamics in Nature and Society vol 2006 Article ID 28756 16pages 2006

[20] J L Varona ldquoGraphic and numerical comparison betweeniterative methodsrdquo The Mathematical Intelligencer vol 24 no1 pp 37ndash46 2002

[21] M Scott B Neta and C Chun ldquoBasin attractors for variousmethodsrdquo Applied Mathematics and Computation vol 218 no6 pp 2584ndash2599 2011

[22] B Neta M Scott and C Chun ldquoBasin attractors for variousmethods for multiple rootsrdquo Applied Mathematics and Compu-tation vol 218 no 9 pp 5043ndash5066 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Advances in Numerical Analysis 7

So we introduce a new one-parameter third ordermethod as

119909119896+1

= 119909119896minus[[

[

(119898 + 1198982

2+

1198621198982

(119898 minus 1)2)

+ (minus(119898 minus 1)2

2minus

2119862119898

119898 minus 1)

1198911015840(119909119896)2

119891 (119909119896) 11989110158401015840 (119909

119896)

+119862((1198911015840 (119909

119896))2

(119891 (119909119896) 11989110158401015840 (119909

119896)))

2

]]

]

119891 (119909119896)

1198911015840 (119909119896)

(35)

where 119862 isin R We call this method the Sharma-Bahl methoddenoted by SBM

4 Numerical Examples

In this section we employ the presented third order methodSBM (35) for 119862 = 12 to solve some nonlinear equationswhich not only illustrate the methods practically but alsoserve to check the validity of the theoretical results we havederived To check the theoretical order of convergence weobtain the computational order of convergence (120588) using theformula [16]

120588 asympln 1003816100381610038161003816(119909119896+1 minus 120572) (119909

119896minus 120572)

1003816100381610038161003816ln 1003816100381610038161003816(119909119896 minus 120572) (119909

119896minus1minus 120572)

1003816100381610038161003816 (36)

The performance is compared with the methods in ourfamilies including the special ones for some arbitrary chosenfunctions with roots of known multiplicity 119898 The methodscompared include the modified Newton method (MNM) (1)the Osada method (OM) (16) Halleyrsquos method (HM) (18)the Sharma-Sharma method (SSM) (24) for 120573 = 32 and theChun-Neta method (CNM) (25)

The test functions along with root 120572 correct up to 28decimal places and its multiplicity 119898 is displayed in Table 1Table 2 shows the values of initial approximation (119909

0) chosen

from both ends to the root and the values of the error |119890119896| =

|119909119896minus 120572| calculated by costing the same total number of

function evaluations (NFE) for each method Table 3 exhibitsthe computational order of convergence (120588) For numericalillustrations in Table 3 we use fixed stopping criterion 120598 =

05 times 10minus28 The NFE is counted as sum of the number ofevaluations of the function plus the number of evaluations ofthe derivativesWe decide to choose 12 NFE for eachmethodThat means that for MNM the error |119890

119896| is calculated at

the sixth iteration whereas for the remaining methods thisis calculated at the fourth iteration All computations areperformed by MATHEMATICA [17] using 600 significantdigits

It can also be observed from the numerical results ofTable 2 that all the presented methods are well behavedand that for most of the functions we tested the methodintroduced in this paper has similar performance comparedto the other methods as expected from methods of the same

order and same computational efficiency However Halleyrsquosmethod (18) and the Chun-Neta method (25) behave betterand these are special cases of our proposed scheme for linearrational function 119867(sdot) The results of Table 3 show that thecomputational order of convergence is in accordance withthe theoretical order of convergenceThe formulae have beenemployed on several other nonlinear equations and resultsare found at par with those presented here A reasonablyclose starting value is necessary for the method to convergeThe condition however practically applies to all iterativemethods for solving nonlinear equations

5 Finding the Basins

In this section we find the basins of attraction of complexroots for the a complex function We consider the simplestcase119891(119911) = (1199112minus1)2 which has roots 119911 = plusmn 1 ofmultiplicity 2Cayley [18] was the first who considered the Newton methodfor the roots of polynomial with iterations over the complexnumbers

We take the initial point as 1199110

isin 119863 where 119863 is arectangular region [minus25 25] times [minus25 25] sub C containingall the roots of 119891(119911) = 0 We consider the stopping criterionfor convergence as 10minus3 up to amaximumof 25 iterationsWetake a grid of 1024 times 1024 points in119863

To generate the pictures we use MATHEMATICA [17]We assign the light to dark colors based on the number ofiterations in which the considered initial point converges to aroot The root 119911 = 1 of 119891(119911) = 0 and the points in the regionconverging to this root are shaded in cyan color and magentacolor is used for the other root 119911 = minus1 in a similar way Ifwe have not obtained the desired tolerance in 25 iterationswe do not continue and we decide that the iterative methodstarting at 119911

0does not converge to any root and such points

are shaded in black color (see [19ndash22])The figures given here clearly show that the mod-

ified Newton method (Figure 1(a)) and Halleyrsquos method(Figure 1(c)) are the best followed by the Osada method(Figure 1(b)) the Chun-Neta method (Figure 1(d)) SBM(Figure 1(f)) and Sharma-Sharma method (Figure 1(e))From these pictures we can guess the behavior and suitabilityof any method depending upon the circumstances Thusconcluding we can say that the linear rational function givesthe best choice of weight function119867(sdot)

6 Conclusions

In this paper a general family of one-point third ordermethods for finding multiple roots of nonlinear equations ispresented This scheme contains some well known methodsand recently developed methods as its particular cases Thetheoretical results have been checked with some numericalexamples comparing ourmethodswith themodifiedNewtonmethod and some third order methodsThe presented basinsof attraction have also demonstrated the strictness of methodfor choice of starting point

8 Advances in Numerical Analysis

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors are grateful to the anonymous reviewers for theirvaluable suggestions and comments on this paper

References

[1] J M Ortega and W C Rheinboldt Iterative Solution of Nonlin-ear Equations in Several Variables Academic Press New YorkNY USA 1970

[2] S C Chapra and R P CanaleNumerical Methods for EngineersMcGraw-Hill Book Company New York NY USA 1988

[3] E Schroder ldquoUeber unendlich vieleAlgorithmen zurAuflosungder GleichungenrdquoMathematische Annalen vol 2 no 2 pp 317ndash365 1870

[4] AM Ostrowski Solution of Equations in Euclidean and BanachSpaces Academic Press New York NY USA 3rd edition 1973

[5] E Hansen and M Patrick ldquoA family of root finding methodsrdquoNumerische Mathematik vol 27 no 3 pp 257ndash269 1977

[6] J F Traub Iterative Methods for the Solution of EquationsPrentice-Hall Englewood Cliffs NJ USA 1964

[7] N Osada ldquoAn optimal multiple root-finding method of orderthreerdquo Journal of Computational and Applied Mathematics vol51 no 1 pp 131ndash133 1994

[8] B Neta ldquoNew third order nonlinear solvers for multiple rootsrdquoApplied Mathematics and Computation vol 202 no 1 pp 162ndash170 2008

[9] C Chun and B Neta ldquoA third-order modification of Newtonrsquosmethod for multiple rootsrdquoAppliedMathematics and Computa-tion vol 211 no 2 pp 474ndash479 2009

[10] C Chun H J Bae and B Neta ldquoNew families of nonlinearthird-order solvers for finding multiple rootsrdquo Computers andMathematics with Applications vol 57 no 9 pp 1574ndash15822009

[11] J Biazar and B Ghanbari ldquoA new third-order family of nonlin-ear solvers formultiple rootsrdquoComputers andMathematics withApplications vol 59 no 10 pp 3315ndash3319 2010

[12] J R Sharma and R Sharma ldquoNew third and fourth ordernonlinear solvers for computing multiple rootsrdquo Applied Math-ematics and Computation vol 217 no 23 pp 9756ndash9764 2011

[13] J R Sharma and R Sharma ldquoModified Chebyshev-Halleytype method and its variants for computing multiple rootsrdquoNumerical Algorithms vol 61 no 4 pp 567ndash578 2012

[14] S Kumar V Kanwar and S Singh ldquoOn some modified familiesof multipoint iterative methods for multiple roots of nonlinearequationsrdquo Applied Mathematics and Computation vol 218 no14 pp 7382ndash7394 2012

[15] W Gautschi Numerical Analysis An Introduction BirkhauserBoston Mass USA 1997

[16] S Weerakoon and T G I Fernando ldquoA variant of Newtonrsquosmethod with accelerated third-order convergencerdquo AppliedMathematics Letters vol 13 no 8 pp 87ndash93 2000

[17] S Wolfram The Mathematica Book Wolfram Media Cham-paign Ill USA 5th edition 2003

[18] A Cayley ldquoThe Newton-Fourier imaginary problemrdquo TheAmerican Journal of Mathematics vol 2 no 1 article 97 1879

[19] M L Sahari and I Djellit ldquoFractal Newton basinsrdquo DiscreteDynamics in Nature and Society vol 2006 Article ID 28756 16pages 2006

[20] J L Varona ldquoGraphic and numerical comparison betweeniterative methodsrdquo The Mathematical Intelligencer vol 24 no1 pp 37ndash46 2002

[21] M Scott B Neta and C Chun ldquoBasin attractors for variousmethodsrdquo Applied Mathematics and Computation vol 218 no6 pp 2584ndash2599 2011

[22] B Neta M Scott and C Chun ldquoBasin attractors for variousmethods for multiple rootsrdquo Applied Mathematics and Compu-tation vol 218 no 9 pp 5043ndash5066 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Advances in Numerical Analysis

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgment

The authors are grateful to the anonymous reviewers for theirvaluable suggestions and comments on this paper

References

[1] J M Ortega and W C Rheinboldt Iterative Solution of Nonlin-ear Equations in Several Variables Academic Press New YorkNY USA 1970

[2] S C Chapra and R P CanaleNumerical Methods for EngineersMcGraw-Hill Book Company New York NY USA 1988

[3] E Schroder ldquoUeber unendlich vieleAlgorithmen zurAuflosungder GleichungenrdquoMathematische Annalen vol 2 no 2 pp 317ndash365 1870

[4] AM Ostrowski Solution of Equations in Euclidean and BanachSpaces Academic Press New York NY USA 3rd edition 1973

[5] E Hansen and M Patrick ldquoA family of root finding methodsrdquoNumerische Mathematik vol 27 no 3 pp 257ndash269 1977

[6] J F Traub Iterative Methods for the Solution of EquationsPrentice-Hall Englewood Cliffs NJ USA 1964

[7] N Osada ldquoAn optimal multiple root-finding method of orderthreerdquo Journal of Computational and Applied Mathematics vol51 no 1 pp 131ndash133 1994

[8] B Neta ldquoNew third order nonlinear solvers for multiple rootsrdquoApplied Mathematics and Computation vol 202 no 1 pp 162ndash170 2008

[9] C Chun and B Neta ldquoA third-order modification of Newtonrsquosmethod for multiple rootsrdquoAppliedMathematics and Computa-tion vol 211 no 2 pp 474ndash479 2009

[10] C Chun H J Bae and B Neta ldquoNew families of nonlinearthird-order solvers for finding multiple rootsrdquo Computers andMathematics with Applications vol 57 no 9 pp 1574ndash15822009

[11] J Biazar and B Ghanbari ldquoA new third-order family of nonlin-ear solvers formultiple rootsrdquoComputers andMathematics withApplications vol 59 no 10 pp 3315ndash3319 2010

[12] J R Sharma and R Sharma ldquoNew third and fourth ordernonlinear solvers for computing multiple rootsrdquo Applied Math-ematics and Computation vol 217 no 23 pp 9756ndash9764 2011

[13] J R Sharma and R Sharma ldquoModified Chebyshev-Halleytype method and its variants for computing multiple rootsrdquoNumerical Algorithms vol 61 no 4 pp 567ndash578 2012

[14] S Kumar V Kanwar and S Singh ldquoOn some modified familiesof multipoint iterative methods for multiple roots of nonlinearequationsrdquo Applied Mathematics and Computation vol 218 no14 pp 7382ndash7394 2012

[15] W Gautschi Numerical Analysis An Introduction BirkhauserBoston Mass USA 1997

[16] S Weerakoon and T G I Fernando ldquoA variant of Newtonrsquosmethod with accelerated third-order convergencerdquo AppliedMathematics Letters vol 13 no 8 pp 87ndash93 2000

[17] S Wolfram The Mathematica Book Wolfram Media Cham-paign Ill USA 5th edition 2003

[18] A Cayley ldquoThe Newton-Fourier imaginary problemrdquo TheAmerican Journal of Mathematics vol 2 no 1 article 97 1879

[19] M L Sahari and I Djellit ldquoFractal Newton basinsrdquo DiscreteDynamics in Nature and Society vol 2006 Article ID 28756 16pages 2006

[20] J L Varona ldquoGraphic and numerical comparison betweeniterative methodsrdquo The Mathematical Intelligencer vol 24 no1 pp 37ndash46 2002

[21] M Scott B Neta and C Chun ldquoBasin attractors for variousmethodsrdquo Applied Mathematics and Computation vol 218 no6 pp 2584ndash2599 2011

[22] B Neta M Scott and C Chun ldquoBasin attractors for variousmethods for multiple rootsrdquo Applied Mathematics and Compu-tation vol 218 no 9 pp 5043ndash5066 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of