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Hindawi Publishing CorporationAdvances in Numerical AnalysisVolume 2013 Article ID 957496 8 pageshttpdxdoiorg1011552013957496

Research ArticleThree New Optimal Fourth-Order Iterative Methods toSolve Nonlinear Equations

Gustavo Fernaacutendez-Torres1 and Juan Vaacutesquez-Aquino2

1 Petroleum Engineering Department UNISTMO 70760 Tehuantepec OAX Mexico2 Applied Mathematics Department UNISTMO 70760 Tehuantepec OAX Mexico

Correspondence should be addressed to Gustavo Fernandez-Torres gusfersandungaunistmoedumx

Received 21 November 2012 Revised 24 January 2013 Accepted 2 February 2013

Academic Editor Michael Ng

Copyright copy 2013 G Fernandez-Torres and J Vasquez-Aquino This is an open access article distributed under the CreativeCommons Attribution License which permits unrestricted use distribution and reproduction in any medium provided theoriginal work is properly cited

We present new modifications to Newtonrsquos method for solving nonlinear equations The analysis of convergence shows that thesemethods have fourth-order convergence Each of the three methods uses three functional evaluations Thus according to Kung-Traubrsquos conjecture these are optimal methods With the previous ideas we extend the analysis to functions with multiple rootsSeveral numerical examples are given to illustrate that the presented methods have better performance compared with Newtonrsquosclassical method and other methods of fourth-order convergence recently published

1 Introduction

One of the most important problems in numerical analysisis solving nonlinear equations To solve these equations wecan use iterative methods such as Newtonrsquos method and itsvariants Newtonrsquos classical method for a single nonlinearequation 119891(119909) = 0 where 119903 is a single root is written as

119909119899+1

= 119909119899minus119891 (119909119899)

1198911015840 (119909119899) (1)

which converges quadratically in some neighborhood of 119903Taking 119910

119899= 119909119899minus 119891(119909

119899)1198911015840(119909

119899) many modifications of

Newtonrsquos method were recently published In [1] Noor andKhan presented a fourth-order optimal method as defined by

119909119899+1

= 119909119899minus119891 (119909119899) minus 119891 (119910

119899)

119891 (119909119899) minus 2119891 (119910

119899)

119891 (119909119899)

1198911015840 (119909119899) (2)

which uses three functional evaluations

In [2] Cordero et al proposed a fourth-order optimalmethod as defined by

119911119899= 119909119899minus119891 (119909119899) + 119891 (119910

119899)

1198911015840 (119909119899)

119908119899= 119911119899minus1198912 (119910119899) (2119891 (119909

119899) + 119891 (119910

119899))

1198912 (119909119899) 1198911015840 (119909

119899)

(3)

which also uses three functional evaluationsChun presented a third-order iterative formula [3] as

defined by

119909119899+1

= 119909119899minus3

2

119891 (119909119899)

1198911015840 (119909119899)+1

2

119891 (119909119899) 1198911015840 (120601 (119909

119899))

11989110158402 (119909119899)

(4)

which uses three functional evaluations where 120601 is anyiterative function of second order

2 Advances in Numerical Analysis

Li et al presented a fifth-order iterative formula in [4] asdefined by

119906119899+1

= 119909119899minus119891 (119909119899) minus 119891 (119909

119899minus 119891 (119909

119899) 1198911015840 (119909

119899))

1198911015840 (119909119899)

119909119899+1

= 119906119899+1

minus119891 (119906119899+1)

1198911015840 (119909119899minus 119891 (119909

119899) 1198911015840 (119909

119899))

(5)

which uses five functional evaluationsThe main goal and motivation in the development of

new methods are to obtain a better computational efficiencyIn other words it is advantageous to obtain the highestpossible convergence order with a fixed number of functionalevaluations per iteration In the case of multipoint methodswithout memory this demand is closely connected with theoptimal order considered in the Kung-Traubrsquos conjecture

Kung-Traubrsquos Conjecture (see [5]) Multipoint iterative meth-ods (without memory) requiring 119899+1 functional evaluationsper iteration have the order of convergence at most 2119899

Multipoint methods which satisfy Kung-Traubrsquos conjec-ture (still unproved) are usually called optimal methodsconsequently 119901 = 2119899 is the optimal order

The computational efficiency of an iterative method oforder 119901 requiring 119899 function evaluations per iteration ismost frequently calculated by Ostrowski-Traubrsquos efficiencyindex [6] 119864 = 1199011119899

On the case of multiple roots the quadratically conver-gent modified Newtonrsquos method [7] is

119909119899+1

= 119909119899minus119898119891 (119909

119899)

1198911015840 (119909119899) (6)

where119898 is the multiplicity of the rootFor this case there are severalmethods recently presented

to approximate the root of the function For example thecubically convergent Halleyrsquos method [8] is a special case ofthe Hansen-Patrickrsquos method [9]

119909119899+1

= 119909119899minus

119891 (119909119899)

((119898 + 1) 2119898)1198911015840 (119909119899) minus 119891 (119909

119899) 11989110158401015840 (119909

119899) 21198911015840 (119909

119899)

(7)

Osada [10] has developed a third-order method using thesecond derivative

119909119899+1

= 119909119899minus1

2119898 (119898 + 1) 119906

119899+1

2(119898 minus 1)

21198911015840 (119909119899)

11989110158401015840 (119909119899) (8)

where 119906119899= 119891(119909

119899)1198911015840(119909

119899)

Another third-order method [11] based on Kingrsquos fifth-order method (for simple roots) [12] is the Euler-Chebyshevrsquosmethod of order three

119909119899+1

= 119909119899minus119898 (3 minus 119898)

2

119891 (119909119899)

1198911015840 (119909119899)+1198982

2

1198912 (119909119899) 11989110158401015840 (119909

119899)

11989110158403 (119909119899)

(9)

Recently Chun and Neta [13] have developed a third-order method using the second derivative

119909119899+1

= 119909119899minus (211989821198912 (119909

119899) 11989110158401015840 (119909

119899)

times (119898 (3 minus 119898)119891 (119909119899) 1198911015840 (119909

119899) 11989110158401015840 (119909

119899)

+(119898 minus 1)211989110158403 (119909

119899))minus1

)

(10)

All previous methods use the second derivative of thefunction to obtain a greater order of convergence Theobjective of the new method is to avoid the use of the secondderivative

The new methods are based on a mixture of LagrangersquosandHermitersquos interpolationsThat is to say not only Hermitersquosinterpolation This is the novelty of the new methods Theinterpolation process is a conventional tool for iterativemethods see [5 7] However this tool has been appliedrecently in several ways For example in [14] Cordero andTorregrosa presented a family of Steffensen-type methodsof fourth-order convergence for solving nonlinear smoothequations by using a linear combination of divided differ-ences to achieve a better approximation to the derivativeZheng et al [15] proposed a general family of Steffensen-type methods with optimal order of convergence by usingNewtonrsquos iteration for the direct Newtonian interpolation In[16] Petkovic et al investigated a general way to constructmultipoint methods for solving nonlinear equations by usinginverse interpolation In [17] Dzunic et al presented anew family of three-point derivative free methods by usinga self-correcting parameter that is calculated applying thesecant-type method in three different ways and Newtonrsquosinterpolatory polynomial of the second degree

The three new methods (for simple roots) in this paperuse three functional evaluations and have fourth-order con-vergence thus they are optimal methods and their efficiencyindex is 119864 = 1587 which is greater than the efficiencyindex of Newtonrsquos method which is 119864 = 1414 In the caseof multiple roots the method developed here is cubicallyconvergent and uses three functional evaluations without theuse of second derivative of the functionThus themethod hasbetter performance than Newtonrsquos modified method and theabove methods with efficiency index 119864 = 14422

2 Development of the Methods

In this paper we consider iterative methods to find a simpleroot 119903 isin 119868 of a nonlinear equation 119891(119909) = 0 where 119891 119868 rarrR is a scalar function for an open interval 119868 We suppose that119891(119909) is sufficiently differentiable and 1198911015840(119909) = 0 for 119909 isin 119868 andsince 119903 is a simple root we can define 119892 = 119891minus1 on 119868 Taking1199090isin 119868 closer to 119903 and supposing that 119909

119899has been chosen we

define

119911119899= 119909119899minus119891 (119909119899)

1198911015840 (119909119899)

119870 = 119891 (119909119899) 119871 = 119891 (119911

119899)

(11)

Advances in Numerical Analysis 3

21 First Method FAM1 Consider the polynomial

1199012(119910) = 119860 (119910 minus 119870) (119910 minus 119871) + 119861 (119910 minus 119870) + 119862 (12)

with the conditions

1199012(119870) = 119892 (119870) = 119909

119899 119901

2(119871) = 119892 (119871) = 119911

119899

11990110158402(119870) = 119892

1015840

(119870) =1

1198911015840 (119909119899)

(13)

Solving simultaneously the conditions (13) and using thecommon representation of divided differences for Hermitersquosinverse interpolation

119892 [119891 (119909119899)] = 119909

119899

119892 [119891 (119909119899) 119891 (119909

119899) 119891 (119911

119899)] =

11198911015840 (119909119899)

119891 (119911119899) minus 119891 (119909

119899)

minus(119911119899minus 119909119899) (119891 (119911

119899) minus 119891 (119909

119899))

119891 (119911119899) minus 119891 (119909

119899)

119892 [119891 (119909119899) 119891 (119909

119899)] =

1

1198911015840 (119909119899)

119892 [119891 (119909119899) 119891 (119911

119899)] =

119911119899minus 119909119899

119891 (119911119899) minus 119891 (119909

119899)

(14)

Consequently we find

119860 = 119892 [119891 (119909119899) 119891 (119909

119899) 119891 (119911

119899)]

119861 = 119892 [119891 (119909119899) 119891 (119911

119899)] 119862 = 119892 [119891 (119909

119899)]

(15)

and the polynomial (12) can be written as

1199012(119910) = 119892 [119891 (119909

119899) 119891 (119909

119899) 119891 (119911

119899)] (119910 minus 119891 (119909

119899))

times (119910 minus 119891 (119911119899)) + 119892 [119891 (119909

119899) 119891 (119911

119899)]

times (119910 minus 119891 (119909119899)) + 119892 [119891 (119909

119899)]

(16)

If we aremaking 119910 = 0 in (16) we have a new iterativemethod(FAM1)

119909119899+1

= 119892 [119891 (119909119899)] minus 119892 [119891 (119909

119899) 119891 (119911

119899)] 119891 (119909

119899)

+ 119892 [119891 (119909119899) 119891 (119909

119899) 119891 (119911

119899)] 119891 (119909

119899) 119891 (119911119899)

(17)

It can be written as

119909119899+1

= 119909119899+119891 (119909119899)

1198911015840 (119909119899)[119891 (119911119899) minus 119891 (119909

119899) minus 1198912 (119911

119899)

(119891 (119911119899) minus 119891 (119909

119899))2

]

119911119899= 119909119899minus119891 (119909119899)

1198911015840 (119909119899)

(18)

which uses three functional evaluations and has fourth-orderconvergence

22 Second Method FAM2 Consider the polynomial

1199013(119910) = 119860(119910 minus 119870)

2

(119910 minus 119871)

+ 119861 (119910 minus 119870) (119910 minus 119871) + 119862 (119910 minus K) + 119863(19)

with the conditions1199013(119870) = 119892 (119870) = 119909

119899 119901

3(119871) = 119892 (119871) = 119911

119899

11990110158403(119870) = 119892

1015840

(119870) =1

1198911015840 (119909119899)

(20)

Taking 119861 = 119860(119871 minus 2119870) we have 1199013(119910) = 119860lowast1199103 + 119861lowast119910 + 119862lowast

Solving simultaneously the conditions (20) and using thecommon representation of divided differences for Hermitersquosinverse interpolation we find

119860 =119892 [119891 (119909

119899) 119891 (119909

119899) 119891 (119911

119899)]

119891 (119911119899) minus 2119891 (119909

119899)

119861 = 119892 [119891 (119909119899) 119891 (119909

119899) 119891 (119911

119899)]

(21)

119862 = 119892 [119891 (119909119899) 119891 (119911

119899)] 119863 = 119892 [119891 (119909

119899)] (22)


1199013(119910) = 119892 [119891 (119909

119899)]

+119892 [119891 (119909

119899) 119891 (119909

119899) 119891 (119911

119899)]

119891 (119911119899) minus 2119891 (119909

119899)

times (119910 minus 119891 (119909119899))2

(119910 minus 119891 (119911119899))

+ 119892 [119891 (119909119899) 119891 (119911

119899)] (119910 minus 119891 (119909

119899))

+ 119892 [119891 (119909119899) 119891 (119909

119899) 119891 (119911

119899)]

times (119910 minus 119891 (119909119899)) (119910 minus 119891 (119911

119899))

(23)

Then if we are making 119910 = 0 in (23) we have our seconditerative method (FAM2)

119909119899+1

= 119892 [119891 (119909119899)] minus 119892 [119891 (119909

119899) 119891 (119911

119899)] 119891 (119909

119899)

+ 119892 [119891 (119909119899) 119891 (119909

119899) 119891 (119911

119899)] 119891 (119909

119899) 119891 (119911119899)

minus119892 [119891 (119909

119899) 119891 (119909

119899) 119891 (119911

119899)]

119891 (119911119899) minus 2119891 (119909

119899)

1198912 (119909119899) 119891 (119911119899)

(24)


119909119899+1

= 119909119899+

1198912 (119909119899)

(119891 (119911119899) minus 119891 (119909

119899)) 1198911015840 (119909

119899)

minus1198912 (119911119899) 119891 (119909

119899) (119891 (119911

119899) minus 3119891 (119909

119899))

(119891 (119911119899) minus 119891 (119909

119899))2

(119891 (119911119899) minus 2119891 (119909

119899)) 1198911015840 (119909

119899)

119911119899= 119909119899minus119891 (119909119899)

1198911015840 (119909119899)

(25)



23 Third Method FAM3 Consider the polynomial

1199013(119910) = 119860(119910 minus 119870)

2

(119910 minus 119871)

+ 119861 (119910 minus 119870) (119910 minus 119871) + 119862 (119910 minus 119870) + 119863(26)

with the conditions

1199013(119870) = 119892 (119870) = 119909

119899 119901

3(119871) = 119892 (119871) = 119911

119899

11990110158403(119870) = 119892

1015840

(119870) =1

1198911015840 (119909119899)

(27)

119901101584010158403(119870) = minus

2119891 (119911119899)

1198912 (119909119899) 1198911015840 (119909

119899) (28)

where we have used an approximation of 11989110158401015840(119909119899) in [2]

11989110158401015840(119909119899) asymp 2119891(119911

119899)11989110158402(119909

119899)1198912(119909

119899) Solving simultaneously the

conditions (27) and (28) and using the common representa-tion of divided differences forHermitersquos inverse interpolationwe have

119860 = (119892 [119891 (119909119899) 119891 (119909

119899)]

119891 (119911119899)

1198912 (119909119899)

+119892 [119891 (119909119899) 119891 (119909

119899) 119891 (119911

119899)] )

times (119891 (119911119899) minus 119891 (119909

119899))minus1

(29)

119861 = 119892 [119891 (119909119899) 119891 (119909

119899) 119891 (119911

119899)]

119862 = 119892 [119891 (119909119899) 119891 (119911

119899)] 119863 = 119892 [119891 (119909

119899)]

(30)

Thus the polynomial (26) can be written as

1199013(119910) = (

119892 [119891 (119909119899) 119891 (119909

119899)] (119891 (119911

119899) 1198912 (119909

119899))

119891 (119911119899) minus 119891 (119909

119899)

+119892 [119891 (119909

119899) 119891 (119909

119899) 119891 (119911

119899)]

119891 (119911119899) minus 119891 (119909

119899)

)

times (119910 minus 119891 (119909119899))2

(119910 minus 119891 (119911119899))

+ 119892 [119891 (119909119899) 119891 (119909

119899) 119891 (119911

119899)]


119899))

+ 119892 [119891 (119909119899) 119891 (119911

119899)] (119910 minus 119891 (119909

119899)) + 119892 [119891 (119909

119899)]

(31)

Making 119910 = 0 in (31) we have

119909119899+1

= 119892 [119891 (119909119899)] minus 119892 [119891 (119909

119899) 119891 (119911

119899)] 119891 (119909

119899)

+ 119892 [119891 (119909119899) 119891 (119909

119899) 119891 (119911

119899)] 119891 (119909

119899) 119891 (119911119899)

minus (119892 [119891 (119909

119899) 119891 (119909

119899)] (119891 (119911

119899) 1198912 (119909

119899))

119891 (119911119899) minus 119891 (119909

119899)

+119892 [119891 (119909

119899) 119891 (119909

119899) 119891 (119911

119899)]

119891 (119911119899) minus 119891 (119909

119899)

)

times 1198912 (119909119899) 119891 (119911119899)

(32)


119909119899+1

= 119909119899+119891 (119911119899) 119891 (119909

119899) minus 1198912 (119909

119899) minus 1198912 (119911

119899)

(119891 (119911119899) minus 119891 (119909

119899))2

119891 (119909119899)

1198911015840 (119909119899)

minus1198913 (119911119899) (119891 (119911

119899) minus 2119891 (119909

119899))

(119891 (119911119899) minus 119891 (119909

119899))3

1198912 (119909119899)

1198911015840 (119909119899)

119911119899= 119909119899minus119891 (119909119899)

1198911015840 (119909119899)

(33)


24 Method FAM4 (Multiple Roots) Consider the polyno-mial

119901119898(119909) = 119860(119909 minus 119909

119899+ 119908)119898

(34)

where 119898 is the multiplicity of the root and 119901119898(119909) verify the

conditions

119901119898(119909119899) = 119891 (119909

119899) 119901

119898(119911119899) = 119891 (119911

119899) (35)

with 119911119899= 119909119899minus 119898(119891(119909

119899)1198911015840(119909

119899))

Solving the system we obtain

119908 =119906119898(119911119899minus 119909119899)

1 minus 119906119898

(36)

where

119906119898= [

119891 (119909119899)

119891 (119910119899)]

1119898

(37)

Thus we have

119909119899+1

= 119909119899minus 119908 (38)

that can be written as

119909119899+1

= 119911119899+119898119891 (119909

119899)

1198911015840 (119909119899)

1

1 minus 119906119898

119911119899= 119909119899minus 119898

119891 (119909119899)

1198911015840 (119909119899) 119906

119898= [

119891 (119909119899)

119891 (119910119899)]

1119898

(39)

which uses three functional evaluations and has third-orderconvergence


3 Analysis of Convergence

Theorem 1 Let 119891 119868subeR rarr R be a sufficiently differentiablefunction and let 119903 isin 119868 be a simple zero of 119891(119909) = 0 in an openinterval 119868 with 1198911015840(119909) = 0 on 119868 If 119909

0isin 119868 is sufficiently close to 119903

then the methods FAM1 FAM2 and FAM3 as defined by (18)(25) and (33) have fourth-order convergence

Proof Following an analogous procedure to find the errorin Lagrangersquos and Hermitersquos interpolations the polynomials(12) (19) and (26) in FAM1 FAM2 and FAM3 respectivelyhave the error

119864 (119910) = 119892 (119910) minus 119901119899(119910) = [

119892101584010158401015840 (120585)

3minus 120573119860] (119910 minus 119870)

2

(119910 minus 119871)

(40)

for some 120585 isin 119868 119860 is the coefficient in (15) (21) (29)that appears in the polynomial 119901

119899(119910) in (12) (19) and (26)

respectivelyThen substituting 119910 = 0 in 119864(119910)

119903 minus 119909119899+1

= minus [119892101584010158401015840 (120585)

3minus 120573119860]1198702119871

= minus [119892101584010158401015840 (120585)

3minus 120573119860]1198912 (119909

119899) 119891 (119911119899)

119909119899+1

minus 119903 = [119892101584010158401015840 (120585)

3minus 120573119860]

times [119891 (119903) + 1198911015840 (1205851) (119909119899minus 119903)]2

times [119891 (119903) + 1198911015840 (1205852) (119911119899minus 119903)]

(41)

with 119909119899+1

minus 119903 = 120598119899+1 119909119899minus 119903 = 120598

119899 Since 119911

119899was taken from

Newtonrsquosmethod we know that 119911119899minus119903 = (11989110158401015840(120585

3)21198911015840(120585

4))1205852119899+

119874(1205852119899+1) Then we have

120598119899+1

asymp [119892101584010158401015840 (120585)

3minus 120573119860]

11989110158402 (1205851) 12058521198991198911015840 (1205852) 11989110158401015840 (120585

3) 1205852119899

21198911015840 (1205854)

120598119899+1

asymp [119892101584010158401015840 (120585)

3minus 120573119860]

11989110158402 (1205851) 1198911015840 (120585

2) 11989110158401015840 (120585

3)

21198911015840 (1205854)

1205854119899

(42)

Now in FAM1 we take 120573 = 0 then

120585119899+1

asymp119892101584010158401015840 (120585)

3

11989110158402 (1205851) 1198911015840 (120585

2) 11989110158401015840 (120585

3)

21198911015840 (1205854)

1205854119899 (43)

In FAM2 and FAM3 we take 120573 = 1 then

120585119899+1

asymp [119892101584010158401015840 (120585)

3minus 119860]

11989110158402 (1205851) 1198911015840 (120585

2) 11989110158401015840 (120585

3)

21198911015840 (1205854)

1205854119899

120585119899+1

1205854119899

asymp [119892101584010158401015840 (120585)

3minus 119860]

11989110158402 (1205851) 1198911015840 (120585

2) 11989110158401015840 (120585

3)

21198911015840 (1205854)

(44)

Thus FAM1 FAM2 and FAM3 have fourth-order conver-gence

Theorem 2 Let 119891 119868 sube R rarr R be a sufficientlydifferentiable function and let 119903 isin 119868 be a zero of 119891(119909) = 0 withmultiplicity119898 in an open interval 119868 If 119909

0isin 119868 is sufficiently close

to 119903 then the method FAM4 defined by (12) (20) is cubicallyconvergent

Proof The proof is based on the error of Lagrangersquos interpo-lation Suppose that 119909

119899has been chosen We can see that

119891 (119909) minus 119901119898(119909) =

11989110158401015840 (1205851) minus 11990110158401015840119898(1205851)

2(119909 minus 119909

119899) (119909 minus 119911

119899) (45)

for some 1205851isin 119868

Taking 119909 = 119903 and expanding 119901119898(119903) around 119909

119896+1 we have

minus1199011015840119898(1205852) (119903 minus 119909

119899+1) =

11989110158401015840 (1205851) minus 11990110158401015840119898(1205851)

2(119903 minus 119909

119899) (119903 minus 119911

119899)

(46)

with 1205851 1205852isin 119868

Since 119911119899= 119909119899minus 119898(119891(119909

119899)1198911015840(119909

119899)) we know that

119911119899minus 119903 =

11989110158401015840 (1205853) minus 11990110158401015840119898(1205853)

21199011015840119898(1205854)

1205982119899 (47)

for some 1205853 1205854isin 119868

Thus

120598119899+1

=11989110158401015840 (1205851) minus 11990110158401015840119898(1205851)

21199011015840119898(1205852)

11989110158401015840 (1205853) minus 11990110158401015840119898(1205853)

21199011015840 (1205854)

1205983119899 (48)

Therefore FAM4 has third-order convergence

Note that 11990110158401015840119898is not zero for119898 ge 2 and this fact allows the

convergence of lim119899rarrinfin

(120598119899+11205983119899)

4 Numerical Analysis

In this section we use numerical examples to comparethe new methods introduced in this paper with Newtonrsquosclassical method (NM) and recent methods of fourth-orderconvergence such as Noorrsquos method (NOM) with 119864 = 1587in [1] Corderorsquos method (CM) with 119864 = 1587 in [2] Chunrsquosthird-order method (CHM) with 119864 = 1442 in [3] andLirsquos fifth-order method (ZM) with 119864 = 1379 in [4] in thecase of simple roots For multiple roots we compare themethod developed here with the quadratically convergentNewtonrsquos modified method (NMM) and with the cubicallyconvergent Halleyrsquos method (HM) Osadarsquos method (OM)Euler-Chebyshevrsquos method (ECM) and Chun-Netarsquos method(CNM) Tables 2 and 4 show the number of iterations (IT) andthe number of functional evaluations (NOFE) The resultsobtained show that the methods presented in this paper aremore efficient

All computations were done using MATLAB 2010 Weaccept an approximate solution rather than the exact rootdepending on the precision (120598) of the computer We use


Table 1 List of functions for a single root

1198911(119909) = 1199093 + 41199092 minus 10 119903 = 13652300134140968457608068290

1198912(119909) = cos(119909) minus 119909 119903 = 073908513321516064165531208767

1198913(119909) = sin(119909) minus (12) 119909 119903 = 18954942670339809471440357381

1198914(119909) = sin2(119909) minus 1199092 + 1 119903 = 14044916482153412260350868178

1198915(119909) = 119909119890119909

2

minus sin2(119909) + 3 cos(119909) + 5 119903 = 12076478271309189270094167584

1198916(119909) = 1199092 minus 119890119909 minus 3119909 + 2 119903 = 0257530285439860760455367304944

1198917(119909) = (119909 minus 1)3 minus 2 119903 = 22599210498948731647672106073

1198918(119909) = (119909 minus 1)2 minus 1 119903 = 2

1198919(119909) = 10119909119890minus119909

2

minus 1 119903 = 16796306104284499

11989110(119909) = (119909 + 2)119890119909 minus 1 119903 = minus04428544010023885831413280000

11989111(119909) = 119890minus119909 + cos(119909) 119903 = 1746139530408012417650703089

Table 2 Comparison of the methods for a single root

119891(119909) 1199090

IT NOFE

NM NOM CHM CM ZM FAM1 FAM2 FAM3 NM NOM CHM CM ZM FAM1 FAM2 FAM3

1198911(119909) 1 6 4 5 4 4 3 3 3 12 12 15 12 12 9 9 9

1198912(119909) 1 5 3 4 2 3 3 3 2 10 9 12 6 9 9 9 6

1198913(119909) 2 5 2 4 3 3 2 3 2 10 6 12 9 9 6 9 6

1198914(119909) 13 5 3 4 4 3 3 3 2 10 9 12 12 9 9 9 6

1198915(119909) -1 6 4 5 4 4 4 4 3 12 12 15 12 12 12 12 9

1198916(119909) 2 6 4 5 4 4 3 4 3 12 12 15 12 12 9 12 9

1198917(119909) 3 7 4 5 4 4 4 3 3 14 12 15 12 12 12 9 9

1198918(119909) 35 6 3 5 3 4 3 3 3 12 9 15 9 12 9 9 9

1198919(119909) 1 6 4 6 4 4 3 3 3 12 12 18 12 12 9 9 9

11989110(119909) 2 9 5 7 6 5 4 4 4 18 15 21 18 15 12 12 12

11989111(119909) 05 5 4 4 4 3 3 3 3 10 12 12 12 9 9 9 9

the following stopping criteria for computer programs (i)|119909119899+1

minus 119909119899| lt 120598 (ii) |119891(119909

119899+1)| lt 120598 Thus when the stopping

criterion is satisfied 119909119899+1

is taken as the exact computed root119903 For numerical illustrations in this section we used the fixedstopping criterion 120598 = 10minus18

We used the functions in Tables 1 and 3The computational results presented in Tables 2 and 4

show that in almost all of cases the presented methodsconverge more rapidly than Newtonrsquos method Newtonrsquosmodified method and those previously presented for thecase of simple and multiple roots The new methods requireless number of functional evaluations This means that thenew methods have better efficiency in computing processthan Newtonrsquos method as compared to other methods and

furthermore themethod FAM3produces the best results Formost of the functions we tested the obtainedmethods behaveat least with equal performance compared to the other knownmethods of the same order

5 Conclusions

In this paper we introduce three new optimal fourth-order iterative methods to solve nonlinear equations Theanalysis of convergence shows that the three new methodshave fourth-order convergence they use three functionalevaluations and thus according to Kung-Traubrsquos conjecturethey are optimal methods In the case of multiple roots themethod developed here is cubically convergent and uses three


Table 3 List of functions for a multiple root

1198911(119909) = (1199093 + 41199092 minus 10)3 119903 = 13652300134140968457608068290

1198912(119909) = (sin2(119909) minus 1199092 + 1)2 119903 = 14044916482153412260350868178

1198913(119909) = (1199092 minus 119890119909 minus 3119909 + 2)5 119903 = 02575302854398607604553673049

1198914(119909) = (cos(119909) minus 119909)3 119903 = 07390851332151606416553120876

1198915(119909) = ((119909 minus 1)3 minus 1)6 119903 = 2

1198916(119909) = (119909119890119909

2

minus sin2(119909) + 3 cos(119909) + 5)4 119903 = minus12076478271309189270094167581198917(119909) = (sin(119909) minus (12) 119909)2 119903 = 18954942670339809471440357381

Table 4 Comparison of the methods for a multiple root

119891(119909) 1199090

IT NOFENMM HM OM ECM CNM FAM4 NMM HM OM ECM CNM FAM4

2 5 3 3 3 3 3 10 9 9 9 9 91198911(119909) 1 5 3 4 3 3 3 10 9 12 9 9 9

minus03 54 5 5 5 5 4 108 15 15 15 15 1223 6 4 4 4 4 4 12 12 12 12 12 12

1198912(119909) 2 6 4 4 4 4 4 12 12 12 12 12 12

15 5 4 4 4 3 3 10 12 12 12 9 90 4 3 3 3 3 2 8 9 9 9 9 6

1198913(119909) 1 4 4 4 4 4 3 8 12 12 12 12 9

minus1 5 4 4 4 4 3 10 12 12 12 12 917 5 4 4 4 4 3 10 12 12 12 12 9

1198914(119909) 1 4 3 3 3 3 3 8 9 9 9 9 9

minus3 99 8 8 8 8 7 198 24 24 24 24 213 6 4 5 5 4 4 12 12 15 15 12 12

1198915(119909) minus1 10 11 24 23 5 6 20 33 72 79 15 18

5 8 9 15 15 5 5 16 27 45 45 15 15minus2 8 5 6 6 6 5 16 15 18 18 18 15

1198916(119909) minus1 5 3 4 3 3 5 10 9 12 9 9 15

minus3 14 9 10 10 10 9 28 27 30 30 30 2717 5 3 4 3 4 4 10 9 12 9 12 12

1198917(119909) 2 4 3 3 3 3 3 8 9 9 9 9 9

3 5 3 3 3 3 3 10 9 9 9 9 9

functional evaluations without the use of second derivativeNumerical analysis shows that these methods have betterperformance as compared with Newtonrsquos classical methodNewtonrsquos modified method and other recent methods ofthird- (multiple roots) and fourth-order (simple roots) con-vergence

Acknowledgments

The authors wishes to acknowledge the valuable participa-tion of Professor Nicole Mercier and Professor Joelle AnnLabrecque in the proofreading of this paper This paperis the result of the research project ldquoAnalisis Numerico deMetodos Iterativos Optimos para la Solucion de EcuacionesNoLinealesrdquo developed at theUniversidad del IstmoCampus

Tehuantepec by Researcher-Professor Gustavo Fernandez-Torres

References

[1] M A Noor and W A Khan ldquoFourth-order iterative methodfree from second derivative for solving nonlinear equationsrdquoAppliedMathematical Sciences vol 6 no 93ndash96 pp 4617ndash46252012

[2] A Cordero J L Hueso E Martınez and J R Torregrosa ldquoNewmodifications of Potra-Ptakrsquos method with optimal fourth andeighth orders of convergencerdquo Journal of Computational andApplied Mathematics vol 234 no 10 pp 2969ndash2976 2010

[3] C Chun ldquoA geometric construction of iterative formulas oforder threerdquoAppliedMathematics Letters vol 23 no 5 pp 512ndash516 2010


[4] Z Li C Peng T Zhou and J Gao ldquoA new Newton-typemethod for solving nonlinear equations with any integer orderof convergencerdquo Journal of Computational Information Systemsvol 7 no 7 pp 2371ndash2378 2011

[5] 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

[6] A M Ostrowski Solution of Equations and Systems of Equa-tions Academic Press New York NY USA 1966

[7] A Ralston and P Rabinowitz A First Course in NumericalAnalysis McGraw-Hill 1978

[8] E Halley ldquoA new exact and easy method of finding the roots ofequations generally and that without any previous reductionrdquoPhilosophical Transactions of the Royal Society of London vol18 pp 136ndash148 1964

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

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

[11] H D Victory and B Neta ldquoA higher order method for multiplezeros of nonlinear functionsrdquo International Journal of ComputerMathematics vol 12 no 3-4 pp 329ndash335 1983

[12] R F King ldquoA family of fourth order methods for nonlinearequationsrdquo SIAM Journal on Numerical Analysis vol 10 pp876ndash879 1973

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

[14] A Cordero and J R Torregrosa ldquoA class of Steffensen typemethods with optimal order of convergencerdquo Applied Mathe-matics and Computation vol 217 no 19 pp 7653ndash7659 2011

[15] Q Zheng J Li and F Huang ldquoAn optimal Steffensen-typefamily for solving nonlinear equationsrdquo Applied Mathematicsand Computation vol 217 no 23 pp 9592ndash9597 2011

[16] M S Petkovic J Dzunic and B Neta ldquoInterpolatory multi-point methods with memory for solving nonlinear equationsrdquoAppliedMathematics and Computation vol 218 no 6 pp 2533ndash2541 2011

[17] J Dzunic M S Petkovic and L D Petkovic ldquoThree-pointmethods with and without memory for solving nonlinearequationsrdquo Applied Mathematics and Computation vol 218 no9 pp 4917ndash4927 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Li et al presented a fifth-order iterative formula in [4] asdefined by

119906119899+1

= 119909119899minus119891 (119909119899) minus 119891 (119909

119899minus 119891 (119909

119899) 1198911015840 (119909

119899))

1198911015840 (119909119899)

119909119899+1

= 119906119899+1

minus119891 (119906119899+1)

1198911015840 (119909119899minus 119891 (119909

119899) 1198911015840 (119909

119899))

(5)

which uses five functional evaluationsThe main goal and motivation in the development of

new methods are to obtain a better computational efficiencyIn other words it is advantageous to obtain the highestpossible convergence order with a fixed number of functionalevaluations per iteration In the case of multipoint methodswithout memory this demand is closely connected with theoptimal order considered in the Kung-Traubrsquos conjecture

Kung-Traubrsquos Conjecture (see [5]) Multipoint iterative meth-ods (without memory) requiring 119899+1 functional evaluationsper iteration have the order of convergence at most 2119899

Multipoint methods which satisfy Kung-Traubrsquos conjec-ture (still unproved) are usually called optimal methodsconsequently 119901 = 2119899 is the optimal order

The computational efficiency of an iterative method oforder 119901 requiring 119899 function evaluations per iteration ismost frequently calculated by Ostrowski-Traubrsquos efficiencyindex [6] 119864 = 1199011119899

On the case of multiple roots the quadratically conver-gent modified Newtonrsquos method [7] is

119909119899+1

= 119909119899minus119898119891 (119909

119899)

1198911015840 (119909119899) (6)

where119898 is the multiplicity of the rootFor this case there are severalmethods recently presented

to approximate the root of the function For example thecubically convergent Halleyrsquos method [8] is a special case ofthe Hansen-Patrickrsquos method [9]

119909119899+1

= 119909119899minus

119891 (119909119899)

((119898 + 1) 2119898)1198911015840 (119909119899) minus 119891 (119909

119899) 11989110158401015840 (119909

119899) 21198911015840 (119909

119899)

(7)

Osada [10] has developed a third-order method using thesecond derivative

119909119899+1

= 119909119899minus1

2119898 (119898 + 1) 119906

119899+1

2(119898 minus 1)

21198911015840 (119909119899)

11989110158401015840 (119909119899) (8)

where 119906119899= 119891(119909

119899)1198911015840(119909

119899)

Another third-order method [11] based on Kingrsquos fifth-order method (for simple roots) [12] is the Euler-Chebyshevrsquosmethod of order three

119909119899+1

= 119909119899minus119898 (3 minus 119898)

2

119891 (119909119899)

1198911015840 (119909119899)+1198982

2

1198912 (119909119899) 11989110158401015840 (119909

119899)

11989110158403 (119909119899)

(9)

Recently Chun and Neta [13] have developed a third-order method using the second derivative

119909119899+1

= 119909119899minus (211989821198912 (119909

119899) 11989110158401015840 (119909

119899)

times (119898 (3 minus 119898)119891 (119909119899) 1198911015840 (119909

119899) 11989110158401015840 (119909

119899)

+(119898 minus 1)211989110158403 (119909

119899))minus1

)

(10)

All previous methods use the second derivative of thefunction to obtain a greater order of convergence Theobjective of the new method is to avoid the use of the secondderivative

The new methods are based on a mixture of LagrangersquosandHermitersquos interpolationsThat is to say not only Hermitersquosinterpolation This is the novelty of the new methods Theinterpolation process is a conventional tool for iterativemethods see [5 7] However this tool has been appliedrecently in several ways For example in [14] Cordero andTorregrosa presented a family of Steffensen-type methodsof fourth-order convergence for solving nonlinear smoothequations by using a linear combination of divided differ-ences to achieve a better approximation to the derivativeZheng et al [15] proposed a general family of Steffensen-type methods with optimal order of convergence by usingNewtonrsquos iteration for the direct Newtonian interpolation In[16] Petkovic et al investigated a general way to constructmultipoint methods for solving nonlinear equations by usinginverse interpolation In [17] Dzunic et al presented anew family of three-point derivative free methods by usinga self-correcting parameter that is calculated applying thesecant-type method in three different ways and Newtonrsquosinterpolatory polynomial of the second degree

The three new methods (for simple roots) in this paperuse three functional evaluations and have fourth-order con-vergence thus they are optimal methods and their efficiencyindex is 119864 = 1587 which is greater than the efficiencyindex of Newtonrsquos method which is 119864 = 1414 In the caseof multiple roots the method developed here is cubicallyconvergent and uses three functional evaluations without theuse of second derivative of the functionThus themethod hasbetter performance than Newtonrsquos modified method and theabove methods with efficiency index 119864 = 14422

2 Development of the Methods

In this paper we consider iterative methods to find a simpleroot 119903 isin 119868 of a nonlinear equation 119891(119909) = 0 where 119891 119868 rarrR is a scalar function for an open interval 119868 We suppose that119891(119909) is sufficiently differentiable and 1198911015840(119909) = 0 for 119909 isin 119868 andsince 119903 is a simple root we can define 119892 = 119891minus1 on 119868 Taking1199090isin 119868 closer to 119903 and supposing that 119909

119899has been chosen we

define

119911119899= 119909119899minus119891 (119909119899)

1198911015840 (119909119899)

119870 = 119891 (119909119899) 119871 = 119891 (119911

119899)

(11)




with the conditions

1199012(119870) = 119892 (119870) = 119909

119899 119901

2(119871) = 119892 (119871) = 119911

119899

11990110158402(119870) = 119892

1015840

(119870) =1

1198911015840 (119909119899)

(13)


119892 [119891 (119909119899)] = 119909

119899

119892 [119891 (119909119899) 119891 (119909

119899) 119891 (119911

119899)] =

11198911015840 (119909119899)

119891 (119911119899) minus 119891 (119909

119899)

minus(119911119899minus 119909119899) (119891 (119911

119899) minus 119891 (119909

119899))

119891 (119911119899) minus 119891 (119909

119899)

119892 [119891 (119909119899) 119891 (119909

119899)] =

1

1198911015840 (119909119899)

119892 [119891 (119909119899) 119891 (119911

119899)] =

119911119899minus 119909119899

119891 (119911119899) minus 119891 (119909

119899)

(14)


119860 = 119892 [119891 (119909119899) 119891 (119909

119899) 119891 (119911

119899)]

119861 = 119892 [119891 (119909119899) 119891 (119911

119899)] 119862 = 119892 [119891 (119909

119899)]

(15)


1199012(119910) = 119892 [119891 (119909

119899) 119891 (119909

119899) 119891 (119911

119899)] (119910 minus 119891 (119909

119899))

times (119910 minus 119891 (119911119899)) + 119892 [119891 (119909

119899) 119891 (119911

119899)]

times (119910 minus 119891 (119909119899)) + 119892 [119891 (119909

119899)]

(16)


119909119899+1

= 119892 [119891 (119909119899)] minus 119892 [119891 (119909

119899) 119891 (119911

119899)] 119891 (119909

119899)

+ 119892 [119891 (119909119899) 119891 (119909

119899) 119891 (119911

119899)] 119891 (119909

119899) 119891 (119911119899)

(17)


119909119899+1

= 119909119899+119891 (119909119899)

1198911015840 (119909119899)[119891 (119911119899) minus 119891 (119909

119899) minus 1198912 (119911

119899)

(119891 (119911119899) minus 119891 (119909

119899))2

]

119911119899= 119909119899minus119891 (119909119899)

1198911015840 (119909119899)

(18)



1199013(119910) = 119860(119910 minus 119870)

2

(119910 minus 119871)



119899 119901

3(119871) = 119892 (119871) = 119911

119899

11990110158403(119870) = 119892

1015840

(119870) =1

1198911015840 (119909119899)

(20)



119860 =119892 [119891 (119909

119899) 119891 (119909

119899) 119891 (119911

119899)]

119891 (119911119899) minus 2119891 (119909

119899)

119861 = 119892 [119891 (119909119899) 119891 (119909

119899) 119891 (119911

119899)]

(21)

119862 = 119892 [119891 (119909119899) 119891 (119911

119899)] 119863 = 119892 [119891 (119909

119899)] (22)


1199013(119910) = 119892 [119891 (119909

119899)]

+119892 [119891 (119909

119899) 119891 (119909

119899) 119891 (119911

119899)]

119891 (119911119899) minus 2119891 (119909

119899)

times (119910 minus 119891 (119909119899))2

(119910 minus 119891 (119911119899))

+ 119892 [119891 (119909119899) 119891 (119911

119899)] (119910 minus 119891 (119909

119899))

+ 119892 [119891 (119909119899) 119891 (119909

119899) 119891 (119911

119899)]


119899))

(23)


119909119899+1

= 119892 [119891 (119909119899)] minus 119892 [119891 (119909

119899) 119891 (119911

119899)] 119891 (119909

119899)

+ 119892 [119891 (119909119899) 119891 (119909

119899) 119891 (119911

119899)] 119891 (119909

119899) 119891 (119911119899)

minus119892 [119891 (119909

119899) 119891 (119909

119899) 119891 (119911

119899)]

119891 (119911119899) minus 2119891 (119909

119899)

1198912 (119909119899) 119891 (119911119899)

(24)


119909119899+1

= 119909119899+

1198912 (119909119899)

(119891 (119911119899) minus 119891 (119909

119899)) 1198911015840 (119909

119899)

minus1198912 (119911119899) 119891 (119909

119899) (119891 (119911

119899) minus 3119891 (119909

119899))

(119891 (119911119899) minus 119891 (119909

119899))2

(119891 (119911119899) minus 2119891 (119909

119899)) 1198911015840 (119909

119899)

119911119899= 119909119899minus119891 (119909119899)

1198911015840 (119909119899)

(25)




1199013(119910) = 119860(119910 minus 119870)

2

(119910 minus 119871)


with the conditions

1199013(119870) = 119892 (119870) = 119909

119899 119901

3(119871) = 119892 (119871) = 119911

119899

11990110158403(119870) = 119892

1015840

(119870) =1

1198911015840 (119909119899)

(27)

119901101584010158403(119870) = minus

2119891 (119911119899)

1198912 (119909119899) 1198911015840 (119909

119899) (28)


11989110158401015840(119909119899) asymp 2119891(119911

119899)11989110158402(119909

119899)1198912(119909



119860 = (119892 [119891 (119909119899) 119891 (119909

119899)]

119891 (119911119899)

1198912 (119909119899)

+119892 [119891 (119909119899) 119891 (119909

119899) 119891 (119911

119899)] )

times (119891 (119911119899) minus 119891 (119909

119899))minus1

(29)

119861 = 119892 [119891 (119909119899) 119891 (119909

119899) 119891 (119911

119899)]

119862 = 119892 [119891 (119909119899) 119891 (119911

119899)] 119863 = 119892 [119891 (119909

119899)]

(30)


1199013(119910) = (

119892 [119891 (119909119899) 119891 (119909

119899)] (119891 (119911

119899) 1198912 (119909

119899))

119891 (119911119899) minus 119891 (119909

119899)

+119892 [119891 (119909

119899) 119891 (119909

119899) 119891 (119911

119899)]

119891 (119911119899) minus 119891 (119909

119899)

)

times (119910 minus 119891 (119909119899))2

(119910 minus 119891 (119911119899))

+ 119892 [119891 (119909119899) 119891 (119909

119899) 119891 (119911

119899)]


119899))

+ 119892 [119891 (119909119899) 119891 (119911

119899)] (119910 minus 119891 (119909

119899)) + 119892 [119891 (119909

119899)]

(31)


119909119899+1

= 119892 [119891 (119909119899)] minus 119892 [119891 (119909

119899) 119891 (119911

119899)] 119891 (119909

119899)

+ 119892 [119891 (119909119899) 119891 (119909

119899) 119891 (119911

119899)] 119891 (119909

119899) 119891 (119911119899)

minus (119892 [119891 (119909

119899) 119891 (119909

119899)] (119891 (119911

119899) 1198912 (119909

119899))

119891 (119911119899) minus 119891 (119909

119899)

+119892 [119891 (119909

119899) 119891 (119909

119899) 119891 (119911

119899)]

119891 (119911119899) minus 119891 (119909

119899)

)

times 1198912 (119909119899) 119891 (119911119899)

(32)


119909119899+1

= 119909119899+119891 (119911119899) 119891 (119909

119899) minus 1198912 (119909

119899) minus 1198912 (119911

119899)

(119891 (119911119899) minus 119891 (119909

119899))2

119891 (119909119899)

1198911015840 (119909119899)

minus1198913 (119911119899) (119891 (119911

119899) minus 2119891 (119909

119899))

(119891 (119911119899) minus 119891 (119909

119899))3

1198912 (119909119899)

1198911015840 (119909119899)

119911119899= 119909119899minus119891 (119909119899)

1198911015840 (119909119899)

(33)



119901119898(119909) = 119860(119909 minus 119909

119899+ 119908)119898

(34)


conditions

119901119898(119909119899) = 119891 (119909

119899) 119901

119898(119911119899) = 119891 (119911

119899) (35)

with 119911119899= 119909119899minus 119898(119891(119909

119899)1198911015840(119909

119899))


119908 =119906119898(119911119899minus 119909119899)

1 minus 119906119898

(36)

where

119906119898= [

119891 (119909119899)

119891 (119910119899)]

1119898

(37)

Thus we have

119909119899+1

= 119909119899minus 119908 (38)


119909119899+1

= 119911119899+119898119891 (119909

119899)

1198911015840 (119909119899)

1

1 minus 119906119898

119911119899= 119909119899minus 119898

119891 (119909119899)

1198911015840 (119909119899) 119906

119898= [

119891 (119909119899)

119891 (119910119899)]

1119898

(39)








119864 (119910) = 119892 (119910) minus 119901119899(119910) = [

119892101584010158401015840 (120585)

3minus 120573119860] (119910 minus 119870)

2

(119910 minus 119871)

(40)


119899(119910) in (12) (19) and (26)


119903 minus 119909119899+1

= minus [119892101584010158401015840 (120585)

3minus 120573119860]1198702119871

= minus [119892101584010158401015840 (120585)

3minus 120573119860]1198912 (119909

119899) 119891 (119911119899)

119909119899+1

minus 119903 = [119892101584010158401015840 (120585)

3minus 120573119860]

times [119891 (119903) + 1198911015840 (1205851) (119909119899minus 119903)]2

times [119891 (119903) + 1198911015840 (1205852) (119911119899minus 119903)]

(41)

with 119909119899+1

minus 119903 = 120598119899+1 119909119899minus 119903 = 120598

119899 Since 119911



3)21198911015840(120585

4))1205852119899+

119874(1205852119899+1) Then we have

120598119899+1

asymp [119892101584010158401015840 (120585)

3minus 120573119860]

11989110158402 (1205851) 12058521198991198911015840 (1205852) 11989110158401015840 (120585

3) 1205852119899

21198911015840 (1205854)

120598119899+1

asymp [119892101584010158401015840 (120585)

3minus 120573119860]

11989110158402 (1205851) 1198911015840 (120585

2) 11989110158401015840 (120585

3)

21198911015840 (1205854)

1205854119899

(42)


120585119899+1

asymp119892101584010158401015840 (120585)

3

11989110158402 (1205851) 1198911015840 (120585

2) 11989110158401015840 (120585

3)

21198911015840 (1205854)

1205854119899 (43)


120585119899+1

asymp [119892101584010158401015840 (120585)

3minus 119860]

11989110158402 (1205851) 1198911015840 (120585

2) 11989110158401015840 (120585

3)

21198911015840 (1205854)

1205854119899

120585119899+1

1205854119899

asymp [119892101584010158401015840 (120585)

3minus 119860]

11989110158402 (1205851) 1198911015840 (120585

2) 11989110158401015840 (120585

3)

21198911015840 (1205854)

(44)







119891 (119909) minus 119901119898(119909) =

11989110158401015840 (1205851) minus 11990110158401015840119898(1205851)

2(119909 minus 119909

119899) (119909 minus 119911

119899) (45)



119896+1 we have

minus1199011015840119898(1205852) (119903 minus 119909

119899+1) =

11989110158401015840 (1205851) minus 11990110158401015840119898(1205851)

2(119903 minus 119909

119899) (119903 minus 119911

119899)

(46)

with 1205851 1205852isin 119868

Since 119911119899= 119909119899minus 119898(119891(119909

119899)1198911015840(119909


119911119899minus 119903 =

11989110158401015840 (1205853) minus 11990110158401015840119898(1205853)

21199011015840119898(1205854)

1205982119899 (47)

for some 1205853 1205854isin 119868

Thus

120598119899+1

=11989110158401015840 (1205851) minus 11990110158401015840119898(1205851)

21199011015840119898(1205852)

11989110158401015840 (1205853) minus 11990110158401015840119898(1205853)

21199011015840 (1205854)

1205983119899 (48)




(120598119899+11205983119899)






1198911(119909) = 1199093 + 41199092 minus 10 119903 = 13652300134140968457608068290

1198912(119909) = cos(119909) minus 119909 119903 = 073908513321516064165531208767

1198913(119909) = sin(119909) minus (12) 119909 119903 = 18954942670339809471440357381

1198914(119909) = sin2(119909) minus 1199092 + 1 119903 = 14044916482153412260350868178

1198915(119909) = 119909119890119909

2

minus sin2(119909) + 3 cos(119909) + 5 119903 = 12076478271309189270094167584

1198916(119909) = 1199092 minus 119890119909 minus 3119909 + 2 119903 = 0257530285439860760455367304944

1198917(119909) = (119909 minus 1)3 minus 2 119903 = 22599210498948731647672106073

1198918(119909) = (119909 minus 1)2 minus 1 119903 = 2

1198919(119909) = 10119909119890minus119909

2

minus 1 119903 = 16796306104284499

11989110(119909) = (119909 + 2)119890119909 minus 1 119903 = minus04428544010023885831413280000

11989111(119909) = 119890minus119909 + cos(119909) 119903 = 1746139530408012417650703089


119891(119909) 1199090

IT NOFE


1198911(119909) 1 6 4 5 4 4 3 3 3 12 12 15 12 12 9 9 9

1198912(119909) 1 5 3 4 2 3 3 3 2 10 9 12 6 9 9 9 6

1198913(119909) 2 5 2 4 3 3 2 3 2 10 6 12 9 9 6 9 6

1198914(119909) 13 5 3 4 4 3 3 3 2 10 9 12 12 9 9 9 6

1198915(119909) -1 6 4 5 4 4 4 4 3 12 12 15 12 12 12 12 9

1198916(119909) 2 6 4 5 4 4 3 4 3 12 12 15 12 12 9 12 9

1198917(119909) 3 7 4 5 4 4 4 3 3 14 12 15 12 12 12 9 9

1198918(119909) 35 6 3 5 3 4 3 3 3 12 9 15 9 12 9 9 9

1198919(119909) 1 6 4 6 4 4 3 3 3 12 12 18 12 12 9 9 9

11989110(119909) 2 9 5 7 6 5 4 4 4 18 15 21 18 15 12 12 12

11989111(119909) 05 5 4 4 4 3 3 3 3 10 12 12 12 9 9 9 9


minus 119909119899| lt 120598 (ii) |119891(119909







5 Conclusions




1198911(119909) = (1199093 + 41199092 minus 10)3 119903 = 13652300134140968457608068290

1198912(119909) = (sin2(119909) minus 1199092 + 1)2 119903 = 14044916482153412260350868178

1198913(119909) = (1199092 minus 119890119909 minus 3119909 + 2)5 119903 = 02575302854398607604553673049

1198914(119909) = (cos(119909) minus 119909)3 119903 = 07390851332151606416553120876

1198915(119909) = ((119909 minus 1)3 minus 1)6 119903 = 2

1198916(119909) = (119909119890119909

2



119891(119909) 1199090


2 5 3 3 3 3 3 10 9 9 9 9 91198911(119909) 1 5 3 4 3 3 3 10 9 12 9 9 9

minus03 54 5 5 5 5 4 108 15 15 15 15 1223 6 4 4 4 4 4 12 12 12 12 12 12

1198912(119909) 2 6 4 4 4 4 4 12 12 12 12 12 12

15 5 4 4 4 3 3 10 12 12 12 9 90 4 3 3 3 3 2 8 9 9 9 9 6

1198913(119909) 1 4 4 4 4 4 3 8 12 12 12 12 9

minus1 5 4 4 4 4 3 10 12 12 12 12 917 5 4 4 4 4 3 10 12 12 12 12 9

1198914(119909) 1 4 3 3 3 3 3 8 9 9 9 9 9

minus3 99 8 8 8 8 7 198 24 24 24 24 213 6 4 5 5 4 4 12 12 15 15 12 12

1198915(119909) minus1 10 11 24 23 5 6 20 33 72 79 15 18

5 8 9 15 15 5 5 16 27 45 45 15 15minus2 8 5 6 6 6 5 16 15 18 18 18 15

1198916(119909) minus1 5 3 4 3 3 5 10 9 12 9 9 15

minus3 14 9 10 10 10 9 28 27 30 30 30 2717 5 3 4 3 4 4 10 9 12 9 12 12

1198917(119909) 2 4 3 3 3 3 3 8 9 9 9 9 9

3 5 3 3 3 3 3 10 9 9 9 9 9


Acknowledgments



References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












with the conditions

1199012(119870) = 119892 (119870) = 119909

119899 119901

2(119871) = 119892 (119871) = 119911

119899

11990110158402(119870) = 119892

1015840

(119870) =1

1198911015840 (119909119899)

(13)


119892 [119891 (119909119899)] = 119909

119899

119892 [119891 (119909119899) 119891 (119909

119899) 119891 (119911

119899)] =

11198911015840 (119909119899)

119891 (119911119899) minus 119891 (119909

119899)

minus(119911119899minus 119909119899) (119891 (119911

119899) minus 119891 (119909

119899))

119891 (119911119899) minus 119891 (119909

119899)

119892 [119891 (119909119899) 119891 (119909

119899)] =

1

1198911015840 (119909119899)

119892 [119891 (119909119899) 119891 (119911

119899)] =

119911119899minus 119909119899

119891 (119911119899) minus 119891 (119909

119899)

(14)


119860 = 119892 [119891 (119909119899) 119891 (119909

119899) 119891 (119911

119899)]

119861 = 119892 [119891 (119909119899) 119891 (119911

119899)] 119862 = 119892 [119891 (119909

119899)]

(15)


1199012(119910) = 119892 [119891 (119909

119899) 119891 (119909

119899) 119891 (119911

119899)] (119910 minus 119891 (119909

119899))

times (119910 minus 119891 (119911119899)) + 119892 [119891 (119909

119899) 119891 (119911

119899)]

times (119910 minus 119891 (119909119899)) + 119892 [119891 (119909

119899)]

(16)


119909119899+1

= 119892 [119891 (119909119899)] minus 119892 [119891 (119909

119899) 119891 (119911

119899)] 119891 (119909

119899)

+ 119892 [119891 (119909119899) 119891 (119909

119899) 119891 (119911

119899)] 119891 (119909

119899) 119891 (119911119899)

(17)


119909119899+1

= 119909119899+119891 (119909119899)

1198911015840 (119909119899)[119891 (119911119899) minus 119891 (119909

119899) minus 1198912 (119911

119899)

(119891 (119911119899) minus 119891 (119909

119899))2

]

119911119899= 119909119899minus119891 (119909119899)

1198911015840 (119909119899)

(18)



1199013(119910) = 119860(119910 minus 119870)

2

(119910 minus 119871)



119899 119901

3(119871) = 119892 (119871) = 119911

119899

11990110158403(119870) = 119892

1015840

(119870) =1

1198911015840 (119909119899)

(20)



119860 =119892 [119891 (119909

119899) 119891 (119909

119899) 119891 (119911

119899)]

119891 (119911119899) minus 2119891 (119909

119899)

119861 = 119892 [119891 (119909119899) 119891 (119909

119899) 119891 (119911

119899)]

(21)

119862 = 119892 [119891 (119909119899) 119891 (119911

119899)] 119863 = 119892 [119891 (119909

119899)] (22)


1199013(119910) = 119892 [119891 (119909

119899)]

+119892 [119891 (119909

119899) 119891 (119909

119899) 119891 (119911

119899)]

119891 (119911119899) minus 2119891 (119909

119899)

times (119910 minus 119891 (119909119899))2

(119910 minus 119891 (119911119899))

+ 119892 [119891 (119909119899) 119891 (119911

119899)] (119910 minus 119891 (119909

119899))

+ 119892 [119891 (119909119899) 119891 (119909

119899) 119891 (119911

119899)]


119899))

(23)


119909119899+1

= 119892 [119891 (119909119899)] minus 119892 [119891 (119909

119899) 119891 (119911

119899)] 119891 (119909

119899)

+ 119892 [119891 (119909119899) 119891 (119909

119899) 119891 (119911

119899)] 119891 (119909

119899) 119891 (119911119899)

minus119892 [119891 (119909

119899) 119891 (119909

119899) 119891 (119911

119899)]

119891 (119911119899) minus 2119891 (119909

119899)

1198912 (119909119899) 119891 (119911119899)

(24)


119909119899+1

= 119909119899+

1198912 (119909119899)

(119891 (119911119899) minus 119891 (119909

119899)) 1198911015840 (119909

119899)

minus1198912 (119911119899) 119891 (119909

119899) (119891 (119911

119899) minus 3119891 (119909

119899))

(119891 (119911119899) minus 119891 (119909

119899))2

(119891 (119911119899) minus 2119891 (119909

119899)) 1198911015840 (119909

119899)

119911119899= 119909119899minus119891 (119909119899)

1198911015840 (119909119899)

(25)




1199013(119910) = 119860(119910 minus 119870)

2

(119910 minus 119871)


with the conditions

1199013(119870) = 119892 (119870) = 119909

119899 119901

3(119871) = 119892 (119871) = 119911

119899

11990110158403(119870) = 119892

1015840

(119870) =1

1198911015840 (119909119899)

(27)

119901101584010158403(119870) = minus

2119891 (119911119899)

1198912 (119909119899) 1198911015840 (119909

119899) (28)


11989110158401015840(119909119899) asymp 2119891(119911

119899)11989110158402(119909

119899)1198912(119909



119860 = (119892 [119891 (119909119899) 119891 (119909

119899)]

119891 (119911119899)

1198912 (119909119899)

+119892 [119891 (119909119899) 119891 (119909

119899) 119891 (119911

119899)] )

times (119891 (119911119899) minus 119891 (119909

119899))minus1

(29)

119861 = 119892 [119891 (119909119899) 119891 (119909

119899) 119891 (119911

119899)]

119862 = 119892 [119891 (119909119899) 119891 (119911

119899)] 119863 = 119892 [119891 (119909

119899)]

(30)


1199013(119910) = (

119892 [119891 (119909119899) 119891 (119909

119899)] (119891 (119911

119899) 1198912 (119909

119899))

119891 (119911119899) minus 119891 (119909

119899)

+119892 [119891 (119909

119899) 119891 (119909

119899) 119891 (119911

119899)]

119891 (119911119899) minus 119891 (119909

119899)

)

times (119910 minus 119891 (119909119899))2

(119910 minus 119891 (119911119899))

+ 119892 [119891 (119909119899) 119891 (119909

119899) 119891 (119911

119899)]


119899))

+ 119892 [119891 (119909119899) 119891 (119911

119899)] (119910 minus 119891 (119909

119899)) + 119892 [119891 (119909

119899)]

(31)


119909119899+1

= 119892 [119891 (119909119899)] minus 119892 [119891 (119909

119899) 119891 (119911

119899)] 119891 (119909

119899)

+ 119892 [119891 (119909119899) 119891 (119909

119899) 119891 (119911

119899)] 119891 (119909

119899) 119891 (119911119899)

minus (119892 [119891 (119909

119899) 119891 (119909

119899)] (119891 (119911

119899) 1198912 (119909

119899))

119891 (119911119899) minus 119891 (119909

119899)

+119892 [119891 (119909

119899) 119891 (119909

119899) 119891 (119911

119899)]

119891 (119911119899) minus 119891 (119909

119899)

)

times 1198912 (119909119899) 119891 (119911119899)

(32)


119909119899+1

= 119909119899+119891 (119911119899) 119891 (119909

119899) minus 1198912 (119909

119899) minus 1198912 (119911

119899)

(119891 (119911119899) minus 119891 (119909

119899))2

119891 (119909119899)

1198911015840 (119909119899)

minus1198913 (119911119899) (119891 (119911

119899) minus 2119891 (119909

119899))

(119891 (119911119899) minus 119891 (119909

119899))3

1198912 (119909119899)

1198911015840 (119909119899)

119911119899= 119909119899minus119891 (119909119899)

1198911015840 (119909119899)

(33)



119901119898(119909) = 119860(119909 minus 119909

119899+ 119908)119898

(34)


conditions

119901119898(119909119899) = 119891 (119909

119899) 119901

119898(119911119899) = 119891 (119911

119899) (35)

with 119911119899= 119909119899minus 119898(119891(119909

119899)1198911015840(119909

119899))


119908 =119906119898(119911119899minus 119909119899)

1 minus 119906119898

(36)

where

119906119898= [

119891 (119909119899)

119891 (119910119899)]

1119898

(37)

Thus we have

119909119899+1

= 119909119899minus 119908 (38)


119909119899+1

= 119911119899+119898119891 (119909

119899)

1198911015840 (119909119899)

1

1 minus 119906119898

119911119899= 119909119899minus 119898

119891 (119909119899)

1198911015840 (119909119899) 119906

119898= [

119891 (119909119899)

119891 (119910119899)]

1119898

(39)








119864 (119910) = 119892 (119910) minus 119901119899(119910) = [

119892101584010158401015840 (120585)

3minus 120573119860] (119910 minus 119870)

2

(119910 minus 119871)

(40)


119899(119910) in (12) (19) and (26)


119903 minus 119909119899+1

= minus [119892101584010158401015840 (120585)

3minus 120573119860]1198702119871

= minus [119892101584010158401015840 (120585)

3minus 120573119860]1198912 (119909

119899) 119891 (119911119899)

119909119899+1

minus 119903 = [119892101584010158401015840 (120585)

3minus 120573119860]

times [119891 (119903) + 1198911015840 (1205851) (119909119899minus 119903)]2

times [119891 (119903) + 1198911015840 (1205852) (119911119899minus 119903)]

(41)

with 119909119899+1

minus 119903 = 120598119899+1 119909119899minus 119903 = 120598

119899 Since 119911



3)21198911015840(120585

4))1205852119899+

119874(1205852119899+1) Then we have

120598119899+1

asymp [119892101584010158401015840 (120585)

3minus 120573119860]

11989110158402 (1205851) 12058521198991198911015840 (1205852) 11989110158401015840 (120585

3) 1205852119899

21198911015840 (1205854)

120598119899+1

asymp [119892101584010158401015840 (120585)

3minus 120573119860]

11989110158402 (1205851) 1198911015840 (120585

2) 11989110158401015840 (120585

3)

21198911015840 (1205854)

1205854119899

(42)


120585119899+1

asymp119892101584010158401015840 (120585)

3

11989110158402 (1205851) 1198911015840 (120585

2) 11989110158401015840 (120585

3)

21198911015840 (1205854)

1205854119899 (43)


120585119899+1

asymp [119892101584010158401015840 (120585)

3minus 119860]

11989110158402 (1205851) 1198911015840 (120585

2) 11989110158401015840 (120585

3)

21198911015840 (1205854)

1205854119899

120585119899+1

1205854119899

asymp [119892101584010158401015840 (120585)

3minus 119860]

11989110158402 (1205851) 1198911015840 (120585

2) 11989110158401015840 (120585

3)

21198911015840 (1205854)

(44)







119891 (119909) minus 119901119898(119909) =

11989110158401015840 (1205851) minus 11990110158401015840119898(1205851)

2(119909 minus 119909

119899) (119909 minus 119911

119899) (45)



119896+1 we have

minus1199011015840119898(1205852) (119903 minus 119909

119899+1) =

11989110158401015840 (1205851) minus 11990110158401015840119898(1205851)

2(119903 minus 119909

119899) (119903 minus 119911

119899)

(46)

with 1205851 1205852isin 119868

Since 119911119899= 119909119899minus 119898(119891(119909

119899)1198911015840(119909


119911119899minus 119903 =

11989110158401015840 (1205853) minus 11990110158401015840119898(1205853)

21199011015840119898(1205854)

1205982119899 (47)

for some 1205853 1205854isin 119868

Thus

120598119899+1

=11989110158401015840 (1205851) minus 11990110158401015840119898(1205851)

21199011015840119898(1205852)

11989110158401015840 (1205853) minus 11990110158401015840119898(1205853)

21199011015840 (1205854)

1205983119899 (48)




(120598119899+11205983119899)






1198911(119909) = 1199093 + 41199092 minus 10 119903 = 13652300134140968457608068290

1198912(119909) = cos(119909) minus 119909 119903 = 073908513321516064165531208767

1198913(119909) = sin(119909) minus (12) 119909 119903 = 18954942670339809471440357381

1198914(119909) = sin2(119909) minus 1199092 + 1 119903 = 14044916482153412260350868178

1198915(119909) = 119909119890119909

2

minus sin2(119909) + 3 cos(119909) + 5 119903 = 12076478271309189270094167584

1198916(119909) = 1199092 minus 119890119909 minus 3119909 + 2 119903 = 0257530285439860760455367304944

1198917(119909) = (119909 minus 1)3 minus 2 119903 = 22599210498948731647672106073

1198918(119909) = (119909 minus 1)2 minus 1 119903 = 2

1198919(119909) = 10119909119890minus119909

2

minus 1 119903 = 16796306104284499

11989110(119909) = (119909 + 2)119890119909 minus 1 119903 = minus04428544010023885831413280000

11989111(119909) = 119890minus119909 + cos(119909) 119903 = 1746139530408012417650703089


119891(119909) 1199090

IT NOFE


1198911(119909) 1 6 4 5 4 4 3 3 3 12 12 15 12 12 9 9 9

1198912(119909) 1 5 3 4 2 3 3 3 2 10 9 12 6 9 9 9 6

1198913(119909) 2 5 2 4 3 3 2 3 2 10 6 12 9 9 6 9 6

1198914(119909) 13 5 3 4 4 3 3 3 2 10 9 12 12 9 9 9 6

1198915(119909) -1 6 4 5 4 4 4 4 3 12 12 15 12 12 12 12 9

1198916(119909) 2 6 4 5 4 4 3 4 3 12 12 15 12 12 9 12 9

1198917(119909) 3 7 4 5 4 4 4 3 3 14 12 15 12 12 12 9 9

1198918(119909) 35 6 3 5 3 4 3 3 3 12 9 15 9 12 9 9 9

1198919(119909) 1 6 4 6 4 4 3 3 3 12 12 18 12 12 9 9 9

11989110(119909) 2 9 5 7 6 5 4 4 4 18 15 21 18 15 12 12 12

11989111(119909) 05 5 4 4 4 3 3 3 3 10 12 12 12 9 9 9 9


minus 119909119899| lt 120598 (ii) |119891(119909







5 Conclusions




1198911(119909) = (1199093 + 41199092 minus 10)3 119903 = 13652300134140968457608068290

1198912(119909) = (sin2(119909) minus 1199092 + 1)2 119903 = 14044916482153412260350868178

1198913(119909) = (1199092 minus 119890119909 minus 3119909 + 2)5 119903 = 02575302854398607604553673049

1198914(119909) = (cos(119909) minus 119909)3 119903 = 07390851332151606416553120876

1198915(119909) = ((119909 minus 1)3 minus 1)6 119903 = 2

1198916(119909) = (119909119890119909

2



119891(119909) 1199090


2 5 3 3 3 3 3 10 9 9 9 9 91198911(119909) 1 5 3 4 3 3 3 10 9 12 9 9 9

minus03 54 5 5 5 5 4 108 15 15 15 15 1223 6 4 4 4 4 4 12 12 12 12 12 12

1198912(119909) 2 6 4 4 4 4 4 12 12 12 12 12 12

15 5 4 4 4 3 3 10 12 12 12 9 90 4 3 3 3 3 2 8 9 9 9 9 6

1198913(119909) 1 4 4 4 4 4 3 8 12 12 12 12 9

minus1 5 4 4 4 4 3 10 12 12 12 12 917 5 4 4 4 4 3 10 12 12 12 12 9

1198914(119909) 1 4 3 3 3 3 3 8 9 9 9 9 9

minus3 99 8 8 8 8 7 198 24 24 24 24 213 6 4 5 5 4 4 12 12 15 15 12 12

1198915(119909) minus1 10 11 24 23 5 6 20 33 72 79 15 18

5 8 9 15 15 5 5 16 27 45 45 15 15minus2 8 5 6 6 6 5 16 15 18 18 18 15

1198916(119909) minus1 5 3 4 3 3 5 10 9 12 9 9 15

minus3 14 9 10 10 10 9 28 27 30 30 30 2717 5 3 4 3 4 4 10 9 12 9 12 12

1198917(119909) 2 4 3 3 3 3 3 8 9 9 9 9 9

3 5 3 3 3 3 3 10 9 9 9 9 9


Acknowledgments



References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1199013(119910) = 119860(119910 minus 119870)

2

(119910 minus 119871)


with the conditions

1199013(119870) = 119892 (119870) = 119909

119899 119901

3(119871) = 119892 (119871) = 119911

119899

11990110158403(119870) = 119892

1015840

(119870) =1

1198911015840 (119909119899)

(27)

119901101584010158403(119870) = minus

2119891 (119911119899)

1198912 (119909119899) 1198911015840 (119909

119899) (28)


11989110158401015840(119909119899) asymp 2119891(119911

119899)11989110158402(119909

119899)1198912(119909



119860 = (119892 [119891 (119909119899) 119891 (119909

119899)]

119891 (119911119899)

1198912 (119909119899)

+119892 [119891 (119909119899) 119891 (119909

119899) 119891 (119911

119899)] )

times (119891 (119911119899) minus 119891 (119909

119899))minus1

(29)

119861 = 119892 [119891 (119909119899) 119891 (119909

119899) 119891 (119911

119899)]

119862 = 119892 [119891 (119909119899) 119891 (119911

119899)] 119863 = 119892 [119891 (119909

119899)]

(30)


1199013(119910) = (

119892 [119891 (119909119899) 119891 (119909

119899)] (119891 (119911

119899) 1198912 (119909

119899))

119891 (119911119899) minus 119891 (119909

119899)

+119892 [119891 (119909

119899) 119891 (119909

119899) 119891 (119911

119899)]

119891 (119911119899) minus 119891 (119909

119899)

)

times (119910 minus 119891 (119909119899))2

(119910 minus 119891 (119911119899))

+ 119892 [119891 (119909119899) 119891 (119909

119899) 119891 (119911

119899)]


119899))

+ 119892 [119891 (119909119899) 119891 (119911

119899)] (119910 minus 119891 (119909

119899)) + 119892 [119891 (119909

119899)]

(31)


119909119899+1

= 119892 [119891 (119909119899)] minus 119892 [119891 (119909

119899) 119891 (119911

119899)] 119891 (119909

119899)

+ 119892 [119891 (119909119899) 119891 (119909

119899) 119891 (119911

119899)] 119891 (119909

119899) 119891 (119911119899)

minus (119892 [119891 (119909

119899) 119891 (119909

119899)] (119891 (119911

119899) 1198912 (119909

119899))

119891 (119911119899) minus 119891 (119909

119899)

+119892 [119891 (119909

119899) 119891 (119909

119899) 119891 (119911

119899)]

119891 (119911119899) minus 119891 (119909

119899)

)

times 1198912 (119909119899) 119891 (119911119899)

(32)


119909119899+1

= 119909119899+119891 (119911119899) 119891 (119909

119899) minus 1198912 (119909

119899) minus 1198912 (119911

119899)

(119891 (119911119899) minus 119891 (119909

119899))2

119891 (119909119899)

1198911015840 (119909119899)

minus1198913 (119911119899) (119891 (119911

119899) minus 2119891 (119909

119899))

(119891 (119911119899) minus 119891 (119909

119899))3

1198912 (119909119899)

1198911015840 (119909119899)

119911119899= 119909119899minus119891 (119909119899)

1198911015840 (119909119899)

(33)



119901119898(119909) = 119860(119909 minus 119909

119899+ 119908)119898

(34)


conditions

119901119898(119909119899) = 119891 (119909

119899) 119901

119898(119911119899) = 119891 (119911

119899) (35)

with 119911119899= 119909119899minus 119898(119891(119909

119899)1198911015840(119909

119899))


119908 =119906119898(119911119899minus 119909119899)

1 minus 119906119898

(36)

where

119906119898= [

119891 (119909119899)

119891 (119910119899)]

1119898

(37)

Thus we have

119909119899+1

= 119909119899minus 119908 (38)


119909119899+1

= 119911119899+119898119891 (119909

119899)

1198911015840 (119909119899)

1

1 minus 119906119898

119911119899= 119909119899minus 119898

119891 (119909119899)

1198911015840 (119909119899) 119906

119898= [

119891 (119909119899)

119891 (119910119899)]

1119898

(39)








119864 (119910) = 119892 (119910) minus 119901119899(119910) = [

119892101584010158401015840 (120585)

3minus 120573119860] (119910 minus 119870)

2

(119910 minus 119871)

(40)


119899(119910) in (12) (19) and (26)


119903 minus 119909119899+1

= minus [119892101584010158401015840 (120585)

3minus 120573119860]1198702119871

= minus [119892101584010158401015840 (120585)

3minus 120573119860]1198912 (119909

119899) 119891 (119911119899)

119909119899+1

minus 119903 = [119892101584010158401015840 (120585)

3minus 120573119860]

times [119891 (119903) + 1198911015840 (1205851) (119909119899minus 119903)]2

times [119891 (119903) + 1198911015840 (1205852) (119911119899minus 119903)]

(41)

with 119909119899+1

minus 119903 = 120598119899+1 119909119899minus 119903 = 120598

119899 Since 119911



3)21198911015840(120585

4))1205852119899+

119874(1205852119899+1) Then we have

120598119899+1

asymp [119892101584010158401015840 (120585)

3minus 120573119860]

11989110158402 (1205851) 12058521198991198911015840 (1205852) 11989110158401015840 (120585

3) 1205852119899

21198911015840 (1205854)

120598119899+1

asymp [119892101584010158401015840 (120585)

3minus 120573119860]

11989110158402 (1205851) 1198911015840 (120585

2) 11989110158401015840 (120585

3)

21198911015840 (1205854)

1205854119899

(42)


120585119899+1

asymp119892101584010158401015840 (120585)

3

11989110158402 (1205851) 1198911015840 (120585

2) 11989110158401015840 (120585

3)

21198911015840 (1205854)

1205854119899 (43)


120585119899+1

asymp [119892101584010158401015840 (120585)

3minus 119860]

11989110158402 (1205851) 1198911015840 (120585

2) 11989110158401015840 (120585

3)

21198911015840 (1205854)

1205854119899

120585119899+1

1205854119899

asymp [119892101584010158401015840 (120585)

3minus 119860]

11989110158402 (1205851) 1198911015840 (120585

2) 11989110158401015840 (120585

3)

21198911015840 (1205854)

(44)







119891 (119909) minus 119901119898(119909) =

11989110158401015840 (1205851) minus 11990110158401015840119898(1205851)

2(119909 minus 119909

119899) (119909 minus 119911

119899) (45)



119896+1 we have

minus1199011015840119898(1205852) (119903 minus 119909

119899+1) =

11989110158401015840 (1205851) minus 11990110158401015840119898(1205851)

2(119903 minus 119909

119899) (119903 minus 119911

119899)

(46)

with 1205851 1205852isin 119868

Since 119911119899= 119909119899minus 119898(119891(119909

119899)1198911015840(119909


119911119899minus 119903 =

11989110158401015840 (1205853) minus 11990110158401015840119898(1205853)

21199011015840119898(1205854)

1205982119899 (47)

for some 1205853 1205854isin 119868

Thus

120598119899+1

=11989110158401015840 (1205851) minus 11990110158401015840119898(1205851)

21199011015840119898(1205852)

11989110158401015840 (1205853) minus 11990110158401015840119898(1205853)

21199011015840 (1205854)

1205983119899 (48)




(120598119899+11205983119899)






1198911(119909) = 1199093 + 41199092 minus 10 119903 = 13652300134140968457608068290

1198912(119909) = cos(119909) minus 119909 119903 = 073908513321516064165531208767

1198913(119909) = sin(119909) minus (12) 119909 119903 = 18954942670339809471440357381

1198914(119909) = sin2(119909) minus 1199092 + 1 119903 = 14044916482153412260350868178

1198915(119909) = 119909119890119909

2

minus sin2(119909) + 3 cos(119909) + 5 119903 = 12076478271309189270094167584

1198916(119909) = 1199092 minus 119890119909 minus 3119909 + 2 119903 = 0257530285439860760455367304944

1198917(119909) = (119909 minus 1)3 minus 2 119903 = 22599210498948731647672106073

1198918(119909) = (119909 minus 1)2 minus 1 119903 = 2

1198919(119909) = 10119909119890minus119909

2

minus 1 119903 = 16796306104284499

11989110(119909) = (119909 + 2)119890119909 minus 1 119903 = minus04428544010023885831413280000

11989111(119909) = 119890minus119909 + cos(119909) 119903 = 1746139530408012417650703089


119891(119909) 1199090

IT NOFE


1198911(119909) 1 6 4 5 4 4 3 3 3 12 12 15 12 12 9 9 9

1198912(119909) 1 5 3 4 2 3 3 3 2 10 9 12 6 9 9 9 6

1198913(119909) 2 5 2 4 3 3 2 3 2 10 6 12 9 9 6 9 6

1198914(119909) 13 5 3 4 4 3 3 3 2 10 9 12 12 9 9 9 6

1198915(119909) -1 6 4 5 4 4 4 4 3 12 12 15 12 12 12 12 9

1198916(119909) 2 6 4 5 4 4 3 4 3 12 12 15 12 12 9 12 9

1198917(119909) 3 7 4 5 4 4 4 3 3 14 12 15 12 12 12 9 9

1198918(119909) 35 6 3 5 3 4 3 3 3 12 9 15 9 12 9 9 9

1198919(119909) 1 6 4 6 4 4 3 3 3 12 12 18 12 12 9 9 9

11989110(119909) 2 9 5 7 6 5 4 4 4 18 15 21 18 15 12 12 12

11989111(119909) 05 5 4 4 4 3 3 3 3 10 12 12 12 9 9 9 9


minus 119909119899| lt 120598 (ii) |119891(119909







5 Conclusions




1198911(119909) = (1199093 + 41199092 minus 10)3 119903 = 13652300134140968457608068290

1198912(119909) = (sin2(119909) minus 1199092 + 1)2 119903 = 14044916482153412260350868178

1198913(119909) = (1199092 minus 119890119909 minus 3119909 + 2)5 119903 = 02575302854398607604553673049

1198914(119909) = (cos(119909) minus 119909)3 119903 = 07390851332151606416553120876

1198915(119909) = ((119909 minus 1)3 minus 1)6 119903 = 2

1198916(119909) = (119909119890119909

2



119891(119909) 1199090


2 5 3 3 3 3 3 10 9 9 9 9 91198911(119909) 1 5 3 4 3 3 3 10 9 12 9 9 9

minus03 54 5 5 5 5 4 108 15 15 15 15 1223 6 4 4 4 4 4 12 12 12 12 12 12

1198912(119909) 2 6 4 4 4 4 4 12 12 12 12 12 12

15 5 4 4 4 3 3 10 12 12 12 9 90 4 3 3 3 3 2 8 9 9 9 9 6

1198913(119909) 1 4 4 4 4 4 3 8 12 12 12 12 9

minus1 5 4 4 4 4 3 10 12 12 12 12 917 5 4 4 4 4 3 10 12 12 12 12 9

1198914(119909) 1 4 3 3 3 3 3 8 9 9 9 9 9

minus3 99 8 8 8 8 7 198 24 24 24 24 213 6 4 5 5 4 4 12 12 15 15 12 12

1198915(119909) minus1 10 11 24 23 5 6 20 33 72 79 15 18

5 8 9 15 15 5 5 16 27 45 45 15 15minus2 8 5 6 6 6 5 16 15 18 18 18 15

1198916(119909) minus1 5 3 4 3 3 5 10 9 12 9 9 15

minus3 14 9 10 10 10 9 28 27 30 30 30 2717 5 3 4 3 4 4 10 9 12 9 12 12

1198917(119909) 2 4 3 3 3 3 3 8 9 9 9 9 9

3 5 3 3 3 3 3 10 9 9 9 9 9


Acknowledgments



References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra















119864 (119910) = 119892 (119910) minus 119901119899(119910) = [

119892101584010158401015840 (120585)

3minus 120573119860] (119910 minus 119870)

2

(119910 minus 119871)

(40)


119899(119910) in (12) (19) and (26)


119903 minus 119909119899+1

= minus [119892101584010158401015840 (120585)

3minus 120573119860]1198702119871

= minus [119892101584010158401015840 (120585)

3minus 120573119860]1198912 (119909

119899) 119891 (119911119899)

119909119899+1

minus 119903 = [119892101584010158401015840 (120585)

3minus 120573119860]

times [119891 (119903) + 1198911015840 (1205851) (119909119899minus 119903)]2

times [119891 (119903) + 1198911015840 (1205852) (119911119899minus 119903)]

(41)

with 119909119899+1

minus 119903 = 120598119899+1 119909119899minus 119903 = 120598

119899 Since 119911



3)21198911015840(120585

4))1205852119899+

119874(1205852119899+1) Then we have

120598119899+1

asymp [119892101584010158401015840 (120585)

3minus 120573119860]

11989110158402 (1205851) 12058521198991198911015840 (1205852) 11989110158401015840 (120585

3) 1205852119899

21198911015840 (1205854)

120598119899+1

asymp [119892101584010158401015840 (120585)

3minus 120573119860]

11989110158402 (1205851) 1198911015840 (120585

2) 11989110158401015840 (120585

3)

21198911015840 (1205854)

1205854119899

(42)


120585119899+1

asymp119892101584010158401015840 (120585)

3

11989110158402 (1205851) 1198911015840 (120585

2) 11989110158401015840 (120585

3)

21198911015840 (1205854)

1205854119899 (43)


120585119899+1

asymp [119892101584010158401015840 (120585)

3minus 119860]

11989110158402 (1205851) 1198911015840 (120585

2) 11989110158401015840 (120585

3)

21198911015840 (1205854)

1205854119899

120585119899+1

1205854119899

asymp [119892101584010158401015840 (120585)

3minus 119860]

11989110158402 (1205851) 1198911015840 (120585

2) 11989110158401015840 (120585

3)

21198911015840 (1205854)

(44)







119891 (119909) minus 119901119898(119909) =

11989110158401015840 (1205851) minus 11990110158401015840119898(1205851)

2(119909 minus 119909

119899) (119909 minus 119911

119899) (45)



119896+1 we have

minus1199011015840119898(1205852) (119903 minus 119909

119899+1) =

11989110158401015840 (1205851) minus 11990110158401015840119898(1205851)

2(119903 minus 119909

119899) (119903 minus 119911

119899)

(46)

with 1205851 1205852isin 119868

Since 119911119899= 119909119899minus 119898(119891(119909

119899)1198911015840(119909


119911119899minus 119903 =

11989110158401015840 (1205853) minus 11990110158401015840119898(1205853)

21199011015840119898(1205854)

1205982119899 (47)

for some 1205853 1205854isin 119868

Thus

120598119899+1

=11989110158401015840 (1205851) minus 11990110158401015840119898(1205851)

21199011015840119898(1205852)

11989110158401015840 (1205853) minus 11990110158401015840119898(1205853)

21199011015840 (1205854)

1205983119899 (48)




(120598119899+11205983119899)






1198911(119909) = 1199093 + 41199092 minus 10 119903 = 13652300134140968457608068290

1198912(119909) = cos(119909) minus 119909 119903 = 073908513321516064165531208767

1198913(119909) = sin(119909) minus (12) 119909 119903 = 18954942670339809471440357381

1198914(119909) = sin2(119909) minus 1199092 + 1 119903 = 14044916482153412260350868178

1198915(119909) = 119909119890119909

2

minus sin2(119909) + 3 cos(119909) + 5 119903 = 12076478271309189270094167584

1198916(119909) = 1199092 minus 119890119909 minus 3119909 + 2 119903 = 0257530285439860760455367304944

1198917(119909) = (119909 minus 1)3 minus 2 119903 = 22599210498948731647672106073

1198918(119909) = (119909 minus 1)2 minus 1 119903 = 2

1198919(119909) = 10119909119890minus119909

2

minus 1 119903 = 16796306104284499

11989110(119909) = (119909 + 2)119890119909 minus 1 119903 = minus04428544010023885831413280000

11989111(119909) = 119890minus119909 + cos(119909) 119903 = 1746139530408012417650703089


119891(119909) 1199090

IT NOFE


1198911(119909) 1 6 4 5 4 4 3 3 3 12 12 15 12 12 9 9 9

1198912(119909) 1 5 3 4 2 3 3 3 2 10 9 12 6 9 9 9 6

1198913(119909) 2 5 2 4 3 3 2 3 2 10 6 12 9 9 6 9 6

1198914(119909) 13 5 3 4 4 3 3 3 2 10 9 12 12 9 9 9 6

1198915(119909) -1 6 4 5 4 4 4 4 3 12 12 15 12 12 12 12 9

1198916(119909) 2 6 4 5 4 4 3 4 3 12 12 15 12 12 9 12 9

1198917(119909) 3 7 4 5 4 4 4 3 3 14 12 15 12 12 12 9 9

1198918(119909) 35 6 3 5 3 4 3 3 3 12 9 15 9 12 9 9 9

1198919(119909) 1 6 4 6 4 4 3 3 3 12 12 18 12 12 9 9 9

11989110(119909) 2 9 5 7 6 5 4 4 4 18 15 21 18 15 12 12 12

11989111(119909) 05 5 4 4 4 3 3 3 3 10 12 12 12 9 9 9 9


minus 119909119899| lt 120598 (ii) |119891(119909







5 Conclusions




1198911(119909) = (1199093 + 41199092 minus 10)3 119903 = 13652300134140968457608068290

1198912(119909) = (sin2(119909) minus 1199092 + 1)2 119903 = 14044916482153412260350868178

1198913(119909) = (1199092 minus 119890119909 minus 3119909 + 2)5 119903 = 02575302854398607604553673049

1198914(119909) = (cos(119909) minus 119909)3 119903 = 07390851332151606416553120876

1198915(119909) = ((119909 minus 1)3 minus 1)6 119903 = 2

1198916(119909) = (119909119890119909

2



119891(119909) 1199090


2 5 3 3 3 3 3 10 9 9 9 9 91198911(119909) 1 5 3 4 3 3 3 10 9 12 9 9 9

minus03 54 5 5 5 5 4 108 15 15 15 15 1223 6 4 4 4 4 4 12 12 12 12 12 12

1198912(119909) 2 6 4 4 4 4 4 12 12 12 12 12 12

15 5 4 4 4 3 3 10 12 12 12 9 90 4 3 3 3 3 2 8 9 9 9 9 6

1198913(119909) 1 4 4 4 4 4 3 8 12 12 12 12 9

minus1 5 4 4 4 4 3 10 12 12 12 12 917 5 4 4 4 4 3 10 12 12 12 12 9

1198914(119909) 1 4 3 3 3 3 3 8 9 9 9 9 9

minus3 99 8 8 8 8 7 198 24 24 24 24 213 6 4 5 5 4 4 12 12 15 15 12 12

1198915(119909) minus1 10 11 24 23 5 6 20 33 72 79 15 18

5 8 9 15 15 5 5 16 27 45 45 15 15minus2 8 5 6 6 6 5 16 15 18 18 18 15

1198916(119909) minus1 5 3 4 3 3 5 10 9 12 9 9 15

minus3 14 9 10 10 10 9 28 27 30 30 30 2717 5 3 4 3 4 4 10 9 12 9 12 12

1198917(119909) 2 4 3 3 3 3 3 8 9 9 9 9 9

3 5 3 3 3 3 3 10 9 9 9 9 9


Acknowledgments



References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1198911(119909) = 1199093 + 41199092 minus 10 119903 = 13652300134140968457608068290

1198912(119909) = cos(119909) minus 119909 119903 = 073908513321516064165531208767

1198913(119909) = sin(119909) minus (12) 119909 119903 = 18954942670339809471440357381

1198914(119909) = sin2(119909) minus 1199092 + 1 119903 = 14044916482153412260350868178

1198915(119909) = 119909119890119909

2

minus sin2(119909) + 3 cos(119909) + 5 119903 = 12076478271309189270094167584

1198916(119909) = 1199092 minus 119890119909 minus 3119909 + 2 119903 = 0257530285439860760455367304944

1198917(119909) = (119909 minus 1)3 minus 2 119903 = 22599210498948731647672106073

1198918(119909) = (119909 minus 1)2 minus 1 119903 = 2

1198919(119909) = 10119909119890minus119909

2

minus 1 119903 = 16796306104284499

11989110(119909) = (119909 + 2)119890119909 minus 1 119903 = minus04428544010023885831413280000

11989111(119909) = 119890minus119909 + cos(119909) 119903 = 1746139530408012417650703089


119891(119909) 1199090

IT NOFE


1198911(119909) 1 6 4 5 4 4 3 3 3 12 12 15 12 12 9 9 9

1198912(119909) 1 5 3 4 2 3 3 3 2 10 9 12 6 9 9 9 6

1198913(119909) 2 5 2 4 3 3 2 3 2 10 6 12 9 9 6 9 6

1198914(119909) 13 5 3 4 4 3 3 3 2 10 9 12 12 9 9 9 6

1198915(119909) -1 6 4 5 4 4 4 4 3 12 12 15 12 12 12 12 9

1198916(119909) 2 6 4 5 4 4 3 4 3 12 12 15 12 12 9 12 9

1198917(119909) 3 7 4 5 4 4 4 3 3 14 12 15 12 12 12 9 9

1198918(119909) 35 6 3 5 3 4 3 3 3 12 9 15 9 12 9 9 9

1198919(119909) 1 6 4 6 4 4 3 3 3 12 12 18 12 12 9 9 9

11989110(119909) 2 9 5 7 6 5 4 4 4 18 15 21 18 15 12 12 12

11989111(119909) 05 5 4 4 4 3 3 3 3 10 12 12 12 9 9 9 9


minus 119909119899| lt 120598 (ii) |119891(119909







5 Conclusions




1198911(119909) = (1199093 + 41199092 minus 10)3 119903 = 13652300134140968457608068290

1198912(119909) = (sin2(119909) minus 1199092 + 1)2 119903 = 14044916482153412260350868178

1198913(119909) = (1199092 minus 119890119909 minus 3119909 + 2)5 119903 = 02575302854398607604553673049

1198914(119909) = (cos(119909) minus 119909)3 119903 = 07390851332151606416553120876

1198915(119909) = ((119909 minus 1)3 minus 1)6 119903 = 2

1198916(119909) = (119909119890119909

2



119891(119909) 1199090


2 5 3 3 3 3 3 10 9 9 9 9 91198911(119909) 1 5 3 4 3 3 3 10 9 12 9 9 9

minus03 54 5 5 5 5 4 108 15 15 15 15 1223 6 4 4 4 4 4 12 12 12 12 12 12

1198912(119909) 2 6 4 4 4 4 4 12 12 12 12 12 12

15 5 4 4 4 3 3 10 12 12 12 9 90 4 3 3 3 3 2 8 9 9 9 9 6

1198913(119909) 1 4 4 4 4 4 3 8 12 12 12 12 9

minus1 5 4 4 4 4 3 10 12 12 12 12 917 5 4 4 4 4 3 10 12 12 12 12 9

1198914(119909) 1 4 3 3 3 3 3 8 9 9 9 9 9

minus3 99 8 8 8 8 7 198 24 24 24 24 213 6 4 5 5 4 4 12 12 15 15 12 12

1198915(119909) minus1 10 11 24 23 5 6 20 33 72 79 15 18

5 8 9 15 15 5 5 16 27 45 45 15 15minus2 8 5 6 6 6 5 16 15 18 18 18 15

1198916(119909) minus1 5 3 4 3 3 5 10 9 12 9 9 15

minus3 14 9 10 10 10 9 28 27 30 30 30 2717 5 3 4 3 4 4 10 9 12 9 12 12

1198917(119909) 2 4 3 3 3 3 3 8 9 9 9 9 9

3 5 3 3 3 3 3 10 9 9 9 9 9


Acknowledgments



References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1198911(119909) = (1199093 + 41199092 minus 10)3 119903 = 13652300134140968457608068290

1198912(119909) = (sin2(119909) minus 1199092 + 1)2 119903 = 14044916482153412260350868178

1198913(119909) = (1199092 minus 119890119909 minus 3119909 + 2)5 119903 = 02575302854398607604553673049

1198914(119909) = (cos(119909) minus 119909)3 119903 = 07390851332151606416553120876

1198915(119909) = ((119909 minus 1)3 minus 1)6 119903 = 2

1198916(119909) = (119909119890119909

2



119891(119909) 1199090


2 5 3 3 3 3 3 10 9 9 9 9 91198911(119909) 1 5 3 4 3 3 3 10 9 12 9 9 9

minus03 54 5 5 5 5 4 108 15 15 15 15 1223 6 4 4 4 4 4 12 12 12 12 12 12

1198912(119909) 2 6 4 4 4 4 4 12 12 12 12 12 12

15 5 4 4 4 3 3 10 12 12 12 9 90 4 3 3 3 3 2 8 9 9 9 9 6

1198913(119909) 1 4 4 4 4 4 3 8 12 12 12 12 9

minus1 5 4 4 4 4 3 10 12 12 12 12 917 5 4 4 4 4 3 10 12 12 12 12 9

1198914(119909) 1 4 3 3 3 3 3 8 9 9 9 9 9

minus3 99 8 8 8 8 7 198 24 24 24 24 213 6 4 5 5 4 4 12 12 15 15 12 12

1198915(119909) minus1 10 11 24 23 5 6 20 33 72 79 15 18

5 8 9 15 15 5 5 16 27 45 45 15 15minus2 8 5 6 6 6 5 16 15 18 18 18 15

1198916(119909) minus1 5 3 4 3 3 5 10 9 12 9 9 15

minus3 14 9 10 10 10 9 28 27 30 30 30 2717 5 3 4 3 4 4 10 9 12 9 12 12

1198917(119909) 2 4 3 3 3 3 3 8 9 9 9 9 9

3 5 3 3 3 3 3 10 9 9 9 9 9


Acknowledgments



References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Three New Optimal Fourth-Order …downloads.hindawi.com/journals/ana/2013/957496.pdf · Research Article Three New Optimal Fourth-Order ... [email protected]