Research ArticleEfficient Iterative Methods with and without Memory PossessingHigh Efficiency Indices

T Lotfi1 F Soleymani2 Z Noori1 A KJlJccedilman3 and F Khaksar Haghani2

1 Department of Mathematics Islamic Azad University Hamedan Branch Hamedan Iran2Department of Mathematics Islamic Azad University Shahrekord Branch Shahrekord Iran3Department of Mathematics and Institute for Mathematical Research Universiti Putra Malaysia 43400 Serdang Malaysia

Correspondence should be addressed to A Kılıcman akilicupmedumy

Received 2 June 2014 Accepted 20 July 2014 Published 3 September 2014

Academic Editor Krzysztof Cieplinski

Copyright copy 2014 T Lotfi et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Two families of derivative-free methods without memory for approximating a simple zero of a nonlinear equation are presentedThe proposed schemes have an accelerator parameter with the property that it can increase the convergence rate without anynew functional evaluations In this way we construct a method with memory that increases considerably efficiency index from814

asymp 1681 to 1214 asymp 1861 Numerical examples and comparison with the existing methods are included to confirm theoreticalresults and high computational efficiency

1 Preliminaries

The main goal and motivation in constructing iterativemethods for solving nonlinear equations is to obtain as highas possible order of convergencewithminimal computationalcost (see eg [1ndash3]) Hence many researchers (see eg [45]) have paid much attention to construct optimal multipointmethods without memory based on the unproved conjectureof Kung and Traub [6] which indicates that any multipointiterative method without memory using 119899 + 1 functionalevaluations can reach the optimal order 2119899

Let 120572 be a simple real zero of a real function 119891 119863 sube

R rarr R and let 1199090be an initial approximation to 120572 In

many practical situations it is preferable to avoid calculationsof derivatives of 119891 This makes the construction of iterativederivative-free methods important [7]

This paper follows two main goals First developingsome optimal three-step derivative-free families of methodswithout memory And second developing the proposedfamilies to methods with memory in a way that reachesconvergence R-orders 6 and 12 without any new functionalevaluation

The main idea in methods with memory is based on theuse of suitable two-valued functions and the variation of a free

parameter in each iterative step This parameter is calculatedusing information from the current and previous iteration sothat the developedmethodsmay be regarded asmethodswithmemory following Traubrsquos classification [8] A supplementalinspiration for studying methods with memory stands upfrom an astonishing fact that such classes of iterativemethodshave been considered in the literature very rarely despite theirhigh efficiency indices

The paper is organized as follows First two families ofoptimal methods with orders four and eight consuming threeand four function evaluations per iteration respectively areproposed in Section 2 Then in Section 3 we state methodswith memory of very high computational efficiencies Theincrease of convergence speed is achieved without additionalfunction evaluations which is the main advantage of themethods with memory Section 4 is assigned to numericalresults connected to the order of the methods with andwithout memory Concluding remarks are given in Section 5

2 Construction of New FamiliesThis section deals with constructing newmultipointmethodsfor solving nonlinear equations The discussions are dividedinto two subsections Let the scalar function119891 119863 sub R rarr R

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2014 Article ID 912796 9 pageshttpdxdoiorg1011552014912796

2 Discrete Dynamics in Nature and Society

and 119891(120572) = 0 = 1198911015840(120572) = 119888

1 that is 120572 is a simple zero of

119891(119909) = 0

21 New Two-Step Methods without Memory In this subsec-tion we start with the two-step scheme

119910 = 119909 minus119891 (119909)

1198911015840 (119909) 119909 = 119910 minus

119891 (119910)

1198911015840 (119910) (1)

Note that we omit the iteration index 119896 for the sakeof simplicity only The order of convergence of scheme(1) is four but it does not contain any contributions Wesubstitute derivatives in the first and second steps by suitableapproximations that use available data thus we introduce anapproximation as follows

Using the points 119909 and119908 = 119909+ 120574119891(119909) (120574 is a nonzero realconstant) we can apply Lagrangersquos interpolatory polynomialfor approximating 1198911015840(119909)

1198751(119905) =

119905 minus 119908

119909 minus 119908119891 (119909) +

119905 minus 119909

119908 minus 119909119891 (119908) (2)

Fix 1198911015840(119909) asymp 1198751015840

1(119909) and by setting 119905 = 119909 we have

1198751015840

1(119909) =

119891 (119909)

119909 minus 119908+119891 (119908)

119908 minus 119909=119891 (119909) minus 119891 (119908)

119909 minus 119908= 119891 [119909 119908] (3)

Also Lagrangersquos interpolatory polynomial at points 119909 119908and 119910 for approximating 1198911015840(119910) can be given as follows

1198752(119905) =

(119905 minus 119908) (119905 minus 119910)

(119909 minus 119908) (119909 minus 119910)119891 (119909) +

(119905 minus 119909) (119905 minus 119910)

(119908 minus 119909) (119908 minus 119910)119891 (119908)

+(119905 minus 119908) (119905 minus 119909)

(119910 minus 119908) (119910 minus 119909)119891 (119910)

(4)

We obtain

1198911015840(119910) asymp 119875

1015840

2(119910)

=2119910 minus 119908 minus 119910

(119909 minus 119908) (119909 minus 119910)119891 (119909)

+2119910 minus 119909 minus 119910

(119908 minus 119909) (119908 minus 119910)119891 (119908)

+2119910 minus 119908 minus 119909

(119910 minus 119908) (119910 minus 119909)119891 (119910)

(5)

Finally we set above approximations in the denominator of(1) and so our derivative-free two-step iterative method isderived in what follows

119910 = 119909 minus119891 (119909)

119891 [119909 119908] 119908 = 119909 + 120574119891 (119909)

119911 = 119910 minus119891 (119910)

1198751015840

2(119910)

(6)

Now to check the convergence order of (6) we avoidretyping the widely practiced approach in the literature andput forward the following self-explained Mathematica code

f[e ] = f1a lowast (e + sum4

k=2 ck lowast ek)ew = e + 120574f[e] (lowast119890119908 = 119908 minus 120572lowast)

f[x y ] =f[x] minus f[y]

x minus y

p1[t ] =t minus ew

e minus ewf[e] +

t minus e

ew minus ef[ew]

dp1[t ] =f[e]

e minus ew+

f[ew]

ew minus e

ey = e minus Series[f[e]p11015840[ew]

e 0 3]FullSimplify (lowastey = y minus 120572lowast)

Out[a] (1 + f1a120574)c2e2 + O[e]3

p2[t ] =(t minus ew)(t minus ey)

(e minus ew)(e minus ey)f[e]

+(t minus e)(t minus ey)

(ew minus e)(ew minus ey)f[ew]

+(t minus ew)(t minus e)

(ey minus ew)(ey minus e)f[ey]

dp2[t ] =2t minus ew minus ey

(e minus ew)(e minus ey)f[e]

+2t minus e minus ey

(ew minus e)(ew minus ey)f[ew]

+2t minus ew minus e

(ey minus ew)(ey minus e)f[ey]

ez = ey minusf[ey]

dp2[ey]FullSimplify

Out[b] (1 + f1a120574)2c2(c22minus c3)e4 + O[e]5

Considering Out[b] of the aboveMathematica programwe observe that the order of convergence of the family (6)is four and so we can state a convergence theorem in whatfollows

Theorem 1 If an initial approximation 1199090is sufficiently close

to a simple zero 120572 of 119891 then the convergence order of the two-step approach (6) is equal to four And its error equation is givenby

119890119896+1

= (1 + 1205741198911015840(120572))2

1198882(1198882

2minus 1198883) 1198904

119896+ 119874 (119890

5

119896) (7)

22 New Three-Step Family Now we construct a three-stepuniparametric family of methods based on the two-stepmethod (6)We start from a three-step schemewhere the firsttwo steps are given by (6) and also the third step is Newtonrsquosmethod that is

119910 = 119909 minus119891 (119909)

119891 [119909 119908] 119908 = 119909 + 120574119891 (119909)

119911 = 119910 minus119891 (119910)

1198751015840

2(119905)

119909 = 119911 minus119891 (119911)

1198911015840 (119911)

(8)

The derivative 1198911015840(119911) in the third step of (8) should be

substituted by a suitable approximation in order to obtain ashigh as possible of convergence order and tomake the schemeoptimal To provide this approximation we apply Lagrangersquos

Discrete Dynamics in Nature and Society 3

interpolatory polynomial at the points 119908 = 119909 + 120574119891(119909) 119909 119910and 119911 that is

1198753(119905) =

(119905 minus 119908) (119905 minus 119910) (119905 minus 119911)

(119909 minus 119908) (119909 minus 119910) (119909 minus 119911)119891 (119909)

+(119905 minus 119909) (119905 minus 119910) (119905 minus 119911)

(119908 minus 119909) (119908 minus 119910) (119908 minus 119911)119891 (119908)

+(119905 minus 119908) (119905 minus 119909) (119905 minus 119911)

(119910 minus 119908) (119910 minus 119909) (119910 minus 119911)119891 (119910)

+(119905 minus 119908) (119905 minus 119909) (119905 minus 119910)

(119911 minus 119908) (119911 minus 119909) (119911 minus 119910)119891 (119911)

(9)

It is obvious that 1198753(119911) = 119891(119911) By differentiating (9) and

setting 119905 = 119911 we obtain

1198751015840

3(119911) =

119908119910 + 119908119911 + 119910119911 minus 2 (119908 + 119910 + 119911) 119911 + 31199112

((119909 minus 119908) (119909 minus 119910) (119909 minus 119911))119891 (119909)

+119909119910 + 119909119911 + 119910119911 minus 2 (119909 + 119910 + 119911) 119911 + 3119911

2

(119908 minus 119909) (119908 minus 119910) (119908 minus 119911)119891 (119908)

+119909119908 + 119909119911 + 119908119911 minus 2 (119909 + 119908 + 119911) 119911 + 3119911

2

(119910 minus 119908) (119910 minus 119909) (119910 minus 119911)119891 (119910)

+119909119908 + 119909119910 + 119908119910 minus 2 (119909 + 119908 + 119910) 119911 + 3119911

2

(119911 minus 119908) (119911 minus 119909) (119911 minus 119910)119891 (119911)

(10)

By substituting 1198911015840(119911) asymp 1198751015840

3(119911) in (8) we have

119910 = 119909 minus119891 (119909)

119891 [119909 119908] 119908 = 119909 + 120574119891 (119909)

119911 = 119910 minus119891 (119910)

1198751015840

2(119910)

119909 = 119911 minus119891 (119911)

1198751015840

3(119911)

(11)

where 11987510158402(119910) is defined by (5) and 1198751015840

3(119911) given by (10)

In the following theorem we state suitable conditions forderiving an optimal three-step scheme without memory

Theorem 2 Let 119891 119863 sub R rarr R be a scalar functionwhich has the simple root 120572 in the open interval 119863 also initialapproximation 119909

0is sufficiently close to a simple zero 120572 of

119891 The three-step iterative method (11) has eighth order andsatisfies the following error equation119890119896+1

= (1 + 1205741198911015840(120572)4) 1198882

2(1198882

2minus 1198883) (1198883

2minus 11988821198883+ 1198884) 1198908

119896+ 119874 (119890

9

119896)

(12)

Proof We are going to employ symbolic computation in thecomputational software package Mathematica We write thefollowing

119891 (119909119896) = 1198911015840(120572) (119890

119896+ 11988821198902

119896+ 11988831198903

119896+ 11988841198904

119896+ 11988851198905

119896+ 11988861198906

119896

+11988871198907

119896+ 11988881198908

119896+ 119874 (119890

9

119896))

(13)

Now by introducing the abbreviations 119890 = 119909 minus 120572 119890119908 = 119908 minus 120572119890119910 = 119910minus120572 119890119911 = 119911minus120572 and 119888

119896= 119891(119896)(120572)(119896119891

1015840(120572)) we provide

the following Mathematica program in order to obtain theconvergence order of (11)

ProgramWritten in Mathematica Consider the following

f[e ] = f1a lowast (e + sum4

k=2 ck lowast ek)

ew = e + 120574f[e]

f[x y ] =f[x] minus f[y]

x minus y

p1[t ] =t minus ew

e minus ewf[e] +

t minus e

ew minus ef[ew]

dp1[t ] =f[e]

e minus ew+

f[ew]

ew minus e

ey = e minus Series[f[e]p11015840[ew]

e 0 8]FullSimplify

Out[a] (1 + 120574f1a)c2e2 + O[e]3

p2[t ] =(t minus ew)(t minus ey)

(e minus ew)(e minus ey)f[e]

+(t minus e)(t minus ey)

(ew minus e)(ew minus ey)f[ew]

+(t minus ew)(t minus e)

(ey minus ew)(ey minus e)f[ey]

dp2[t ] =2t minus ew minus ey

(e minus ew)(e minus ey)f[e]

+2t minus e minus ey

(ew minus e)(ew minus ey)f[ew]

+2t minus ew minus e

(ey minus ew)(ey minus e)f[ey]

ez = ey minusf[ey]

dp2[ey]FullSimplify

Out[b] (1 + 120574f1a)2c2(c22minus c3)e4 + O[e]5

dp3[t ]

=ewey + ewez + eyez minus 2(ew + ey + ez)t + 3t2

(e minus ew)(e minus ey)(e minus ez)f[e]

+eey + eez + eyez minus 2(e + ey + ez)t + 3t2

(ew minus e)(ew minus ey)(ew minus ez)f[ew]

+eew + eez + ewez minus 2(e + ew + ez)t + 3t2

(ey minus ew)(ey minus e)(ey minus ez)f[ey]

+eew + eey + ewey minus 2(e + ew + ey)t + 3t2

(ez minus ew)(ez minus e)(ez minus ey)f[ez]

ex = ez minusf[ez]

dp3[ez]FullSimplify

Out[c] (1 + 120574f1a)4c22(c22minus c3)(c32minus c2c3+ c4)e8

+O[e]9

As a result the proof of the theorem is finished Accordingto Out[c] it possesses eighth order

Error equations (7) and (12) indicate that the orders ofmethods (6) and (11) are four and eight respectively

In the next section we will modify the proposedmethodsand introduce new methods with memory With the useof accelerator parameters the order of convergence willsignificantly increase

4 Discrete Dynamics in Nature and Society

3 Extension to Methods with MemoryThis section is concernedwith extraction of efficientmethodswith memory from (6) and (11) through a careful inspectionof their error equations containing the parameter 120574 whichcan be approximated in such a way that increases the localorder of convergence

Toward this goal we set 120574 = 120574119896as the iteration proceeds by

the formula 120574119896= minus11198911015840(120572) for 119896 = 1 2 where 1198911015840(120572) is an

approximation of 1198911015840(120572) We therefore use the approximation

120574119896= minus

1

1198911015840 (120572)

= minus1

1198731015840

3(119909119896) (14)

for (6) and the following one

120574119896= minus

1

1198911015840 (120572)

= minus1

1198731015840

4(119909119896) (15)

for (11) Here we consider Newtonrsquos interpolating polynomialof third degree as the method for approximating 119891

1015840(120572) in

two-step method (6) and Newtonrsquos interpolating polynomialof fourth degree for approximating 119891

1015840(120572) in the three-

step method (11) where 1198733(119905) is the Newtonrsquos interpola-

tion polynomial of third degree set through four availableapproximations 119909

119896 119909119896minus1

119910119896minus1

119908119896minus1

and1198734(119905) is the Newtonrsquos

interpolation polynomial of fourth degree set through fiveavailable approximations 119909

119896 119911119896minus1

119910119896minus1

119908119896minus1

119909119896minus1

Consider

1198731015840

3(119909119896) = [

119889

1198891199051198733(119905)]

119905=119909119896

= [119889

119889119905(119891 (119909119896) + 119891 [119909

119896 119909119896minus1

] (119905 minus 119909119896)

+ 119891 [119909119896 119909119896minus1

119910119896minus1

] (119905 minus 119909119896) (119905 minus 119909

119896minus1)

+ 119891 [119909119896 119909119896minus1

119910119896minus1

119908119896minus1

] (119905 minus 119909119896)

times (119905 minus 119909119896minus1

) (119905 minus 119910119896minus1

)) ]

119905=119909119896

= 119891 [119909119896 119909119896minus1

] + 119891 [119909119896 119909119896minus1

119910119896minus1

] (119909119896minus 119909119896minus1

)

+ 119891 [119909119896 119909119896minus1

119910119896minus1

119908119896minus1

] (119909119896minus 119909119896minus1

) (119909119896minus 119910119896minus1

)

(16)

1198731015840

4(119909119896)

= [119889

1198891199051198734(119905)]

119905=119909119896

= [119889

119889119905(119891 (119909119896) + 119891 [119909

119896 119911119896minus1

] (119905 minus 119909119896)

+ 119891 [119909119896 119911119896minus1

119910119896minus1

] (119905 minus 119909119896) (119905 minus 119911

119896minus1)

+ 119891 [119909119896 119911119896minus1

119910119896minus1

119909119896minus1

] (119905 minus 119909119896)

times (119905 minus 119911119896minus1

) (119905 minus 119910119896minus1

)

+ 119891 [119909119896 119911119896minus1

119910119896minus1

119909119896minus1

119908119896minus1

]

times(119905minus119909119896)(119905 minus 119911

119896minus1)(119905minus 119910

119896minus1)(119905minus 119909

119896minus1))]

119905=119909119896

= 119891 [119909119896 119911119896minus1

] + 119891 [119909119896 119911119896minus1

119910119896minus1

] (119909119896minus 119911119896minus1

)

+ 119891 [119909119896 119911119896minus1

119910119896minus1

119909119896minus1

] (119909119896minus 119911119896minus1

)(119909119896minus 119910119896minus1

)

+ 119891 [119909119896 119911119896minus1

119910119896minus1

119909119896minus1

119908119896minus1

] (119909119896minus 119911119896minus1

)

times (119909119896minus 119910119896minus1

) (119909119896minus 119909119896minus1

)

(17)

Note that a divided difference of order 119898 defined recur-sively as

119891 [1199090 1199091 119909

119898]

=119891 [1199091 119909

119898] minus 119891 [119909

0 1199091 119909

119898minus1]

119909119898minus 1199090

119898 ⩾ 2

(18)

has been used throughout this paper Hence the with mem-ory developments of (6) and (11) can be presented as follows

119910119896= 119909119896minus

119891 (119909119896)

119891 [119909119896 119908119896] 119908119896= 119909119896+ 120574119896119891 (119909119896)

119909119896+1

= 119910119896minus

119891 (119910119896)

1198751015840

2(119910119896)

(19)

119910119896= 119909119896minus

119891 (119909119896)

119891 [119909119896 119908119896] 119908119896= 119909119896+ 120574119896119891 (119909119896)

119911119896= 119910119896minus119891 (119910119896)

1198751015840

2(119905119896)

119909 = 119909119896+1

= 119911119896minus

119891 (119911119896)

1198751015840

3(119911119896)

(20)

Remark 3 Accelerating methods obtained by recursivelycalculated free parameter may also be called self-acceleratingmethodsThe initial value 120574

0should be chosen before starting

the iterative process for example using one of the waysproposed in [8]

Here we attempt to prove that themethods withmemory(19) and (20) have convergence orders six and twelve providedthat we use accelerator 120574

119896as in (14) and (15) For ease of

continuing analysis we introduce the convenient notation asfollows If the sequence 119909

119896 converges to the zero 120572 of119891with

the order 119901 then we write 119890119896+1

sim 119890119901

119896 where 119890

119896= 119909119896minus 120572

The following lemma will play a crucial role in improvingthe convergence order of the methods with memory to beproposed in this paper

Lemma 4 If 120574119896= minus1119873

1015840

3(119909119896) 119890119896= 119909119896minus 120572 119890119896119910

= 119910119896minus 120572 and

119890119896119908

= 119908119896minus 120572 then the following relation holds

1 + 1205741198961198911015840(120572) sim 119888

4119890119896minus1

119890119896minus1119908

119890119896minus1119910

sim 119890119896minus1

119890119896minus1119908

119890119896minus1119910

(21)

Discrete Dynamics in Nature and Society 5

Proof Following the same terminology as in Theorem 2 andthe symbolic software Mathematica it would be easy toobtain (21) via writing the following code

ClearAll[Globallsquo lowast ]A[t ] = InterpolatingPolynomial[e fxew fw ey fy e1 fx1 t]SimplifyApproximation = minus1A1015840[e1]Simplifyfx = f1a lowast (e + c2 lowast e2 + c3 lowast e3 + c4 lowast e4)fw = f1a lowast (ew + c2 lowast ew2 + c3 lowast ew3

+ c4 lowast ew4)fy = f1a lowast (ey + c2 lowast ey2 + c3 lowast ey3

+ c4 lowast ey4)fx1 = f1a lowast (e1 + c2 lowast e12 + c3 lowast e13

+ c4 lowast e14)b = Series[Approximation e 0 2 ew 0 2ey 0 2 e1 0 2]SimplifyCollect[Series[1 + b lowast f1a e 0 1 ew 0 1ey 0 1 e1 0 0]e ew ey e1 Simplify]

The proof is complete

In order to obtain the R-order of convergence [9] ofthe method with memory (19) we establish the followingtheorem

Theorem 5 If an initial approximation 1199090is sufficiently close

to the zero 120572 of 119891(119909) and the parameter 120574119896in the iterative

scheme (19) is recursively calculated by the forms given in (14)then the R-order of convergence for (19) is at least six

Proof Let 119909119896 be a sequence of approximations generated

by the iterative method with memory (19) If this sequenceconverges to the zero 120572 of 119891 with the order 119901 then we write

119890119896+1

sim 119890119901

119896 119890

119896= 119909119896minus 120572 (22)

Thus

119890119896+1

sim (119890119901

119896minus1)119901

= 1198901199012

119896minus1 (23)

Moreover assume that the iterative sequences119908119896and 119910119896have

the orders 1199011and 119901

2 respectively Then (22) gives

119890119896119908

sim 1198901199011

119896sim (119890119901

119896minus1)1199011= 1198901199011199011

119896minus1 (24)

119890119896119910

sim 1198901199012

119896sim (119890119901

119896minus1)1199012= 1198901199011199012

119896minus1 (25)

Since

119890119896119908

sim (1 + 1205741198961198911015840(120572)) 119890

119896 (26)

119890119896119910

sim (1 + 1205741198961198911015840(120572)) 119890

2

119896 (27)

119890119896+1

sim (1 + 1205741198961198911015840(120572))2

1198904

119896 (28)

using Lemma 4 and (27) induce

119890119896119908

sim (1 + 1205741198961198911015840(120572)) 119890

119896sim (119890119896minus1

119890119896minus1119910

119890119896minus1119908

) 119890119896= 119890119901+1199011+1199012+1

119896minus1

(29)

119890119896119910

sim (1 + 1205741198961198911015840(120572)) 119890

2

119896sim (119890119896minus1

119890119896minus1119910

119890119896minus1119908

) 1198902

119896= 1198902119901+1199011+1199012+1

119896minus1

(30)

119890119896+1

sim (1 + 1205741198961198911015840(120572))2

1198904

119896sim (119890119896minus1

119890119896minus1119910

119890119896minus1119908

)2

1198904

119896

= 1198904119901+21199011+21199012+2

119896minus1

(31)

Matching the powers of 119890119896minus1

on the right hand sides of (24)ndash(29) (25)ndash(30) and (23)ndash(31) one can obtain

1199011199011minus 119901 minus 119901

1minus 1199012minus 1 = 0

1199011199012minus 2119901 minus 119901

1minus 1199012minus 1 = 0

1199012minus 4119901 minus 2119901

1minus 21199012minus 2 = 0

(32)

The nontrivial solution of this system is 1199011= 2 119901

2= 3 and

119901 = 6 This completes the proof

Using symbolic computations and Taylor expansions it iseasy to derive the following lemma

Lemma 6 Assuming (15) and (17) we have

1 + 1205741198961198911015840(120572) sim 119890

119896minus1119890119896minus1119908

119890119896minus1119910

119890119896minus1119911

(33)

where 120574119896= minus1119873

1015840

4(119909119896) 119890119896= 119909119896minus120572 119890119896119911

= 119911119896minus120572 119890119896119910

= 119910119896minus120572

and 119890119896119908

= 119908119896minus 120572

Proof The proof of this lemma is similar to Lemma 4 It ishence omitted

Similarly for the three-step method with memory (20)we have the following theorem

Theorem 7 If an initial approximation 1199090is sufficiently close

to the zero 120572 of 119891(119909) and the parameter 120574119896in the iterative

scheme (20) is recursively calculated by the forms given in (15)then the order of convergence for (20) is at least twelve

Proof Let 119909119896 be a sequence of approximations generated

by the iterative method with memory (20) If this sequenceconverges to the zero 120572 of 119891 with the order 119901 then we write

119890119896+1

sim 119890119901

119896 119890

119896= 119909119896minus 120572 (34)

So

119890119896+1

sim (119890119901

119896minus1)119901

= 1198901199012

119896minus1 (35)

Moreover assume that the iterative sequences 119908119896 119910119896 and 119911

119896

have the orders 1199011 1199012 and 119901

3 respectively Then (34) gives

119890119896119908

sim 1198901199011

119896sim (119890119901

119896minus1)1199011= 1198901199011199011

119896minus1 (36)

119890119896119910

sim 1198901199012

119896sim (119890119901

119896minus1)1199012= 1198901199011199012

119896minus1 (37)

119890119896119911

sim 1198901199013

119896sim (119890119901

119896minus1)1199013= 1198901199011199013

119896minus1 (38)

6 Discrete Dynamics in Nature and Society

Since

119890119896119908

sim (1 + 1205741198961198911015840(120572)) 119890

119896 (39)

119890119896119910

sim (1 + 1205741198961198911015840(120572)) 119890

2

119896 (40)

119890119896119911

sim (1 + 1205741198961198911015840(120572))2

1198904

119896 (41)

119890119896+1

sim (1 + 1205741198961198911015840(120572))4

1198908

119896 (42)

by Lemma 6 and (40) we obtain

119890119896119908

sim (1 + 1205741198961198911015840(120572)) 119890

119896sim (119890119896minus1

119890119896minus1119910

119890119896minus1119908

119890119896minus1119911

) 119890119896

= 119890119901+1199011+1199012+1199013+1

119896minus1

(43)

119890119896119910

sim (1 + 1205741198961198911015840(120572)) 119890

2

119896sim (119890119896minus1

119890119896minus1119910

119890119896minus1119908

119890119896minus1119911

) 1198902

119896

= 1198902119901+1199011+1199012+1199013+1

119896minus1

(44)

119890119896119911

sim (1 + 1205741198961198911015840(120572)) 119890

2

119896sim (119890119896minus1

119890119896minus1119910

119890119896minus1119908

119890119896minus1119911

)2

1198904

119896

= 1198904119901+21199011+21199012+21199013+2

119896minus1

(45)

119890119896+1

sim (1 + 1205741198961198911015840(120572))2

1198904

119896sim (119890119896minus1

119890119896minus1119910

119890119896minus1119908

119890119896minus1119911

)4

1198908

119896

= 1198908119901+41199011+41199012+41199013+4

119896minus1

(46)

Matching the powers of 119890119896minus1

on the right hand sides of (36)ndash(43) (37)ndash(44) (38)ndash(45) and (35)ndash(46) one can obtain

1199011199011minus 119901 minus 119901

1minus 1199012minus 1199013minus 1 = 0

1199011199012minus 2119901 minus 119901

1minus 1199012minus 1199013minus 1 = 0

1199011199013minus 4119901 minus 2119901

1minus 21199012minus 21199013minus 2 = 0

1199012minus 8119901 minus 4119901

1minus 41199012minus 41199013minus 2 = 0

(47)

This system has the solutions 1199011= 2 119901

2= 3 119901

3= 6 and

119901 = 12 The proof is complete

Remark 8 The advantage of the proposedmethods is in theirhigher computational efficiency indices We emphasize thatthe increase of the R-order of convergence has been obtainedwithout any additional function evaluations which pointsto very high computational efficiency Indeed the efficiencyindex 1214 asymp 1861 of the proposed three-step twelfth-ordermethod with memory is higher than the efficiency index613

asymp 1817 of (19) 814 asymp 1682 of the optimal three-pointmethod (11) and 413 asymp 1587 of (6)

Remark 9 We observe that the methods (19) and (20) withmemory are considerably accelerated (up to 50) in contrastto the corresponding method (11) without memory

4 Numerical Experiments

In this section we test our proposed methods and comparetheir results with some other methods of the same order of

convergenceThe results are reported using the programmingpackage Mathematica 8 in multiple precision arithmeticenvironment We have considered 1000 digits floating pointarithmetic so as to minimize the round-off errors as muchas possible The errors |119909

119896minus 120572| denote approximations to

the sought zeros and 119886(minus119887) stands for 119886 times 10minus119887 Moreover

coc indicates the computational order of convergence and iscomputed by

coc =log (1003816100381610038161003816119891 (119909

119896) 119891 (119909

119896minus1)1003816100381610038161003816)

log (1003816100381610038161003816119891 (119909119896minus1

) 119891 (119909119896minus2

)1003816100381610038161003816) (48)

It is assumed that the initial estimate 1205740should be chosen

before starting the iterative process and also 1199090is given

suitablySeveral iterative methods of optimal orders four and

eight for comparing with our proposed methods have beenchosen as comes next

Derivative-free Kung-Traubrsquos two-step method (KT4) [6]is as follows

119910119896= 119909119896minus

119891 (119909119896)

119891 [119909119896 119908119896] 119908119896= 119909119896+ 120574119891 (119909

119896)

119909119896+1

= 119910119896minus

119891 (119910119896) 119891 (119908

119896)

[119891 (119908119896) minus 119891 (119910

119896)] 119891 [119909

119896 119910119896]

(49)

Two-step method by Zheng et al (ZLH4) [10] is asfollows

119910119896= 119909119896minus

119891 (119909119896)

119891 [119909119896 119908119896] 119908119896= 119909119896+ 120574119891 (119909

119896)

119909119896+1

= 119910119896minus

119891 (119910119896)

119891 [119910119896 119908119896] + 119891 [119910

119896 119909119896 119908119896] (119910119896minus 119909119896)

(50)

Two-step method by Lotfi and Tavakoli (LT4) [11] is asfollows

119910119896= 119909119896minus

119891 (119909119896)

119891 [119909119896 119908119896]

119909119896+1

= 119910119896minus 119867 (119905

119896 119906119896)

119891 (119910119896)

119891 [119910119896 119908119896]

119905119896=119891 (119910119896)

119891 (119909119896) 119906

119896=119891 (119908119896)

119891 (119909119896)

(51)

where119867(119905119896 119906119896) = 1 + 119905

119896

Discrete Dynamics in Nature and Society 7

Table 1 1198911(119909) = sin(120587119909)119890(119909

2+119909 cos(119909)minus1)

+ 119909 log(119909 sin(119909) + 1) 120572 = 0 and 1199090= 06

Methods |1199091minus 120572| |119909

2minus 120572| |119909

3minus 120572| coc

Without memoryNew method (6) 120574 = minus1 06742(minus2) 01587(minus10) 04307(minus45) 40049KT4 120574 = minus001 06012(minus1) 01486(minus4) 06094(minus19) 40171ZLH4 120574 = minus001 05882(minus1) 01320(minus5) 08291(minus24) 39365LT 120574 = minus001 06178(minus1) 03334(minus4) 02739(minus17) 40368

With memoryNew method (19) 120574 = minus1 06742(minus2) 03177(minus11) 08372(minus62) 54213KT4 120574 = minus1 02102(minus1) 09935(minus9) 02326(minus48) 54031ZLH4 120574 = minus001 05882(minus1) 02204(minus8) 01503(minus45) 50216LT4 120574 = minus1 01752(minus1) 04934(minus9) 05220(minus50) 54214

Derivative-free Kung-Traubrsquos three-step method (KT8)[6] is as follows

119910119896= 119909119896minus

119891 (119909119896)

119891 [119909119896 119908119896] 119908119896= 119909119896+ 120574119891 (119909

119896)

119911119896= 119910119896minus

119891 (119910119896) 119891 (119908

119896)

[119891 (119908119896) minus 119891 (119910

119896)] 119891 [119909

119896 119910119896]

119909119896+1

= 119911119896minus119891 (119910119896) 119891 (119908

119896) (119910119896minus 119909119896+ 119891 (119909

119896) 119891 [119909

119896 119911119896])

[119891 (119910119896) minus 119891 (119911

119896)] [119891 (119908

119896) minus 119891 (119911

119896)]

+119891 (119910119896)

119891 [119910119896 119911119896]

(52)

Three-step methods developed by Zheng et al (ZLH8)[10] is as follows

119910119896= 119909119896minus

119891 (119909119896)

119891 [119909119896 119908119896] 119908119896= 119909119896+ 120574119891 (119909

119896)

119911119896= 119910119896minus

119891 (119910119896)

119891 [119910119896 119908119896] + 119891 [119910

119896 119909119896 119908119896] (119910119896minus 119909119896)

119909119896+1

= 119911119896

minus ( (119891 (119911119896))

times (119891 [119911119896 119910119896] + 119891 [119911

119896 119910119896 119909119896] (119911119896minus 119910119896)

+119891 [119911119896 119910119896 119909119896 119908119896] (119911119896minus 119910119896) (119911119896minus 119909119896))minus1

)

(53)

Three-step method by Lotfi and Tavakoli (LT8) [11] is asfollows

119910119896= 119909119896minus

119891 (119909119896)

119891 [119909119896 119908119896]

119911119896= 119910119896minus 119867 (119905

119896 119906119896)

119891 (119910119896)

119891 [119910119896 119908119896]

119905119896=119891 (119910119896)

119891 (119909119896) 119906

119896=119891 (119908119896)

119891 (119909119896)

119909119896+1

= 119911119896minus 119866 (119905

119896 119904119896)119882 (V

119896 119904119896)

119891 (119911119896)

119891 [119911119896 119908119896]

119904119896=119891 (119911119896)

119891 (119910119896) V

119896=119891 (119911119896)

119891 (119909119896)

(54)

where119882(119904119896 V119896) = 1+119904

2

119896+V2119896119866(119905119896 119904119896) = 1+119905

119896+119904119896+2119905119896119904119896+(minus1minus

120601119896)1199053

119896and (120601

119896= 1(1 + 120574

119896119891[119909119896 119908119896])) and119867(119905

119896 119906119896) = 1 + 119905

119896

are the weight functionsIn Tables 1 2 and 3 our two-step proposed classes (6)

and (19) have been compared with optimal two-stepmethodsKT4 ZLH4 and LT4 We observe that all these methodsbehave very well practically and confirm their theoreticalresults

Also Tables 4 5 and 6 present numerical results for ourthree-step classes (11) and (20) and methods KT8 ZLH8 andLT8 It is also clear that all these methods behave very wellpractically and confirm their relevant theories

We remark the importance of the choice of initial guessesIf they are chosen sufficiently close to the sought roots thenthe expected (theoretical) convergence speed will be reachedin practice otherwise all iterative root-finding methodsshow slower convergence especially at the beginning of theiterative process Hence a special attention should be paidto finding good initial approximations We note that efficientways for the determination of initial approximations of greataccuracy were discussed thoroughly in the works [12ndash14]

5 Conclusions

We have constructed two families of iterative methods with-out memory which are optimal in the sense of Kung andTraubrsquos conjecture in this paper Our proposed methods donot need any derivative

In addition they contain an accelerator parameter whichraises convergence order without any new functional evalu-ations In other words the efficiency index of the three-stepwith memory hit 1214 asymp 1861

We finalize this work by suggesting some points forfuture researches first developing the proposed methods forsome matrix functions such as the ones in [15 16] second

8 Discrete Dynamics in Nature and Society

Table 2 1198912(119909) = 119890

minus5119909(119909 minus 2)(119909

10+ 119909 + 2) 120572 = 2 and 119909

0= 22

Methods |1199091minus 120572| |119909

2minus 120572| |119909

3minus 120572| coc

Without memoryNew method (6) 120574 = minus001 09445(minus3) 01550(minus13) 01289(minus56) 39945KT4 120574 = minus001 01125(minus2) 03033(minus13) 01891(minus55) 39932ZLH4 120574 = minus01 09349(minus3) 01478(minus13) 01074(minus56) 39939LT 120574 = minus001 01343(minus2) 05965(minus13) 02821(minus54) 39918

With memoryNew method (19) 120574 = 1 01057(minus2) 01102(minus19) 08335(minus109) 52480KT4 120574 = 1 01267(minus2) 02180(minus19) 03580(minus107) 52364ZLH4 120574 = minus001 09445(minus3) 02868(minus20) 08014(minus112) 52264LT4 120574 = minus1 01343(minus2) 01072(minus19) 01150(minus108) 52035

Table 3 1198913(119909) = 119890

1199093minus119909

minus cos(1199092 minus 1) + 1199093+ 1 120572 = minus1 and 119909

0= minus165

Methods |1199091minus 120572| |119909

2minus 120572| |119909

3minus 120572| coc

Without memoryNew method (6) 120574 = minus001 06849(minus3) 08725(minus13) 02284(minus52) 40003KT4 120574 = minus001 06610(minus3) 08682(minus13) 02564(minus52) 40004ZLH4 120574 = 01 01214(minus2) 02388(minus12) 03560(minus51) 40004LT4 120574 = minus001 06398(minus3) 08549(minus13) 02714(minus52) 40002

With memoryNew method (19) 120574 = minus1 04838(minus2) 01348(minus11) 01230(minus64) 55506KT4 120574 = 1 01170(minus1) 03517(minus7) 08936(minus41) 60860ZLH4 120574 = minus001 06849(minus3) 07630(minus17) 01662(minus92) 54226LT4 120574 = minus001 06398(minus3) 06666(minus17) 07373(minus93) 54324

Table 4 1198911(119909) = sin(120587119909)119890(119909

2+119909 cos(119909)minus1)

+ 119909 log(119909 sin(119909) + 1) 120572 = 0 and 1199090= 06

Methods |1199091minus 120572| |119909

2minus 120572| |119909

3minus 120572| coc

Without memoryNew method (11) 120574 = minus1 02345(minus3) 01042(minus32) 01593(minus267) 79999KT8 120574 = minus1 03101(minus3) 02671(minus31) 08196(minus256) 79999ZLH8 120574 = minus001 01742(minus2) 01558(minus22) 06471(minus183) 79999LT8 120574 = minus1 05758(minus3) 07106(minus29) 03880(minus236) 79998

With memoryNew method (20) 120574 = minus01 03440(minus3) 01220(minus41) 08636(minus503) 119935KT8 120574 = minus001 09773(minus3) 02960(minus33) 01505(minus399) 120021ZLH8 120574 = minus001 01742(minus2) 03012(minus33) 04438(minus402) 119900LT8 120574 = minus01 07107(minus4) 02040(minus49) 04971(minus596) 120024

Table 5 1198912(119909) = 119890

minus5119909(119909 minus 2)(119909

10+ 119909 + 2) 120572 = 2 and 119909

0= 22

Methods |1199091minus 120572| |119909

2minus 120572| |119909

3minus 120572| coc

Without memoryNew method (11) 120574 = minus1 03124(minus6) 09972(minus56) 01074(minus451) 80000KT8 120574 = minus1 08055(minus6) 01413(minus52) 01268(minus426) 80000ZLH8 120574 = 1 04955(minus6) 05798(minus54) 02038(minus437) 80000LT8 120574 = minus1 04052(minus6) 05819(minus55) 01053(minus445) 80000

With memoryNew method (20) 120574 = minus1 03124(minus6) 01213(minus81) 03093(minus985) 119822KT8 120574 = minus1 08055(minus6) 02641(minus78) 09143(minus945) 119538ZLH8 120574 = minus001 03943(minus6) 01889(minus80) 06308(minus971) 119817LT8 120574 = minus001 04532(minus6) 04774(minus76) 05848(minus853) 111023

Discrete Dynamics in Nature and Society 9

Table 6 1198913(119909) = 119890

1199093minus119909

minus cos(1199092 minus 1) + 1199093+ 1 120572 = minus1 and 119909

0= minus165

Methods |1199091minus 120572| |119909

2minus 120572| |119909

3minus 120572| coc

Without memoryNew method (11) 120574 = minus1 02768(minus3) 03670(minus27) 03499(minus218) 80000KT8 120574 = minus1 02915(minus4) 05324(minus34) 06583(minus272) 80000ZLH8 120574 = 1 02322(minus1) 05044(minus11) 02253(minus88) 80081LT 120574 = minus1 01094(minus2) 01454(minus22) 01323(minus181) 80014

With memoryNew method (20) 120574 = minus1 02768(minus3) 02660(minus42) 01805(minus509) 119734KT8 120574 = minus1 02915(minus4) 05214(minus53) 01157(minus642) 120961ZLH8 120574 = minus001 06052(minus5) 02355(minus65) 04206(minus786) 119310LT8 120574 = minus1 01094(minus2) 03522(minus34) 01717(minus415) 121081

exploring its dynamic or basins of attractions and lastlyextending the developedmethods with memory using two orthree accelerators

Conflict of Interests

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

Acknowledgments

The first author thanks Islamic Azad University HamedanBranch for the financial support of the present research Theauthors are also thankful for insightful comments of three ref-erees whom helped with the readability and reliability of thepresent paperThe fourth author also gratefully acknowledgesthat this research was partially supported by the UniversityPutra Malaysia under the GP-IBT Grant Scheme havingproject number GP-IBT20139420100

References

[1] N Kyurkchiev and A Iliev ldquoOn some multipoint methodsarising from optimal in the sense of Kung-Traub algorithmsrdquoBiomath vol 2 no 1 Article ID 1305155 7 pages 2013

[2] T Lotfi F Soleymani S Sharifi S Shateyi and F K HaghanildquoMulti-point iterative methods for finding all the simple zerosin an intervalrdquo Journal of AppliedMathematics vol 2014 ArticleID 601205 14 pages 2014

[3] J F Traub G W Wasilkowski and H Wozniakowski Informa-tion Uncertainty Complexity Addison-Wesley 1983

[4] M S Rhee and Y I Kim ldquoA general class of optimal fourth-order multiple-root finders without memory for nonlinearequationsrdquo Applied Mathematical Sciences vol 7 pp 5537ndash55512013

[5] H Wozniakowski ldquoMaximal order of multipoint iterationsusing n evaluationsrdquo in Analytic Computational Complexity JF Traub Ed pp 75ndash107 Academic Press New York NY USA1976

[6] 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

[7] Y H Geum and Y I Kim ldquoAn optimal family of fast 16th-orderderivative-free multipoint simple-root finders for nonlinear

equationsrdquo Journal ofOptimizationTheory andApplications vol160 no 2 pp 608ndash622 2014

[8] J F Traub Iterative Methods for the Solution of EquationsPrentice-Hall New York NY USA 1964

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

[10] 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

[11] T Lotfi and E Tavakoli ldquoOn a new efficient Steffensen-likeiterative class by applying a suitable self-accelerator parameterrdquoThe Scientific World Journal vol 2014 Article ID 769758 9pages 2014

[12] F Soleymani and S Shateyi ldquoTwo optimal eighth-orderderivative-free classes of iterative methodsrdquo Abstract andApplied Analysis vol 2012 Article ID 318165 14 pages 2012

[13] F Soleymani and D K R Babajee ldquoComputing multiple zerosusing a class of quartically convergent methodsrdquo AlexandriaEngineering Journal vol 52 no 3 pp 531ndash541 2013

[14] F Soleymani ldquoSome high-order iterativemethods for finding allthe real zerosrdquoThai Journal of Mathematics vol 12 pp 313ndash3272014

[15] F Soleymani E Tohidi S Shateyi and F K Haghani ldquoSomematrix iterations for computing matrix sign functionrdquo Journalof Applied Mathematics vol 2014 Article ID 425654 9 pages2014

[16] F Soleymani P S Stanimirovic S Shateyi and F K HaghanildquoApproximating the matrix sign function using a novel iterativemethodrdquo Abstract and Applied Analysis vol 2014 Article ID105301 9 pages 2014

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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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

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Discrete Dynamics in Nature and Society

and 119891(120572) = 0 = 1198911015840(120572) = 119888

1 that is 120572 is a simple zero of

119891(119909) = 0

21 New Two-Step Methods without Memory In this subsec-tion we start with the two-step scheme

119910 = 119909 minus119891 (119909)

1198911015840 (119909) 119909 = 119910 minus

119891 (119910)

1198911015840 (119910) (1)

Note that we omit the iteration index 119896 for the sakeof simplicity only The order of convergence of scheme(1) is four but it does not contain any contributions Wesubstitute derivatives in the first and second steps by suitableapproximations that use available data thus we introduce anapproximation as follows

Using the points 119909 and119908 = 119909+ 120574119891(119909) (120574 is a nonzero realconstant) we can apply Lagrangersquos interpolatory polynomialfor approximating 1198911015840(119909)

1198751(119905) =

119905 minus 119908

119909 minus 119908119891 (119909) +

119905 minus 119909

119908 minus 119909119891 (119908) (2)

Fix 1198911015840(119909) asymp 1198751015840

1(119909) and by setting 119905 = 119909 we have

1198751015840

1(119909) =

119891 (119909)

119909 minus 119908+119891 (119908)

119908 minus 119909=119891 (119909) minus 119891 (119908)

119909 minus 119908= 119891 [119909 119908] (3)

Also Lagrangersquos interpolatory polynomial at points 119909 119908and 119910 for approximating 1198911015840(119910) can be given as follows

1198752(119905) =

(119905 minus 119908) (119905 minus 119910)

(119909 minus 119908) (119909 minus 119910)119891 (119909) +

(119905 minus 119909) (119905 minus 119910)

(119908 minus 119909) (119908 minus 119910)119891 (119908)

+(119905 minus 119908) (119905 minus 119909)

(119910 minus 119908) (119910 minus 119909)119891 (119910)

(4)

We obtain

1198911015840(119910) asymp 119875

1015840

2(119910)

=2119910 minus 119908 minus 119910

(119909 minus 119908) (119909 minus 119910)119891 (119909)

+2119910 minus 119909 minus 119910

(119908 minus 119909) (119908 minus 119910)119891 (119908)

+2119910 minus 119908 minus 119909

(119910 minus 119908) (119910 minus 119909)119891 (119910)

(5)

Finally we set above approximations in the denominator of(1) and so our derivative-free two-step iterative method isderived in what follows

119910 = 119909 minus119891 (119909)

119891 [119909 119908] 119908 = 119909 + 120574119891 (119909)

119911 = 119910 minus119891 (119910)

1198751015840

2(119910)

(6)

Now to check the convergence order of (6) we avoidretyping the widely practiced approach in the literature andput forward the following self-explained Mathematica code

f[e ] = f1a lowast (e + sum4

k=2 ck lowast ek)ew = e + 120574f[e] (lowast119890119908 = 119908 minus 120572lowast)

f[x y ] =f[x] minus f[y]

x minus y

p1[t ] =t minus ew

e minus ewf[e] +

t minus e

ew minus ef[ew]

dp1[t ] =f[e]

e minus ew+

f[ew]

ew minus e

ey = e minus Series[f[e]p11015840[ew]

e 0 3]FullSimplify (lowastey = y minus 120572lowast)

Out[a] (1 + f1a120574)c2e2 + O[e]3

p2[t ] =(t minus ew)(t minus ey)

(e minus ew)(e minus ey)f[e]

+(t minus e)(t minus ey)

(ew minus e)(ew minus ey)f[ew]

+(t minus ew)(t minus e)

(ey minus ew)(ey minus e)f[ey]

dp2[t ] =2t minus ew minus ey

(e minus ew)(e minus ey)f[e]

+2t minus e minus ey

(ew minus e)(ew minus ey)f[ew]

+2t minus ew minus e

(ey minus ew)(ey minus e)f[ey]

ez = ey minusf[ey]

dp2[ey]FullSimplify

Out[b] (1 + f1a120574)2c2(c22minus c3)e4 + O[e]5

Considering Out[b] of the aboveMathematica programwe observe that the order of convergence of the family (6)is four and so we can state a convergence theorem in whatfollows

Theorem 1 If an initial approximation 1199090is sufficiently close

to a simple zero 120572 of 119891 then the convergence order of the two-step approach (6) is equal to four And its error equation is givenby

119890119896+1

= (1 + 1205741198911015840(120572))2

1198882(1198882

2minus 1198883) 1198904

119896+ 119874 (119890

5

119896) (7)

22 New Three-Step Family Now we construct a three-stepuniparametric family of methods based on the two-stepmethod (6)We start from a three-step schemewhere the firsttwo steps are given by (6) and also the third step is Newtonrsquosmethod that is

119910 = 119909 minus119891 (119909)

119891 [119909 119908] 119908 = 119909 + 120574119891 (119909)

119911 = 119910 minus119891 (119910)

1198751015840

2(119905)

119909 = 119911 minus119891 (119911)

1198911015840 (119911)

(8)

The derivative 1198911015840(119911) in the third step of (8) should be

substituted by a suitable approximation in order to obtain ashigh as possible of convergence order and tomake the schemeoptimal To provide this approximation we apply Lagrangersquos

Discrete Dynamics in Nature and Society 3

interpolatory polynomial at the points 119908 = 119909 + 120574119891(119909) 119909 119910and 119911 that is

1198753(119905) =

(119905 minus 119908) (119905 minus 119910) (119905 minus 119911)

(119909 minus 119908) (119909 minus 119910) (119909 minus 119911)119891 (119909)

+(119905 minus 119909) (119905 minus 119910) (119905 minus 119911)

(119908 minus 119909) (119908 minus 119910) (119908 minus 119911)119891 (119908)

+(119905 minus 119908) (119905 minus 119909) (119905 minus 119911)

(119910 minus 119908) (119910 minus 119909) (119910 minus 119911)119891 (119910)

+(119905 minus 119908) (119905 minus 119909) (119905 minus 119910)

(119911 minus 119908) (119911 minus 119909) (119911 minus 119910)119891 (119911)

(9)

It is obvious that 1198753(119911) = 119891(119911) By differentiating (9) and

setting 119905 = 119911 we obtain

1198751015840

3(119911) =

119908119910 + 119908119911 + 119910119911 minus 2 (119908 + 119910 + 119911) 119911 + 31199112

((119909 minus 119908) (119909 minus 119910) (119909 minus 119911))119891 (119909)

+119909119910 + 119909119911 + 119910119911 minus 2 (119909 + 119910 + 119911) 119911 + 3119911

2

(119908 minus 119909) (119908 minus 119910) (119908 minus 119911)119891 (119908)

+119909119908 + 119909119911 + 119908119911 minus 2 (119909 + 119908 + 119911) 119911 + 3119911

2

(119910 minus 119908) (119910 minus 119909) (119910 minus 119911)119891 (119910)

+119909119908 + 119909119910 + 119908119910 minus 2 (119909 + 119908 + 119910) 119911 + 3119911

2

(119911 minus 119908) (119911 minus 119909) (119911 minus 119910)119891 (119911)

(10)

By substituting 1198911015840(119911) asymp 1198751015840

3(119911) in (8) we have

119910 = 119909 minus119891 (119909)

119891 [119909 119908] 119908 = 119909 + 120574119891 (119909)

119911 = 119910 minus119891 (119910)

1198751015840

2(119910)

119909 = 119911 minus119891 (119911)

1198751015840

3(119911)

(11)

where 11987510158402(119910) is defined by (5) and 1198751015840

3(119911) given by (10)

In the following theorem we state suitable conditions forderiving an optimal three-step scheme without memory

Theorem 2 Let 119891 119863 sub R rarr R be a scalar functionwhich has the simple root 120572 in the open interval 119863 also initialapproximation 119909

0is sufficiently close to a simple zero 120572 of

119891 The three-step iterative method (11) has eighth order andsatisfies the following error equation119890119896+1

= (1 + 1205741198911015840(120572)4) 1198882

2(1198882

2minus 1198883) (1198883

2minus 11988821198883+ 1198884) 1198908

119896+ 119874 (119890

9

119896)

(12)

Proof We are going to employ symbolic computation in thecomputational software package Mathematica We write thefollowing

119891 (119909119896) = 1198911015840(120572) (119890

119896+ 11988821198902

119896+ 11988831198903

119896+ 11988841198904

119896+ 11988851198905

119896+ 11988861198906

119896

+11988871198907

119896+ 11988881198908

119896+ 119874 (119890

9

119896))

(13)

Now by introducing the abbreviations 119890 = 119909 minus 120572 119890119908 = 119908 minus 120572119890119910 = 119910minus120572 119890119911 = 119911minus120572 and 119888

119896= 119891(119896)(120572)(119896119891

1015840(120572)) we provide

the following Mathematica program in order to obtain theconvergence order of (11)

ProgramWritten in Mathematica Consider the following

f[e ] = f1a lowast (e + sum4

k=2 ck lowast ek)

ew = e + 120574f[e]

f[x y ] =f[x] minus f[y]

x minus y

p1[t ] =t minus ew

e minus ewf[e] +

t minus e

ew minus ef[ew]

dp1[t ] =f[e]

e minus ew+

f[ew]

ew minus e

ey = e minus Series[f[e]p11015840[ew]

e 0 8]FullSimplify

Out[a] (1 + 120574f1a)c2e2 + O[e]3

p2[t ] =(t minus ew)(t minus ey)

(e minus ew)(e minus ey)f[e]

+(t minus e)(t minus ey)

(ew minus e)(ew minus ey)f[ew]

+(t minus ew)(t minus e)

(ey minus ew)(ey minus e)f[ey]

dp2[t ] =2t minus ew minus ey

(e minus ew)(e minus ey)f[e]

+2t minus e minus ey

(ew minus e)(ew minus ey)f[ew]

+2t minus ew minus e

(ey minus ew)(ey minus e)f[ey]

ez = ey minusf[ey]

dp2[ey]FullSimplify

Out[b] (1 + 120574f1a)2c2(c22minus c3)e4 + O[e]5

dp3[t ]

=ewey + ewez + eyez minus 2(ew + ey + ez)t + 3t2

(e minus ew)(e minus ey)(e minus ez)f[e]

+eey + eez + eyez minus 2(e + ey + ez)t + 3t2

(ew minus e)(ew minus ey)(ew minus ez)f[ew]

+eew + eez + ewez minus 2(e + ew + ez)t + 3t2

(ey minus ew)(ey minus e)(ey minus ez)f[ey]

+eew + eey + ewey minus 2(e + ew + ey)t + 3t2

(ez minus ew)(ez minus e)(ez minus ey)f[ez]

ex = ez minusf[ez]

dp3[ez]FullSimplify

Out[c] (1 + 120574f1a)4c22(c22minus c3)(c32minus c2c3+ c4)e8

+O[e]9

As a result the proof of the theorem is finished Accordingto Out[c] it possesses eighth order

Error equations (7) and (12) indicate that the orders ofmethods (6) and (11) are four and eight respectively

In the next section we will modify the proposedmethodsand introduce new methods with memory With the useof accelerator parameters the order of convergence willsignificantly increase

4 Discrete Dynamics in Nature and Society

3 Extension to Methods with MemoryThis section is concernedwith extraction of efficientmethodswith memory from (6) and (11) through a careful inspectionof their error equations containing the parameter 120574 whichcan be approximated in such a way that increases the localorder of convergence

Toward this goal we set 120574 = 120574119896as the iteration proceeds by

the formula 120574119896= minus11198911015840(120572) for 119896 = 1 2 where 1198911015840(120572) is an

approximation of 1198911015840(120572) We therefore use the approximation

120574119896= minus

1

1198911015840 (120572)

= minus1

1198731015840

3(119909119896) (14)

for (6) and the following one

120574119896= minus

1

1198911015840 (120572)

= minus1

1198731015840

4(119909119896) (15)

for (11) Here we consider Newtonrsquos interpolating polynomialof third degree as the method for approximating 119891

1015840(120572) in

two-step method (6) and Newtonrsquos interpolating polynomialof fourth degree for approximating 119891

1015840(120572) in the three-

step method (11) where 1198733(119905) is the Newtonrsquos interpola-

tion polynomial of third degree set through four availableapproximations 119909

119896 119909119896minus1

119910119896minus1

119908119896minus1

and1198734(119905) is the Newtonrsquos

interpolation polynomial of fourth degree set through fiveavailable approximations 119909

119896 119911119896minus1

119910119896minus1

119908119896minus1

119909119896minus1

Consider

1198731015840

3(119909119896) = [

119889

1198891199051198733(119905)]

119905=119909119896

= [119889

119889119905(119891 (119909119896) + 119891 [119909

119896 119909119896minus1

] (119905 minus 119909119896)

+ 119891 [119909119896 119909119896minus1

119910119896minus1

] (119905 minus 119909119896) (119905 minus 119909

119896minus1)

+ 119891 [119909119896 119909119896minus1

119910119896minus1

119908119896minus1

] (119905 minus 119909119896)

times (119905 minus 119909119896minus1

) (119905 minus 119910119896minus1

)) ]

119905=119909119896

= 119891 [119909119896 119909119896minus1

] + 119891 [119909119896 119909119896minus1

119910119896minus1

] (119909119896minus 119909119896minus1

)

+ 119891 [119909119896 119909119896minus1

119910119896minus1

119908119896minus1

] (119909119896minus 119909119896minus1

) (119909119896minus 119910119896minus1

)

(16)

1198731015840

4(119909119896)

= [119889

1198891199051198734(119905)]

119905=119909119896

= [119889

119889119905(119891 (119909119896) + 119891 [119909

119896 119911119896minus1

] (119905 minus 119909119896)

+ 119891 [119909119896 119911119896minus1

119910119896minus1

] (119905 minus 119909119896) (119905 minus 119911

119896minus1)

+ 119891 [119909119896 119911119896minus1

119910119896minus1

119909119896minus1

] (119905 minus 119909119896)

times (119905 minus 119911119896minus1

) (119905 minus 119910119896minus1

)

+ 119891 [119909119896 119911119896minus1

119910119896minus1

119909119896minus1

119908119896minus1

]

times(119905minus119909119896)(119905 minus 119911

119896minus1)(119905minus 119910

119896minus1)(119905minus 119909

119896minus1))]

119905=119909119896

= 119891 [119909119896 119911119896minus1

] + 119891 [119909119896 119911119896minus1

119910119896minus1

] (119909119896minus 119911119896minus1

)

+ 119891 [119909119896 119911119896minus1

119910119896minus1

119909119896minus1

] (119909119896minus 119911119896minus1

)(119909119896minus 119910119896minus1

)

+ 119891 [119909119896 119911119896minus1

119910119896minus1

119909119896minus1

119908119896minus1

] (119909119896minus 119911119896minus1

)

times (119909119896minus 119910119896minus1

) (119909119896minus 119909119896minus1

)

(17)

Note that a divided difference of order 119898 defined recur-sively as

119891 [1199090 1199091 119909

119898]

=119891 [1199091 119909

119898] minus 119891 [119909

0 1199091 119909

119898minus1]

119909119898minus 1199090

119898 ⩾ 2

(18)

has been used throughout this paper Hence the with mem-ory developments of (6) and (11) can be presented as follows

119910119896= 119909119896minus

119891 (119909119896)

119891 [119909119896 119908119896] 119908119896= 119909119896+ 120574119896119891 (119909119896)

119909119896+1

= 119910119896minus

119891 (119910119896)

1198751015840

2(119910119896)

(19)

119910119896= 119909119896minus

119891 (119909119896)

119891 [119909119896 119908119896] 119908119896= 119909119896+ 120574119896119891 (119909119896)

119911119896= 119910119896minus119891 (119910119896)

1198751015840

2(119905119896)

119909 = 119909119896+1

= 119911119896minus

119891 (119911119896)

1198751015840

3(119911119896)

(20)

Remark 3 Accelerating methods obtained by recursivelycalculated free parameter may also be called self-acceleratingmethodsThe initial value 120574

0should be chosen before starting

the iterative process for example using one of the waysproposed in [8]

Here we attempt to prove that themethods withmemory(19) and (20) have convergence orders six and twelve providedthat we use accelerator 120574

119896as in (14) and (15) For ease of

continuing analysis we introduce the convenient notation asfollows If the sequence 119909

119896 converges to the zero 120572 of119891with

the order 119901 then we write 119890119896+1

sim 119890119901

119896 where 119890

119896= 119909119896minus 120572

The following lemma will play a crucial role in improvingthe convergence order of the methods with memory to beproposed in this paper

Lemma 4 If 120574119896= minus1119873

1015840

3(119909119896) 119890119896= 119909119896minus 120572 119890119896119910

= 119910119896minus 120572 and

119890119896119908

= 119908119896minus 120572 then the following relation holds

1 + 1205741198961198911015840(120572) sim 119888

4119890119896minus1

119890119896minus1119908

119890119896minus1119910

sim 119890119896minus1

119890119896minus1119908

119890119896minus1119910

(21)

Discrete Dynamics in Nature and Society 5

Proof Following the same terminology as in Theorem 2 andthe symbolic software Mathematica it would be easy toobtain (21) via writing the following code

ClearAll[Globallsquo lowast ]A[t ] = InterpolatingPolynomial[e fxew fw ey fy e1 fx1 t]SimplifyApproximation = minus1A1015840[e1]Simplifyfx = f1a lowast (e + c2 lowast e2 + c3 lowast e3 + c4 lowast e4)fw = f1a lowast (ew + c2 lowast ew2 + c3 lowast ew3

+ c4 lowast ew4)fy = f1a lowast (ey + c2 lowast ey2 + c3 lowast ey3

+ c4 lowast ey4)fx1 = f1a lowast (e1 + c2 lowast e12 + c3 lowast e13

+ c4 lowast e14)b = Series[Approximation e 0 2 ew 0 2ey 0 2 e1 0 2]SimplifyCollect[Series[1 + b lowast f1a e 0 1 ew 0 1ey 0 1 e1 0 0]e ew ey e1 Simplify]

The proof is complete

In order to obtain the R-order of convergence [9] ofthe method with memory (19) we establish the followingtheorem

Theorem 5 If an initial approximation 1199090is sufficiently close

to the zero 120572 of 119891(119909) and the parameter 120574119896in the iterative

scheme (19) is recursively calculated by the forms given in (14)then the R-order of convergence for (19) is at least six

Proof Let 119909119896 be a sequence of approximations generated

by the iterative method with memory (19) If this sequenceconverges to the zero 120572 of 119891 with the order 119901 then we write

119890119896+1

sim 119890119901

119896 119890

119896= 119909119896minus 120572 (22)

Thus

119890119896+1

sim (119890119901

119896minus1)119901

= 1198901199012

119896minus1 (23)

Moreover assume that the iterative sequences119908119896and 119910119896have

the orders 1199011and 119901

2 respectively Then (22) gives

119890119896119908

sim 1198901199011

119896sim (119890119901

119896minus1)1199011= 1198901199011199011

119896minus1 (24)

119890119896119910

sim 1198901199012

119896sim (119890119901

119896minus1)1199012= 1198901199011199012

119896minus1 (25)

Since

119890119896119908

sim (1 + 1205741198961198911015840(120572)) 119890

119896 (26)

119890119896119910

sim (1 + 1205741198961198911015840(120572)) 119890

2

119896 (27)

119890119896+1

sim (1 + 1205741198961198911015840(120572))2

1198904

119896 (28)

using Lemma 4 and (27) induce

119890119896119908

sim (1 + 1205741198961198911015840(120572)) 119890

119896sim (119890119896minus1

119890119896minus1119910

119890119896minus1119908

) 119890119896= 119890119901+1199011+1199012+1

119896minus1

(29)

119890119896119910

sim (1 + 1205741198961198911015840(120572)) 119890

2

119896sim (119890119896minus1

119890119896minus1119910

119890119896minus1119908

) 1198902

119896= 1198902119901+1199011+1199012+1

119896minus1

(30)

119890119896+1

sim (1 + 1205741198961198911015840(120572))2

1198904

119896sim (119890119896minus1

119890119896minus1119910

119890119896minus1119908

)2

1198904

119896

= 1198904119901+21199011+21199012+2

119896minus1

(31)

Matching the powers of 119890119896minus1

on the right hand sides of (24)ndash(29) (25)ndash(30) and (23)ndash(31) one can obtain

1199011199011minus 119901 minus 119901

1minus 1199012minus 1 = 0

1199011199012minus 2119901 minus 119901

1minus 1199012minus 1 = 0

1199012minus 4119901 minus 2119901

1minus 21199012minus 2 = 0

(32)

The nontrivial solution of this system is 1199011= 2 119901

2= 3 and

119901 = 6 This completes the proof

Using symbolic computations and Taylor expansions it iseasy to derive the following lemma

Lemma 6 Assuming (15) and (17) we have

1 + 1205741198961198911015840(120572) sim 119890

119896minus1119890119896minus1119908

119890119896minus1119910

119890119896minus1119911

(33)

where 120574119896= minus1119873

1015840

4(119909119896) 119890119896= 119909119896minus120572 119890119896119911

= 119911119896minus120572 119890119896119910

= 119910119896minus120572

and 119890119896119908

= 119908119896minus 120572

Proof The proof of this lemma is similar to Lemma 4 It ishence omitted

Similarly for the three-step method with memory (20)we have the following theorem

Theorem 7 If an initial approximation 1199090is sufficiently close

to the zero 120572 of 119891(119909) and the parameter 120574119896in the iterative

scheme (20) is recursively calculated by the forms given in (15)then the order of convergence for (20) is at least twelve

Proof Let 119909119896 be a sequence of approximations generated

by the iterative method with memory (20) If this sequenceconverges to the zero 120572 of 119891 with the order 119901 then we write

119890119896+1

sim 119890119901

119896 119890

119896= 119909119896minus 120572 (34)

So

119890119896+1

sim (119890119901

119896minus1)119901

= 1198901199012

119896minus1 (35)

Moreover assume that the iterative sequences 119908119896 119910119896 and 119911

119896

have the orders 1199011 1199012 and 119901

3 respectively Then (34) gives

119890119896119908

sim 1198901199011

119896sim (119890119901

119896minus1)1199011= 1198901199011199011

119896minus1 (36)

119890119896119910

sim 1198901199012

119896sim (119890119901

119896minus1)1199012= 1198901199011199012

119896minus1 (37)

119890119896119911

sim 1198901199013

119896sim (119890119901

119896minus1)1199013= 1198901199011199013

119896minus1 (38)

6 Discrete Dynamics in Nature and Society

Since

119890119896119908

sim (1 + 1205741198961198911015840(120572)) 119890

119896 (39)

119890119896119910

sim (1 + 1205741198961198911015840(120572)) 119890

2

119896 (40)

119890119896119911

sim (1 + 1205741198961198911015840(120572))2

1198904

119896 (41)

119890119896+1

sim (1 + 1205741198961198911015840(120572))4

1198908

119896 (42)

by Lemma 6 and (40) we obtain

119890119896119908

sim (1 + 1205741198961198911015840(120572)) 119890

119896sim (119890119896minus1

119890119896minus1119910

119890119896minus1119908

119890119896minus1119911

) 119890119896

= 119890119901+1199011+1199012+1199013+1

119896minus1

(43)

119890119896119910

sim (1 + 1205741198961198911015840(120572)) 119890

2

119896sim (119890119896minus1

119890119896minus1119910

119890119896minus1119908

119890119896minus1119911

) 1198902

119896

= 1198902119901+1199011+1199012+1199013+1

119896minus1

(44)

119890119896119911

sim (1 + 1205741198961198911015840(120572)) 119890

2

119896sim (119890119896minus1

119890119896minus1119910

119890119896minus1119908

119890119896minus1119911

)2

1198904

119896

= 1198904119901+21199011+21199012+21199013+2

119896minus1

(45)

119890119896+1

sim (1 + 1205741198961198911015840(120572))2

1198904

119896sim (119890119896minus1

119890119896minus1119910

119890119896minus1119908

119890119896minus1119911

)4

1198908

119896

= 1198908119901+41199011+41199012+41199013+4

119896minus1

(46)

Matching the powers of 119890119896minus1

on the right hand sides of (36)ndash(43) (37)ndash(44) (38)ndash(45) and (35)ndash(46) one can obtain

1199011199011minus 119901 minus 119901

1minus 1199012minus 1199013minus 1 = 0

1199011199012minus 2119901 minus 119901

1minus 1199012minus 1199013minus 1 = 0

1199011199013minus 4119901 minus 2119901

1minus 21199012minus 21199013minus 2 = 0

1199012minus 8119901 minus 4119901

1minus 41199012minus 41199013minus 2 = 0

(47)

This system has the solutions 1199011= 2 119901

2= 3 119901

3= 6 and

119901 = 12 The proof is complete

Remark 8 The advantage of the proposedmethods is in theirhigher computational efficiency indices We emphasize thatthe increase of the R-order of convergence has been obtainedwithout any additional function evaluations which pointsto very high computational efficiency Indeed the efficiencyindex 1214 asymp 1861 of the proposed three-step twelfth-ordermethod with memory is higher than the efficiency index613

asymp 1817 of (19) 814 asymp 1682 of the optimal three-pointmethod (11) and 413 asymp 1587 of (6)

Remark 9 We observe that the methods (19) and (20) withmemory are considerably accelerated (up to 50) in contrastto the corresponding method (11) without memory

4 Numerical Experiments

In this section we test our proposed methods and comparetheir results with some other methods of the same order of

convergenceThe results are reported using the programmingpackage Mathematica 8 in multiple precision arithmeticenvironment We have considered 1000 digits floating pointarithmetic so as to minimize the round-off errors as muchas possible The errors |119909

119896minus 120572| denote approximations to

the sought zeros and 119886(minus119887) stands for 119886 times 10minus119887 Moreover

coc indicates the computational order of convergence and iscomputed by

coc =log (1003816100381610038161003816119891 (119909

119896) 119891 (119909

119896minus1)1003816100381610038161003816)

log (1003816100381610038161003816119891 (119909119896minus1

) 119891 (119909119896minus2

)1003816100381610038161003816) (48)

It is assumed that the initial estimate 1205740should be chosen

before starting the iterative process and also 1199090is given

suitablySeveral iterative methods of optimal orders four and

eight for comparing with our proposed methods have beenchosen as comes next

Derivative-free Kung-Traubrsquos two-step method (KT4) [6]is as follows

119910119896= 119909119896minus

119891 (119909119896)

119891 [119909119896 119908119896] 119908119896= 119909119896+ 120574119891 (119909

119896)

119909119896+1

= 119910119896minus

119891 (119910119896) 119891 (119908

119896)

[119891 (119908119896) minus 119891 (119910

119896)] 119891 [119909

119896 119910119896]

(49)

Two-step method by Zheng et al (ZLH4) [10] is asfollows

119910119896= 119909119896minus

119891 (119909119896)

119891 [119909119896 119908119896] 119908119896= 119909119896+ 120574119891 (119909

119896)

119909119896+1

= 119910119896minus

119891 (119910119896)

119891 [119910119896 119908119896] + 119891 [119910

119896 119909119896 119908119896] (119910119896minus 119909119896)

(50)

Two-step method by Lotfi and Tavakoli (LT4) [11] is asfollows

119910119896= 119909119896minus

119891 (119909119896)

119891 [119909119896 119908119896]

119909119896+1

= 119910119896minus 119867 (119905

119896 119906119896)

119891 (119910119896)

119891 [119910119896 119908119896]

119905119896=119891 (119910119896)

119891 (119909119896) 119906

119896=119891 (119908119896)

119891 (119909119896)

(51)

where119867(119905119896 119906119896) = 1 + 119905

119896

Discrete Dynamics in Nature and Society 7

Table 1 1198911(119909) = sin(120587119909)119890(119909

2+119909 cos(119909)minus1)

+ 119909 log(119909 sin(119909) + 1) 120572 = 0 and 1199090= 06

Methods |1199091minus 120572| |119909

2minus 120572| |119909

3minus 120572| coc

Without memoryNew method (6) 120574 = minus1 06742(minus2) 01587(minus10) 04307(minus45) 40049KT4 120574 = minus001 06012(minus1) 01486(minus4) 06094(minus19) 40171ZLH4 120574 = minus001 05882(minus1) 01320(minus5) 08291(minus24) 39365LT 120574 = minus001 06178(minus1) 03334(minus4) 02739(minus17) 40368

With memoryNew method (19) 120574 = minus1 06742(minus2) 03177(minus11) 08372(minus62) 54213KT4 120574 = minus1 02102(minus1) 09935(minus9) 02326(minus48) 54031ZLH4 120574 = minus001 05882(minus1) 02204(minus8) 01503(minus45) 50216LT4 120574 = minus1 01752(minus1) 04934(minus9) 05220(minus50) 54214

Derivative-free Kung-Traubrsquos three-step method (KT8)[6] is as follows

119910119896= 119909119896minus

119891 (119909119896)

119891 [119909119896 119908119896] 119908119896= 119909119896+ 120574119891 (119909

119896)

119911119896= 119910119896minus

119891 (119910119896) 119891 (119908

119896)

[119891 (119908119896) minus 119891 (119910

119896)] 119891 [119909

119896 119910119896]

119909119896+1

= 119911119896minus119891 (119910119896) 119891 (119908

119896) (119910119896minus 119909119896+ 119891 (119909

119896) 119891 [119909

119896 119911119896])

[119891 (119910119896) minus 119891 (119911

119896)] [119891 (119908

119896) minus 119891 (119911

119896)]

+119891 (119910119896)

119891 [119910119896 119911119896]

(52)

Three-step methods developed by Zheng et al (ZLH8)[10] is as follows

119910119896= 119909119896minus

119891 (119909119896)

119891 [119909119896 119908119896] 119908119896= 119909119896+ 120574119891 (119909

119896)

119911119896= 119910119896minus

119891 (119910119896)

119891 [119910119896 119908119896] + 119891 [119910

119896 119909119896 119908119896] (119910119896minus 119909119896)

119909119896+1

= 119911119896

minus ( (119891 (119911119896))

times (119891 [119911119896 119910119896] + 119891 [119911

119896 119910119896 119909119896] (119911119896minus 119910119896)

+119891 [119911119896 119910119896 119909119896 119908119896] (119911119896minus 119910119896) (119911119896minus 119909119896))minus1

)

(53)

Three-step method by Lotfi and Tavakoli (LT8) [11] is asfollows

119910119896= 119909119896minus

119891 (119909119896)

119891 [119909119896 119908119896]

119911119896= 119910119896minus 119867 (119905

119896 119906119896)

119891 (119910119896)

119891 [119910119896 119908119896]

119905119896=119891 (119910119896)

119891 (119909119896) 119906

119896=119891 (119908119896)

119891 (119909119896)

119909119896+1

= 119911119896minus 119866 (119905

119896 119904119896)119882 (V

119896 119904119896)

119891 (119911119896)

119891 [119911119896 119908119896]

119904119896=119891 (119911119896)

119891 (119910119896) V

119896=119891 (119911119896)

119891 (119909119896)

(54)

where119882(119904119896 V119896) = 1+119904

2

119896+V2119896119866(119905119896 119904119896) = 1+119905

119896+119904119896+2119905119896119904119896+(minus1minus

120601119896)1199053

119896and (120601

119896= 1(1 + 120574

119896119891[119909119896 119908119896])) and119867(119905

119896 119906119896) = 1 + 119905

119896

are the weight functionsIn Tables 1 2 and 3 our two-step proposed classes (6)

and (19) have been compared with optimal two-stepmethodsKT4 ZLH4 and LT4 We observe that all these methodsbehave very well practically and confirm their theoreticalresults

Also Tables 4 5 and 6 present numerical results for ourthree-step classes (11) and (20) and methods KT8 ZLH8 andLT8 It is also clear that all these methods behave very wellpractically and confirm their relevant theories

We remark the importance of the choice of initial guessesIf they are chosen sufficiently close to the sought roots thenthe expected (theoretical) convergence speed will be reachedin practice otherwise all iterative root-finding methodsshow slower convergence especially at the beginning of theiterative process Hence a special attention should be paidto finding good initial approximations We note that efficientways for the determination of initial approximations of greataccuracy were discussed thoroughly in the works [12ndash14]

5 Conclusions

We have constructed two families of iterative methods with-out memory which are optimal in the sense of Kung andTraubrsquos conjecture in this paper Our proposed methods donot need any derivative

In addition they contain an accelerator parameter whichraises convergence order without any new functional evalu-ations In other words the efficiency index of the three-stepwith memory hit 1214 asymp 1861

We finalize this work by suggesting some points forfuture researches first developing the proposed methods forsome matrix functions such as the ones in [15 16] second

8 Discrete Dynamics in Nature and Society

Table 2 1198912(119909) = 119890

minus5119909(119909 minus 2)(119909

10+ 119909 + 2) 120572 = 2 and 119909

0= 22

Methods |1199091minus 120572| |119909

2minus 120572| |119909

3minus 120572| coc

Without memoryNew method (6) 120574 = minus001 09445(minus3) 01550(minus13) 01289(minus56) 39945KT4 120574 = minus001 01125(minus2) 03033(minus13) 01891(minus55) 39932ZLH4 120574 = minus01 09349(minus3) 01478(minus13) 01074(minus56) 39939LT 120574 = minus001 01343(minus2) 05965(minus13) 02821(minus54) 39918

With memoryNew method (19) 120574 = 1 01057(minus2) 01102(minus19) 08335(minus109) 52480KT4 120574 = 1 01267(minus2) 02180(minus19) 03580(minus107) 52364ZLH4 120574 = minus001 09445(minus3) 02868(minus20) 08014(minus112) 52264LT4 120574 = minus1 01343(minus2) 01072(minus19) 01150(minus108) 52035

Table 3 1198913(119909) = 119890

1199093minus119909

minus cos(1199092 minus 1) + 1199093+ 1 120572 = minus1 and 119909

0= minus165

Methods |1199091minus 120572| |119909

2minus 120572| |119909

3minus 120572| coc

Without memoryNew method (6) 120574 = minus001 06849(minus3) 08725(minus13) 02284(minus52) 40003KT4 120574 = minus001 06610(minus3) 08682(minus13) 02564(minus52) 40004ZLH4 120574 = 01 01214(minus2) 02388(minus12) 03560(minus51) 40004LT4 120574 = minus001 06398(minus3) 08549(minus13) 02714(minus52) 40002

With memoryNew method (19) 120574 = minus1 04838(minus2) 01348(minus11) 01230(minus64) 55506KT4 120574 = 1 01170(minus1) 03517(minus7) 08936(minus41) 60860ZLH4 120574 = minus001 06849(minus3) 07630(minus17) 01662(minus92) 54226LT4 120574 = minus001 06398(minus3) 06666(minus17) 07373(minus93) 54324

Table 4 1198911(119909) = sin(120587119909)119890(119909

2+119909 cos(119909)minus1)

+ 119909 log(119909 sin(119909) + 1) 120572 = 0 and 1199090= 06

Methods |1199091minus 120572| |119909

2minus 120572| |119909

3minus 120572| coc

Without memoryNew method (11) 120574 = minus1 02345(minus3) 01042(minus32) 01593(minus267) 79999KT8 120574 = minus1 03101(minus3) 02671(minus31) 08196(minus256) 79999ZLH8 120574 = minus001 01742(minus2) 01558(minus22) 06471(minus183) 79999LT8 120574 = minus1 05758(minus3) 07106(minus29) 03880(minus236) 79998

With memoryNew method (20) 120574 = minus01 03440(minus3) 01220(minus41) 08636(minus503) 119935KT8 120574 = minus001 09773(minus3) 02960(minus33) 01505(minus399) 120021ZLH8 120574 = minus001 01742(minus2) 03012(minus33) 04438(minus402) 119900LT8 120574 = minus01 07107(minus4) 02040(minus49) 04971(minus596) 120024

Table 5 1198912(119909) = 119890

minus5119909(119909 minus 2)(119909

10+ 119909 + 2) 120572 = 2 and 119909

0= 22

Methods |1199091minus 120572| |119909

2minus 120572| |119909

3minus 120572| coc

Without memoryNew method (11) 120574 = minus1 03124(minus6) 09972(minus56) 01074(minus451) 80000KT8 120574 = minus1 08055(minus6) 01413(minus52) 01268(minus426) 80000ZLH8 120574 = 1 04955(minus6) 05798(minus54) 02038(minus437) 80000LT8 120574 = minus1 04052(minus6) 05819(minus55) 01053(minus445) 80000

With memoryNew method (20) 120574 = minus1 03124(minus6) 01213(minus81) 03093(minus985) 119822KT8 120574 = minus1 08055(minus6) 02641(minus78) 09143(minus945) 119538ZLH8 120574 = minus001 03943(minus6) 01889(minus80) 06308(minus971) 119817LT8 120574 = minus001 04532(minus6) 04774(minus76) 05848(minus853) 111023

Discrete Dynamics in Nature and Society 9

Table 6 1198913(119909) = 119890

1199093minus119909

minus cos(1199092 minus 1) + 1199093+ 1 120572 = minus1 and 119909

0= minus165

Methods |1199091minus 120572| |119909

2minus 120572| |119909

3minus 120572| coc

Without memoryNew method (11) 120574 = minus1 02768(minus3) 03670(minus27) 03499(minus218) 80000KT8 120574 = minus1 02915(minus4) 05324(minus34) 06583(minus272) 80000ZLH8 120574 = 1 02322(minus1) 05044(minus11) 02253(minus88) 80081LT 120574 = minus1 01094(minus2) 01454(minus22) 01323(minus181) 80014

With memoryNew method (20) 120574 = minus1 02768(minus3) 02660(minus42) 01805(minus509) 119734KT8 120574 = minus1 02915(minus4) 05214(minus53) 01157(minus642) 120961ZLH8 120574 = minus001 06052(minus5) 02355(minus65) 04206(minus786) 119310LT8 120574 = minus1 01094(minus2) 03522(minus34) 01717(minus415) 121081

exploring its dynamic or basins of attractions and lastlyextending the developedmethods with memory using two orthree accelerators

Conflict of Interests

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

Acknowledgments

The first author thanks Islamic Azad University HamedanBranch for the financial support of the present research Theauthors are also thankful for insightful comments of three ref-erees whom helped with the readability and reliability of thepresent paperThe fourth author also gratefully acknowledgesthat this research was partially supported by the UniversityPutra Malaysia under the GP-IBT Grant Scheme havingproject number GP-IBT20139420100

References

[1] N Kyurkchiev and A Iliev ldquoOn some multipoint methodsarising from optimal in the sense of Kung-Traub algorithmsrdquoBiomath vol 2 no 1 Article ID 1305155 7 pages 2013

[2] T Lotfi F Soleymani S Sharifi S Shateyi and F K HaghanildquoMulti-point iterative methods for finding all the simple zerosin an intervalrdquo Journal of AppliedMathematics vol 2014 ArticleID 601205 14 pages 2014

[3] J F Traub G W Wasilkowski and H Wozniakowski Informa-tion Uncertainty Complexity Addison-Wesley 1983

[4] M S Rhee and Y I Kim ldquoA general class of optimal fourth-order multiple-root finders without memory for nonlinearequationsrdquo Applied Mathematical Sciences vol 7 pp 5537ndash55512013

[5] H Wozniakowski ldquoMaximal order of multipoint iterationsusing n evaluationsrdquo in Analytic Computational Complexity JF Traub Ed pp 75ndash107 Academic Press New York NY USA1976

[6] 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

[7] Y H Geum and Y I Kim ldquoAn optimal family of fast 16th-orderderivative-free multipoint simple-root finders for nonlinear

equationsrdquo Journal ofOptimizationTheory andApplications vol160 no 2 pp 608ndash622 2014

[8] J F Traub Iterative Methods for the Solution of EquationsPrentice-Hall New York NY USA 1964

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

[10] 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

[11] T Lotfi and E Tavakoli ldquoOn a new efficient Steffensen-likeiterative class by applying a suitable self-accelerator parameterrdquoThe Scientific World Journal vol 2014 Article ID 769758 9pages 2014

[12] F Soleymani and S Shateyi ldquoTwo optimal eighth-orderderivative-free classes of iterative methodsrdquo Abstract andApplied Analysis vol 2012 Article ID 318165 14 pages 2012

[13] F Soleymani and D K R Babajee ldquoComputing multiple zerosusing a class of quartically convergent methodsrdquo AlexandriaEngineering Journal vol 52 no 3 pp 531ndash541 2013

[14] F Soleymani ldquoSome high-order iterativemethods for finding allthe real zerosrdquoThai Journal of Mathematics vol 12 pp 313ndash3272014

[15] F Soleymani E Tohidi S Shateyi and F K Haghani ldquoSomematrix iterations for computing matrix sign functionrdquo Journalof Applied Mathematics vol 2014 Article ID 425654 9 pages2014

[16] F Soleymani P S Stanimirovic S Shateyi and F K HaghanildquoApproximating the matrix sign function using a novel iterativemethodrdquo Abstract and Applied Analysis vol 2014 Article ID105301 9 pages 2014

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

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 3

interpolatory polynomial at the points 119908 = 119909 + 120574119891(119909) 119909 119910and 119911 that is

1198753(119905) =

(119905 minus 119908) (119905 minus 119910) (119905 minus 119911)

(119909 minus 119908) (119909 minus 119910) (119909 minus 119911)119891 (119909)

+(119905 minus 119909) (119905 minus 119910) (119905 minus 119911)

(119908 minus 119909) (119908 minus 119910) (119908 minus 119911)119891 (119908)

+(119905 minus 119908) (119905 minus 119909) (119905 minus 119911)

(119910 minus 119908) (119910 minus 119909) (119910 minus 119911)119891 (119910)

+(119905 minus 119908) (119905 minus 119909) (119905 minus 119910)

(119911 minus 119908) (119911 minus 119909) (119911 minus 119910)119891 (119911)

(9)

It is obvious that 1198753(119911) = 119891(119911) By differentiating (9) and

setting 119905 = 119911 we obtain

1198751015840

3(119911) =

119908119910 + 119908119911 + 119910119911 minus 2 (119908 + 119910 + 119911) 119911 + 31199112

((119909 minus 119908) (119909 minus 119910) (119909 minus 119911))119891 (119909)

+119909119910 + 119909119911 + 119910119911 minus 2 (119909 + 119910 + 119911) 119911 + 3119911

2

(119908 minus 119909) (119908 minus 119910) (119908 minus 119911)119891 (119908)

+119909119908 + 119909119911 + 119908119911 minus 2 (119909 + 119908 + 119911) 119911 + 3119911

2

(119910 minus 119908) (119910 minus 119909) (119910 minus 119911)119891 (119910)

+119909119908 + 119909119910 + 119908119910 minus 2 (119909 + 119908 + 119910) 119911 + 3119911

2

(119911 minus 119908) (119911 minus 119909) (119911 minus 119910)119891 (119911)

(10)

By substituting 1198911015840(119911) asymp 1198751015840

3(119911) in (8) we have

119910 = 119909 minus119891 (119909)

119891 [119909 119908] 119908 = 119909 + 120574119891 (119909)

119911 = 119910 minus119891 (119910)

1198751015840

2(119910)

119909 = 119911 minus119891 (119911)

1198751015840

3(119911)

(11)

where 11987510158402(119910) is defined by (5) and 1198751015840

3(119911) given by (10)

In the following theorem we state suitable conditions forderiving an optimal three-step scheme without memory

Theorem 2 Let 119891 119863 sub R rarr R be a scalar functionwhich has the simple root 120572 in the open interval 119863 also initialapproximation 119909

0is sufficiently close to a simple zero 120572 of

119891 The three-step iterative method (11) has eighth order andsatisfies the following error equation119890119896+1

= (1 + 1205741198911015840(120572)4) 1198882

2(1198882

2minus 1198883) (1198883

2minus 11988821198883+ 1198884) 1198908

119896+ 119874 (119890

9

119896)

(12)

Proof We are going to employ symbolic computation in thecomputational software package Mathematica We write thefollowing

119891 (119909119896) = 1198911015840(120572) (119890

119896+ 11988821198902

119896+ 11988831198903

119896+ 11988841198904

119896+ 11988851198905

119896+ 11988861198906

119896

+11988871198907

119896+ 11988881198908

119896+ 119874 (119890

9

119896))

(13)

Now by introducing the abbreviations 119890 = 119909 minus 120572 119890119908 = 119908 minus 120572119890119910 = 119910minus120572 119890119911 = 119911minus120572 and 119888

119896= 119891(119896)(120572)(119896119891

1015840(120572)) we provide

the following Mathematica program in order to obtain theconvergence order of (11)

ProgramWritten in Mathematica Consider the following

f[e ] = f1a lowast (e + sum4

k=2 ck lowast ek)

ew = e + 120574f[e]

f[x y ] =f[x] minus f[y]

x minus y

p1[t ] =t minus ew

e minus ewf[e] +

t minus e

ew minus ef[ew]

dp1[t ] =f[e]

e minus ew+

f[ew]

ew minus e

ey = e minus Series[f[e]p11015840[ew]

e 0 8]FullSimplify

Out[a] (1 + 120574f1a)c2e2 + O[e]3

p2[t ] =(t minus ew)(t minus ey)

(e minus ew)(e minus ey)f[e]

+(t minus e)(t minus ey)

(ew minus e)(ew minus ey)f[ew]

+(t minus ew)(t minus e)

(ey minus ew)(ey minus e)f[ey]

dp2[t ] =2t minus ew minus ey

(e minus ew)(e minus ey)f[e]

+2t minus e minus ey

(ew minus e)(ew minus ey)f[ew]

+2t minus ew minus e

(ey minus ew)(ey minus e)f[ey]

ez = ey minusf[ey]

dp2[ey]FullSimplify

Out[b] (1 + 120574f1a)2c2(c22minus c3)e4 + O[e]5

dp3[t ]

=ewey + ewez + eyez minus 2(ew + ey + ez)t + 3t2

(e minus ew)(e minus ey)(e minus ez)f[e]

+eey + eez + eyez minus 2(e + ey + ez)t + 3t2

(ew minus e)(ew minus ey)(ew minus ez)f[ew]

+eew + eez + ewez minus 2(e + ew + ez)t + 3t2

(ey minus ew)(ey minus e)(ey minus ez)f[ey]

+eew + eey + ewey minus 2(e + ew + ey)t + 3t2

(ez minus ew)(ez minus e)(ez minus ey)f[ez]

ex = ez minusf[ez]

dp3[ez]FullSimplify

Out[c] (1 + 120574f1a)4c22(c22minus c3)(c32minus c2c3+ c4)e8

+O[e]9

As a result the proof of the theorem is finished Accordingto Out[c] it possesses eighth order

Error equations (7) and (12) indicate that the orders ofmethods (6) and (11) are four and eight respectively

In the next section we will modify the proposedmethodsand introduce new methods with memory With the useof accelerator parameters the order of convergence willsignificantly increase

4 Discrete Dynamics in Nature and Society

3 Extension to Methods with MemoryThis section is concernedwith extraction of efficientmethodswith memory from (6) and (11) through a careful inspectionof their error equations containing the parameter 120574 whichcan be approximated in such a way that increases the localorder of convergence

Toward this goal we set 120574 = 120574119896as the iteration proceeds by

the formula 120574119896= minus11198911015840(120572) for 119896 = 1 2 where 1198911015840(120572) is an

approximation of 1198911015840(120572) We therefore use the approximation

120574119896= minus

1

1198911015840 (120572)

= minus1

1198731015840

3(119909119896) (14)

for (6) and the following one

120574119896= minus

1

1198911015840 (120572)

= minus1

1198731015840

4(119909119896) (15)

for (11) Here we consider Newtonrsquos interpolating polynomialof third degree as the method for approximating 119891

1015840(120572) in

two-step method (6) and Newtonrsquos interpolating polynomialof fourth degree for approximating 119891

1015840(120572) in the three-

step method (11) where 1198733(119905) is the Newtonrsquos interpola-

tion polynomial of third degree set through four availableapproximations 119909

119896 119909119896minus1

119910119896minus1

119908119896minus1

and1198734(119905) is the Newtonrsquos

interpolation polynomial of fourth degree set through fiveavailable approximations 119909

119896 119911119896minus1

119910119896minus1

119908119896minus1

119909119896minus1

Consider

1198731015840

3(119909119896) = [

119889

1198891199051198733(119905)]

119905=119909119896

= [119889

119889119905(119891 (119909119896) + 119891 [119909

119896 119909119896minus1

] (119905 minus 119909119896)

+ 119891 [119909119896 119909119896minus1

119910119896minus1

] (119905 minus 119909119896) (119905 minus 119909

119896minus1)

+ 119891 [119909119896 119909119896minus1

119910119896minus1

119908119896minus1

] (119905 minus 119909119896)

times (119905 minus 119909119896minus1

) (119905 minus 119910119896minus1

)) ]

119905=119909119896

= 119891 [119909119896 119909119896minus1

] + 119891 [119909119896 119909119896minus1

119910119896minus1

] (119909119896minus 119909119896minus1

)

+ 119891 [119909119896 119909119896minus1

119910119896minus1

119908119896minus1

] (119909119896minus 119909119896minus1

) (119909119896minus 119910119896minus1

)

(16)

1198731015840

4(119909119896)

= [119889

1198891199051198734(119905)]

119905=119909119896

= [119889

119889119905(119891 (119909119896) + 119891 [119909

119896 119911119896minus1

] (119905 minus 119909119896)

+ 119891 [119909119896 119911119896minus1

119910119896minus1

] (119905 minus 119909119896) (119905 minus 119911

119896minus1)

+ 119891 [119909119896 119911119896minus1

119910119896minus1

119909119896minus1

] (119905 minus 119909119896)

times (119905 minus 119911119896minus1

) (119905 minus 119910119896minus1

)

+ 119891 [119909119896 119911119896minus1

119910119896minus1

119909119896minus1

119908119896minus1

]

times(119905minus119909119896)(119905 minus 119911

119896minus1)(119905minus 119910

119896minus1)(119905minus 119909

119896minus1))]

119905=119909119896

= 119891 [119909119896 119911119896minus1

] + 119891 [119909119896 119911119896minus1

119910119896minus1

] (119909119896minus 119911119896minus1

)

+ 119891 [119909119896 119911119896minus1

119910119896minus1

119909119896minus1

] (119909119896minus 119911119896minus1

)(119909119896minus 119910119896minus1

)

+ 119891 [119909119896 119911119896minus1

119910119896minus1

119909119896minus1

119908119896minus1

] (119909119896minus 119911119896minus1

)

times (119909119896minus 119910119896minus1

) (119909119896minus 119909119896minus1

)

(17)

Note that a divided difference of order 119898 defined recur-sively as

119891 [1199090 1199091 119909

119898]

=119891 [1199091 119909

119898] minus 119891 [119909

0 1199091 119909

119898minus1]

119909119898minus 1199090

119898 ⩾ 2

(18)

has been used throughout this paper Hence the with mem-ory developments of (6) and (11) can be presented as follows

119910119896= 119909119896minus

119891 (119909119896)

119891 [119909119896 119908119896] 119908119896= 119909119896+ 120574119896119891 (119909119896)

119909119896+1

= 119910119896minus

119891 (119910119896)

1198751015840

2(119910119896)

(19)

119910119896= 119909119896minus

119891 (119909119896)

119891 [119909119896 119908119896] 119908119896= 119909119896+ 120574119896119891 (119909119896)

119911119896= 119910119896minus119891 (119910119896)

1198751015840

2(119905119896)

119909 = 119909119896+1

= 119911119896minus

119891 (119911119896)

1198751015840

3(119911119896)

(20)

Remark 3 Accelerating methods obtained by recursivelycalculated free parameter may also be called self-acceleratingmethodsThe initial value 120574

0should be chosen before starting

the iterative process for example using one of the waysproposed in [8]

Here we attempt to prove that themethods withmemory(19) and (20) have convergence orders six and twelve providedthat we use accelerator 120574

119896as in (14) and (15) For ease of

continuing analysis we introduce the convenient notation asfollows If the sequence 119909

119896 converges to the zero 120572 of119891with

the order 119901 then we write 119890119896+1

sim 119890119901

119896 where 119890

119896= 119909119896minus 120572

The following lemma will play a crucial role in improvingthe convergence order of the methods with memory to beproposed in this paper

Lemma 4 If 120574119896= minus1119873

1015840

3(119909119896) 119890119896= 119909119896minus 120572 119890119896119910

= 119910119896minus 120572 and

119890119896119908

= 119908119896minus 120572 then the following relation holds

1 + 1205741198961198911015840(120572) sim 119888

4119890119896minus1

119890119896minus1119908

119890119896minus1119910

sim 119890119896minus1

119890119896minus1119908

119890119896minus1119910

(21)

Discrete Dynamics in Nature and Society 5

Proof Following the same terminology as in Theorem 2 andthe symbolic software Mathematica it would be easy toobtain (21) via writing the following code

ClearAll[Globallsquo lowast ]A[t ] = InterpolatingPolynomial[e fxew fw ey fy e1 fx1 t]SimplifyApproximation = minus1A1015840[e1]Simplifyfx = f1a lowast (e + c2 lowast e2 + c3 lowast e3 + c4 lowast e4)fw = f1a lowast (ew + c2 lowast ew2 + c3 lowast ew3

+ c4 lowast ew4)fy = f1a lowast (ey + c2 lowast ey2 + c3 lowast ey3

+ c4 lowast ey4)fx1 = f1a lowast (e1 + c2 lowast e12 + c3 lowast e13

+ c4 lowast e14)b = Series[Approximation e 0 2 ew 0 2ey 0 2 e1 0 2]SimplifyCollect[Series[1 + b lowast f1a e 0 1 ew 0 1ey 0 1 e1 0 0]e ew ey e1 Simplify]

The proof is complete

In order to obtain the R-order of convergence [9] ofthe method with memory (19) we establish the followingtheorem

Theorem 5 If an initial approximation 1199090is sufficiently close

to the zero 120572 of 119891(119909) and the parameter 120574119896in the iterative

scheme (19) is recursively calculated by the forms given in (14)then the R-order of convergence for (19) is at least six

Proof Let 119909119896 be a sequence of approximations generated

by the iterative method with memory (19) If this sequenceconverges to the zero 120572 of 119891 with the order 119901 then we write

119890119896+1

sim 119890119901

119896 119890

119896= 119909119896minus 120572 (22)

Thus

119890119896+1

sim (119890119901

119896minus1)119901

= 1198901199012

119896minus1 (23)

Moreover assume that the iterative sequences119908119896and 119910119896have

the orders 1199011and 119901

2 respectively Then (22) gives

119890119896119908

sim 1198901199011

119896sim (119890119901

119896minus1)1199011= 1198901199011199011

119896minus1 (24)

119890119896119910

sim 1198901199012

119896sim (119890119901

119896minus1)1199012= 1198901199011199012

119896minus1 (25)

Since

119890119896119908

sim (1 + 1205741198961198911015840(120572)) 119890

119896 (26)

119890119896119910

sim (1 + 1205741198961198911015840(120572)) 119890

2

119896 (27)

119890119896+1

sim (1 + 1205741198961198911015840(120572))2

1198904

119896 (28)

using Lemma 4 and (27) induce

119890119896119908

sim (1 + 1205741198961198911015840(120572)) 119890

119896sim (119890119896minus1

119890119896minus1119910

119890119896minus1119908

) 119890119896= 119890119901+1199011+1199012+1

119896minus1

(29)

119890119896119910

sim (1 + 1205741198961198911015840(120572)) 119890

2

119896sim (119890119896minus1

119890119896minus1119910

119890119896minus1119908

) 1198902

119896= 1198902119901+1199011+1199012+1

119896minus1

(30)

119890119896+1

sim (1 + 1205741198961198911015840(120572))2

1198904

119896sim (119890119896minus1

119890119896minus1119910

119890119896minus1119908

)2

1198904

119896

= 1198904119901+21199011+21199012+2

119896minus1

(31)

Matching the powers of 119890119896minus1

on the right hand sides of (24)ndash(29) (25)ndash(30) and (23)ndash(31) one can obtain

1199011199011minus 119901 minus 119901

1minus 1199012minus 1 = 0

1199011199012minus 2119901 minus 119901

1minus 1199012minus 1 = 0

1199012minus 4119901 minus 2119901

1minus 21199012minus 2 = 0

(32)

The nontrivial solution of this system is 1199011= 2 119901

2= 3 and

119901 = 6 This completes the proof

Using symbolic computations and Taylor expansions it iseasy to derive the following lemma

Lemma 6 Assuming (15) and (17) we have

1 + 1205741198961198911015840(120572) sim 119890

119896minus1119890119896minus1119908

119890119896minus1119910

119890119896minus1119911

(33)

where 120574119896= minus1119873

1015840

4(119909119896) 119890119896= 119909119896minus120572 119890119896119911

= 119911119896minus120572 119890119896119910

= 119910119896minus120572

and 119890119896119908

= 119908119896minus 120572

Proof The proof of this lemma is similar to Lemma 4 It ishence omitted

Similarly for the three-step method with memory (20)we have the following theorem

Theorem 7 If an initial approximation 1199090is sufficiently close

to the zero 120572 of 119891(119909) and the parameter 120574119896in the iterative

scheme (20) is recursively calculated by the forms given in (15)then the order of convergence for (20) is at least twelve

Proof Let 119909119896 be a sequence of approximations generated

by the iterative method with memory (20) If this sequenceconverges to the zero 120572 of 119891 with the order 119901 then we write

119890119896+1

sim 119890119901

119896 119890

119896= 119909119896minus 120572 (34)

So

119890119896+1

sim (119890119901

119896minus1)119901

= 1198901199012

119896minus1 (35)

Moreover assume that the iterative sequences 119908119896 119910119896 and 119911

119896

have the orders 1199011 1199012 and 119901

3 respectively Then (34) gives

119890119896119908

sim 1198901199011

119896sim (119890119901

119896minus1)1199011= 1198901199011199011

119896minus1 (36)

119890119896119910

sim 1198901199012

119896sim (119890119901

119896minus1)1199012= 1198901199011199012

119896minus1 (37)

119890119896119911

sim 1198901199013

119896sim (119890119901

119896minus1)1199013= 1198901199011199013

119896minus1 (38)

6 Discrete Dynamics in Nature and Society

Since

119890119896119908

sim (1 + 1205741198961198911015840(120572)) 119890

119896 (39)

119890119896119910

sim (1 + 1205741198961198911015840(120572)) 119890

2

119896 (40)

119890119896119911

sim (1 + 1205741198961198911015840(120572))2

1198904

119896 (41)

119890119896+1

sim (1 + 1205741198961198911015840(120572))4

1198908

119896 (42)

by Lemma 6 and (40) we obtain

119890119896119908

sim (1 + 1205741198961198911015840(120572)) 119890

119896sim (119890119896minus1

119890119896minus1119910

119890119896minus1119908

119890119896minus1119911

) 119890119896

= 119890119901+1199011+1199012+1199013+1

119896minus1

(43)

119890119896119910

sim (1 + 1205741198961198911015840(120572)) 119890

2

119896sim (119890119896minus1

119890119896minus1119910

119890119896minus1119908

119890119896minus1119911

) 1198902

119896

= 1198902119901+1199011+1199012+1199013+1

119896minus1

(44)

119890119896119911

sim (1 + 1205741198961198911015840(120572)) 119890

2

119896sim (119890119896minus1

119890119896minus1119910

119890119896minus1119908

119890119896minus1119911

)2

1198904

119896

= 1198904119901+21199011+21199012+21199013+2

119896minus1

(45)

119890119896+1

sim (1 + 1205741198961198911015840(120572))2

1198904

119896sim (119890119896minus1

119890119896minus1119910

119890119896minus1119908

119890119896minus1119911

)4

1198908

119896

= 1198908119901+41199011+41199012+41199013+4

119896minus1

(46)

Matching the powers of 119890119896minus1

on the right hand sides of (36)ndash(43) (37)ndash(44) (38)ndash(45) and (35)ndash(46) one can obtain

1199011199011minus 119901 minus 119901

1minus 1199012minus 1199013minus 1 = 0

1199011199012minus 2119901 minus 119901

1minus 1199012minus 1199013minus 1 = 0

1199011199013minus 4119901 minus 2119901

1minus 21199012minus 21199013minus 2 = 0

1199012minus 8119901 minus 4119901

1minus 41199012minus 41199013minus 2 = 0

(47)

This system has the solutions 1199011= 2 119901

2= 3 119901

3= 6 and

119901 = 12 The proof is complete

Remark 8 The advantage of the proposedmethods is in theirhigher computational efficiency indices We emphasize thatthe increase of the R-order of convergence has been obtainedwithout any additional function evaluations which pointsto very high computational efficiency Indeed the efficiencyindex 1214 asymp 1861 of the proposed three-step twelfth-ordermethod with memory is higher than the efficiency index613

asymp 1817 of (19) 814 asymp 1682 of the optimal three-pointmethod (11) and 413 asymp 1587 of (6)

Remark 9 We observe that the methods (19) and (20) withmemory are considerably accelerated (up to 50) in contrastto the corresponding method (11) without memory

4 Numerical Experiments

In this section we test our proposed methods and comparetheir results with some other methods of the same order of

convergenceThe results are reported using the programmingpackage Mathematica 8 in multiple precision arithmeticenvironment We have considered 1000 digits floating pointarithmetic so as to minimize the round-off errors as muchas possible The errors |119909

119896minus 120572| denote approximations to

the sought zeros and 119886(minus119887) stands for 119886 times 10minus119887 Moreover

coc indicates the computational order of convergence and iscomputed by

coc =log (1003816100381610038161003816119891 (119909

119896) 119891 (119909

119896minus1)1003816100381610038161003816)

log (1003816100381610038161003816119891 (119909119896minus1

) 119891 (119909119896minus2

)1003816100381610038161003816) (48)

It is assumed that the initial estimate 1205740should be chosen

before starting the iterative process and also 1199090is given

suitablySeveral iterative methods of optimal orders four and

eight for comparing with our proposed methods have beenchosen as comes next

Derivative-free Kung-Traubrsquos two-step method (KT4) [6]is as follows

119910119896= 119909119896minus

119891 (119909119896)

119891 [119909119896 119908119896] 119908119896= 119909119896+ 120574119891 (119909

119896)

119909119896+1

= 119910119896minus

119891 (119910119896) 119891 (119908

119896)

[119891 (119908119896) minus 119891 (119910

119896)] 119891 [119909

119896 119910119896]

(49)

Two-step method by Zheng et al (ZLH4) [10] is asfollows

119910119896= 119909119896minus

119891 (119909119896)

119891 [119909119896 119908119896] 119908119896= 119909119896+ 120574119891 (119909

119896)

119909119896+1

= 119910119896minus

119891 (119910119896)

119891 [119910119896 119908119896] + 119891 [119910

119896 119909119896 119908119896] (119910119896minus 119909119896)

(50)

Two-step method by Lotfi and Tavakoli (LT4) [11] is asfollows

119910119896= 119909119896minus

119891 (119909119896)

119891 [119909119896 119908119896]

119909119896+1

= 119910119896minus 119867 (119905

119896 119906119896)

119891 (119910119896)

119891 [119910119896 119908119896]

119905119896=119891 (119910119896)

119891 (119909119896) 119906

119896=119891 (119908119896)

119891 (119909119896)

(51)

where119867(119905119896 119906119896) = 1 + 119905

119896

Discrete Dynamics in Nature and Society 7

Table 1 1198911(119909) = sin(120587119909)119890(119909

2+119909 cos(119909)minus1)

+ 119909 log(119909 sin(119909) + 1) 120572 = 0 and 1199090= 06

Methods |1199091minus 120572| |119909

2minus 120572| |119909

3minus 120572| coc

Without memoryNew method (6) 120574 = minus1 06742(minus2) 01587(minus10) 04307(minus45) 40049KT4 120574 = minus001 06012(minus1) 01486(minus4) 06094(minus19) 40171ZLH4 120574 = minus001 05882(minus1) 01320(minus5) 08291(minus24) 39365LT 120574 = minus001 06178(minus1) 03334(minus4) 02739(minus17) 40368

With memoryNew method (19) 120574 = minus1 06742(minus2) 03177(minus11) 08372(minus62) 54213KT4 120574 = minus1 02102(minus1) 09935(minus9) 02326(minus48) 54031ZLH4 120574 = minus001 05882(minus1) 02204(minus8) 01503(minus45) 50216LT4 120574 = minus1 01752(minus1) 04934(minus9) 05220(minus50) 54214

Derivative-free Kung-Traubrsquos three-step method (KT8)[6] is as follows

119910119896= 119909119896minus

119891 (119909119896)

119891 [119909119896 119908119896] 119908119896= 119909119896+ 120574119891 (119909

119896)

119911119896= 119910119896minus

119891 (119910119896) 119891 (119908

119896)

[119891 (119908119896) minus 119891 (119910

119896)] 119891 [119909

119896 119910119896]

119909119896+1

= 119911119896minus119891 (119910119896) 119891 (119908

119896) (119910119896minus 119909119896+ 119891 (119909

119896) 119891 [119909

119896 119911119896])

[119891 (119910119896) minus 119891 (119911

119896)] [119891 (119908

119896) minus 119891 (119911

119896)]

+119891 (119910119896)

119891 [119910119896 119911119896]

(52)

Three-step methods developed by Zheng et al (ZLH8)[10] is as follows

119910119896= 119909119896minus

119891 (119909119896)

119891 [119909119896 119908119896] 119908119896= 119909119896+ 120574119891 (119909

119896)

119911119896= 119910119896minus

119891 (119910119896)

119891 [119910119896 119908119896] + 119891 [119910

119896 119909119896 119908119896] (119910119896minus 119909119896)

119909119896+1

= 119911119896

minus ( (119891 (119911119896))

times (119891 [119911119896 119910119896] + 119891 [119911

119896 119910119896 119909119896] (119911119896minus 119910119896)

+119891 [119911119896 119910119896 119909119896 119908119896] (119911119896minus 119910119896) (119911119896minus 119909119896))minus1

)

(53)

Three-step method by Lotfi and Tavakoli (LT8) [11] is asfollows

119910119896= 119909119896minus

119891 (119909119896)

119891 [119909119896 119908119896]

119911119896= 119910119896minus 119867 (119905

119896 119906119896)

119891 (119910119896)

119891 [119910119896 119908119896]

119905119896=119891 (119910119896)

119891 (119909119896) 119906

119896=119891 (119908119896)

119891 (119909119896)

119909119896+1

= 119911119896minus 119866 (119905

119896 119904119896)119882 (V

119896 119904119896)

119891 (119911119896)

119891 [119911119896 119908119896]

119904119896=119891 (119911119896)

119891 (119910119896) V

119896=119891 (119911119896)

119891 (119909119896)

(54)

where119882(119904119896 V119896) = 1+119904

2

119896+V2119896119866(119905119896 119904119896) = 1+119905

119896+119904119896+2119905119896119904119896+(minus1minus

120601119896)1199053

119896and (120601

119896= 1(1 + 120574

119896119891[119909119896 119908119896])) and119867(119905

119896 119906119896) = 1 + 119905

119896

are the weight functionsIn Tables 1 2 and 3 our two-step proposed classes (6)

and (19) have been compared with optimal two-stepmethodsKT4 ZLH4 and LT4 We observe that all these methodsbehave very well practically and confirm their theoreticalresults

Also Tables 4 5 and 6 present numerical results for ourthree-step classes (11) and (20) and methods KT8 ZLH8 andLT8 It is also clear that all these methods behave very wellpractically and confirm their relevant theories

We remark the importance of the choice of initial guessesIf they are chosen sufficiently close to the sought roots thenthe expected (theoretical) convergence speed will be reachedin practice otherwise all iterative root-finding methodsshow slower convergence especially at the beginning of theiterative process Hence a special attention should be paidto finding good initial approximations We note that efficientways for the determination of initial approximations of greataccuracy were discussed thoroughly in the works [12ndash14]

5 Conclusions

We have constructed two families of iterative methods with-out memory which are optimal in the sense of Kung andTraubrsquos conjecture in this paper Our proposed methods donot need any derivative

In addition they contain an accelerator parameter whichraises convergence order without any new functional evalu-ations In other words the efficiency index of the three-stepwith memory hit 1214 asymp 1861

We finalize this work by suggesting some points forfuture researches first developing the proposed methods forsome matrix functions such as the ones in [15 16] second

8 Discrete Dynamics in Nature and Society

Table 2 1198912(119909) = 119890

minus5119909(119909 minus 2)(119909

10+ 119909 + 2) 120572 = 2 and 119909

0= 22

Methods |1199091minus 120572| |119909

2minus 120572| |119909

3minus 120572| coc

Without memoryNew method (6) 120574 = minus001 09445(minus3) 01550(minus13) 01289(minus56) 39945KT4 120574 = minus001 01125(minus2) 03033(minus13) 01891(minus55) 39932ZLH4 120574 = minus01 09349(minus3) 01478(minus13) 01074(minus56) 39939LT 120574 = minus001 01343(minus2) 05965(minus13) 02821(minus54) 39918

With memoryNew method (19) 120574 = 1 01057(minus2) 01102(minus19) 08335(minus109) 52480KT4 120574 = 1 01267(minus2) 02180(minus19) 03580(minus107) 52364ZLH4 120574 = minus001 09445(minus3) 02868(minus20) 08014(minus112) 52264LT4 120574 = minus1 01343(minus2) 01072(minus19) 01150(minus108) 52035

Table 3 1198913(119909) = 119890

1199093minus119909

minus cos(1199092 minus 1) + 1199093+ 1 120572 = minus1 and 119909

0= minus165

Methods |1199091minus 120572| |119909

2minus 120572| |119909

3minus 120572| coc

Without memoryNew method (6) 120574 = minus001 06849(minus3) 08725(minus13) 02284(minus52) 40003KT4 120574 = minus001 06610(minus3) 08682(minus13) 02564(minus52) 40004ZLH4 120574 = 01 01214(minus2) 02388(minus12) 03560(minus51) 40004LT4 120574 = minus001 06398(minus3) 08549(minus13) 02714(minus52) 40002

With memoryNew method (19) 120574 = minus1 04838(minus2) 01348(minus11) 01230(minus64) 55506KT4 120574 = 1 01170(minus1) 03517(minus7) 08936(minus41) 60860ZLH4 120574 = minus001 06849(minus3) 07630(minus17) 01662(minus92) 54226LT4 120574 = minus001 06398(minus3) 06666(minus17) 07373(minus93) 54324

Table 4 1198911(119909) = sin(120587119909)119890(119909

2+119909 cos(119909)minus1)

+ 119909 log(119909 sin(119909) + 1) 120572 = 0 and 1199090= 06

Methods |1199091minus 120572| |119909

2minus 120572| |119909

3minus 120572| coc

Without memoryNew method (11) 120574 = minus1 02345(minus3) 01042(minus32) 01593(minus267) 79999KT8 120574 = minus1 03101(minus3) 02671(minus31) 08196(minus256) 79999ZLH8 120574 = minus001 01742(minus2) 01558(minus22) 06471(minus183) 79999LT8 120574 = minus1 05758(minus3) 07106(minus29) 03880(minus236) 79998

With memoryNew method (20) 120574 = minus01 03440(minus3) 01220(minus41) 08636(minus503) 119935KT8 120574 = minus001 09773(minus3) 02960(minus33) 01505(minus399) 120021ZLH8 120574 = minus001 01742(minus2) 03012(minus33) 04438(minus402) 119900LT8 120574 = minus01 07107(minus4) 02040(minus49) 04971(minus596) 120024

Table 5 1198912(119909) = 119890

minus5119909(119909 minus 2)(119909

10+ 119909 + 2) 120572 = 2 and 119909

0= 22

Methods |1199091minus 120572| |119909

2minus 120572| |119909

3minus 120572| coc

Without memoryNew method (11) 120574 = minus1 03124(minus6) 09972(minus56) 01074(minus451) 80000KT8 120574 = minus1 08055(minus6) 01413(minus52) 01268(minus426) 80000ZLH8 120574 = 1 04955(minus6) 05798(minus54) 02038(minus437) 80000LT8 120574 = minus1 04052(minus6) 05819(minus55) 01053(minus445) 80000

With memoryNew method (20) 120574 = minus1 03124(minus6) 01213(minus81) 03093(minus985) 119822KT8 120574 = minus1 08055(minus6) 02641(minus78) 09143(minus945) 119538ZLH8 120574 = minus001 03943(minus6) 01889(minus80) 06308(minus971) 119817LT8 120574 = minus001 04532(minus6) 04774(minus76) 05848(minus853) 111023

Discrete Dynamics in Nature and Society 9

Table 6 1198913(119909) = 119890

1199093minus119909

minus cos(1199092 minus 1) + 1199093+ 1 120572 = minus1 and 119909

0= minus165

Methods |1199091minus 120572| |119909

2minus 120572| |119909

3minus 120572| coc

Without memoryNew method (11) 120574 = minus1 02768(minus3) 03670(minus27) 03499(minus218) 80000KT8 120574 = minus1 02915(minus4) 05324(minus34) 06583(minus272) 80000ZLH8 120574 = 1 02322(minus1) 05044(minus11) 02253(minus88) 80081LT 120574 = minus1 01094(minus2) 01454(minus22) 01323(minus181) 80014

With memoryNew method (20) 120574 = minus1 02768(minus3) 02660(minus42) 01805(minus509) 119734KT8 120574 = minus1 02915(minus4) 05214(minus53) 01157(minus642) 120961ZLH8 120574 = minus001 06052(minus5) 02355(minus65) 04206(minus786) 119310LT8 120574 = minus1 01094(minus2) 03522(minus34) 01717(minus415) 121081

exploring its dynamic or basins of attractions and lastlyextending the developedmethods with memory using two orthree accelerators

Conflict of Interests

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

Acknowledgments

The first author thanks Islamic Azad University HamedanBranch for the financial support of the present research Theauthors are also thankful for insightful comments of three ref-erees whom helped with the readability and reliability of thepresent paperThe fourth author also gratefully acknowledgesthat this research was partially supported by the UniversityPutra Malaysia under the GP-IBT Grant Scheme havingproject number GP-IBT20139420100

References

[1] N Kyurkchiev and A Iliev ldquoOn some multipoint methodsarising from optimal in the sense of Kung-Traub algorithmsrdquoBiomath vol 2 no 1 Article ID 1305155 7 pages 2013

[2] T Lotfi F Soleymani S Sharifi S Shateyi and F K HaghanildquoMulti-point iterative methods for finding all the simple zerosin an intervalrdquo Journal of AppliedMathematics vol 2014 ArticleID 601205 14 pages 2014

[3] J F Traub G W Wasilkowski and H Wozniakowski Informa-tion Uncertainty Complexity Addison-Wesley 1983

[4] M S Rhee and Y I Kim ldquoA general class of optimal fourth-order multiple-root finders without memory for nonlinearequationsrdquo Applied Mathematical Sciences vol 7 pp 5537ndash55512013

[5] H Wozniakowski ldquoMaximal order of multipoint iterationsusing n evaluationsrdquo in Analytic Computational Complexity JF Traub Ed pp 75ndash107 Academic Press New York NY USA1976

[6] 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

[7] Y H Geum and Y I Kim ldquoAn optimal family of fast 16th-orderderivative-free multipoint simple-root finders for nonlinear

equationsrdquo Journal ofOptimizationTheory andApplications vol160 no 2 pp 608ndash622 2014

[8] J F Traub Iterative Methods for the Solution of EquationsPrentice-Hall New York NY USA 1964

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

[10] 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

[11] T Lotfi and E Tavakoli ldquoOn a new efficient Steffensen-likeiterative class by applying a suitable self-accelerator parameterrdquoThe Scientific World Journal vol 2014 Article ID 769758 9pages 2014

[12] F Soleymani and S Shateyi ldquoTwo optimal eighth-orderderivative-free classes of iterative methodsrdquo Abstract andApplied Analysis vol 2012 Article ID 318165 14 pages 2012

[13] F Soleymani and D K R Babajee ldquoComputing multiple zerosusing a class of quartically convergent methodsrdquo AlexandriaEngineering Journal vol 52 no 3 pp 531ndash541 2013

[14] F Soleymani ldquoSome high-order iterativemethods for finding allthe real zerosrdquoThai Journal of Mathematics vol 12 pp 313ndash3272014

[15] F Soleymani E Tohidi S Shateyi and F K Haghani ldquoSomematrix iterations for computing matrix sign functionrdquo Journalof Applied Mathematics vol 2014 Article ID 425654 9 pages2014

[16] F Soleymani P S Stanimirovic S Shateyi and F K HaghanildquoApproximating the matrix sign function using a novel iterativemethodrdquo Abstract and Applied Analysis vol 2014 Article ID105301 9 pages 2014

Submit your manuscripts athttpwwwhindawicom

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Differential EquationsInternational Journal of

Volume 2014

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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

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Discrete Dynamics in Nature and Society

3 Extension to Methods with MemoryThis section is concernedwith extraction of efficientmethodswith memory from (6) and (11) through a careful inspectionof their error equations containing the parameter 120574 whichcan be approximated in such a way that increases the localorder of convergence

Toward this goal we set 120574 = 120574119896as the iteration proceeds by

the formula 120574119896= minus11198911015840(120572) for 119896 = 1 2 where 1198911015840(120572) is an

approximation of 1198911015840(120572) We therefore use the approximation

120574119896= minus

1

1198911015840 (120572)

= minus1

1198731015840

3(119909119896) (14)

for (6) and the following one

120574119896= minus

1

1198911015840 (120572)

= minus1

1198731015840

4(119909119896) (15)

for (11) Here we consider Newtonrsquos interpolating polynomialof third degree as the method for approximating 119891

1015840(120572) in

two-step method (6) and Newtonrsquos interpolating polynomialof fourth degree for approximating 119891

1015840(120572) in the three-

step method (11) where 1198733(119905) is the Newtonrsquos interpola-

tion polynomial of third degree set through four availableapproximations 119909

119896 119909119896minus1

119910119896minus1

119908119896minus1

and1198734(119905) is the Newtonrsquos

interpolation polynomial of fourth degree set through fiveavailable approximations 119909

119896 119911119896minus1

119910119896minus1

119908119896minus1

119909119896minus1

Consider

1198731015840

3(119909119896) = [

119889

1198891199051198733(119905)]

119905=119909119896

= [119889

119889119905(119891 (119909119896) + 119891 [119909

119896 119909119896minus1

] (119905 minus 119909119896)

+ 119891 [119909119896 119909119896minus1

119910119896minus1

] (119905 minus 119909119896) (119905 minus 119909

119896minus1)

+ 119891 [119909119896 119909119896minus1

119910119896minus1

119908119896minus1

] (119905 minus 119909119896)

times (119905 minus 119909119896minus1

) (119905 minus 119910119896minus1

)) ]

119905=119909119896

= 119891 [119909119896 119909119896minus1

] + 119891 [119909119896 119909119896minus1

119910119896minus1

] (119909119896minus 119909119896minus1

)

+ 119891 [119909119896 119909119896minus1

119910119896minus1

119908119896minus1

] (119909119896minus 119909119896minus1

) (119909119896minus 119910119896minus1

)

(16)

1198731015840

4(119909119896)

= [119889

1198891199051198734(119905)]

119905=119909119896

= [119889

119889119905(119891 (119909119896) + 119891 [119909

119896 119911119896minus1

] (119905 minus 119909119896)

+ 119891 [119909119896 119911119896minus1

119910119896minus1

] (119905 minus 119909119896) (119905 minus 119911

119896minus1)

+ 119891 [119909119896 119911119896minus1

119910119896minus1

119909119896minus1

] (119905 minus 119909119896)

times (119905 minus 119911119896minus1

) (119905 minus 119910119896minus1

)

+ 119891 [119909119896 119911119896minus1

119910119896minus1

119909119896minus1

119908119896minus1

]

times(119905minus119909119896)(119905 minus 119911

119896minus1)(119905minus 119910

119896minus1)(119905minus 119909

119896minus1))]

119905=119909119896

= 119891 [119909119896 119911119896minus1

] + 119891 [119909119896 119911119896minus1

119910119896minus1

] (119909119896minus 119911119896minus1

)

+ 119891 [119909119896 119911119896minus1

119910119896minus1

119909119896minus1

] (119909119896minus 119911119896minus1

)(119909119896minus 119910119896minus1

)

+ 119891 [119909119896 119911119896minus1

119910119896minus1

119909119896minus1

119908119896minus1

] (119909119896minus 119911119896minus1

)

times (119909119896minus 119910119896minus1

) (119909119896minus 119909119896minus1

)

(17)

Note that a divided difference of order 119898 defined recur-sively as

119891 [1199090 1199091 119909

119898]

=119891 [1199091 119909

119898] minus 119891 [119909

0 1199091 119909

119898minus1]

119909119898minus 1199090

119898 ⩾ 2

(18)

has been used throughout this paper Hence the with mem-ory developments of (6) and (11) can be presented as follows

119910119896= 119909119896minus

119891 (119909119896)

119891 [119909119896 119908119896] 119908119896= 119909119896+ 120574119896119891 (119909119896)

119909119896+1

= 119910119896minus

119891 (119910119896)

1198751015840

2(119910119896)

(19)

119910119896= 119909119896minus

119891 (119909119896)

119891 [119909119896 119908119896] 119908119896= 119909119896+ 120574119896119891 (119909119896)

119911119896= 119910119896minus119891 (119910119896)

1198751015840

2(119905119896)

119909 = 119909119896+1

= 119911119896minus

119891 (119911119896)

1198751015840

3(119911119896)

(20)

Remark 3 Accelerating methods obtained by recursivelycalculated free parameter may also be called self-acceleratingmethodsThe initial value 120574

0should be chosen before starting

the iterative process for example using one of the waysproposed in [8]

Here we attempt to prove that themethods withmemory(19) and (20) have convergence orders six and twelve providedthat we use accelerator 120574

119896as in (14) and (15) For ease of

continuing analysis we introduce the convenient notation asfollows If the sequence 119909

119896 converges to the zero 120572 of119891with

the order 119901 then we write 119890119896+1

sim 119890119901

119896 where 119890

119896= 119909119896minus 120572

The following lemma will play a crucial role in improvingthe convergence order of the methods with memory to beproposed in this paper

Lemma 4 If 120574119896= minus1119873

1015840

3(119909119896) 119890119896= 119909119896minus 120572 119890119896119910

= 119910119896minus 120572 and

119890119896119908

= 119908119896minus 120572 then the following relation holds

1 + 1205741198961198911015840(120572) sim 119888

4119890119896minus1

119890119896minus1119908

119890119896minus1119910

sim 119890119896minus1

119890119896minus1119908

119890119896minus1119910

(21)

Discrete Dynamics in Nature and Society 5

Proof Following the same terminology as in Theorem 2 andthe symbolic software Mathematica it would be easy toobtain (21) via writing the following code

ClearAll[Globallsquo lowast ]A[t ] = InterpolatingPolynomial[e fxew fw ey fy e1 fx1 t]SimplifyApproximation = minus1A1015840[e1]Simplifyfx = f1a lowast (e + c2 lowast e2 + c3 lowast e3 + c4 lowast e4)fw = f1a lowast (ew + c2 lowast ew2 + c3 lowast ew3

+ c4 lowast ew4)fy = f1a lowast (ey + c2 lowast ey2 + c3 lowast ey3

+ c4 lowast ey4)fx1 = f1a lowast (e1 + c2 lowast e12 + c3 lowast e13

+ c4 lowast e14)b = Series[Approximation e 0 2 ew 0 2ey 0 2 e1 0 2]SimplifyCollect[Series[1 + b lowast f1a e 0 1 ew 0 1ey 0 1 e1 0 0]e ew ey e1 Simplify]

The proof is complete

In order to obtain the R-order of convergence [9] ofthe method with memory (19) we establish the followingtheorem

Theorem 5 If an initial approximation 1199090is sufficiently close

to the zero 120572 of 119891(119909) and the parameter 120574119896in the iterative

scheme (19) is recursively calculated by the forms given in (14)then the R-order of convergence for (19) is at least six

Proof Let 119909119896 be a sequence of approximations generated

by the iterative method with memory (19) If this sequenceconverges to the zero 120572 of 119891 with the order 119901 then we write

119890119896+1

sim 119890119901

119896 119890

119896= 119909119896minus 120572 (22)

Thus

119890119896+1

sim (119890119901

119896minus1)119901

= 1198901199012

119896minus1 (23)

Moreover assume that the iterative sequences119908119896and 119910119896have

the orders 1199011and 119901

2 respectively Then (22) gives

119890119896119908

sim 1198901199011

119896sim (119890119901

119896minus1)1199011= 1198901199011199011

119896minus1 (24)

119890119896119910

sim 1198901199012

119896sim (119890119901

119896minus1)1199012= 1198901199011199012

119896minus1 (25)

Since

119890119896119908

sim (1 + 1205741198961198911015840(120572)) 119890

119896 (26)

119890119896119910

sim (1 + 1205741198961198911015840(120572)) 119890

2

119896 (27)

119890119896+1

sim (1 + 1205741198961198911015840(120572))2

1198904

119896 (28)

using Lemma 4 and (27) induce

119890119896119908

sim (1 + 1205741198961198911015840(120572)) 119890

119896sim (119890119896minus1

119890119896minus1119910

119890119896minus1119908

) 119890119896= 119890119901+1199011+1199012+1

119896minus1

(29)

119890119896119910

sim (1 + 1205741198961198911015840(120572)) 119890

2

119896sim (119890119896minus1

119890119896minus1119910

119890119896minus1119908

) 1198902

119896= 1198902119901+1199011+1199012+1

119896minus1

(30)

119890119896+1

sim (1 + 1205741198961198911015840(120572))2

1198904

119896sim (119890119896minus1

119890119896minus1119910

119890119896minus1119908

)2

1198904

119896

= 1198904119901+21199011+21199012+2

119896minus1

(31)

Matching the powers of 119890119896minus1

on the right hand sides of (24)ndash(29) (25)ndash(30) and (23)ndash(31) one can obtain

1199011199011minus 119901 minus 119901

1minus 1199012minus 1 = 0

1199011199012minus 2119901 minus 119901

1minus 1199012minus 1 = 0

1199012minus 4119901 minus 2119901

1minus 21199012minus 2 = 0

(32)

The nontrivial solution of this system is 1199011= 2 119901

2= 3 and

119901 = 6 This completes the proof

Using symbolic computations and Taylor expansions it iseasy to derive the following lemma

Lemma 6 Assuming (15) and (17) we have

1 + 1205741198961198911015840(120572) sim 119890

119896minus1119890119896minus1119908

119890119896minus1119910

119890119896minus1119911

(33)

where 120574119896= minus1119873

1015840

4(119909119896) 119890119896= 119909119896minus120572 119890119896119911

= 119911119896minus120572 119890119896119910

= 119910119896minus120572

and 119890119896119908

= 119908119896minus 120572

Proof The proof of this lemma is similar to Lemma 4 It ishence omitted

Similarly for the three-step method with memory (20)we have the following theorem

Theorem 7 If an initial approximation 1199090is sufficiently close

to the zero 120572 of 119891(119909) and the parameter 120574119896in the iterative

scheme (20) is recursively calculated by the forms given in (15)then the order of convergence for (20) is at least twelve

Proof Let 119909119896 be a sequence of approximations generated

by the iterative method with memory (20) If this sequenceconverges to the zero 120572 of 119891 with the order 119901 then we write

119890119896+1

sim 119890119901

119896 119890

119896= 119909119896minus 120572 (34)

So

119890119896+1

sim (119890119901

119896minus1)119901

= 1198901199012

119896minus1 (35)

Moreover assume that the iterative sequences 119908119896 119910119896 and 119911

119896

have the orders 1199011 1199012 and 119901

3 respectively Then (34) gives

119890119896119908

sim 1198901199011

119896sim (119890119901

119896minus1)1199011= 1198901199011199011

119896minus1 (36)

119890119896119910

sim 1198901199012

119896sim (119890119901

119896minus1)1199012= 1198901199011199012

119896minus1 (37)

119890119896119911

sim 1198901199013

119896sim (119890119901

119896minus1)1199013= 1198901199011199013

119896minus1 (38)

6 Discrete Dynamics in Nature and Society

Since

119890119896119908

sim (1 + 1205741198961198911015840(120572)) 119890

119896 (39)

119890119896119910

sim (1 + 1205741198961198911015840(120572)) 119890

2

119896 (40)

119890119896119911

sim (1 + 1205741198961198911015840(120572))2

1198904

119896 (41)

119890119896+1

sim (1 + 1205741198961198911015840(120572))4

1198908

119896 (42)

by Lemma 6 and (40) we obtain

119890119896119908

sim (1 + 1205741198961198911015840(120572)) 119890

119896sim (119890119896minus1

119890119896minus1119910

119890119896minus1119908

119890119896minus1119911

) 119890119896

= 119890119901+1199011+1199012+1199013+1

119896minus1

(43)

119890119896119910

sim (1 + 1205741198961198911015840(120572)) 119890

2

119896sim (119890119896minus1

119890119896minus1119910

119890119896minus1119908

119890119896minus1119911

) 1198902

119896

= 1198902119901+1199011+1199012+1199013+1

119896minus1

(44)

119890119896119911

sim (1 + 1205741198961198911015840(120572)) 119890

2

119896sim (119890119896minus1

119890119896minus1119910

119890119896minus1119908

119890119896minus1119911

)2

1198904

119896

= 1198904119901+21199011+21199012+21199013+2

119896minus1

(45)

119890119896+1

sim (1 + 1205741198961198911015840(120572))2

1198904

119896sim (119890119896minus1

119890119896minus1119910

119890119896minus1119908

119890119896minus1119911

)4

1198908

119896

= 1198908119901+41199011+41199012+41199013+4

119896minus1

(46)

Matching the powers of 119890119896minus1

on the right hand sides of (36)ndash(43) (37)ndash(44) (38)ndash(45) and (35)ndash(46) one can obtain

1199011199011minus 119901 minus 119901

1minus 1199012minus 1199013minus 1 = 0

1199011199012minus 2119901 minus 119901

1minus 1199012minus 1199013minus 1 = 0

1199011199013minus 4119901 minus 2119901

1minus 21199012minus 21199013minus 2 = 0

1199012minus 8119901 minus 4119901

1minus 41199012minus 41199013minus 2 = 0

(47)

This system has the solutions 1199011= 2 119901

2= 3 119901

3= 6 and

119901 = 12 The proof is complete

Remark 8 The advantage of the proposedmethods is in theirhigher computational efficiency indices We emphasize thatthe increase of the R-order of convergence has been obtainedwithout any additional function evaluations which pointsto very high computational efficiency Indeed the efficiencyindex 1214 asymp 1861 of the proposed three-step twelfth-ordermethod with memory is higher than the efficiency index613

asymp 1817 of (19) 814 asymp 1682 of the optimal three-pointmethod (11) and 413 asymp 1587 of (6)

Remark 9 We observe that the methods (19) and (20) withmemory are considerably accelerated (up to 50) in contrastto the corresponding method (11) without memory

4 Numerical Experiments

In this section we test our proposed methods and comparetheir results with some other methods of the same order of

convergenceThe results are reported using the programmingpackage Mathematica 8 in multiple precision arithmeticenvironment We have considered 1000 digits floating pointarithmetic so as to minimize the round-off errors as muchas possible The errors |119909

119896minus 120572| denote approximations to

the sought zeros and 119886(minus119887) stands for 119886 times 10minus119887 Moreover

coc indicates the computational order of convergence and iscomputed by

coc =log (1003816100381610038161003816119891 (119909

119896) 119891 (119909

119896minus1)1003816100381610038161003816)

log (1003816100381610038161003816119891 (119909119896minus1

) 119891 (119909119896minus2

)1003816100381610038161003816) (48)

It is assumed that the initial estimate 1205740should be chosen

before starting the iterative process and also 1199090is given

suitablySeveral iterative methods of optimal orders four and

eight for comparing with our proposed methods have beenchosen as comes next

Derivative-free Kung-Traubrsquos two-step method (KT4) [6]is as follows

119910119896= 119909119896minus

119891 (119909119896)

119891 [119909119896 119908119896] 119908119896= 119909119896+ 120574119891 (119909

119896)

119909119896+1

= 119910119896minus

119891 (119910119896) 119891 (119908

119896)

[119891 (119908119896) minus 119891 (119910

119896)] 119891 [119909

119896 119910119896]

(49)

Two-step method by Zheng et al (ZLH4) [10] is asfollows

119910119896= 119909119896minus

119891 (119909119896)

119891 [119909119896 119908119896] 119908119896= 119909119896+ 120574119891 (119909

119896)

119909119896+1

= 119910119896minus

119891 (119910119896)

119891 [119910119896 119908119896] + 119891 [119910

119896 119909119896 119908119896] (119910119896minus 119909119896)

(50)

Two-step method by Lotfi and Tavakoli (LT4) [11] is asfollows

119910119896= 119909119896minus

119891 (119909119896)

119891 [119909119896 119908119896]

119909119896+1

= 119910119896minus 119867 (119905

119896 119906119896)

119891 (119910119896)

119891 [119910119896 119908119896]

119905119896=119891 (119910119896)

119891 (119909119896) 119906

119896=119891 (119908119896)

119891 (119909119896)

(51)

where119867(119905119896 119906119896) = 1 + 119905

119896

Discrete Dynamics in Nature and Society 7

Table 1 1198911(119909) = sin(120587119909)119890(119909

2+119909 cos(119909)minus1)

+ 119909 log(119909 sin(119909) + 1) 120572 = 0 and 1199090= 06

Methods |1199091minus 120572| |119909

2minus 120572| |119909

3minus 120572| coc

Without memoryNew method (6) 120574 = minus1 06742(minus2) 01587(minus10) 04307(minus45) 40049KT4 120574 = minus001 06012(minus1) 01486(minus4) 06094(minus19) 40171ZLH4 120574 = minus001 05882(minus1) 01320(minus5) 08291(minus24) 39365LT 120574 = minus001 06178(minus1) 03334(minus4) 02739(minus17) 40368

With memoryNew method (19) 120574 = minus1 06742(minus2) 03177(minus11) 08372(minus62) 54213KT4 120574 = minus1 02102(minus1) 09935(minus9) 02326(minus48) 54031ZLH4 120574 = minus001 05882(minus1) 02204(minus8) 01503(minus45) 50216LT4 120574 = minus1 01752(minus1) 04934(minus9) 05220(minus50) 54214

Derivative-free Kung-Traubrsquos three-step method (KT8)[6] is as follows

119910119896= 119909119896minus

119891 (119909119896)

119891 [119909119896 119908119896] 119908119896= 119909119896+ 120574119891 (119909

119896)

119911119896= 119910119896minus

119891 (119910119896) 119891 (119908

119896)

[119891 (119908119896) minus 119891 (119910

119896)] 119891 [119909

119896 119910119896]

119909119896+1

= 119911119896minus119891 (119910119896) 119891 (119908

119896) (119910119896minus 119909119896+ 119891 (119909

119896) 119891 [119909

119896 119911119896])

[119891 (119910119896) minus 119891 (119911

119896)] [119891 (119908

119896) minus 119891 (119911

119896)]

+119891 (119910119896)

119891 [119910119896 119911119896]

(52)

Three-step methods developed by Zheng et al (ZLH8)[10] is as follows

119910119896= 119909119896minus

119891 (119909119896)

119891 [119909119896 119908119896] 119908119896= 119909119896+ 120574119891 (119909

119896)

119911119896= 119910119896minus

119891 (119910119896)

119891 [119910119896 119908119896] + 119891 [119910

119896 119909119896 119908119896] (119910119896minus 119909119896)

119909119896+1

= 119911119896

minus ( (119891 (119911119896))

times (119891 [119911119896 119910119896] + 119891 [119911

119896 119910119896 119909119896] (119911119896minus 119910119896)

+119891 [119911119896 119910119896 119909119896 119908119896] (119911119896minus 119910119896) (119911119896minus 119909119896))minus1

)

(53)

Three-step method by Lotfi and Tavakoli (LT8) [11] is asfollows

119910119896= 119909119896minus

119891 (119909119896)

119891 [119909119896 119908119896]

119911119896= 119910119896minus 119867 (119905

119896 119906119896)

119891 (119910119896)

119891 [119910119896 119908119896]

119905119896=119891 (119910119896)

119891 (119909119896) 119906

119896=119891 (119908119896)

119891 (119909119896)

119909119896+1

= 119911119896minus 119866 (119905

119896 119904119896)119882 (V

119896 119904119896)

119891 (119911119896)

119891 [119911119896 119908119896]

119904119896=119891 (119911119896)

119891 (119910119896) V

119896=119891 (119911119896)

119891 (119909119896)

(54)

where119882(119904119896 V119896) = 1+119904

2

119896+V2119896119866(119905119896 119904119896) = 1+119905

119896+119904119896+2119905119896119904119896+(minus1minus

120601119896)1199053

119896and (120601

119896= 1(1 + 120574

119896119891[119909119896 119908119896])) and119867(119905

119896 119906119896) = 1 + 119905

119896

are the weight functionsIn Tables 1 2 and 3 our two-step proposed classes (6)

and (19) have been compared with optimal two-stepmethodsKT4 ZLH4 and LT4 We observe that all these methodsbehave very well practically and confirm their theoreticalresults

Also Tables 4 5 and 6 present numerical results for ourthree-step classes (11) and (20) and methods KT8 ZLH8 andLT8 It is also clear that all these methods behave very wellpractically and confirm their relevant theories

We remark the importance of the choice of initial guessesIf they are chosen sufficiently close to the sought roots thenthe expected (theoretical) convergence speed will be reachedin practice otherwise all iterative root-finding methodsshow slower convergence especially at the beginning of theiterative process Hence a special attention should be paidto finding good initial approximations We note that efficientways for the determination of initial approximations of greataccuracy were discussed thoroughly in the works [12ndash14]

5 Conclusions

We have constructed two families of iterative methods with-out memory which are optimal in the sense of Kung andTraubrsquos conjecture in this paper Our proposed methods donot need any derivative

In addition they contain an accelerator parameter whichraises convergence order without any new functional evalu-ations In other words the efficiency index of the three-stepwith memory hit 1214 asymp 1861

We finalize this work by suggesting some points forfuture researches first developing the proposed methods forsome matrix functions such as the ones in [15 16] second

8 Discrete Dynamics in Nature and Society

Table 2 1198912(119909) = 119890

minus5119909(119909 minus 2)(119909

10+ 119909 + 2) 120572 = 2 and 119909

0= 22

Methods |1199091minus 120572| |119909

2minus 120572| |119909

3minus 120572| coc

Without memoryNew method (6) 120574 = minus001 09445(minus3) 01550(minus13) 01289(minus56) 39945KT4 120574 = minus001 01125(minus2) 03033(minus13) 01891(minus55) 39932ZLH4 120574 = minus01 09349(minus3) 01478(minus13) 01074(minus56) 39939LT 120574 = minus001 01343(minus2) 05965(minus13) 02821(minus54) 39918

With memoryNew method (19) 120574 = 1 01057(minus2) 01102(minus19) 08335(minus109) 52480KT4 120574 = 1 01267(minus2) 02180(minus19) 03580(minus107) 52364ZLH4 120574 = minus001 09445(minus3) 02868(minus20) 08014(minus112) 52264LT4 120574 = minus1 01343(minus2) 01072(minus19) 01150(minus108) 52035

Table 3 1198913(119909) = 119890

1199093minus119909

minus cos(1199092 minus 1) + 1199093+ 1 120572 = minus1 and 119909

0= minus165

Methods |1199091minus 120572| |119909

2minus 120572| |119909

3minus 120572| coc

Without memoryNew method (6) 120574 = minus001 06849(minus3) 08725(minus13) 02284(minus52) 40003KT4 120574 = minus001 06610(minus3) 08682(minus13) 02564(minus52) 40004ZLH4 120574 = 01 01214(minus2) 02388(minus12) 03560(minus51) 40004LT4 120574 = minus001 06398(minus3) 08549(minus13) 02714(minus52) 40002

With memoryNew method (19) 120574 = minus1 04838(minus2) 01348(minus11) 01230(minus64) 55506KT4 120574 = 1 01170(minus1) 03517(minus7) 08936(minus41) 60860ZLH4 120574 = minus001 06849(minus3) 07630(minus17) 01662(minus92) 54226LT4 120574 = minus001 06398(minus3) 06666(minus17) 07373(minus93) 54324

Table 4 1198911(119909) = sin(120587119909)119890(119909

2+119909 cos(119909)minus1)

+ 119909 log(119909 sin(119909) + 1) 120572 = 0 and 1199090= 06

Methods |1199091minus 120572| |119909

2minus 120572| |119909

3minus 120572| coc

Without memoryNew method (11) 120574 = minus1 02345(minus3) 01042(minus32) 01593(minus267) 79999KT8 120574 = minus1 03101(minus3) 02671(minus31) 08196(minus256) 79999ZLH8 120574 = minus001 01742(minus2) 01558(minus22) 06471(minus183) 79999LT8 120574 = minus1 05758(minus3) 07106(minus29) 03880(minus236) 79998

With memoryNew method (20) 120574 = minus01 03440(minus3) 01220(minus41) 08636(minus503) 119935KT8 120574 = minus001 09773(minus3) 02960(minus33) 01505(minus399) 120021ZLH8 120574 = minus001 01742(minus2) 03012(minus33) 04438(minus402) 119900LT8 120574 = minus01 07107(minus4) 02040(minus49) 04971(minus596) 120024

Table 5 1198912(119909) = 119890

minus5119909(119909 minus 2)(119909

10+ 119909 + 2) 120572 = 2 and 119909

0= 22

Methods |1199091minus 120572| |119909

2minus 120572| |119909

3minus 120572| coc

Without memoryNew method (11) 120574 = minus1 03124(minus6) 09972(minus56) 01074(minus451) 80000KT8 120574 = minus1 08055(minus6) 01413(minus52) 01268(minus426) 80000ZLH8 120574 = 1 04955(minus6) 05798(minus54) 02038(minus437) 80000LT8 120574 = minus1 04052(minus6) 05819(minus55) 01053(minus445) 80000

With memoryNew method (20) 120574 = minus1 03124(minus6) 01213(minus81) 03093(minus985) 119822KT8 120574 = minus1 08055(minus6) 02641(minus78) 09143(minus945) 119538ZLH8 120574 = minus001 03943(minus6) 01889(minus80) 06308(minus971) 119817LT8 120574 = minus001 04532(minus6) 04774(minus76) 05848(minus853) 111023

Discrete Dynamics in Nature and Society 9

Table 6 1198913(119909) = 119890

1199093minus119909

minus cos(1199092 minus 1) + 1199093+ 1 120572 = minus1 and 119909

0= minus165

Methods |1199091minus 120572| |119909

2minus 120572| |119909

3minus 120572| coc

Without memoryNew method (11) 120574 = minus1 02768(minus3) 03670(minus27) 03499(minus218) 80000KT8 120574 = minus1 02915(minus4) 05324(minus34) 06583(minus272) 80000ZLH8 120574 = 1 02322(minus1) 05044(minus11) 02253(minus88) 80081LT 120574 = minus1 01094(minus2) 01454(minus22) 01323(minus181) 80014

With memoryNew method (20) 120574 = minus1 02768(minus3) 02660(minus42) 01805(minus509) 119734KT8 120574 = minus1 02915(minus4) 05214(minus53) 01157(minus642) 120961ZLH8 120574 = minus001 06052(minus5) 02355(minus65) 04206(minus786) 119310LT8 120574 = minus1 01094(minus2) 03522(minus34) 01717(minus415) 121081

exploring its dynamic or basins of attractions and lastlyextending the developedmethods with memory using two orthree accelerators

Conflict of Interests

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

Acknowledgments

The first author thanks Islamic Azad University HamedanBranch for the financial support of the present research Theauthors are also thankful for insightful comments of three ref-erees whom helped with the readability and reliability of thepresent paperThe fourth author also gratefully acknowledgesthat this research was partially supported by the UniversityPutra Malaysia under the GP-IBT Grant Scheme havingproject number GP-IBT20139420100

References

[1] N Kyurkchiev and A Iliev ldquoOn some multipoint methodsarising from optimal in the sense of Kung-Traub algorithmsrdquoBiomath vol 2 no 1 Article ID 1305155 7 pages 2013

[2] T Lotfi F Soleymani S Sharifi S Shateyi and F K HaghanildquoMulti-point iterative methods for finding all the simple zerosin an intervalrdquo Journal of AppliedMathematics vol 2014 ArticleID 601205 14 pages 2014

[3] J F Traub G W Wasilkowski and H Wozniakowski Informa-tion Uncertainty Complexity Addison-Wesley 1983

[4] M S Rhee and Y I Kim ldquoA general class of optimal fourth-order multiple-root finders without memory for nonlinearequationsrdquo Applied Mathematical Sciences vol 7 pp 5537ndash55512013

[5] H Wozniakowski ldquoMaximal order of multipoint iterationsusing n evaluationsrdquo in Analytic Computational Complexity JF Traub Ed pp 75ndash107 Academic Press New York NY USA1976

[6] 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

[7] Y H Geum and Y I Kim ldquoAn optimal family of fast 16th-orderderivative-free multipoint simple-root finders for nonlinear

equationsrdquo Journal ofOptimizationTheory andApplications vol160 no 2 pp 608ndash622 2014

[8] J F Traub Iterative Methods for the Solution of EquationsPrentice-Hall New York NY USA 1964

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

[10] 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

[11] T Lotfi and E Tavakoli ldquoOn a new efficient Steffensen-likeiterative class by applying a suitable self-accelerator parameterrdquoThe Scientific World Journal vol 2014 Article ID 769758 9pages 2014

[12] F Soleymani and S Shateyi ldquoTwo optimal eighth-orderderivative-free classes of iterative methodsrdquo Abstract andApplied Analysis vol 2012 Article ID 318165 14 pages 2012

[13] F Soleymani and D K R Babajee ldquoComputing multiple zerosusing a class of quartically convergent methodsrdquo AlexandriaEngineering Journal vol 52 no 3 pp 531ndash541 2013

[14] F Soleymani ldquoSome high-order iterativemethods for finding allthe real zerosrdquoThai Journal of Mathematics vol 12 pp 313ndash3272014

[15] F Soleymani E Tohidi S Shateyi and F K Haghani ldquoSomematrix iterations for computing matrix sign functionrdquo Journalof Applied Mathematics vol 2014 Article ID 425654 9 pages2014

[16] F Soleymani P S Stanimirovic S Shateyi and F K HaghanildquoApproximating the matrix sign function using a novel iterativemethodrdquo Abstract and Applied Analysis vol 2014 Article ID105301 9 pages 2014

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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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

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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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

Discrete Dynamics in Nature and Society 5

Proof Following the same terminology as in Theorem 2 andthe symbolic software Mathematica it would be easy toobtain (21) via writing the following code

ClearAll[Globallsquo lowast ]A[t ] = InterpolatingPolynomial[e fxew fw ey fy e1 fx1 t]SimplifyApproximation = minus1A1015840[e1]Simplifyfx = f1a lowast (e + c2 lowast e2 + c3 lowast e3 + c4 lowast e4)fw = f1a lowast (ew + c2 lowast ew2 + c3 lowast ew3

+ c4 lowast ew4)fy = f1a lowast (ey + c2 lowast ey2 + c3 lowast ey3

+ c4 lowast ey4)fx1 = f1a lowast (e1 + c2 lowast e12 + c3 lowast e13

+ c4 lowast e14)b = Series[Approximation e 0 2 ew 0 2ey 0 2 e1 0 2]SimplifyCollect[Series[1 + b lowast f1a e 0 1 ew 0 1ey 0 1 e1 0 0]e ew ey e1 Simplify]

The proof is complete

In order to obtain the R-order of convergence [9] ofthe method with memory (19) we establish the followingtheorem

Theorem 5 If an initial approximation 1199090is sufficiently close

to the zero 120572 of 119891(119909) and the parameter 120574119896in the iterative

scheme (19) is recursively calculated by the forms given in (14)then the R-order of convergence for (19) is at least six

Proof Let 119909119896 be a sequence of approximations generated

by the iterative method with memory (19) If this sequenceconverges to the zero 120572 of 119891 with the order 119901 then we write

119890119896+1

sim 119890119901

119896 119890

119896= 119909119896minus 120572 (22)

Thus

119890119896+1

sim (119890119901

119896minus1)119901

= 1198901199012

119896minus1 (23)

Moreover assume that the iterative sequences119908119896and 119910119896have

the orders 1199011and 119901

2 respectively Then (22) gives

119890119896119908

sim 1198901199011

119896sim (119890119901

119896minus1)1199011= 1198901199011199011

119896minus1 (24)

119890119896119910

sim 1198901199012

119896sim (119890119901

119896minus1)1199012= 1198901199011199012

119896minus1 (25)

Since

119890119896119908

sim (1 + 1205741198961198911015840(120572)) 119890

119896 (26)

119890119896119910

sim (1 + 1205741198961198911015840(120572)) 119890

2

119896 (27)

119890119896+1

sim (1 + 1205741198961198911015840(120572))2

1198904

119896 (28)

using Lemma 4 and (27) induce

119890119896119908

sim (1 + 1205741198961198911015840(120572)) 119890

119896sim (119890119896minus1

119890119896minus1119910

119890119896minus1119908

) 119890119896= 119890119901+1199011+1199012+1

119896minus1

(29)

119890119896119910

sim (1 + 1205741198961198911015840(120572)) 119890

2

119896sim (119890119896minus1

119890119896minus1119910

119890119896minus1119908

) 1198902

119896= 1198902119901+1199011+1199012+1

119896minus1

(30)

119890119896+1

sim (1 + 1205741198961198911015840(120572))2

1198904

119896sim (119890119896minus1

119890119896minus1119910

119890119896minus1119908

)2

1198904

119896

= 1198904119901+21199011+21199012+2

119896minus1

(31)

Matching the powers of 119890119896minus1

on the right hand sides of (24)ndash(29) (25)ndash(30) and (23)ndash(31) one can obtain

1199011199011minus 119901 minus 119901

1minus 1199012minus 1 = 0

1199011199012minus 2119901 minus 119901

1minus 1199012minus 1 = 0

1199012minus 4119901 minus 2119901

1minus 21199012minus 2 = 0

(32)

The nontrivial solution of this system is 1199011= 2 119901

2= 3 and

119901 = 6 This completes the proof

Using symbolic computations and Taylor expansions it iseasy to derive the following lemma

Lemma 6 Assuming (15) and (17) we have

1 + 1205741198961198911015840(120572) sim 119890

119896minus1119890119896minus1119908

119890119896minus1119910

119890119896minus1119911

(33)

where 120574119896= minus1119873

1015840

4(119909119896) 119890119896= 119909119896minus120572 119890119896119911

= 119911119896minus120572 119890119896119910

= 119910119896minus120572

and 119890119896119908

= 119908119896minus 120572

Proof The proof of this lemma is similar to Lemma 4 It ishence omitted

Similarly for the three-step method with memory (20)we have the following theorem

Theorem 7 If an initial approximation 1199090is sufficiently close

to the zero 120572 of 119891(119909) and the parameter 120574119896in the iterative

scheme (20) is recursively calculated by the forms given in (15)then the order of convergence for (20) is at least twelve

Proof Let 119909119896 be a sequence of approximations generated

by the iterative method with memory (20) If this sequenceconverges to the zero 120572 of 119891 with the order 119901 then we write

119890119896+1

sim 119890119901

119896 119890

119896= 119909119896minus 120572 (34)

So

119890119896+1

sim (119890119901

119896minus1)119901

= 1198901199012

119896minus1 (35)

Moreover assume that the iterative sequences 119908119896 119910119896 and 119911

119896

have the orders 1199011 1199012 and 119901

3 respectively Then (34) gives

119890119896119908

sim 1198901199011

119896sim (119890119901

119896minus1)1199011= 1198901199011199011

119896minus1 (36)

119890119896119910

sim 1198901199012

119896sim (119890119901

119896minus1)1199012= 1198901199011199012

119896minus1 (37)

119890119896119911

sim 1198901199013

119896sim (119890119901

119896minus1)1199013= 1198901199011199013

119896minus1 (38)

6 Discrete Dynamics in Nature and Society

Since

119890119896119908

sim (1 + 1205741198961198911015840(120572)) 119890

119896 (39)

119890119896119910

sim (1 + 1205741198961198911015840(120572)) 119890

2

119896 (40)

119890119896119911

sim (1 + 1205741198961198911015840(120572))2

1198904

119896 (41)

119890119896+1

sim (1 + 1205741198961198911015840(120572))4

1198908

119896 (42)

by Lemma 6 and (40) we obtain

119890119896119908

sim (1 + 1205741198961198911015840(120572)) 119890

119896sim (119890119896minus1

119890119896minus1119910

119890119896minus1119908

119890119896minus1119911

) 119890119896

= 119890119901+1199011+1199012+1199013+1

119896minus1

(43)

119890119896119910

sim (1 + 1205741198961198911015840(120572)) 119890

2

119896sim (119890119896minus1

119890119896minus1119910

119890119896minus1119908

119890119896minus1119911

) 1198902

119896

= 1198902119901+1199011+1199012+1199013+1

119896minus1

(44)

119890119896119911

sim (1 + 1205741198961198911015840(120572)) 119890

2

119896sim (119890119896minus1

119890119896minus1119910

119890119896minus1119908

119890119896minus1119911

)2

1198904

119896

= 1198904119901+21199011+21199012+21199013+2

119896minus1

(45)

119890119896+1

sim (1 + 1205741198961198911015840(120572))2

1198904

119896sim (119890119896minus1

119890119896minus1119910

119890119896minus1119908

119890119896minus1119911

)4

1198908

119896

= 1198908119901+41199011+41199012+41199013+4

119896minus1

(46)

Matching the powers of 119890119896minus1

on the right hand sides of (36)ndash(43) (37)ndash(44) (38)ndash(45) and (35)ndash(46) one can obtain

1199011199011minus 119901 minus 119901

1minus 1199012minus 1199013minus 1 = 0

1199011199012minus 2119901 minus 119901

1minus 1199012minus 1199013minus 1 = 0

1199011199013minus 4119901 minus 2119901

1minus 21199012minus 21199013minus 2 = 0

1199012minus 8119901 minus 4119901

1minus 41199012minus 41199013minus 2 = 0

(47)

This system has the solutions 1199011= 2 119901

2= 3 119901

3= 6 and

119901 = 12 The proof is complete

Remark 8 The advantage of the proposedmethods is in theirhigher computational efficiency indices We emphasize thatthe increase of the R-order of convergence has been obtainedwithout any additional function evaluations which pointsto very high computational efficiency Indeed the efficiencyindex 1214 asymp 1861 of the proposed three-step twelfth-ordermethod with memory is higher than the efficiency index613

asymp 1817 of (19) 814 asymp 1682 of the optimal three-pointmethod (11) and 413 asymp 1587 of (6)

Remark 9 We observe that the methods (19) and (20) withmemory are considerably accelerated (up to 50) in contrastto the corresponding method (11) without memory

4 Numerical Experiments

In this section we test our proposed methods and comparetheir results with some other methods of the same order of

convergenceThe results are reported using the programmingpackage Mathematica 8 in multiple precision arithmeticenvironment We have considered 1000 digits floating pointarithmetic so as to minimize the round-off errors as muchas possible The errors |119909

119896minus 120572| denote approximations to

the sought zeros and 119886(minus119887) stands for 119886 times 10minus119887 Moreover

coc indicates the computational order of convergence and iscomputed by

coc =log (1003816100381610038161003816119891 (119909

119896) 119891 (119909

119896minus1)1003816100381610038161003816)

log (1003816100381610038161003816119891 (119909119896minus1

) 119891 (119909119896minus2

)1003816100381610038161003816) (48)

It is assumed that the initial estimate 1205740should be chosen

before starting the iterative process and also 1199090is given

suitablySeveral iterative methods of optimal orders four and

eight for comparing with our proposed methods have beenchosen as comes next

Derivative-free Kung-Traubrsquos two-step method (KT4) [6]is as follows

119910119896= 119909119896minus

119891 (119909119896)

119891 [119909119896 119908119896] 119908119896= 119909119896+ 120574119891 (119909

119896)

119909119896+1

= 119910119896minus

119891 (119910119896) 119891 (119908

119896)

[119891 (119908119896) minus 119891 (119910

119896)] 119891 [119909

119896 119910119896]

(49)

Two-step method by Zheng et al (ZLH4) [10] is asfollows

119910119896= 119909119896minus

119891 (119909119896)

119891 [119909119896 119908119896] 119908119896= 119909119896+ 120574119891 (119909

119896)

119909119896+1

= 119910119896minus

119891 (119910119896)

119891 [119910119896 119908119896] + 119891 [119910

119896 119909119896 119908119896] (119910119896minus 119909119896)

(50)

Two-step method by Lotfi and Tavakoli (LT4) [11] is asfollows

119910119896= 119909119896minus

119891 (119909119896)

119891 [119909119896 119908119896]

119909119896+1

= 119910119896minus 119867 (119905

119896 119906119896)

119891 (119910119896)

119891 [119910119896 119908119896]

119905119896=119891 (119910119896)

119891 (119909119896) 119906

119896=119891 (119908119896)

119891 (119909119896)

(51)

where119867(119905119896 119906119896) = 1 + 119905

119896

Discrete Dynamics in Nature and Society 7

Table 1 1198911(119909) = sin(120587119909)119890(119909

2+119909 cos(119909)minus1)

+ 119909 log(119909 sin(119909) + 1) 120572 = 0 and 1199090= 06

Methods |1199091minus 120572| |119909

2minus 120572| |119909

3minus 120572| coc

Without memoryNew method (6) 120574 = minus1 06742(minus2) 01587(minus10) 04307(minus45) 40049KT4 120574 = minus001 06012(minus1) 01486(minus4) 06094(minus19) 40171ZLH4 120574 = minus001 05882(minus1) 01320(minus5) 08291(minus24) 39365LT 120574 = minus001 06178(minus1) 03334(minus4) 02739(minus17) 40368

With memoryNew method (19) 120574 = minus1 06742(minus2) 03177(minus11) 08372(minus62) 54213KT4 120574 = minus1 02102(minus1) 09935(minus9) 02326(minus48) 54031ZLH4 120574 = minus001 05882(minus1) 02204(minus8) 01503(minus45) 50216LT4 120574 = minus1 01752(minus1) 04934(minus9) 05220(minus50) 54214

Derivative-free Kung-Traubrsquos three-step method (KT8)[6] is as follows

119910119896= 119909119896minus

119891 (119909119896)

119891 [119909119896 119908119896] 119908119896= 119909119896+ 120574119891 (119909

119896)

119911119896= 119910119896minus

119891 (119910119896) 119891 (119908

119896)

[119891 (119908119896) minus 119891 (119910

119896)] 119891 [119909

119896 119910119896]

119909119896+1

= 119911119896minus119891 (119910119896) 119891 (119908

119896) (119910119896minus 119909119896+ 119891 (119909

119896) 119891 [119909

119896 119911119896])

[119891 (119910119896) minus 119891 (119911

119896)] [119891 (119908

119896) minus 119891 (119911

119896)]

+119891 (119910119896)

119891 [119910119896 119911119896]

(52)

Three-step methods developed by Zheng et al (ZLH8)[10] is as follows

119910119896= 119909119896minus

119891 (119909119896)

119891 [119909119896 119908119896] 119908119896= 119909119896+ 120574119891 (119909

119896)

119911119896= 119910119896minus

119891 (119910119896)

119891 [119910119896 119908119896] + 119891 [119910

119896 119909119896 119908119896] (119910119896minus 119909119896)

119909119896+1

= 119911119896

minus ( (119891 (119911119896))

times (119891 [119911119896 119910119896] + 119891 [119911

119896 119910119896 119909119896] (119911119896minus 119910119896)

+119891 [119911119896 119910119896 119909119896 119908119896] (119911119896minus 119910119896) (119911119896minus 119909119896))minus1

)

(53)

Three-step method by Lotfi and Tavakoli (LT8) [11] is asfollows

119910119896= 119909119896minus

119891 (119909119896)

119891 [119909119896 119908119896]

119911119896= 119910119896minus 119867 (119905

119896 119906119896)

119891 (119910119896)

119891 [119910119896 119908119896]

119905119896=119891 (119910119896)

119891 (119909119896) 119906

119896=119891 (119908119896)

119891 (119909119896)

119909119896+1

= 119911119896minus 119866 (119905

119896 119904119896)119882 (V

119896 119904119896)

119891 (119911119896)

119891 [119911119896 119908119896]

119904119896=119891 (119911119896)

119891 (119910119896) V

119896=119891 (119911119896)

119891 (119909119896)

(54)

where119882(119904119896 V119896) = 1+119904

2

119896+V2119896119866(119905119896 119904119896) = 1+119905

119896+119904119896+2119905119896119904119896+(minus1minus

120601119896)1199053

119896and (120601

119896= 1(1 + 120574

119896119891[119909119896 119908119896])) and119867(119905

119896 119906119896) = 1 + 119905

119896

are the weight functionsIn Tables 1 2 and 3 our two-step proposed classes (6)

and (19) have been compared with optimal two-stepmethodsKT4 ZLH4 and LT4 We observe that all these methodsbehave very well practically and confirm their theoreticalresults

Also Tables 4 5 and 6 present numerical results for ourthree-step classes (11) and (20) and methods KT8 ZLH8 andLT8 It is also clear that all these methods behave very wellpractically and confirm their relevant theories

We remark the importance of the choice of initial guessesIf they are chosen sufficiently close to the sought roots thenthe expected (theoretical) convergence speed will be reachedin practice otherwise all iterative root-finding methodsshow slower convergence especially at the beginning of theiterative process Hence a special attention should be paidto finding good initial approximations We note that efficientways for the determination of initial approximations of greataccuracy were discussed thoroughly in the works [12ndash14]

5 Conclusions

We have constructed two families of iterative methods with-out memory which are optimal in the sense of Kung andTraubrsquos conjecture in this paper Our proposed methods donot need any derivative

In addition they contain an accelerator parameter whichraises convergence order without any new functional evalu-ations In other words the efficiency index of the three-stepwith memory hit 1214 asymp 1861

We finalize this work by suggesting some points forfuture researches first developing the proposed methods forsome matrix functions such as the ones in [15 16] second

8 Discrete Dynamics in Nature and Society

Table 2 1198912(119909) = 119890

minus5119909(119909 minus 2)(119909

10+ 119909 + 2) 120572 = 2 and 119909

0= 22

Methods |1199091minus 120572| |119909

2minus 120572| |119909

3minus 120572| coc

Without memoryNew method (6) 120574 = minus001 09445(minus3) 01550(minus13) 01289(minus56) 39945KT4 120574 = minus001 01125(minus2) 03033(minus13) 01891(minus55) 39932ZLH4 120574 = minus01 09349(minus3) 01478(minus13) 01074(minus56) 39939LT 120574 = minus001 01343(minus2) 05965(minus13) 02821(minus54) 39918

With memoryNew method (19) 120574 = 1 01057(minus2) 01102(minus19) 08335(minus109) 52480KT4 120574 = 1 01267(minus2) 02180(minus19) 03580(minus107) 52364ZLH4 120574 = minus001 09445(minus3) 02868(minus20) 08014(minus112) 52264LT4 120574 = minus1 01343(minus2) 01072(minus19) 01150(minus108) 52035

Table 3 1198913(119909) = 119890

1199093minus119909

minus cos(1199092 minus 1) + 1199093+ 1 120572 = minus1 and 119909

0= minus165

Methods |1199091minus 120572| |119909

2minus 120572| |119909

3minus 120572| coc

Without memoryNew method (6) 120574 = minus001 06849(minus3) 08725(minus13) 02284(minus52) 40003KT4 120574 = minus001 06610(minus3) 08682(minus13) 02564(minus52) 40004ZLH4 120574 = 01 01214(minus2) 02388(minus12) 03560(minus51) 40004LT4 120574 = minus001 06398(minus3) 08549(minus13) 02714(minus52) 40002

With memoryNew method (19) 120574 = minus1 04838(minus2) 01348(minus11) 01230(minus64) 55506KT4 120574 = 1 01170(minus1) 03517(minus7) 08936(minus41) 60860ZLH4 120574 = minus001 06849(minus3) 07630(minus17) 01662(minus92) 54226LT4 120574 = minus001 06398(minus3) 06666(minus17) 07373(minus93) 54324

Table 4 1198911(119909) = sin(120587119909)119890(119909

2+119909 cos(119909)minus1)

+ 119909 log(119909 sin(119909) + 1) 120572 = 0 and 1199090= 06

Methods |1199091minus 120572| |119909

2minus 120572| |119909

3minus 120572| coc

Without memoryNew method (11) 120574 = minus1 02345(minus3) 01042(minus32) 01593(minus267) 79999KT8 120574 = minus1 03101(minus3) 02671(minus31) 08196(minus256) 79999ZLH8 120574 = minus001 01742(minus2) 01558(minus22) 06471(minus183) 79999LT8 120574 = minus1 05758(minus3) 07106(minus29) 03880(minus236) 79998

With memoryNew method (20) 120574 = minus01 03440(minus3) 01220(minus41) 08636(minus503) 119935KT8 120574 = minus001 09773(minus3) 02960(minus33) 01505(minus399) 120021ZLH8 120574 = minus001 01742(minus2) 03012(minus33) 04438(minus402) 119900LT8 120574 = minus01 07107(minus4) 02040(minus49) 04971(minus596) 120024

Table 5 1198912(119909) = 119890

minus5119909(119909 minus 2)(119909

10+ 119909 + 2) 120572 = 2 and 119909

0= 22

Methods |1199091minus 120572| |119909

2minus 120572| |119909

3minus 120572| coc

Without memoryNew method (11) 120574 = minus1 03124(minus6) 09972(minus56) 01074(minus451) 80000KT8 120574 = minus1 08055(minus6) 01413(minus52) 01268(minus426) 80000ZLH8 120574 = 1 04955(minus6) 05798(minus54) 02038(minus437) 80000LT8 120574 = minus1 04052(minus6) 05819(minus55) 01053(minus445) 80000

With memoryNew method (20) 120574 = minus1 03124(minus6) 01213(minus81) 03093(minus985) 119822KT8 120574 = minus1 08055(minus6) 02641(minus78) 09143(minus945) 119538ZLH8 120574 = minus001 03943(minus6) 01889(minus80) 06308(minus971) 119817LT8 120574 = minus001 04532(minus6) 04774(minus76) 05848(minus853) 111023

Discrete Dynamics in Nature and Society 9

Table 6 1198913(119909) = 119890

1199093minus119909

minus cos(1199092 minus 1) + 1199093+ 1 120572 = minus1 and 119909

0= minus165

Methods |1199091minus 120572| |119909

2minus 120572| |119909

3minus 120572| coc

Without memoryNew method (11) 120574 = minus1 02768(minus3) 03670(minus27) 03499(minus218) 80000KT8 120574 = minus1 02915(minus4) 05324(minus34) 06583(minus272) 80000ZLH8 120574 = 1 02322(minus1) 05044(minus11) 02253(minus88) 80081LT 120574 = minus1 01094(minus2) 01454(minus22) 01323(minus181) 80014

With memoryNew method (20) 120574 = minus1 02768(minus3) 02660(minus42) 01805(minus509) 119734KT8 120574 = minus1 02915(minus4) 05214(minus53) 01157(minus642) 120961ZLH8 120574 = minus001 06052(minus5) 02355(minus65) 04206(minus786) 119310LT8 120574 = minus1 01094(minus2) 03522(minus34) 01717(minus415) 121081

exploring its dynamic or basins of attractions and lastlyextending the developedmethods with memory using two orthree accelerators

Conflict of Interests

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

Acknowledgments

The first author thanks Islamic Azad University HamedanBranch for the financial support of the present research Theauthors are also thankful for insightful comments of three ref-erees whom helped with the readability and reliability of thepresent paperThe fourth author also gratefully acknowledgesthat this research was partially supported by the UniversityPutra Malaysia under the GP-IBT Grant Scheme havingproject number GP-IBT20139420100

References

[1] N Kyurkchiev and A Iliev ldquoOn some multipoint methodsarising from optimal in the sense of Kung-Traub algorithmsrdquoBiomath vol 2 no 1 Article ID 1305155 7 pages 2013

[2] T Lotfi F Soleymani S Sharifi S Shateyi and F K HaghanildquoMulti-point iterative methods for finding all the simple zerosin an intervalrdquo Journal of AppliedMathematics vol 2014 ArticleID 601205 14 pages 2014

[3] J F Traub G W Wasilkowski and H Wozniakowski Informa-tion Uncertainty Complexity Addison-Wesley 1983

[4] M S Rhee and Y I Kim ldquoA general class of optimal fourth-order multiple-root finders without memory for nonlinearequationsrdquo Applied Mathematical Sciences vol 7 pp 5537ndash55512013

[5] H Wozniakowski ldquoMaximal order of multipoint iterationsusing n evaluationsrdquo in Analytic Computational Complexity JF Traub Ed pp 75ndash107 Academic Press New York NY USA1976

[6] 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

[7] Y H Geum and Y I Kim ldquoAn optimal family of fast 16th-orderderivative-free multipoint simple-root finders for nonlinear

equationsrdquo Journal ofOptimizationTheory andApplications vol160 no 2 pp 608ndash622 2014

[8] J F Traub Iterative Methods for the Solution of EquationsPrentice-Hall New York NY USA 1964

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

[10] 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

[11] T Lotfi and E Tavakoli ldquoOn a new efficient Steffensen-likeiterative class by applying a suitable self-accelerator parameterrdquoThe Scientific World Journal vol 2014 Article ID 769758 9pages 2014

[12] F Soleymani and S Shateyi ldquoTwo optimal eighth-orderderivative-free classes of iterative methodsrdquo Abstract andApplied Analysis vol 2012 Article ID 318165 14 pages 2012

[13] F Soleymani and D K R Babajee ldquoComputing multiple zerosusing a class of quartically convergent methodsrdquo AlexandriaEngineering Journal vol 52 no 3 pp 531ndash541 2013

[14] F Soleymani ldquoSome high-order iterativemethods for finding allthe real zerosrdquoThai Journal of Mathematics vol 12 pp 313ndash3272014

[15] F Soleymani E Tohidi S Shateyi and F K Haghani ldquoSomematrix iterations for computing matrix sign functionrdquo Journalof Applied Mathematics vol 2014 Article ID 425654 9 pages2014

[16] F Soleymani P S Stanimirovic S Shateyi and F K HaghanildquoApproximating the matrix sign function using a novel iterativemethodrdquo Abstract and Applied Analysis vol 2014 Article ID105301 9 pages 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Discrete Dynamics in Nature and Society

Since

119890119896119908

sim (1 + 1205741198961198911015840(120572)) 119890

119896 (39)

119890119896119910

sim (1 + 1205741198961198911015840(120572)) 119890

2

119896 (40)

119890119896119911

sim (1 + 1205741198961198911015840(120572))2

1198904

119896 (41)

119890119896+1

sim (1 + 1205741198961198911015840(120572))4

1198908

119896 (42)

by Lemma 6 and (40) we obtain

119890119896119908

sim (1 + 1205741198961198911015840(120572)) 119890

119896sim (119890119896minus1

119890119896minus1119910

119890119896minus1119908

119890119896minus1119911

) 119890119896

= 119890119901+1199011+1199012+1199013+1

119896minus1

(43)

119890119896119910

sim (1 + 1205741198961198911015840(120572)) 119890

2

119896sim (119890119896minus1

119890119896minus1119910

119890119896minus1119908

119890119896minus1119911

) 1198902

119896

= 1198902119901+1199011+1199012+1199013+1

119896minus1

(44)

119890119896119911

sim (1 + 1205741198961198911015840(120572)) 119890

2

119896sim (119890119896minus1

119890119896minus1119910

119890119896minus1119908

119890119896minus1119911

)2

1198904

119896

= 1198904119901+21199011+21199012+21199013+2

119896minus1

(45)

119890119896+1

sim (1 + 1205741198961198911015840(120572))2

1198904

119896sim (119890119896minus1

119890119896minus1119910

119890119896minus1119908

119890119896minus1119911

)4

1198908

119896

= 1198908119901+41199011+41199012+41199013+4

119896minus1

(46)

Matching the powers of 119890119896minus1

on the right hand sides of (36)ndash(43) (37)ndash(44) (38)ndash(45) and (35)ndash(46) one can obtain

1199011199011minus 119901 minus 119901

1minus 1199012minus 1199013minus 1 = 0

1199011199012minus 2119901 minus 119901

1minus 1199012minus 1199013minus 1 = 0

1199011199013minus 4119901 minus 2119901

1minus 21199012minus 21199013minus 2 = 0

1199012minus 8119901 minus 4119901

1minus 41199012minus 41199013minus 2 = 0

(47)

This system has the solutions 1199011= 2 119901

2= 3 119901

3= 6 and

119901 = 12 The proof is complete

Remark 8 The advantage of the proposedmethods is in theirhigher computational efficiency indices We emphasize thatthe increase of the R-order of convergence has been obtainedwithout any additional function evaluations which pointsto very high computational efficiency Indeed the efficiencyindex 1214 asymp 1861 of the proposed three-step twelfth-ordermethod with memory is higher than the efficiency index613

asymp 1817 of (19) 814 asymp 1682 of the optimal three-pointmethod (11) and 413 asymp 1587 of (6)

Remark 9 We observe that the methods (19) and (20) withmemory are considerably accelerated (up to 50) in contrastto the corresponding method (11) without memory

4 Numerical Experiments

In this section we test our proposed methods and comparetheir results with some other methods of the same order of

convergenceThe results are reported using the programmingpackage Mathematica 8 in multiple precision arithmeticenvironment We have considered 1000 digits floating pointarithmetic so as to minimize the round-off errors as muchas possible The errors |119909

119896minus 120572| denote approximations to

the sought zeros and 119886(minus119887) stands for 119886 times 10minus119887 Moreover

coc indicates the computational order of convergence and iscomputed by

coc =log (1003816100381610038161003816119891 (119909

119896) 119891 (119909

119896minus1)1003816100381610038161003816)

log (1003816100381610038161003816119891 (119909119896minus1

) 119891 (119909119896minus2

)1003816100381610038161003816) (48)

It is assumed that the initial estimate 1205740should be chosen

before starting the iterative process and also 1199090is given

suitablySeveral iterative methods of optimal orders four and

eight for comparing with our proposed methods have beenchosen as comes next

Derivative-free Kung-Traubrsquos two-step method (KT4) [6]is as follows

119910119896= 119909119896minus

119891 (119909119896)

119891 [119909119896 119908119896] 119908119896= 119909119896+ 120574119891 (119909

119896)

119909119896+1

= 119910119896minus

119891 (119910119896) 119891 (119908

119896)

[119891 (119908119896) minus 119891 (119910

119896)] 119891 [119909

119896 119910119896]

(49)

Two-step method by Zheng et al (ZLH4) [10] is asfollows

119910119896= 119909119896minus

119891 (119909119896)

119891 [119909119896 119908119896] 119908119896= 119909119896+ 120574119891 (119909

119896)

119909119896+1

= 119910119896minus

119891 (119910119896)

119891 [119910119896 119908119896] + 119891 [119910

119896 119909119896 119908119896] (119910119896minus 119909119896)

(50)

Two-step method by Lotfi and Tavakoli (LT4) [11] is asfollows

119910119896= 119909119896minus

119891 (119909119896)

119891 [119909119896 119908119896]

119909119896+1

= 119910119896minus 119867 (119905

119896 119906119896)

119891 (119910119896)

119891 [119910119896 119908119896]

119905119896=119891 (119910119896)

119891 (119909119896) 119906

119896=119891 (119908119896)

119891 (119909119896)

(51)

where119867(119905119896 119906119896) = 1 + 119905

119896

Discrete Dynamics in Nature and Society 7

Table 1 1198911(119909) = sin(120587119909)119890(119909

2+119909 cos(119909)minus1)

+ 119909 log(119909 sin(119909) + 1) 120572 = 0 and 1199090= 06

Methods |1199091minus 120572| |119909

2minus 120572| |119909

3minus 120572| coc

Without memoryNew method (6) 120574 = minus1 06742(minus2) 01587(minus10) 04307(minus45) 40049KT4 120574 = minus001 06012(minus1) 01486(minus4) 06094(minus19) 40171ZLH4 120574 = minus001 05882(minus1) 01320(minus5) 08291(minus24) 39365LT 120574 = minus001 06178(minus1) 03334(minus4) 02739(minus17) 40368

With memoryNew method (19) 120574 = minus1 06742(minus2) 03177(minus11) 08372(minus62) 54213KT4 120574 = minus1 02102(minus1) 09935(minus9) 02326(minus48) 54031ZLH4 120574 = minus001 05882(minus1) 02204(minus8) 01503(minus45) 50216LT4 120574 = minus1 01752(minus1) 04934(minus9) 05220(minus50) 54214

Derivative-free Kung-Traubrsquos three-step method (KT8)[6] is as follows

119910119896= 119909119896minus

119891 (119909119896)

119891 [119909119896 119908119896] 119908119896= 119909119896+ 120574119891 (119909

119896)

119911119896= 119910119896minus

119891 (119910119896) 119891 (119908

119896)

[119891 (119908119896) minus 119891 (119910

119896)] 119891 [119909

119896 119910119896]

119909119896+1

= 119911119896minus119891 (119910119896) 119891 (119908

119896) (119910119896minus 119909119896+ 119891 (119909

119896) 119891 [119909

119896 119911119896])

[119891 (119910119896) minus 119891 (119911

119896)] [119891 (119908

119896) minus 119891 (119911

119896)]

+119891 (119910119896)

119891 [119910119896 119911119896]

(52)

Three-step methods developed by Zheng et al (ZLH8)[10] is as follows

119910119896= 119909119896minus

119891 (119909119896)

119891 [119909119896 119908119896] 119908119896= 119909119896+ 120574119891 (119909

119896)

119911119896= 119910119896minus

119891 (119910119896)

119891 [119910119896 119908119896] + 119891 [119910

119896 119909119896 119908119896] (119910119896minus 119909119896)

119909119896+1

= 119911119896

minus ( (119891 (119911119896))

times (119891 [119911119896 119910119896] + 119891 [119911

119896 119910119896 119909119896] (119911119896minus 119910119896)

+119891 [119911119896 119910119896 119909119896 119908119896] (119911119896minus 119910119896) (119911119896minus 119909119896))minus1

)

(53)

Three-step method by Lotfi and Tavakoli (LT8) [11] is asfollows

119910119896= 119909119896minus

119891 (119909119896)

119891 [119909119896 119908119896]

119911119896= 119910119896minus 119867 (119905

119896 119906119896)

119891 (119910119896)

119891 [119910119896 119908119896]

119905119896=119891 (119910119896)

119891 (119909119896) 119906

119896=119891 (119908119896)

119891 (119909119896)

119909119896+1

= 119911119896minus 119866 (119905

119896 119904119896)119882 (V

119896 119904119896)

119891 (119911119896)

119891 [119911119896 119908119896]

119904119896=119891 (119911119896)

119891 (119910119896) V

119896=119891 (119911119896)

119891 (119909119896)

(54)

where119882(119904119896 V119896) = 1+119904

2

119896+V2119896119866(119905119896 119904119896) = 1+119905

119896+119904119896+2119905119896119904119896+(minus1minus

120601119896)1199053

119896and (120601

119896= 1(1 + 120574

119896119891[119909119896 119908119896])) and119867(119905

119896 119906119896) = 1 + 119905

119896

are the weight functionsIn Tables 1 2 and 3 our two-step proposed classes (6)

and (19) have been compared with optimal two-stepmethodsKT4 ZLH4 and LT4 We observe that all these methodsbehave very well practically and confirm their theoreticalresults

Also Tables 4 5 and 6 present numerical results for ourthree-step classes (11) and (20) and methods KT8 ZLH8 andLT8 It is also clear that all these methods behave very wellpractically and confirm their relevant theories

We remark the importance of the choice of initial guessesIf they are chosen sufficiently close to the sought roots thenthe expected (theoretical) convergence speed will be reachedin practice otherwise all iterative root-finding methodsshow slower convergence especially at the beginning of theiterative process Hence a special attention should be paidto finding good initial approximations We note that efficientways for the determination of initial approximations of greataccuracy were discussed thoroughly in the works [12ndash14]

5 Conclusions

We have constructed two families of iterative methods with-out memory which are optimal in the sense of Kung andTraubrsquos conjecture in this paper Our proposed methods donot need any derivative

In addition they contain an accelerator parameter whichraises convergence order without any new functional evalu-ations In other words the efficiency index of the three-stepwith memory hit 1214 asymp 1861

We finalize this work by suggesting some points forfuture researches first developing the proposed methods forsome matrix functions such as the ones in [15 16] second

8 Discrete Dynamics in Nature and Society

Table 2 1198912(119909) = 119890

minus5119909(119909 minus 2)(119909

10+ 119909 + 2) 120572 = 2 and 119909

0= 22

Methods |1199091minus 120572| |119909

2minus 120572| |119909

3minus 120572| coc

Without memoryNew method (6) 120574 = minus001 09445(minus3) 01550(minus13) 01289(minus56) 39945KT4 120574 = minus001 01125(minus2) 03033(minus13) 01891(minus55) 39932ZLH4 120574 = minus01 09349(minus3) 01478(minus13) 01074(minus56) 39939LT 120574 = minus001 01343(minus2) 05965(minus13) 02821(minus54) 39918

With memoryNew method (19) 120574 = 1 01057(minus2) 01102(minus19) 08335(minus109) 52480KT4 120574 = 1 01267(minus2) 02180(minus19) 03580(minus107) 52364ZLH4 120574 = minus001 09445(minus3) 02868(minus20) 08014(minus112) 52264LT4 120574 = minus1 01343(minus2) 01072(minus19) 01150(minus108) 52035

Table 3 1198913(119909) = 119890

1199093minus119909

minus cos(1199092 minus 1) + 1199093+ 1 120572 = minus1 and 119909

0= minus165

Methods |1199091minus 120572| |119909

2minus 120572| |119909

3minus 120572| coc

Without memoryNew method (6) 120574 = minus001 06849(minus3) 08725(minus13) 02284(minus52) 40003KT4 120574 = minus001 06610(minus3) 08682(minus13) 02564(minus52) 40004ZLH4 120574 = 01 01214(minus2) 02388(minus12) 03560(minus51) 40004LT4 120574 = minus001 06398(minus3) 08549(minus13) 02714(minus52) 40002

With memoryNew method (19) 120574 = minus1 04838(minus2) 01348(minus11) 01230(minus64) 55506KT4 120574 = 1 01170(minus1) 03517(minus7) 08936(minus41) 60860ZLH4 120574 = minus001 06849(minus3) 07630(minus17) 01662(minus92) 54226LT4 120574 = minus001 06398(minus3) 06666(minus17) 07373(minus93) 54324

Table 4 1198911(119909) = sin(120587119909)119890(119909

2+119909 cos(119909)minus1)

+ 119909 log(119909 sin(119909) + 1) 120572 = 0 and 1199090= 06

Methods |1199091minus 120572| |119909

2minus 120572| |119909

3minus 120572| coc

Without memoryNew method (11) 120574 = minus1 02345(minus3) 01042(minus32) 01593(minus267) 79999KT8 120574 = minus1 03101(minus3) 02671(minus31) 08196(minus256) 79999ZLH8 120574 = minus001 01742(minus2) 01558(minus22) 06471(minus183) 79999LT8 120574 = minus1 05758(minus3) 07106(minus29) 03880(minus236) 79998

With memoryNew method (20) 120574 = minus01 03440(minus3) 01220(minus41) 08636(minus503) 119935KT8 120574 = minus001 09773(minus3) 02960(minus33) 01505(minus399) 120021ZLH8 120574 = minus001 01742(minus2) 03012(minus33) 04438(minus402) 119900LT8 120574 = minus01 07107(minus4) 02040(minus49) 04971(minus596) 120024

Table 5 1198912(119909) = 119890

minus5119909(119909 minus 2)(119909

10+ 119909 + 2) 120572 = 2 and 119909

0= 22

Methods |1199091minus 120572| |119909

2minus 120572| |119909

3minus 120572| coc

Without memoryNew method (11) 120574 = minus1 03124(minus6) 09972(minus56) 01074(minus451) 80000KT8 120574 = minus1 08055(minus6) 01413(minus52) 01268(minus426) 80000ZLH8 120574 = 1 04955(minus6) 05798(minus54) 02038(minus437) 80000LT8 120574 = minus1 04052(minus6) 05819(minus55) 01053(minus445) 80000

With memoryNew method (20) 120574 = minus1 03124(minus6) 01213(minus81) 03093(minus985) 119822KT8 120574 = minus1 08055(minus6) 02641(minus78) 09143(minus945) 119538ZLH8 120574 = minus001 03943(minus6) 01889(minus80) 06308(minus971) 119817LT8 120574 = minus001 04532(minus6) 04774(minus76) 05848(minus853) 111023

Discrete Dynamics in Nature and Society 9

Table 6 1198913(119909) = 119890

1199093minus119909

minus cos(1199092 minus 1) + 1199093+ 1 120572 = minus1 and 119909

0= minus165

Methods |1199091minus 120572| |119909

2minus 120572| |119909

3minus 120572| coc

Without memoryNew method (11) 120574 = minus1 02768(minus3) 03670(minus27) 03499(minus218) 80000KT8 120574 = minus1 02915(minus4) 05324(minus34) 06583(minus272) 80000ZLH8 120574 = 1 02322(minus1) 05044(minus11) 02253(minus88) 80081LT 120574 = minus1 01094(minus2) 01454(minus22) 01323(minus181) 80014

With memoryNew method (20) 120574 = minus1 02768(minus3) 02660(minus42) 01805(minus509) 119734KT8 120574 = minus1 02915(minus4) 05214(minus53) 01157(minus642) 120961ZLH8 120574 = minus001 06052(minus5) 02355(minus65) 04206(minus786) 119310LT8 120574 = minus1 01094(minus2) 03522(minus34) 01717(minus415) 121081

exploring its dynamic or basins of attractions and lastlyextending the developedmethods with memory using two orthree accelerators

Conflict of Interests

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

Acknowledgments

The first author thanks Islamic Azad University HamedanBranch for the financial support of the present research Theauthors are also thankful for insightful comments of three ref-erees whom helped with the readability and reliability of thepresent paperThe fourth author also gratefully acknowledgesthat this research was partially supported by the UniversityPutra Malaysia under the GP-IBT Grant Scheme havingproject number GP-IBT20139420100

References

[1] N Kyurkchiev and A Iliev ldquoOn some multipoint methodsarising from optimal in the sense of Kung-Traub algorithmsrdquoBiomath vol 2 no 1 Article ID 1305155 7 pages 2013

[2] T Lotfi F Soleymani S Sharifi S Shateyi and F K HaghanildquoMulti-point iterative methods for finding all the simple zerosin an intervalrdquo Journal of AppliedMathematics vol 2014 ArticleID 601205 14 pages 2014

[3] J F Traub G W Wasilkowski and H Wozniakowski Informa-tion Uncertainty Complexity Addison-Wesley 1983

[4] M S Rhee and Y I Kim ldquoA general class of optimal fourth-order multiple-root finders without memory for nonlinearequationsrdquo Applied Mathematical Sciences vol 7 pp 5537ndash55512013

[5] H Wozniakowski ldquoMaximal order of multipoint iterationsusing n evaluationsrdquo in Analytic Computational Complexity JF Traub Ed pp 75ndash107 Academic Press New York NY USA1976

[6] 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

[7] Y H Geum and Y I Kim ldquoAn optimal family of fast 16th-orderderivative-free multipoint simple-root finders for nonlinear

equationsrdquo Journal ofOptimizationTheory andApplications vol160 no 2 pp 608ndash622 2014

[8] J F Traub Iterative Methods for the Solution of EquationsPrentice-Hall New York NY USA 1964

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

[10] 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

[11] T Lotfi and E Tavakoli ldquoOn a new efficient Steffensen-likeiterative class by applying a suitable self-accelerator parameterrdquoThe Scientific World Journal vol 2014 Article ID 769758 9pages 2014

[12] F Soleymani and S Shateyi ldquoTwo optimal eighth-orderderivative-free classes of iterative methodsrdquo Abstract andApplied Analysis vol 2012 Article ID 318165 14 pages 2012

[13] F Soleymani and D K R Babajee ldquoComputing multiple zerosusing a class of quartically convergent methodsrdquo AlexandriaEngineering Journal vol 52 no 3 pp 531ndash541 2013

[14] F Soleymani ldquoSome high-order iterativemethods for finding allthe real zerosrdquoThai Journal of Mathematics vol 12 pp 313ndash3272014

[15] F Soleymani E Tohidi S Shateyi and F K Haghani ldquoSomematrix iterations for computing matrix sign functionrdquo Journalof Applied Mathematics vol 2014 Article ID 425654 9 pages2014

[16] F Soleymani P S Stanimirovic S Shateyi and F K HaghanildquoApproximating the matrix sign function using a novel iterativemethodrdquo Abstract and Applied Analysis vol 2014 Article ID105301 9 pages 2014

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

Discrete Dynamics in Nature and Society 7

Table 1 1198911(119909) = sin(120587119909)119890(119909

2+119909 cos(119909)minus1)

+ 119909 log(119909 sin(119909) + 1) 120572 = 0 and 1199090= 06

Methods |1199091minus 120572| |119909

2minus 120572| |119909

3minus 120572| coc

Without memoryNew method (6) 120574 = minus1 06742(minus2) 01587(minus10) 04307(minus45) 40049KT4 120574 = minus001 06012(minus1) 01486(minus4) 06094(minus19) 40171ZLH4 120574 = minus001 05882(minus1) 01320(minus5) 08291(minus24) 39365LT 120574 = minus001 06178(minus1) 03334(minus4) 02739(minus17) 40368

With memoryNew method (19) 120574 = minus1 06742(minus2) 03177(minus11) 08372(minus62) 54213KT4 120574 = minus1 02102(minus1) 09935(minus9) 02326(minus48) 54031ZLH4 120574 = minus001 05882(minus1) 02204(minus8) 01503(minus45) 50216LT4 120574 = minus1 01752(minus1) 04934(minus9) 05220(minus50) 54214

Derivative-free Kung-Traubrsquos three-step method (KT8)[6] is as follows

119910119896= 119909119896minus

119891 (119909119896)

119891 [119909119896 119908119896] 119908119896= 119909119896+ 120574119891 (119909

119896)

119911119896= 119910119896minus

119891 (119910119896) 119891 (119908

119896)

[119891 (119908119896) minus 119891 (119910

119896)] 119891 [119909

119896 119910119896]

119909119896+1

= 119911119896minus119891 (119910119896) 119891 (119908

119896) (119910119896minus 119909119896+ 119891 (119909

119896) 119891 [119909

119896 119911119896])

[119891 (119910119896) minus 119891 (119911

119896)] [119891 (119908

119896) minus 119891 (119911

119896)]

+119891 (119910119896)

119891 [119910119896 119911119896]

(52)

Three-step methods developed by Zheng et al (ZLH8)[10] is as follows

119910119896= 119909119896minus

119891 (119909119896)

119891 [119909119896 119908119896] 119908119896= 119909119896+ 120574119891 (119909

119896)

119911119896= 119910119896minus

119891 (119910119896)

119891 [119910119896 119908119896] + 119891 [119910

119896 119909119896 119908119896] (119910119896minus 119909119896)

119909119896+1

= 119911119896

minus ( (119891 (119911119896))

times (119891 [119911119896 119910119896] + 119891 [119911

119896 119910119896 119909119896] (119911119896minus 119910119896)

+119891 [119911119896 119910119896 119909119896 119908119896] (119911119896minus 119910119896) (119911119896minus 119909119896))minus1

)

(53)

Three-step method by Lotfi and Tavakoli (LT8) [11] is asfollows

119910119896= 119909119896minus

119891 (119909119896)

119891 [119909119896 119908119896]

119911119896= 119910119896minus 119867 (119905

119896 119906119896)

119891 (119910119896)

119891 [119910119896 119908119896]

119905119896=119891 (119910119896)

119891 (119909119896) 119906

119896=119891 (119908119896)

119891 (119909119896)

119909119896+1

= 119911119896minus 119866 (119905

119896 119904119896)119882 (V

119896 119904119896)

119891 (119911119896)

119891 [119911119896 119908119896]

119904119896=119891 (119911119896)

119891 (119910119896) V

119896=119891 (119911119896)

119891 (119909119896)

(54)

where119882(119904119896 V119896) = 1+119904

2

119896+V2119896119866(119905119896 119904119896) = 1+119905

119896+119904119896+2119905119896119904119896+(minus1minus

120601119896)1199053

119896and (120601

119896= 1(1 + 120574

119896119891[119909119896 119908119896])) and119867(119905

119896 119906119896) = 1 + 119905

119896

are the weight functionsIn Tables 1 2 and 3 our two-step proposed classes (6)

and (19) have been compared with optimal two-stepmethodsKT4 ZLH4 and LT4 We observe that all these methodsbehave very well practically and confirm their theoreticalresults

Also Tables 4 5 and 6 present numerical results for ourthree-step classes (11) and (20) and methods KT8 ZLH8 andLT8 It is also clear that all these methods behave very wellpractically and confirm their relevant theories

We remark the importance of the choice of initial guessesIf they are chosen sufficiently close to the sought roots thenthe expected (theoretical) convergence speed will be reachedin practice otherwise all iterative root-finding methodsshow slower convergence especially at the beginning of theiterative process Hence a special attention should be paidto finding good initial approximations We note that efficientways for the determination of initial approximations of greataccuracy were discussed thoroughly in the works [12ndash14]

5 Conclusions

We have constructed two families of iterative methods with-out memory which are optimal in the sense of Kung andTraubrsquos conjecture in this paper Our proposed methods donot need any derivative

In addition they contain an accelerator parameter whichraises convergence order without any new functional evalu-ations In other words the efficiency index of the three-stepwith memory hit 1214 asymp 1861

We finalize this work by suggesting some points forfuture researches first developing the proposed methods forsome matrix functions such as the ones in [15 16] second

8 Discrete Dynamics in Nature and Society

Table 2 1198912(119909) = 119890

minus5119909(119909 minus 2)(119909

10+ 119909 + 2) 120572 = 2 and 119909

0= 22

Methods |1199091minus 120572| |119909

2minus 120572| |119909

3minus 120572| coc

Without memoryNew method (6) 120574 = minus001 09445(minus3) 01550(minus13) 01289(minus56) 39945KT4 120574 = minus001 01125(minus2) 03033(minus13) 01891(minus55) 39932ZLH4 120574 = minus01 09349(minus3) 01478(minus13) 01074(minus56) 39939LT 120574 = minus001 01343(minus2) 05965(minus13) 02821(minus54) 39918

With memoryNew method (19) 120574 = 1 01057(minus2) 01102(minus19) 08335(minus109) 52480KT4 120574 = 1 01267(minus2) 02180(minus19) 03580(minus107) 52364ZLH4 120574 = minus001 09445(minus3) 02868(minus20) 08014(minus112) 52264LT4 120574 = minus1 01343(minus2) 01072(minus19) 01150(minus108) 52035

Table 3 1198913(119909) = 119890

1199093minus119909

minus cos(1199092 minus 1) + 1199093+ 1 120572 = minus1 and 119909

0= minus165

Methods |1199091minus 120572| |119909

2minus 120572| |119909

3minus 120572| coc

Without memoryNew method (6) 120574 = minus001 06849(minus3) 08725(minus13) 02284(minus52) 40003KT4 120574 = minus001 06610(minus3) 08682(minus13) 02564(minus52) 40004ZLH4 120574 = 01 01214(minus2) 02388(minus12) 03560(minus51) 40004LT4 120574 = minus001 06398(minus3) 08549(minus13) 02714(minus52) 40002

With memoryNew method (19) 120574 = minus1 04838(minus2) 01348(minus11) 01230(minus64) 55506KT4 120574 = 1 01170(minus1) 03517(minus7) 08936(minus41) 60860ZLH4 120574 = minus001 06849(minus3) 07630(minus17) 01662(minus92) 54226LT4 120574 = minus001 06398(minus3) 06666(minus17) 07373(minus93) 54324

Table 4 1198911(119909) = sin(120587119909)119890(119909

2+119909 cos(119909)minus1)

+ 119909 log(119909 sin(119909) + 1) 120572 = 0 and 1199090= 06

Methods |1199091minus 120572| |119909

2minus 120572| |119909

3minus 120572| coc

Without memoryNew method (11) 120574 = minus1 02345(minus3) 01042(minus32) 01593(minus267) 79999KT8 120574 = minus1 03101(minus3) 02671(minus31) 08196(minus256) 79999ZLH8 120574 = minus001 01742(minus2) 01558(minus22) 06471(minus183) 79999LT8 120574 = minus1 05758(minus3) 07106(minus29) 03880(minus236) 79998

With memoryNew method (20) 120574 = minus01 03440(minus3) 01220(minus41) 08636(minus503) 119935KT8 120574 = minus001 09773(minus3) 02960(minus33) 01505(minus399) 120021ZLH8 120574 = minus001 01742(minus2) 03012(minus33) 04438(minus402) 119900LT8 120574 = minus01 07107(minus4) 02040(minus49) 04971(minus596) 120024

Table 5 1198912(119909) = 119890

minus5119909(119909 minus 2)(119909

10+ 119909 + 2) 120572 = 2 and 119909

0= 22

Methods |1199091minus 120572| |119909

2minus 120572| |119909

3minus 120572| coc

Without memoryNew method (11) 120574 = minus1 03124(minus6) 09972(minus56) 01074(minus451) 80000KT8 120574 = minus1 08055(minus6) 01413(minus52) 01268(minus426) 80000ZLH8 120574 = 1 04955(minus6) 05798(minus54) 02038(minus437) 80000LT8 120574 = minus1 04052(minus6) 05819(minus55) 01053(minus445) 80000

With memoryNew method (20) 120574 = minus1 03124(minus6) 01213(minus81) 03093(minus985) 119822KT8 120574 = minus1 08055(minus6) 02641(minus78) 09143(minus945) 119538ZLH8 120574 = minus001 03943(minus6) 01889(minus80) 06308(minus971) 119817LT8 120574 = minus001 04532(minus6) 04774(minus76) 05848(minus853) 111023

Discrete Dynamics in Nature and Society 9

Table 6 1198913(119909) = 119890

1199093minus119909

minus cos(1199092 minus 1) + 1199093+ 1 120572 = minus1 and 119909

0= minus165

Methods |1199091minus 120572| |119909

2minus 120572| |119909

3minus 120572| coc

Without memoryNew method (11) 120574 = minus1 02768(minus3) 03670(minus27) 03499(minus218) 80000KT8 120574 = minus1 02915(minus4) 05324(minus34) 06583(minus272) 80000ZLH8 120574 = 1 02322(minus1) 05044(minus11) 02253(minus88) 80081LT 120574 = minus1 01094(minus2) 01454(minus22) 01323(minus181) 80014

With memoryNew method (20) 120574 = minus1 02768(minus3) 02660(minus42) 01805(minus509) 119734KT8 120574 = minus1 02915(minus4) 05214(minus53) 01157(minus642) 120961ZLH8 120574 = minus001 06052(minus5) 02355(minus65) 04206(minus786) 119310LT8 120574 = minus1 01094(minus2) 03522(minus34) 01717(minus415) 121081

exploring its dynamic or basins of attractions and lastlyextending the developedmethods with memory using two orthree accelerators

Conflict of Interests

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

Acknowledgments

The first author thanks Islamic Azad University HamedanBranch for the financial support of the present research Theauthors are also thankful for insightful comments of three ref-erees whom helped with the readability and reliability of thepresent paperThe fourth author also gratefully acknowledgesthat this research was partially supported by the UniversityPutra Malaysia under the GP-IBT Grant Scheme havingproject number GP-IBT20139420100

References

[1] N Kyurkchiev and A Iliev ldquoOn some multipoint methodsarising from optimal in the sense of Kung-Traub algorithmsrdquoBiomath vol 2 no 1 Article ID 1305155 7 pages 2013

[2] T Lotfi F Soleymani S Sharifi S Shateyi and F K HaghanildquoMulti-point iterative methods for finding all the simple zerosin an intervalrdquo Journal of AppliedMathematics vol 2014 ArticleID 601205 14 pages 2014

[3] J F Traub G W Wasilkowski and H Wozniakowski Informa-tion Uncertainty Complexity Addison-Wesley 1983

[4] M S Rhee and Y I Kim ldquoA general class of optimal fourth-order multiple-root finders without memory for nonlinearequationsrdquo Applied Mathematical Sciences vol 7 pp 5537ndash55512013

[5] H Wozniakowski ldquoMaximal order of multipoint iterationsusing n evaluationsrdquo in Analytic Computational Complexity JF Traub Ed pp 75ndash107 Academic Press New York NY USA1976

[6] 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

[7] Y H Geum and Y I Kim ldquoAn optimal family of fast 16th-orderderivative-free multipoint simple-root finders for nonlinear

equationsrdquo Journal ofOptimizationTheory andApplications vol160 no 2 pp 608ndash622 2014

[8] J F Traub Iterative Methods for the Solution of EquationsPrentice-Hall New York NY USA 1964

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

[10] 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

[11] T Lotfi and E Tavakoli ldquoOn a new efficient Steffensen-likeiterative class by applying a suitable self-accelerator parameterrdquoThe Scientific World Journal vol 2014 Article ID 769758 9pages 2014

[12] F Soleymani and S Shateyi ldquoTwo optimal eighth-orderderivative-free classes of iterative methodsrdquo Abstract andApplied Analysis vol 2012 Article ID 318165 14 pages 2012

[13] F Soleymani and D K R Babajee ldquoComputing multiple zerosusing a class of quartically convergent methodsrdquo AlexandriaEngineering Journal vol 52 no 3 pp 531ndash541 2013

[14] F Soleymani ldquoSome high-order iterativemethods for finding allthe real zerosrdquoThai Journal of Mathematics vol 12 pp 313ndash3272014

[15] F Soleymani E Tohidi S Shateyi and F K Haghani ldquoSomematrix iterations for computing matrix sign functionrdquo Journalof Applied Mathematics vol 2014 Article ID 425654 9 pages2014

[16] F Soleymani P S Stanimirovic S Shateyi and F K HaghanildquoApproximating the matrix sign function using a novel iterativemethodrdquo Abstract and Applied Analysis vol 2014 Article ID105301 9 pages 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Discrete Dynamics in Nature and Society

Table 2 1198912(119909) = 119890

minus5119909(119909 minus 2)(119909

10+ 119909 + 2) 120572 = 2 and 119909

0= 22

Methods |1199091minus 120572| |119909

2minus 120572| |119909

3minus 120572| coc

Without memoryNew method (6) 120574 = minus001 09445(minus3) 01550(minus13) 01289(minus56) 39945KT4 120574 = minus001 01125(minus2) 03033(minus13) 01891(minus55) 39932ZLH4 120574 = minus01 09349(minus3) 01478(minus13) 01074(minus56) 39939LT 120574 = minus001 01343(minus2) 05965(minus13) 02821(minus54) 39918

With memoryNew method (19) 120574 = 1 01057(minus2) 01102(minus19) 08335(minus109) 52480KT4 120574 = 1 01267(minus2) 02180(minus19) 03580(minus107) 52364ZLH4 120574 = minus001 09445(minus3) 02868(minus20) 08014(minus112) 52264LT4 120574 = minus1 01343(minus2) 01072(minus19) 01150(minus108) 52035

Table 3 1198913(119909) = 119890

1199093minus119909

minus cos(1199092 minus 1) + 1199093+ 1 120572 = minus1 and 119909

0= minus165

Methods |1199091minus 120572| |119909

2minus 120572| |119909

3minus 120572| coc

Without memoryNew method (6) 120574 = minus001 06849(minus3) 08725(minus13) 02284(minus52) 40003KT4 120574 = minus001 06610(minus3) 08682(minus13) 02564(minus52) 40004ZLH4 120574 = 01 01214(minus2) 02388(minus12) 03560(minus51) 40004LT4 120574 = minus001 06398(minus3) 08549(minus13) 02714(minus52) 40002

With memoryNew method (19) 120574 = minus1 04838(minus2) 01348(minus11) 01230(minus64) 55506KT4 120574 = 1 01170(minus1) 03517(minus7) 08936(minus41) 60860ZLH4 120574 = minus001 06849(minus3) 07630(minus17) 01662(minus92) 54226LT4 120574 = minus001 06398(minus3) 06666(minus17) 07373(minus93) 54324

Table 4 1198911(119909) = sin(120587119909)119890(119909

2+119909 cos(119909)minus1)

+ 119909 log(119909 sin(119909) + 1) 120572 = 0 and 1199090= 06

Methods |1199091minus 120572| |119909

2minus 120572| |119909

3minus 120572| coc

Without memoryNew method (11) 120574 = minus1 02345(minus3) 01042(minus32) 01593(minus267) 79999KT8 120574 = minus1 03101(minus3) 02671(minus31) 08196(minus256) 79999ZLH8 120574 = minus001 01742(minus2) 01558(minus22) 06471(minus183) 79999LT8 120574 = minus1 05758(minus3) 07106(minus29) 03880(minus236) 79998

With memoryNew method (20) 120574 = minus01 03440(minus3) 01220(minus41) 08636(minus503) 119935KT8 120574 = minus001 09773(minus3) 02960(minus33) 01505(minus399) 120021ZLH8 120574 = minus001 01742(minus2) 03012(minus33) 04438(minus402) 119900LT8 120574 = minus01 07107(minus4) 02040(minus49) 04971(minus596) 120024

Table 5 1198912(119909) = 119890

minus5119909(119909 minus 2)(119909

10+ 119909 + 2) 120572 = 2 and 119909

0= 22

Methods |1199091minus 120572| |119909

2minus 120572| |119909

3minus 120572| coc

Without memoryNew method (11) 120574 = minus1 03124(minus6) 09972(minus56) 01074(minus451) 80000KT8 120574 = minus1 08055(minus6) 01413(minus52) 01268(minus426) 80000ZLH8 120574 = 1 04955(minus6) 05798(minus54) 02038(minus437) 80000LT8 120574 = minus1 04052(minus6) 05819(minus55) 01053(minus445) 80000

With memoryNew method (20) 120574 = minus1 03124(minus6) 01213(minus81) 03093(minus985) 119822KT8 120574 = minus1 08055(minus6) 02641(minus78) 09143(minus945) 119538ZLH8 120574 = minus001 03943(minus6) 01889(minus80) 06308(minus971) 119817LT8 120574 = minus001 04532(minus6) 04774(minus76) 05848(minus853) 111023

Discrete Dynamics in Nature and Society 9

Table 6 1198913(119909) = 119890

1199093minus119909

minus cos(1199092 minus 1) + 1199093+ 1 120572 = minus1 and 119909

0= minus165

Methods |1199091minus 120572| |119909

2minus 120572| |119909

3minus 120572| coc

Without memoryNew method (11) 120574 = minus1 02768(minus3) 03670(minus27) 03499(minus218) 80000KT8 120574 = minus1 02915(minus4) 05324(minus34) 06583(minus272) 80000ZLH8 120574 = 1 02322(minus1) 05044(minus11) 02253(minus88) 80081LT 120574 = minus1 01094(minus2) 01454(minus22) 01323(minus181) 80014

With memoryNew method (20) 120574 = minus1 02768(minus3) 02660(minus42) 01805(minus509) 119734KT8 120574 = minus1 02915(minus4) 05214(minus53) 01157(minus642) 120961ZLH8 120574 = minus001 06052(minus5) 02355(minus65) 04206(minus786) 119310LT8 120574 = minus1 01094(minus2) 03522(minus34) 01717(minus415) 121081

exploring its dynamic or basins of attractions and lastlyextending the developedmethods with memory using two orthree accelerators

Conflict of Interests

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

Acknowledgments

The first author thanks Islamic Azad University HamedanBranch for the financial support of the present research Theauthors are also thankful for insightful comments of three ref-erees whom helped with the readability and reliability of thepresent paperThe fourth author also gratefully acknowledgesthat this research was partially supported by the UniversityPutra Malaysia under the GP-IBT Grant Scheme havingproject number GP-IBT20139420100

References

[1] N Kyurkchiev and A Iliev ldquoOn some multipoint methodsarising from optimal in the sense of Kung-Traub algorithmsrdquoBiomath vol 2 no 1 Article ID 1305155 7 pages 2013

[2] T Lotfi F Soleymani S Sharifi S Shateyi and F K HaghanildquoMulti-point iterative methods for finding all the simple zerosin an intervalrdquo Journal of AppliedMathematics vol 2014 ArticleID 601205 14 pages 2014

[3] J F Traub G W Wasilkowski and H Wozniakowski Informa-tion Uncertainty Complexity Addison-Wesley 1983

[4] M S Rhee and Y I Kim ldquoA general class of optimal fourth-order multiple-root finders without memory for nonlinearequationsrdquo Applied Mathematical Sciences vol 7 pp 5537ndash55512013

[5] H Wozniakowski ldquoMaximal order of multipoint iterationsusing n evaluationsrdquo in Analytic Computational Complexity JF Traub Ed pp 75ndash107 Academic Press New York NY USA1976

[6] 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

[7] Y H Geum and Y I Kim ldquoAn optimal family of fast 16th-orderderivative-free multipoint simple-root finders for nonlinear

equationsrdquo Journal ofOptimizationTheory andApplications vol160 no 2 pp 608ndash622 2014

[8] J F Traub Iterative Methods for the Solution of EquationsPrentice-Hall New York NY USA 1964

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

[10] 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

[11] T Lotfi and E Tavakoli ldquoOn a new efficient Steffensen-likeiterative class by applying a suitable self-accelerator parameterrdquoThe Scientific World Journal vol 2014 Article ID 769758 9pages 2014

[12] F Soleymani and S Shateyi ldquoTwo optimal eighth-orderderivative-free classes of iterative methodsrdquo Abstract andApplied Analysis vol 2012 Article ID 318165 14 pages 2012

[13] F Soleymani and D K R Babajee ldquoComputing multiple zerosusing a class of quartically convergent methodsrdquo AlexandriaEngineering Journal vol 52 no 3 pp 531ndash541 2013

[14] F Soleymani ldquoSome high-order iterativemethods for finding allthe real zerosrdquoThai Journal of Mathematics vol 12 pp 313ndash3272014

[15] F Soleymani E Tohidi S Shateyi and F K Haghani ldquoSomematrix iterations for computing matrix sign functionrdquo Journalof Applied Mathematics vol 2014 Article ID 425654 9 pages2014

[16] F Soleymani P S Stanimirovic S Shateyi and F K HaghanildquoApproximating the matrix sign function using a novel iterativemethodrdquo Abstract and Applied Analysis vol 2014 Article ID105301 9 pages 2014

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

Discrete Dynamics in Nature and Society 9

Table 6 1198913(119909) = 119890

1199093minus119909

minus cos(1199092 minus 1) + 1199093+ 1 120572 = minus1 and 119909

0= minus165

Methods |1199091minus 120572| |119909

2minus 120572| |119909

3minus 120572| coc

Without memoryNew method (11) 120574 = minus1 02768(minus3) 03670(minus27) 03499(minus218) 80000KT8 120574 = minus1 02915(minus4) 05324(minus34) 06583(minus272) 80000ZLH8 120574 = 1 02322(minus1) 05044(minus11) 02253(minus88) 80081LT 120574 = minus1 01094(minus2) 01454(minus22) 01323(minus181) 80014

With memoryNew method (20) 120574 = minus1 02768(minus3) 02660(minus42) 01805(minus509) 119734KT8 120574 = minus1 02915(minus4) 05214(minus53) 01157(minus642) 120961ZLH8 120574 = minus001 06052(minus5) 02355(minus65) 04206(minus786) 119310LT8 120574 = minus1 01094(minus2) 03522(minus34) 01717(minus415) 121081

exploring its dynamic or basins of attractions and lastlyextending the developedmethods with memory using two orthree accelerators

Conflict of Interests

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

Acknowledgments

The first author thanks Islamic Azad University HamedanBranch for the financial support of the present research Theauthors are also thankful for insightful comments of three ref-erees whom helped with the readability and reliability of thepresent paperThe fourth author also gratefully acknowledgesthat this research was partially supported by the UniversityPutra Malaysia under the GP-IBT Grant Scheme havingproject number GP-IBT20139420100

References

[1] N Kyurkchiev and A Iliev ldquoOn some multipoint methodsarising from optimal in the sense of Kung-Traub algorithmsrdquoBiomath vol 2 no 1 Article ID 1305155 7 pages 2013

[2] T Lotfi F Soleymani S Sharifi S Shateyi and F K HaghanildquoMulti-point iterative methods for finding all the simple zerosin an intervalrdquo Journal of AppliedMathematics vol 2014 ArticleID 601205 14 pages 2014

[3] J F Traub G W Wasilkowski and H Wozniakowski Informa-tion Uncertainty Complexity Addison-Wesley 1983

[4] M S Rhee and Y I Kim ldquoA general class of optimal fourth-order multiple-root finders without memory for nonlinearequationsrdquo Applied Mathematical Sciences vol 7 pp 5537ndash55512013

[5] H Wozniakowski ldquoMaximal order of multipoint iterationsusing n evaluationsrdquo in Analytic Computational Complexity JF Traub Ed pp 75ndash107 Academic Press New York NY USA1976

[6] 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

[7] Y H Geum and Y I Kim ldquoAn optimal family of fast 16th-orderderivative-free multipoint simple-root finders for nonlinear

equationsrdquo Journal ofOptimizationTheory andApplications vol160 no 2 pp 608ndash622 2014

[8] J F Traub Iterative Methods for the Solution of EquationsPrentice-Hall New York NY USA 1964

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

[10] 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

[11] T Lotfi and E Tavakoli ldquoOn a new efficient Steffensen-likeiterative class by applying a suitable self-accelerator parameterrdquoThe Scientific World Journal vol 2014 Article ID 769758 9pages 2014

[12] F Soleymani and S Shateyi ldquoTwo optimal eighth-orderderivative-free classes of iterative methodsrdquo Abstract andApplied Analysis vol 2012 Article ID 318165 14 pages 2012

[13] F Soleymani and D K R Babajee ldquoComputing multiple zerosusing a class of quartically convergent methodsrdquo AlexandriaEngineering Journal vol 52 no 3 pp 531ndash541 2013

[14] F Soleymani ldquoSome high-order iterativemethods for finding allthe real zerosrdquoThai Journal of Mathematics vol 12 pp 313ndash3272014

[15] F Soleymani E Tohidi S Shateyi and F K Haghani ldquoSomematrix iterations for computing matrix sign functionrdquo Journalof Applied Mathematics vol 2014 Article ID 425654 9 pages2014

[16] F Soleymani P S Stanimirovic S Shateyi and F K HaghanildquoApproximating the matrix sign function using a novel iterativemethodrdquo Abstract and Applied Analysis vol 2014 Article ID105301 9 pages 2014

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