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Hindawi Publishing CorporationJournal of MathematicsVolume 2013 Article ID 404635 3 pageshttpdxdoiorg1011552013404635
Research ArticleIntroduction to Higher-Order Iterative Methods for FindingMultiple Roots of Nonlinear Equations
Rajinder Thukral
Padeacute Research Centre 39 Deanswood Hill Leeds West Yorkshire LS17 5JS UK
Correspondence should be addressed to Rajinder ukral rthukralhotmailcouk
Received 28 September 2012 Accepted 19 November 2012
Academic Editor Petr Ekel
Copyright copy 2013 Rajinder ukral is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
e introduce two higher-order iterative methods for nding multiple eros of nonlinear equations Per iteration the new methodsrequire three evaluations of function and one of its rst derivatives It is proved that the two methods have a convergence of orderve or six
1 Introduction
Solving nonlinear equations is one of the most importantproblems in numerical analysis In this paper we consideriterative methods to nd a multiple root 120572120572 of multiplicity 119898119898that is 119891119891(119895119895119895(120572120572119895 120572 120572120572 119895119895 120572 120572120572 120572120572120572 120572119898119898 120572 120572 and 119891119891(119898119898119895(120572120572119895 120572 120572 of anonlinear equation
119891119891 (119909119909119895 120572 120572120572 (1)
where 119891119891 119891 119891119891 119891 119891 119891 119891 is a scalar function on an openinterval 119891119891 and it is sufficiently smooth in a neighbourhood of120572120572 In recent years somemodications of the Newtonmethodfor multiple roots have been proposed and analysed [1ndash8]However there are not many methods known to handle thecase of multiple roots Hence we present two higher-ordermethods for nding multiple eros of a nonlinear equationand only use four evaluations of the function per iterationIn addition the new methods have a better efficiency indexthan the third- and fourth-order methods given in [1ndash3 7]in view of this fact the new methods are signicantly betterwhen compared with the established methods
e well-known Newtons method for nding multipleroots is given by
119909119909119899119899119899120572 120572 119909119909119899119899 120572 119898119898119891119891 1007649100764911990911990911989911989910076651007665
119891119891prime 1007649100764911990911990911989911989910076651007665120572 (2)
which converges quadratically [4] For the purpose of thispaper we use (2) to construct new higher-order methods
2 Development of theMethods andConvergence Analysis
In this section we dene new higher-ordermethods In orderto establish the order of convergence of the new methods westate the three essential denitions
Denition 1 Let 119891119891(119909119909119895 be a real function with a simple root120572120572 and let 119909119909119899119899 be a sequence of real numbers that convergetowards 120572120572 e order of convergence 119901119901 is given by
lim119899119899119891119899
119909119909119899119899119899120572 120572 12057212057210076491007649119909119909119899119899 120572 12057212057210076651007665119901119901
120572 120577120577 120572 120572120572 (3)
where 120577120577 is the asymptotic error constant and 119901119901 119901 119891119899
Denition Let 119896119896 be the number of function evaluationsof the new method e efficiency of the new method ismeasured by the concept of efficiency index [9 10] anddened as
119901119901120572119896119896120572 (4)
where 119901119901 is the order of convergence of the method
Denition 3 Kung-Traub conjectured that the multipointiterative methods without memory requiring 119896119896 119899 120572 functionevaluations per iteration have the order of convergence of atmost
119901119901 120572 119901119896119896 (5)
2 Journal of Mathematics
Multipoint methods that satisfy the Kung-Traub conjectureare usually called optimal methods [11]
Recently ukral [8] presented a fourth-order iterativemethod given by
119910119910119899119899 = 119909119909119899119899 minus 119898119898119891119891 1007649100764911990911990911989911989910076651007665
119891119891prime 1007649100764911990911990911989911989910076651007665 (6)
119909119909119899119899119899119899 = 119909119909119899119899 minus 11989811989810076551007655310055761005576119894119894=11989911989411989410076541007654
119891119891 1007649100764911991011991011989911989910076651007665119891119891 1007649100764911990911990911989911989910076651007665
10076701007670119894119894119894119898119898
1007671100767110076551007655119891119891 1007649100764911990911990911989911989910076651007665
119891119891prime 100764910076491199091199091198991198991007665100766510076711007671 (7)
In fact the new methods presented in this paper are theextension of the above scheme To develop the higher-ordermethods we use the above two steps and introduce the thirdstep with a new parameter First we dene the sixth-ordermethod and then followed by the h-order method
21 New Sixth-Order Method e new sixth-order methodfor nding multiple root of a nonlinear equation is expressedas
119910119910119899119899 = 119909119909119899119899 minus 119898119898119891119891 1007649100764911990911990911989911989910076651007665
119891119891prime 1007649100764911990911990911989911989910076651007665 (8)
119911119911119899119899 = 119909119909119899119899 minus 11989811989810076551007655310055761005576119894119894=11989911989411989410076541007654
119891119891 1007649100764911991011991011989911989910076651007665119891119891 1007649100764911990911990911989911989910076651007665
10076701007670119894119894119894119898119898
1007671100767110076551007655119891119891 1007649100764911990911990911989911989910076651007665
119891119891prime 100764910076491199091199091198991198991007665100766510076711007671 (9)
119909119909119899119899119899119899 = 119911119911119899119899 minus 11989811989810076551007655310055761005576119894119894=11989911989411989410076541007654
119891119891 1007649100764911991011991011989911989910076651007665119891119891 1007649100764911990911990911989911989910076651007665
10076701007670119894119894119894119898119898
100767110076712
10076541007654119891119891 1007649100764911991111991111989911989910076651007665119891119891 1007649100764911990911990911989911989910076651007665
10076701007670119898119898minus119899
times 10076551007655119891119891 1007649100764911990911990911989911989910076651007665
119891119891prime 100764910076491199091199091198991198991007665100766510076711007671
(10)
where 119899119899 119899 119899 1199091199090 is the initial value provided that thedenominators of (6) and (7) are not equal to zero
eorem 4 Let 120572120572 119899 120572120572 be a multiple root of multiplicity 119898119898 ofa sufficiently differentiable function 119891119891 119891 120572120572 119891 119891 119891 119891 for anopen interval 120572120572 If 1199091199090 is sufficiently close to 120572120572 then the order ofconverence of the new method dened by (10) is six
Proof Let 120572120572 be a multiple root of multicity119898119898 of a sufficientlysmooth function119891119891119891119909119909119891 119890119890 = 119909119909 minus 120572120572 and 119890119890 = 119910119910 minus 120572120572 where 119910119910 isdened in (6)
Using the Taylor expansion of 119891119891119891119909119909119891 and119891119891prime119891119909119909119891 about 120572120572 wehave
119891119891 1007649100764911990911990911989911989910076651007665 = 10076551007655119891119891119891119898119898119891 119891120572120572119891119898119898119898
10076711007671 119890119890119898119898119899119899 10077141007714119899 119899 119888119888119899119890119890119899119899 119899 11988811988821198901198902119899119899 119899 1198881198883119890119890
3119899119899 119899 ⋯10077301007730
(11)
119891119891prime 1007649100764911990911990911989911989910076651007665 = 10076551007655119891119891119891119898119898119891 119891120572120572119891119891119898119898 minus 119899119891 119898
10076711007671 119890119890119898119898minus119899119899119899
times 10077161007716119899 119899 10076521007652119898119898 119899 119899119898119898
10076681007668 119888119888119899119890119890119899119899 119899 10076521007652119898119898 119899 2119898119898
10076681007668 11988811988821198901198902119899119899 119899 ⋯10077321007732
(12)
where 119899119899 119899 119899 and
119888119888119896119896 =119898119898119898 119891119891119891119898119898119899119896119896119891 119891120572120572119891
119891119898119898 119899 119896119896119891 119898 119891119891119891119898119898119891 119891120572120572119891 (13)
From (6) we have
119910119910119899119899 = 119890119890119899119899 minus 119898119898119891119891 1007649100764911990911990911989911989910076651007665
119891119891prime 1007649100764911990911990911989911989910076651007665
= 119890119890119899119899 minus 119890119890119899119899 10077181007718119899 minus119888119888119899119898119898119890119890119899119899 119899
119891119898119898 119899 119899119891 1198881198882119899 minus 211989811989811988811988821198981198982 1198901198902119899119899 119899 ⋯10077341007734
(14)
e expansion of 119891119891119891119910119910119899119899119891 and about 120572120572 is given by
119891119891 1007649100764911991011991011989911989910076651007665 = 10076551007655119891119891119891119898119898119891 119891120572120572119891
1198981198981198981007671100767110065951006595119890119890119898119898119899119899 10077141007714119899 119899 11988811988811989910065941006594119890119890119899119899 119899 119888119888210065941006594119890119890
2119899119899 119899 119888119888310065941006594119890119890
3119899119899 119899 ⋯10077301007730
(15)
By using (11) and (15) we have
10076541007654119891119891 1007649100764911991011991011989911989910076651007665119891119891 1007649100764911990911990911989911989910076651007665
10076701007670119898119898minus119899
= 119890119890119899119899 1007652100765211988811988811989911989811989810076681007668
times 10077141007714119899 119899 1007650100765021198981198981198881198882 minus 119891119898119898 119899 2119891 119888119888211989910076661007666119898119898minus1198991198901198902119899119899
119899 2minus119899 100765010076501007650100765021198981198982 119899 7119898119898 119899 710076661007666 1198881198883119899 minus 2119898119898 1198913119898119898 119899 7119891
times1198881198881198991198881198882 119899 61198981198982119888119888310076661007666119898119898minus31198901198903119899119899 119899 ⋯10077301007730
(16)
Since from (7) we have
119911119911119899119899 = 119890119890119899119899 minus 11989811989810076551007655310055761005576119894119894=11989911989411989410076541007654
119891119891 1007649100764911991011991011989911989910076651007665119891119891 1007649100764911990911990911989911989910076651007665
10076701007670119894119894119894119898119898
1007671100767110076551007655119891119891 1007649100764911990911990911989911989910076651007665
119891119891prime 100764910076491199091199091198991198991007665100766510076711007671
119891119891 1007649100764911991111991111989911989910076651007665 = 10076551007655119891119891119891119898119898119891 119891120572120572119891119898119898119898
10076711007671 119911119911119898119898119899119899 10077131007713119899 119899 119888119888119899119911119911119899119899 119899 ⋯10077291007729
(17)
As before we expand 119891119891119891119911119911119899119899119891 119891119891119891119891119911119911119899119899119891119891119891119891119909119909119899119899119891minus119899119891
119898119898minus119899
and substitutethe appropriate expressions in (10) Aer simplication weobtain the error equation
119890119890119899119899119899119899 = 2minus2119898119898minus5119888119888119899 100765010076501198981198981198881198882119899 119899 31198881198882119899 minus 2119898119898119888119888210076661007666
times 100765010076501198981198981198881198882119899 119899 1198881198882119899 minus 2119898119898119888119888210076661007666 1198901198906119899119899 119899 ⋯
(18)
e error equation (18) establishes the sixth-order conver-gence of the new method dened by (10)
Journal of Mathematics 3
22NewFih-OrderMethod enewh-ordermethod fornding multiple root of a nonlinear equation is expressed as
119910119910119899119899 = 119909119909119899119899 minus 119898119898119891119891 1007649100764911990911990911989911989910076651007665
119891119891prime 1007649100764911990911990911989911989910076651007665 (19)
119911119911119899119899 = 119909119909119899119899 minus 11989811989810076551007655310055761005576119894119894=11989411989411989410076541007654119891119891 1007649100764911991011991011989911989910076651007665119891119891 1007649100764911990911990911989911989910076651007665
10076701007670119894119894119894119898119898
1007671100767110076551007655119891119891 1007649100764911990911990911989911989910076651007665
119891119891prime 100764910076491199091199091198991198991007665100766510076711007671 (20)
119909119909119899119899119899119894 = 119911119911119899119899 minus 11989811989810076541007654119891119891 1007649100764911991111991111989911989910076651007665119891119891 1007649100764911990911990911989911989910076651007665
10076701007670119898119898minus119894
10076551007655119891119891 1007649100764911990911990911989911989910076651007665
119891119891prime 100764910076491199091199091198991198991007665100766510076711007671 (21)
where 119899119899 119899 119899 1199091199090 is the initial value provided that thedenominators of (19)ndash(21) are not equal to zero
eorem 5 Let 120572120572 119899 120572120572 be a multiple root of multiplicity 119898119898 ofa sufficiently differentiable function 119891119891 119891 120572120572 119891 119891 119891 119891 for anopen interval 120572120572 If 1199091199090 is sufficiently close to 120572120572 then the order ofconvergence of the new method dened by (21) is ve
Proof Substituting appropriate expressions in (21) and aersimplication we obtain the error equation
119890119890119899119899119899119894 = 119898119898minus41198881198882119894 100765010076501198981198981198881198882119894 119899 3119888119888
2119894 minus 2119898119898119888119888210076661007666 119890119890
5119899119899 119899 ⋯ (22)
e error equation (22) establishes the h-order conver-gence of the new method dened by (21)
3 Conclusion
In this paper we have introduced two new higher-ordermethods for solving nonlinear equations with multiple rootsConvergence analysis proves that the new methods preservetheir order of convergencee h-order method presentedin this paper was actually the rst improvement of thefourth-order method recently introduced in [8] and furtherimprovement was made to achieve the sixth-order methodSimply by introducing new parameters we have achievedhigher order of convergence e purpose of this paper isto introduce higher order of convergence methods Since wehave veried the order of convergence of the new methodswe have not illustrated the numerical performance of thesemethods Finally we conjecture that these new higher-orderiterative methods may be extended to an optimal eighth-order of convergence and may be considered a very goodalternative to the classical methods given in [1ndash8]
Acknowledgment
e author is grateful to the reviewer for his helpful com-ments on this paper
References
[1] S Kumar Kanwar and S Singh ldquoOn some modied familiesof multipoint iterative methods for multiple roots of nonlinearequationsrdquoAppliedMathematics and Computation vol 218 no14 pp 7382ndash7394 2012
[2] S G Li L Z Cheng and BNeta ldquoSome fourth-order nonlinearsolvers with closed formulae for multiple rootsrdquo Computers andMathematics with Applications vol 59 no 1 pp 126ndash135 2010
[3] J Peng and Z Zeng ldquoAn approach nding multiple roots ofnonlinear equations or polynomialsrdquo Advanced Research onElectronic Commerce Web Application and Communicationvol 149 pp 83ndash87 2012
[4] Schrder ldquoUeber unendlich viele Algorithmen zur Au-sung der Gleichungenrdquo Mathematische Annalen vol 2 no 2pp 317ndash365 1870
[5] J R Sharma and R Sharma ldquoModied Jarratt method for com-puting multiple rootsrdquo Applied Mathematics and Computationvol 217 no 2 pp 878ndash881 2010
[6] L Shengguo L Xiangke and C Lizhi ldquoA new fourth-orderiterative method for nding multiple roots of nonlinear equa-tionsrdquo Applied Mathematics and Computation vol 215 no 3pp 1288ndash1292 2009
[7] R ukral ldquoA new third-order iterative method for solvingnonlinear equations with multiple rootsrdquo International Journalof Mathematics and Computation vol 6 pp 61ndash68 2010
[8] Rajinder ukral ldquoNew fourth-order nonlinear solver formultiple rootsrdquo Advances in Numerical Analysis Submitted inAdvances in Numerical Analysis
[9] W Gautschi Numerical Analysis An Introduction BirkhaumluserBoston Mass USA 1997
[10] J F Traub Iterative Methods for Solution of Equations ChelseaPublishing New York NY USA 1977
[11] H T Kung and J F Traub ldquoOptimal order of one-point andmultipoint iterationrdquo Journal of the Association for ComputingMachinery vol 21 pp 643ndash651 1974
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2 Journal of Mathematics
Multipoint methods that satisfy the Kung-Traub conjectureare usually called optimal methods [11]
Recently ukral [8] presented a fourth-order iterativemethod given by
119910119910119899119899 = 119909119909119899119899 minus 119898119898119891119891 1007649100764911990911990911989911989910076651007665
119891119891prime 1007649100764911990911990911989911989910076651007665 (6)
119909119909119899119899119899119899 = 119909119909119899119899 minus 11989811989810076551007655310055761005576119894119894=11989911989411989410076541007654
119891119891 1007649100764911991011991011989911989910076651007665119891119891 1007649100764911990911990911989911989910076651007665
10076701007670119894119894119894119898119898
1007671100767110076551007655119891119891 1007649100764911990911990911989911989910076651007665
119891119891prime 100764910076491199091199091198991198991007665100766510076711007671 (7)
In fact the new methods presented in this paper are theextension of the above scheme To develop the higher-ordermethods we use the above two steps and introduce the thirdstep with a new parameter First we dene the sixth-ordermethod and then followed by the h-order method
21 New Sixth-Order Method e new sixth-order methodfor nding multiple root of a nonlinear equation is expressedas
119910119910119899119899 = 119909119909119899119899 minus 119898119898119891119891 1007649100764911990911990911989911989910076651007665
119891119891prime 1007649100764911990911990911989911989910076651007665 (8)
119911119911119899119899 = 119909119909119899119899 minus 11989811989810076551007655310055761005576119894119894=11989911989411989410076541007654
119891119891 1007649100764911991011991011989911989910076651007665119891119891 1007649100764911990911990911989911989910076651007665
10076701007670119894119894119894119898119898
1007671100767110076551007655119891119891 1007649100764911990911990911989911989910076651007665
119891119891prime 100764910076491199091199091198991198991007665100766510076711007671 (9)
119909119909119899119899119899119899 = 119911119911119899119899 minus 11989811989810076551007655310055761005576119894119894=11989911989411989410076541007654
119891119891 1007649100764911991011991011989911989910076651007665119891119891 1007649100764911990911990911989911989910076651007665
10076701007670119894119894119894119898119898
100767110076712
10076541007654119891119891 1007649100764911991111991111989911989910076651007665119891119891 1007649100764911990911990911989911989910076651007665
10076701007670119898119898minus119899
times 10076551007655119891119891 1007649100764911990911990911989911989910076651007665
119891119891prime 100764910076491199091199091198991198991007665100766510076711007671
(10)
where 119899119899 119899 119899 1199091199090 is the initial value provided that thedenominators of (6) and (7) are not equal to zero
eorem 4 Let 120572120572 119899 120572120572 be a multiple root of multiplicity 119898119898 ofa sufficiently differentiable function 119891119891 119891 120572120572 119891 119891 119891 119891 for anopen interval 120572120572 If 1199091199090 is sufficiently close to 120572120572 then the order ofconverence of the new method dened by (10) is six
Proof Let 120572120572 be a multiple root of multicity119898119898 of a sufficientlysmooth function119891119891119891119909119909119891 119890119890 = 119909119909 minus 120572120572 and 119890119890 = 119910119910 minus 120572120572 where 119910119910 isdened in (6)
Using the Taylor expansion of 119891119891119891119909119909119891 and119891119891prime119891119909119909119891 about 120572120572 wehave
119891119891 1007649100764911990911990911989911989910076651007665 = 10076551007655119891119891119891119898119898119891 119891120572120572119891119898119898119898
10076711007671 119890119890119898119898119899119899 10077141007714119899 119899 119888119888119899119890119890119899119899 119899 11988811988821198901198902119899119899 119899 1198881198883119890119890
3119899119899 119899 ⋯10077301007730
(11)
119891119891prime 1007649100764911990911990911989911989910076651007665 = 10076551007655119891119891119891119898119898119891 119891120572120572119891119891119898119898 minus 119899119891 119898
10076711007671 119890119890119898119898minus119899119899119899
times 10077161007716119899 119899 10076521007652119898119898 119899 119899119898119898
10076681007668 119888119888119899119890119890119899119899 119899 10076521007652119898119898 119899 2119898119898
10076681007668 11988811988821198901198902119899119899 119899 ⋯10077321007732
(12)
where 119899119899 119899 119899 and
119888119888119896119896 =119898119898119898 119891119891119891119898119898119899119896119896119891 119891120572120572119891
119891119898119898 119899 119896119896119891 119898 119891119891119891119898119898119891 119891120572120572119891 (13)
From (6) we have
119910119910119899119899 = 119890119890119899119899 minus 119898119898119891119891 1007649100764911990911990911989911989910076651007665
119891119891prime 1007649100764911990911990911989911989910076651007665
= 119890119890119899119899 minus 119890119890119899119899 10077181007718119899 minus119888119888119899119898119898119890119890119899119899 119899
119891119898119898 119899 119899119891 1198881198882119899 minus 211989811989811988811988821198981198982 1198901198902119899119899 119899 ⋯10077341007734
(14)
e expansion of 119891119891119891119910119910119899119899119891 and about 120572120572 is given by
119891119891 1007649100764911991011991011989911989910076651007665 = 10076551007655119891119891119891119898119898119891 119891120572120572119891
1198981198981198981007671100767110065951006595119890119890119898119898119899119899 10077141007714119899 119899 11988811988811989910065941006594119890119890119899119899 119899 119888119888210065941006594119890119890
2119899119899 119899 119888119888310065941006594119890119890
3119899119899 119899 ⋯10077301007730
(15)
By using (11) and (15) we have
10076541007654119891119891 1007649100764911991011991011989911989910076651007665119891119891 1007649100764911990911990911989911989910076651007665
10076701007670119898119898minus119899
= 119890119890119899119899 1007652100765211988811988811989911989811989810076681007668
times 10077141007714119899 119899 1007650100765021198981198981198881198882 minus 119891119898119898 119899 2119891 119888119888211989910076661007666119898119898minus1198991198901198902119899119899
119899 2minus119899 100765010076501007650100765021198981198982 119899 7119898119898 119899 710076661007666 1198881198883119899 minus 2119898119898 1198913119898119898 119899 7119891
times1198881198881198991198881198882 119899 61198981198982119888119888310076661007666119898119898minus31198901198903119899119899 119899 ⋯10077301007730
(16)
Since from (7) we have
119911119911119899119899 = 119890119890119899119899 minus 11989811989810076551007655310055761005576119894119894=11989911989411989410076541007654
119891119891 1007649100764911991011991011989911989910076651007665119891119891 1007649100764911990911990911989911989910076651007665
10076701007670119894119894119894119898119898
1007671100767110076551007655119891119891 1007649100764911990911990911989911989910076651007665
119891119891prime 100764910076491199091199091198991198991007665100766510076711007671
119891119891 1007649100764911991111991111989911989910076651007665 = 10076551007655119891119891119891119898119898119891 119891120572120572119891119898119898119898
10076711007671 119911119911119898119898119899119899 10077131007713119899 119899 119888119888119899119911119911119899119899 119899 ⋯10077291007729
(17)
As before we expand 119891119891119891119911119911119899119899119891 119891119891119891119891119911119911119899119899119891119891119891119891119909119909119899119899119891minus119899119891
119898119898minus119899
and substitutethe appropriate expressions in (10) Aer simplication weobtain the error equation
119890119890119899119899119899119899 = 2minus2119898119898minus5119888119888119899 100765010076501198981198981198881198882119899 119899 31198881198882119899 minus 2119898119898119888119888210076661007666
times 100765010076501198981198981198881198882119899 119899 1198881198882119899 minus 2119898119898119888119888210076661007666 1198901198906119899119899 119899 ⋯
(18)
e error equation (18) establishes the sixth-order conver-gence of the new method dened by (10)
Journal of Mathematics 3
22NewFih-OrderMethod enewh-ordermethod fornding multiple root of a nonlinear equation is expressed as
119910119910119899119899 = 119909119909119899119899 minus 119898119898119891119891 1007649100764911990911990911989911989910076651007665
119891119891prime 1007649100764911990911990911989911989910076651007665 (19)
119911119911119899119899 = 119909119909119899119899 minus 11989811989810076551007655310055761005576119894119894=11989411989411989410076541007654119891119891 1007649100764911991011991011989911989910076651007665119891119891 1007649100764911990911990911989911989910076651007665
10076701007670119894119894119894119898119898
1007671100767110076551007655119891119891 1007649100764911990911990911989911989910076651007665
119891119891prime 100764910076491199091199091198991198991007665100766510076711007671 (20)
119909119909119899119899119899119894 = 119911119911119899119899 minus 11989811989810076541007654119891119891 1007649100764911991111991111989911989910076651007665119891119891 1007649100764911990911990911989911989910076651007665
10076701007670119898119898minus119894
10076551007655119891119891 1007649100764911990911990911989911989910076651007665
119891119891prime 100764910076491199091199091198991198991007665100766510076711007671 (21)
where 119899119899 119899 119899 1199091199090 is the initial value provided that thedenominators of (19)ndash(21) are not equal to zero
eorem 5 Let 120572120572 119899 120572120572 be a multiple root of multiplicity 119898119898 ofa sufficiently differentiable function 119891119891 119891 120572120572 119891 119891 119891 119891 for anopen interval 120572120572 If 1199091199090 is sufficiently close to 120572120572 then the order ofconvergence of the new method dened by (21) is ve
Proof Substituting appropriate expressions in (21) and aersimplication we obtain the error equation
119890119890119899119899119899119894 = 119898119898minus41198881198882119894 100765010076501198981198981198881198882119894 119899 3119888119888
2119894 minus 2119898119898119888119888210076661007666 119890119890
5119899119899 119899 ⋯ (22)
e error equation (22) establishes the h-order conver-gence of the new method dened by (21)
3 Conclusion
In this paper we have introduced two new higher-ordermethods for solving nonlinear equations with multiple rootsConvergence analysis proves that the new methods preservetheir order of convergencee h-order method presentedin this paper was actually the rst improvement of thefourth-order method recently introduced in [8] and furtherimprovement was made to achieve the sixth-order methodSimply by introducing new parameters we have achievedhigher order of convergence e purpose of this paper isto introduce higher order of convergence methods Since wehave veried the order of convergence of the new methodswe have not illustrated the numerical performance of thesemethods Finally we conjecture that these new higher-orderiterative methods may be extended to an optimal eighth-order of convergence and may be considered a very goodalternative to the classical methods given in [1ndash8]
Acknowledgment
e author is grateful to the reviewer for his helpful com-ments on this paper
References
[1] S Kumar Kanwar and S Singh ldquoOn some modied familiesof multipoint iterative methods for multiple roots of nonlinearequationsrdquoAppliedMathematics and Computation vol 218 no14 pp 7382ndash7394 2012
[2] S G Li L Z Cheng and BNeta ldquoSome fourth-order nonlinearsolvers with closed formulae for multiple rootsrdquo Computers andMathematics with Applications vol 59 no 1 pp 126ndash135 2010
[3] J Peng and Z Zeng ldquoAn approach nding multiple roots ofnonlinear equations or polynomialsrdquo Advanced Research onElectronic Commerce Web Application and Communicationvol 149 pp 83ndash87 2012
[4] Schrder ldquoUeber unendlich viele Algorithmen zur Au-sung der Gleichungenrdquo Mathematische Annalen vol 2 no 2pp 317ndash365 1870
[5] J R Sharma and R Sharma ldquoModied Jarratt method for com-puting multiple rootsrdquo Applied Mathematics and Computationvol 217 no 2 pp 878ndash881 2010
[6] L Shengguo L Xiangke and C Lizhi ldquoA new fourth-orderiterative method for nding multiple roots of nonlinear equa-tionsrdquo Applied Mathematics and Computation vol 215 no 3pp 1288ndash1292 2009
[7] R ukral ldquoA new third-order iterative method for solvingnonlinear equations with multiple rootsrdquo International Journalof Mathematics and Computation vol 6 pp 61ndash68 2010
[8] Rajinder ukral ldquoNew fourth-order nonlinear solver formultiple rootsrdquo Advances in Numerical Analysis Submitted inAdvances in Numerical Analysis
[9] W Gautschi Numerical Analysis An Introduction BirkhaumluserBoston Mass USA 1997
[10] J F Traub Iterative Methods for Solution of Equations ChelseaPublishing New York NY USA 1977
[11] H T Kung and J F Traub ldquoOptimal order of one-point andmultipoint iterationrdquo Journal of the Association for ComputingMachinery vol 21 pp 643ndash651 1974
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
Journal of Mathematics 3
22NewFih-OrderMethod enewh-ordermethod fornding multiple root of a nonlinear equation is expressed as
119910119910119899119899 = 119909119909119899119899 minus 119898119898119891119891 1007649100764911990911990911989911989910076651007665
119891119891prime 1007649100764911990911990911989911989910076651007665 (19)
119911119911119899119899 = 119909119909119899119899 minus 11989811989810076551007655310055761005576119894119894=11989411989411989410076541007654119891119891 1007649100764911991011991011989911989910076651007665119891119891 1007649100764911990911990911989911989910076651007665
10076701007670119894119894119894119898119898
1007671100767110076551007655119891119891 1007649100764911990911990911989911989910076651007665
119891119891prime 100764910076491199091199091198991198991007665100766510076711007671 (20)
119909119909119899119899119899119894 = 119911119911119899119899 minus 11989811989810076541007654119891119891 1007649100764911991111991111989911989910076651007665119891119891 1007649100764911990911990911989911989910076651007665
10076701007670119898119898minus119894
10076551007655119891119891 1007649100764911990911990911989911989910076651007665
119891119891prime 100764910076491199091199091198991198991007665100766510076711007671 (21)
where 119899119899 119899 119899 1199091199090 is the initial value provided that thedenominators of (19)ndash(21) are not equal to zero
eorem 5 Let 120572120572 119899 120572120572 be a multiple root of multiplicity 119898119898 ofa sufficiently differentiable function 119891119891 119891 120572120572 119891 119891 119891 119891 for anopen interval 120572120572 If 1199091199090 is sufficiently close to 120572120572 then the order ofconvergence of the new method dened by (21) is ve
Proof Substituting appropriate expressions in (21) and aersimplication we obtain the error equation
119890119890119899119899119899119894 = 119898119898minus41198881198882119894 100765010076501198981198981198881198882119894 119899 3119888119888
2119894 minus 2119898119898119888119888210076661007666 119890119890
5119899119899 119899 ⋯ (22)
e error equation (22) establishes the h-order conver-gence of the new method dened by (21)
3 Conclusion
In this paper we have introduced two new higher-ordermethods for solving nonlinear equations with multiple rootsConvergence analysis proves that the new methods preservetheir order of convergencee h-order method presentedin this paper was actually the rst improvement of thefourth-order method recently introduced in [8] and furtherimprovement was made to achieve the sixth-order methodSimply by introducing new parameters we have achievedhigher order of convergence e purpose of this paper isto introduce higher order of convergence methods Since wehave veried the order of convergence of the new methodswe have not illustrated the numerical performance of thesemethods Finally we conjecture that these new higher-orderiterative methods may be extended to an optimal eighth-order of convergence and may be considered a very goodalternative to the classical methods given in [1ndash8]
Acknowledgment
e author is grateful to the reviewer for his helpful com-ments on this paper
References
[1] S Kumar Kanwar and S Singh ldquoOn some modied familiesof multipoint iterative methods for multiple roots of nonlinearequationsrdquoAppliedMathematics and Computation vol 218 no14 pp 7382ndash7394 2012
[2] S G Li L Z Cheng and BNeta ldquoSome fourth-order nonlinearsolvers with closed formulae for multiple rootsrdquo Computers andMathematics with Applications vol 59 no 1 pp 126ndash135 2010
[3] J Peng and Z Zeng ldquoAn approach nding multiple roots ofnonlinear equations or polynomialsrdquo Advanced Research onElectronic Commerce Web Application and Communicationvol 149 pp 83ndash87 2012
[4] Schrder ldquoUeber unendlich viele Algorithmen zur Au-sung der Gleichungenrdquo Mathematische Annalen vol 2 no 2pp 317ndash365 1870
[5] J R Sharma and R Sharma ldquoModied Jarratt method for com-puting multiple rootsrdquo Applied Mathematics and Computationvol 217 no 2 pp 878ndash881 2010
[6] L Shengguo L Xiangke and C Lizhi ldquoA new fourth-orderiterative method for nding multiple roots of nonlinear equa-tionsrdquo Applied Mathematics and Computation vol 215 no 3pp 1288ndash1292 2009
[7] R ukral ldquoA new third-order iterative method for solvingnonlinear equations with multiple rootsrdquo International Journalof Mathematics and Computation vol 6 pp 61ndash68 2010
[8] Rajinder ukral ldquoNew fourth-order nonlinear solver formultiple rootsrdquo Advances in Numerical Analysis Submitted inAdvances in Numerical Analysis
[9] W Gautschi Numerical Analysis An Introduction BirkhaumluserBoston Mass USA 1997
[10] J F Traub Iterative Methods for Solution of Equations ChelseaPublishing New York NY USA 1977
[11] H T Kung and J F Traub ldquoOptimal order of one-point andmultipoint iterationrdquo Journal of the Association for ComputingMachinery vol 21 pp 643ndash651 1974
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