Research Article A New Sixth-Order Steffensen-Type Iterative Method for Solving Nonlinear Equations Tahereh Eftekhari Faculty of Mathematics, University of Sistan and Baluchestan, Zahedan 987-98155, Iran Correspondence should be addressed to Tahereh Eſtekhari; t.eſt[email protected] Received 11 November 2013; Revised 6 January 2014; Accepted 9 January 2014; Published 12 February 2014 Academic Editor: Ahmed Zayed Copyright © 2014 Tahereh Eſtekhari. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Based on iterative method proposed by Basto et al. (2006), we present a new derivative-free iterative method for solving nonlinear equations. e aim of this paper is to develop a new method to find the approximation of the root of the nonlinear equation () = 0. is method has the efficiency index which equals 6 1/4 = 1.5651. e benefit of this method is that this method does not need to calculate any derivative. Several examples illustrate that the efficiency of the new method is better than that of previous methods. 1. Introduction Solving nonlinear equations is one of the most important and challenging problems in scientific and engineering applica- tions. In this paper, we consider an iterative method to find the root of a nonlinear equation () = 0. Newton’s method is the best known iterative method for solving nonlinear equations [1], given by +1 = − ( ) ( ) , = 0, 1, 2, . . . (1) which converges quadratically. But it has a major weakness, one has to calculate the derivative of () at each approxima- tion. Frequently, ( ) is far more difficult to evaluate and needs more arithmetic operations to calculate than (). It is well known that the forward-difference approxima- tion for ( ) at is () ≈ ( + ℎ) − () ℎ . (2) If the derivative ( ) is replaced by the forward-difference approximation with ℎ = ( ) that is ( )≈ ( + ( )) − ( ) ( ) , (3) the Newton’s method becomes +1 = − ( ) 2 ( + ( )) − ( ) , = 0, 1, 2, . . . (4) which is the famous Steffensen’s method [2]. e Steffensen’s method is based on forward-difference approximation to derivative. is method is a tough competitor of Newton’s method. Both methods are of quadratic convergence, and both require two functions evaluation per iteration, but in contrast to Newton’s method, Steffensen’s method is derivative-free. Based on this method, many derivative-free iterative methods have been proposed. In [3], a sixth-order derivative-free iterative method has been proposed by Cordero et al. as follows: = − 2( ) 2 ( + ( )) − ( − ( )) , = − − 2 ( ) − ( ) ( ), +1 = − − 2 ( ) − ( ) ( ), (5) which has efficiency index 1.430. Hindawi Publishing Corporation International Journal of Analysis Volume 2014, Article ID 685796, 5 pages http://dx.doi.org/10.1155/2014/685796
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Research ArticleA New Sixth-Order Steffensen-Type Iterative Method forSolving Nonlinear Equations
Tahereh Eftekhari
Faculty of Mathematics University of Sistan and Baluchestan Zahedan 987-98155 Iran
Correspondence should be addressed to Tahereh Eftekhari teftekhari2009gmailcom
Received 11 November 2013 Revised 6 January 2014 Accepted 9 January 2014 Published 12 February 2014
Academic Editor Ahmed Zayed
Copyright copy 2014 Tahereh Eftekhari This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Based on iterative method proposed by Basto et al (2006) we present a new derivative-free iterative method for solving nonlinearequations The aim of this paper is to develop a new method to find the approximation of the root 120572 of the nonlinear equation119891(119909) = 0 This method has the efficiency index which equals 614 = 15651 The benefit of this method is that this method doesnot need to calculate any derivative Several examples illustrate that the efficiency of the newmethod is better than that of previousmethods
1 Introduction
Solving nonlinear equations is one of themost important andchallenging problems in scientific and engineering applica-tions In this paper we consider an iterative method to findthe root of a nonlinear equation 119891(119909) = 0
Newtonrsquos method is the best known iterative method forsolving nonlinear equations [1] given by
119909119899+1= 119909119899minus
119891 (119909119899)
1198911015840(119909119899)
119899 = 0 1 2 (1)
which converges quadratically But it has a major weaknessone has to calculate the derivative of 119891(119909) at each approxima-tion Frequently 1198911015840(119909
119899) is far more difficult to evaluate and
needs more arithmetic operations to calculate than 119891(119909)It is well known that the forward-difference approxima-
tion for 1198911015840(119909119899) at 119909 is
1198911015840(119909) asymp
119891 (119909 + ℎ) minus 119891 (119909)
ℎ
(2)
If the derivative 1198911015840(119909119899) is replaced by the forward-difference
approximation with ℎ = 119891(119909119899) that is
1198911015840(119909119899) asymp
119891 (119909119899+ 119891 (119909
119899)) minus 119891 (119909
119899)
119891 (119909119899)
(3)
the Newtonrsquos method becomes
119909119899+1= 119909119899minus
119891(119909119899)2
119891 (119909119899+ 119891 (119909
119899)) minus 119891 (119909
119899)
119899 = 0 1 2 (4)
which is the famous Steffensenrsquos method [2] The Steffensenrsquosmethod is based on forward-difference approximation toderivative This method is a tough competitor of Newtonrsquosmethod Both methods are of quadratic convergence andboth require two functions evaluation per iteration butin contrast to Newtonrsquos method Steffensenrsquos method isderivative-free Based on this method many derivative-freeiterative methods have been proposed
In [3] a sixth-order derivative-free iterative method hasbeen proposed by Cordero et al as follows
119910119899= 119909119899minus
2119891(119909119899)2
119891 (119909119899+ 119891 (119909
119899)) minus 119891 (119909
119899minus 119891 (119909
119899))
119911119899= 119910119899minus
119910119899minus 119909119899
2119891 (119910119899) minus 119891 (119909
119899)
119891 (119910119899)
119909119899+1= 119911119899minus
119910119899minus 119909119899
2119891 (119910119899) minus 119891 (119909
119899)
119891 (119911119899)
(5)
which has efficiency index 1430
Hindawi Publishing CorporationInternational Journal of AnalysisVolume 2014 Article ID 685796 5 pageshttpdxdoiorg1011552014685796
2 International Journal of Analysis
Soleymani andHosseinabadi [4] suggested another sixth-order derivative-free scheme in the form below
119910119899= 119909119899minus
119891 (119909119899)
119891 [119909119899 119908119899]
119911119899= 119909119899minus
119891 (119909119899)
119891 [119909119899 119908119899]
(1 +
119891 (119910119899)
119891 (119909119899)
(1 + 2
119891 (119910119899)
119891 (119909119899)
))
119909119899+1= 119911119899minus
119891 (119911119899)
119891 [119910119899 119911119899]
(1 minus
1 + 119891 [119909119899 119908119899]
119891 [119909119899 119908119899]
119891 (119911119899)
119891 (119908119899)
)
(6)
wherein the efficiency index is 15651In [5] a sixth-order derivative-free algorithm has been
derived by Yasmin and Iftikhar which is written as
119910119899= 119909119899minus
119891 (119909119899)
119891 [119909119899 119908119899]
119911119899= 119910119899minus 119891 (119910
119899)
times (
1
119891 [119909119899 119908119899]
minus
4 (119891 (119909119899) + 119891 (119910
119899))
119891 [119909119899 119908119899] + 119891 [119909
119899 119908119899] (119891 (119909
119899) minus 119891 (119910
119899))
)
119909119899+1= 119911119899minus
119891 (119911119899)
119891 [119910119899 119911119899] minus 119891 [119909
119899 119910119899] + 119891 [119909
119899 119911119899]
(7)
which has efficiency index 15651Theprocesses of removing the derivatives usually increase
the number of function evaluations per iteration In ourmethod we used the technique of composition of Newtonrsquosmethod with the known methods which not only increasethe order of themethod as high as possible but also reduce thenumber of function evaluations and improve the efficiencyindex of the composed method In view of this fact thenew steffensen-like methods are significantly better whencompared with the established methods described above
This paper is organized as follows some basic definitionsrelevant to the present work are presented in Section 2 InSection 3 we present a new three-step sixth-order iterativemethod for solving nonlinear equations The new method isfree from derivative We prove that the order of convergenceof the new method is six Numerical examples show betterperformance of our method in Section 4 Section 5 is a shortconclusion
2 Basic Definitions
Definition 1 Let 119891(119909) be a real function with a simple root 120572and let 119909
119899119899isinN be a sequence of real numbers that converges
towards 120572 Then we say that the order of convergence of thesequence is 119901 if there exists a number 119901 isin R+ such that
lim119899rarrinfin
119909119899+1minus 120572
(119909119899minus 120572)119901= 119862 (8)
where for some 119862 = 0 119862 is known as the asymptotic errorconstant
If 119901 = 1 2 or 3 the sequence is said to have linearconvergence quadratic convergence or cubic convergencerespectively
Definition 2 Let 119890119899= 119909119899minus 120572 be the error in the 119899th iteration
One calls the relation
119890119899+1= 119862119890119901
119899+ 119874 (119890
119901+1
119899) (9)
the error equation
Definition 3 Let 119902 be the number of function evaluationsof the new method The efficiency of the new method ismeasured by the concept of efficiency index [6] and definedas
119864 = 1199011119902 (10)
where 119901 is the order of the method
Definition 4 Suppose that119909119899+1
119909119899 and 119909
119899minus1are three succes-
sive iterations closer to the root 120572 Then the computationalorder of convergence [7] is approximated by using (9) asfollows
120588 asymp
119871119899
10038161003816100381610038161003816(119909119899+1minus 120572) (119909
119899minus 120572)minus110038161003816100381610038161003816
119871119899
10038161003816100381610038161003816(119909119899minus 120572) (119909
119899minus1minus 120572)minus110038161003816100381610038161003816
(11)
where 119899 isin N
3 Iterative Method and Convergence Analyses
Consider the following iterativemethod proposed by Basto etal [8] to construct a new sixth-order method
119909119899+1= 119909119899minus
119891 (119909119899)
1198911015840(119909119899)
minus
[119891 (119909119899)]2
11989110158401015840(119909119899)
2[1198911015840(119909119899)]3
minus 2119891 (119909119899) 1198911015840(119909119899) 11989110158401015840(119909119899)
(12)
This method is third-order and the efficiency index is313= 14422First we replace 11989110158401015840(119909
119899) from (12) with a finite difference
between the first derivatives [9] that is
11989110158401015840(119909119899) =
1198911015840(119910119899) minus 1198911015840(119909119899)
119910119899minus 119909119899
(13)
where
119910119899= 119909119899minus
119891 (119909119899)
1198911015840(119909119899)
(14)
using (13) and (14) in (12) we obtained an equivalent form
119910119899= 119909119899minus
119891 (119909119899)
1198911015840(119909119899)
119909119899+1= 119910119899+
119891 (119909119899) (1198911015840(119910119899) minus 1198911015840(119909119899))
21198911015840(119909119899) 1198911015840(119910119899)
(15)
International Journal of Analysis 3
By combining the method (15) with Newtonrsquos method weobtain a new three-step iterative algorithm without memoryas follows
119910119899= 119909119899minus
119891 (119909119899)
1198911015840(119909119899)
119911119899= 119910119899+
119891 (119909119899) (1198911015840(119910119899) minus 1198911015840(119909119899))
21198911015840(119909119899) 1198911015840(119910119899)
119909119899+1= 119911119899minus
119891 (119911119899)
1198911015840(119911119899)
(16)
which is required of three evaluations of the first derivativeof the function To remedy these derivatives firstly weapproximate 1198911015840(119909
119899) by the divided difference of order one
1198911015840(119909119899) asymp 119891 [119908
119899 119909119899] =
119891 (119908119899) minus 119891 (119909
119899)
119908119899minus 119909119899
(17)
where 119908119899= 119909119899+ 119891(119909
119899)
Secondly approximate 1198911015840(119911119899) by the linear combination
of divided differences
1198911015840(119911119899) asymp 119891 [119910
119899 119911119899] minus 119891 [119909
119899 119910119899] + 119891 [119909
119899 119911119899] (18)
and for 1198911015840(119910119899) we use the following approximation [10]
1198911015840(119910119899) asymp
1198911015840(119909119899) (119891 (119909
119899) minus 119891 (119910
119899))
119891 (119909119899) + 119891 (119910
119899)
(19)
Thus our new three-step derivative-free iterative algorithmwithout memory is given as
119910119899= 119909119899minus
119891 (119909119899)
119891 [119909119899 119908119899]
119911119899= 119910119899minus
119891 (119909119899) 119891 (119910
119899)
119891 [119909119899 119908119899] (119891 (119909
119899) minus 119891 (119910
119899))
119909119899+1= 119911119899minus
119891 (119911119899)
119891 [119910119899 119911119899] minus 119891 [119909
119899 119910119899] + 119891 [119909
119899 119911119899]
(20)
where 119908119899= 119909119899+ 119891(119909
119899)
Theorem 5 demonstrates its convergence analysis
Theorem 5 Let 120572 isin 119868 be a simple root of a sufficientlydifferentiable function 119891 119868 sube R rarr R in an open interval 119868If the initial approximation 119909
0is sufficiently close to 120572 then the
derivative-freemethod defined by (20) has order of convergencesix
Proof Let 120572 be the simple root of 119891(119909) that is 119891(120572) = 01198911015840(120572) = 0 and the error equation is 119890
119899= 119909119899minus 120572
By Taylorrsquos expansion of 119891(119909119899) about 119909 = 120572 and putting
119891(120572) = 0 we have
119891 (119909119899) = 1198881119890119899+ 11988821198902
119899+ 11988831198903
119899+ 11988841198904
119899+ 11988851198905
119899+ 11988861198906
119899+ 119874 (119890
7
119899)
(21)
Table 1 The examples considered in this study
Test functions Zeros1198911(119909) = 119909
2minus 119890119909minus 3119909 + 2 120572 asymp 025753028543986079
1198912(119909) = 119909119890
1199092
minus sin2(119909) + 3 cos(119909) + 5 120572 asymp minus1207647827130919
1198913(119909) = sin(119909)119890119909 + Ln(1199092 + 1) 120572 asymp 0
1198914(119909) = 10119909119890
minus1199092
minus 1 120572 asymp 16796306104284499
1198915(119909) = cos(119909) minus 119909 120572 asymp 073908513321516064
1198916(119909) = 119890
minus1199092+119909+2minus 1 120572 asymp minus1000000000000000
1198917(119909) = Ln(1199092 + 119909 + 2) minus 119909 + 1 120572 asymp 41525907367571583
where
119888119896=
119891119896(120572)
119896
119896 = 1 2 3 (22)
Expanding the Taylor series of119891(119908119899) about the solution 120572 we
have
119891 (119908119899) =
infin
sum
119894=1
119888119894(119890119899+ 119891 (119909
119899))119894
(23)
substituting 119891(119909119899) given by (21) gives us
119891 (119908119899) = 1198881(1 + 1198881) 119890119899+ (1 + 119888
1(3 + 1198881)) 11988821198902
119899
+ (2 (1 + 1198881) 1198882
2+ 11988811198883+ (1 + 119888
1)3
1198883) 1198903
119899
+ sdot sdot sdot + 119874 (1198907
119899)
(24)
Note that to save the space we onlywrite someof the obtainedterms for the error equations and show the others by
Substituting (21) and (24) in the first step of (20) gives us
119910119899minus 120572 = 119909
119899minus 120572 minus
119891 (119909119899)
119891 [119908119899 119909119899]
= (1 +
1
1198881
) 11988821198902
119899
+
minus (2 + 1198881(2 + 1198881)) 1198882
2+ 1198881(1 + 1198881) (2 + 119888
1) 1198883
1198882
1
1198903
119899
+ sdot sdot sdot + 119874 (1198907
119899)
(25)
Using the Taylor expansion of 119891(119910119899) about the solution 120572 we
have
119891 (119910119899) =
infin
sum
119894=1
119888119894(119910119899minus 120572)119894
(26)
substituting (25) into the preceding equation we have
119891 (119910119899) = (1 + 119888
1) 11988821198902
119899
+ ((minus2 minus
2
1198881
minus 1198881) 1198882
2+ (1 + 119888
1) (2 + 119888
1) 1198883) 1198903
119899
+ sdot sdot sdot + 119874 (1198907
119899)
(27)
4 International Journal of Analysis
Table 2 Comparison of various iterative methods
New method (CM) (SHM) (YIM) (SM)1198911 1199090= 0
1003816100381610038161003816119909119899minus 1205721003816100381610038161003816
023119890 minus 291 046119890 minus 165 036119890 minus 267 045119890 minus 155 055119890 minus 80
1003816100381610038161003816119891 (119909119899)1003816100381610038161003816
088119890 minus 291 018119890 minus 164 014119890 minus 266 017119890 minus 154 021119890 minus 79
COC 6000000 6000006 6000000 5999992 2000000
TNFE 12 15 12 12 12
1198912 1199090= minus12
1003816100381610038161003816119909119899minus 1205721003816100381610038161003816
036119890 minus 345 034119890 minus 180 012119890 minus 157 067119890 minus 190 032119890 minus 40
1003816100381610038161003816119891 (119909119899)1003816100381610038161003816
073119890 minus 344 069119890 minus 179 024119890 minus 156 014119890 minus 188 065119890 minus 39
COC 6000000 5999994 6000004 6000000 2000000
TNFE 12 15 12 12 12
1198913 1199090= 01
1003816100381610038161003816119909119899minus 1205721003816100381610038161003816
043119890 minus 145 041119890 minus 158 052119890 minus 121 013119890 minus 88 071119890 minus 32
1003816100381610038161003816119891 (119909119899)1003816100381610038161003816
043119890 minus 145 041119890 minus 158 052119890 minus 121 013119890 minus 88 071119890 minus 32
COC 5999990 5999996 5999862 5999567 2000000
TNFE 12 15 12 12 12
1198914 1199090= 18
1003816100381610038161003816119909119899minus 1205721003816100381610038161003816
013119890 minus 188 012119890 minus 280 099119890 minus 171 043119890 minus 93 038119890 minus 48
1003816100381610038161003816119891 (119909119899)1003816100381610038161003816
035119890 minus 188 035119890 minus 280 027119890 minus 170 012119890 minus 92 011119890 minus 47
COC 6000000 6000000 5999998 5999951 2000000
TNFE 12 15 12 12 12
1198915 1199090= 1
1003816100381610038161003816119909119899minus 1205721003816100381610038161003816
021119890 minus 265 014119890 minus 204 031119890 minus 257 014119890 minus 133 054119890 minus 88
1003816100381610038161003816119891 (119909119899)1003816100381610038161003816
036119890 minus 265 024119890 minus 204 0515119890 minus 257 023119890 minus 133 091119890 minus 88
COC 6000000 6000000 6000000 6000024 2000000
TNFE 12 15 12 12 12
1198916 1199090= minus075
1003816100381610038161003816119909119899minus 1205721003816100381610038161003816
048119890 minus 109 011119890 minus 104 087119890 minus 74 018119890 minus 22 011119890 minus 12
1003816100381610038161003816119891 (119909119899)1003816100381610038161003816
015119890 minus 108 034119890 minus 104 026119890 minus 73 054119890 minus 22 034119890 minus 12
COC 5999904 5999991 5996977 5920939 1999927
TNFE 12 15 12 12 12
1198917 1199090= 32
1003816100381610038161003816119909119899minus 1205721003816100381610038161003816
014119890 minus 265 041119890 minus 199 061119890 minus 316 012119890 minus 168 021119890 minus 98
1003816100381610038161003816119891 (119909119899)1003816100381610038161003816
084119890 minus 266 024119890 minus 199 037119890 minus 316 072119890 minus 169 013119890 minus 98
COC 6000000 6000000 6000000 6000001 2000000
TNFE 12 15 12 12 12
Using (21) (24) (25) and (27) in the second step of (20) givesus
119911119899minus 120572 =
(1 + 1198881) 1198882
2
1198882
1
1198903
119899
+
minus (3 + 1198881(3 + 1198881)) 1198883
2+ 1198881(1 + 1198881) (3 + 119888
1) 11988821198883
1198883
1
1198904
119899
+ sdot sdot sdot + 119874 (1198907
119899)
(28)
This shows that at the end of the second step the methodis of third order convergence Therefore the third step is
introduced to achieve the higher orderThe Taylor expansionabout the simple root for 119891(119911
119899) is given as follows
119891 (119911119899) =
(1 + 1198881) 1198882
2
1198881
1198903
119899
+
minus (3 + 1198881(3 + 1198881)) 1198883
2+ 1198881(1 + 1198881) (3 + 119888
1) 11988821198883
1198882
1
1198904
119899
+ sdot sdot sdot + 119874 (1198907
119899)
(29)Using (27) and (29) in
119891 [119910119899 119911119899] =
119891 (119910119899) minus 119891 (119911
119899)
119910119899minus 119911119899
(30)
International Journal of Analysis 5
we get
119891 [119910119899 119911119899] = 1198881+
(1 + 1198881) 1198882
2
1198881
1198902
119899+ sdot sdot sdot + 119874 (119890
7
119899) (31)
Similarly
119891 [119909119899 119910119899] = 1198881+ 1198882119890119899
+ ((1 +
1
1198881
) 1198882
2+ 1198883) 1198902
119899+ sdot sdot sdot + 119874 (119890
7
119899)
119891 [119909119899 119911119899] = 1198881+ 1198882119890119899+ 11988831198902
119899+ sdot sdot sdot + 119874 (119890
7
119899)
(32)
Combining the above terms we have
1198911015840(119911119899) = 119891 [119910
119899 119911119899] minus 119891 [119909
119899 119910119899] + 119891 [119909
119899 119911119899]
= 1198881+
(1 + 1198881) 1198882(21198882
2minus 11988811198883)
1198882
1
1198903
119899+ sdot sdot sdot + 119874 (119890
7
119899)
(33)
Now dividing (33) by (29) and using the last step of (20) wehave
119890119899+1=
(1 + 1198881)2
1198883
2(1198882
2minus 11988811198883)
1198885
1
1198906
119899+ 119874 (119890
7
119899) (34)
This proves that our first proposed method defined by (20) isa sixth-order derivative-free algorithm and satisfies the aboveerror equation This completes the proof
Nowwe discuss the efficiency index of the newmethod byusing Definition 3 as 1199011119902 where 119901 is the order of the methodand 119902 is the number of function evaluations per iterationrequired of the method It is easy to know that the numberof function evaluations per iteration required by the methoddefined in algorithm 1 is four So the efficiency index is 614 =15651
4 Numerical Results
In this section we test the effectiveness of our new methodWe have used second-order method of Steffensen (SM) [2]sixth-order method of Cordero et al (CM) [3] six-ordermethod of Soleymani and Hosseinabadi (SHM) [4] andsix-order method of Yasmin and Iftikhar (YIM) [5] forcomparison with our method to find the simple root ofnonlinear equations The test functions of 119891(119909) are listed inTable 1
Numerical computations reported here have been carriedout in a 119872119886119905ℎ119890119898119886119905119894119888119886 80 environment Table 2 shows thedifference of the root 120572 and the approximation 119909
119899to 120572
where 120572 is the exact root computed with 800 significant digits(Digits = 800) The absolute values of the function (|119891(119909
119899)|)
the difference between the approximated root 119909119899and the
exact root 120572 the number of function evaluations (TNFE)and the computational order of convergence (COC) are alsoshown in Table 2 Here COC is defined in Definition 4
5 Conclusions
We have obtained a new Steffensen-type iterative method forsolving nonlinear equations The convergence order of thismethod is six and consists of four evaluations of the functionper iteration so it has an efficiency index equal to 614 =15651 Numerical examples also show that the numericalresults of our new method in equal iterations improve theresults of other existing methods
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The author is grateful to the referees for their comments andsuggestions that helped to improve the paper
References
[1] W Bi H Ren and Q Wu ldquoNew family of seventh-ordermethods for nonlinear equationsrdquo Applied Mathematics andComputation vol 203 no 1 pp 408ndash412 2008
[2] D Kincaid and W Cheney Numerical Analysis BrooksColePacific Grove Calif USA 2nd edition 1996
[3] A Cordero J L Hueso E Martınez and J R TorregrosaldquoSteffensen type methods for solving nonlinear equationsrdquoJournal of Computational andAppliedMathematics vol 236 no12 pp 3058ndash3064 2012
[4] F Soleymani and V Hosseinabadi ldquoNew third- and sixth-orderderivativefree techniques for nonlinear equationsrdquo Journal ofMathematics Research vol 3 no 2 pp 107ndash112 2011
[5] N Yasmin and S Iftikhar ldquoSome new Steffensen like three-stepmethods for solving nonlinear equationsrdquo International Journalof Pure and Applied Mathematics vol 82 pp 557ndash572 2013
[6] W Gautschi Numerical Analysis Birkhauser Boston MassUSA 1997
[7] S Weerakoon and T G I Fernando ldquoA variant of Newtonrsquosmethod with accelerated third-order convergencerdquo AppliedMathematics Letters vol 13 no 8 pp 87ndash93 2000
[8] M Basto V Semiao and F L Calheiros ldquoA new iterativemethod to compute nonlinear equationsrdquo Applied Mathematicsand Computation vol 173 no 1 pp 468ndash483 2006
[9] J Kou Y Li and XWang ldquoModified Halleyrsquos method free fromsecond derivativerdquo Applied Mathematics and Computation vol183 no 1 pp 704ndash708 2006
[10] M Dehghan and M Hajarian ldquoSome derivative free quadraticand cubic convergence iterative formulas for solving nonlinearequationsrdquoComputational andAppliedMathematics vol 29 no1 pp 19ndash30 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 International Journal of Analysis
Soleymani andHosseinabadi [4] suggested another sixth-order derivative-free scheme in the form below
119910119899= 119909119899minus
119891 (119909119899)
119891 [119909119899 119908119899]
119911119899= 119909119899minus
119891 (119909119899)
119891 [119909119899 119908119899]
(1 +
119891 (119910119899)
119891 (119909119899)
(1 + 2
119891 (119910119899)
119891 (119909119899)
))
119909119899+1= 119911119899minus
119891 (119911119899)
119891 [119910119899 119911119899]
(1 minus
1 + 119891 [119909119899 119908119899]
119891 [119909119899 119908119899]
119891 (119911119899)
119891 (119908119899)
)
(6)
wherein the efficiency index is 15651In [5] a sixth-order derivative-free algorithm has been
derived by Yasmin and Iftikhar which is written as
119910119899= 119909119899minus
119891 (119909119899)
119891 [119909119899 119908119899]
119911119899= 119910119899minus 119891 (119910
119899)
times (
1
119891 [119909119899 119908119899]
minus
4 (119891 (119909119899) + 119891 (119910
119899))
119891 [119909119899 119908119899] + 119891 [119909
119899 119908119899] (119891 (119909
119899) minus 119891 (119910
119899))
)
119909119899+1= 119911119899minus
119891 (119911119899)
119891 [119910119899 119911119899] minus 119891 [119909
119899 119910119899] + 119891 [119909
119899 119911119899]
(7)
which has efficiency index 15651Theprocesses of removing the derivatives usually increase
the number of function evaluations per iteration In ourmethod we used the technique of composition of Newtonrsquosmethod with the known methods which not only increasethe order of themethod as high as possible but also reduce thenumber of function evaluations and improve the efficiencyindex of the composed method In view of this fact thenew steffensen-like methods are significantly better whencompared with the established methods described above
This paper is organized as follows some basic definitionsrelevant to the present work are presented in Section 2 InSection 3 we present a new three-step sixth-order iterativemethod for solving nonlinear equations The new method isfree from derivative We prove that the order of convergenceof the new method is six Numerical examples show betterperformance of our method in Section 4 Section 5 is a shortconclusion
2 Basic Definitions
Definition 1 Let 119891(119909) be a real function with a simple root 120572and let 119909
119899119899isinN be a sequence of real numbers that converges
towards 120572 Then we say that the order of convergence of thesequence is 119901 if there exists a number 119901 isin R+ such that
lim119899rarrinfin
119909119899+1minus 120572
(119909119899minus 120572)119901= 119862 (8)
where for some 119862 = 0 119862 is known as the asymptotic errorconstant
If 119901 = 1 2 or 3 the sequence is said to have linearconvergence quadratic convergence or cubic convergencerespectively
Definition 2 Let 119890119899= 119909119899minus 120572 be the error in the 119899th iteration
One calls the relation
119890119899+1= 119862119890119901
119899+ 119874 (119890
119901+1
119899) (9)
the error equation
Definition 3 Let 119902 be the number of function evaluationsof the new method The efficiency of the new method ismeasured by the concept of efficiency index [6] and definedas
119864 = 1199011119902 (10)
where 119901 is the order of the method
Definition 4 Suppose that119909119899+1
119909119899 and 119909
119899minus1are three succes-
sive iterations closer to the root 120572 Then the computationalorder of convergence [7] is approximated by using (9) asfollows
120588 asymp
119871119899
10038161003816100381610038161003816(119909119899+1minus 120572) (119909
119899minus 120572)minus110038161003816100381610038161003816
119871119899
10038161003816100381610038161003816(119909119899minus 120572) (119909
119899minus1minus 120572)minus110038161003816100381610038161003816
(11)
where 119899 isin N
3 Iterative Method and Convergence Analyses
Consider the following iterativemethod proposed by Basto etal [8] to construct a new sixth-order method
119909119899+1= 119909119899minus
119891 (119909119899)
1198911015840(119909119899)
minus
[119891 (119909119899)]2
11989110158401015840(119909119899)
2[1198911015840(119909119899)]3
minus 2119891 (119909119899) 1198911015840(119909119899) 11989110158401015840(119909119899)
(12)
This method is third-order and the efficiency index is313= 14422First we replace 11989110158401015840(119909
119899) from (12) with a finite difference
between the first derivatives [9] that is
11989110158401015840(119909119899) =
1198911015840(119910119899) minus 1198911015840(119909119899)
119910119899minus 119909119899
(13)
where
119910119899= 119909119899minus
119891 (119909119899)
1198911015840(119909119899)
(14)
using (13) and (14) in (12) we obtained an equivalent form
119910119899= 119909119899minus
119891 (119909119899)
1198911015840(119909119899)
119909119899+1= 119910119899+
119891 (119909119899) (1198911015840(119910119899) minus 1198911015840(119909119899))
21198911015840(119909119899) 1198911015840(119910119899)
(15)
International Journal of Analysis 3
By combining the method (15) with Newtonrsquos method weobtain a new three-step iterative algorithm without memoryas follows
119910119899= 119909119899minus
119891 (119909119899)
1198911015840(119909119899)
119911119899= 119910119899+
119891 (119909119899) (1198911015840(119910119899) minus 1198911015840(119909119899))
21198911015840(119909119899) 1198911015840(119910119899)
119909119899+1= 119911119899minus
119891 (119911119899)
1198911015840(119911119899)
(16)
which is required of three evaluations of the first derivativeof the function To remedy these derivatives firstly weapproximate 1198911015840(119909
119899) by the divided difference of order one
1198911015840(119909119899) asymp 119891 [119908
119899 119909119899] =
119891 (119908119899) minus 119891 (119909
119899)
119908119899minus 119909119899
(17)
where 119908119899= 119909119899+ 119891(119909
119899)
Secondly approximate 1198911015840(119911119899) by the linear combination
of divided differences
1198911015840(119911119899) asymp 119891 [119910
119899 119911119899] minus 119891 [119909
119899 119910119899] + 119891 [119909
119899 119911119899] (18)
and for 1198911015840(119910119899) we use the following approximation [10]
1198911015840(119910119899) asymp
1198911015840(119909119899) (119891 (119909
119899) minus 119891 (119910
119899))
119891 (119909119899) + 119891 (119910
119899)
(19)
Thus our new three-step derivative-free iterative algorithmwithout memory is given as
119910119899= 119909119899minus
119891 (119909119899)
119891 [119909119899 119908119899]
119911119899= 119910119899minus
119891 (119909119899) 119891 (119910
119899)
119891 [119909119899 119908119899] (119891 (119909
119899) minus 119891 (119910
119899))
119909119899+1= 119911119899minus
119891 (119911119899)
119891 [119910119899 119911119899] minus 119891 [119909
119899 119910119899] + 119891 [119909
119899 119911119899]
(20)
where 119908119899= 119909119899+ 119891(119909
119899)
Theorem 5 demonstrates its convergence analysis
Theorem 5 Let 120572 isin 119868 be a simple root of a sufficientlydifferentiable function 119891 119868 sube R rarr R in an open interval 119868If the initial approximation 119909
0is sufficiently close to 120572 then the
derivative-freemethod defined by (20) has order of convergencesix
Proof Let 120572 be the simple root of 119891(119909) that is 119891(120572) = 01198911015840(120572) = 0 and the error equation is 119890
119899= 119909119899minus 120572
By Taylorrsquos expansion of 119891(119909119899) about 119909 = 120572 and putting
119891(120572) = 0 we have
119891 (119909119899) = 1198881119890119899+ 11988821198902
119899+ 11988831198903
119899+ 11988841198904
119899+ 11988851198905
119899+ 11988861198906
119899+ 119874 (119890
7
119899)
(21)
Table 1 The examples considered in this study
Test functions Zeros1198911(119909) = 119909
2minus 119890119909minus 3119909 + 2 120572 asymp 025753028543986079
1198912(119909) = 119909119890
1199092
minus sin2(119909) + 3 cos(119909) + 5 120572 asymp minus1207647827130919
1198913(119909) = sin(119909)119890119909 + Ln(1199092 + 1) 120572 asymp 0
1198914(119909) = 10119909119890
minus1199092
minus 1 120572 asymp 16796306104284499
1198915(119909) = cos(119909) minus 119909 120572 asymp 073908513321516064
1198916(119909) = 119890
minus1199092+119909+2minus 1 120572 asymp minus1000000000000000
1198917(119909) = Ln(1199092 + 119909 + 2) minus 119909 + 1 120572 asymp 41525907367571583
where
119888119896=
119891119896(120572)
119896
119896 = 1 2 3 (22)
Expanding the Taylor series of119891(119908119899) about the solution 120572 we
have
119891 (119908119899) =
infin
sum
119894=1
119888119894(119890119899+ 119891 (119909
119899))119894
(23)
substituting 119891(119909119899) given by (21) gives us
119891 (119908119899) = 1198881(1 + 1198881) 119890119899+ (1 + 119888
1(3 + 1198881)) 11988821198902
119899
+ (2 (1 + 1198881) 1198882
2+ 11988811198883+ (1 + 119888
1)3
1198883) 1198903
119899
+ sdot sdot sdot + 119874 (1198907
119899)
(24)
Note that to save the space we onlywrite someof the obtainedterms for the error equations and show the others by
Substituting (21) and (24) in the first step of (20) gives us
119910119899minus 120572 = 119909
119899minus 120572 minus
119891 (119909119899)
119891 [119908119899 119909119899]
= (1 +
1
1198881
) 11988821198902
119899
+
minus (2 + 1198881(2 + 1198881)) 1198882
2+ 1198881(1 + 1198881) (2 + 119888
1) 1198883
1198882
1
1198903
119899
+ sdot sdot sdot + 119874 (1198907
119899)
(25)
Using the Taylor expansion of 119891(119910119899) about the solution 120572 we
have
119891 (119910119899) =
infin
sum
119894=1
119888119894(119910119899minus 120572)119894
(26)
substituting (25) into the preceding equation we have
119891 (119910119899) = (1 + 119888
1) 11988821198902
119899
+ ((minus2 minus
2
1198881
minus 1198881) 1198882
2+ (1 + 119888
1) (2 + 119888
1) 1198883) 1198903
119899
+ sdot sdot sdot + 119874 (1198907
119899)
(27)
4 International Journal of Analysis
Table 2 Comparison of various iterative methods
New method (CM) (SHM) (YIM) (SM)1198911 1199090= 0
1003816100381610038161003816119909119899minus 1205721003816100381610038161003816
023119890 minus 291 046119890 minus 165 036119890 minus 267 045119890 minus 155 055119890 minus 80
1003816100381610038161003816119891 (119909119899)1003816100381610038161003816
088119890 minus 291 018119890 minus 164 014119890 minus 266 017119890 minus 154 021119890 minus 79
COC 6000000 6000006 6000000 5999992 2000000
TNFE 12 15 12 12 12
1198912 1199090= minus12
1003816100381610038161003816119909119899minus 1205721003816100381610038161003816
036119890 minus 345 034119890 minus 180 012119890 minus 157 067119890 minus 190 032119890 minus 40
1003816100381610038161003816119891 (119909119899)1003816100381610038161003816
073119890 minus 344 069119890 minus 179 024119890 minus 156 014119890 minus 188 065119890 minus 39
COC 6000000 5999994 6000004 6000000 2000000
TNFE 12 15 12 12 12
1198913 1199090= 01
1003816100381610038161003816119909119899minus 1205721003816100381610038161003816
043119890 minus 145 041119890 minus 158 052119890 minus 121 013119890 minus 88 071119890 minus 32
1003816100381610038161003816119891 (119909119899)1003816100381610038161003816
043119890 minus 145 041119890 minus 158 052119890 minus 121 013119890 minus 88 071119890 minus 32
COC 5999990 5999996 5999862 5999567 2000000
TNFE 12 15 12 12 12
1198914 1199090= 18
1003816100381610038161003816119909119899minus 1205721003816100381610038161003816
013119890 minus 188 012119890 minus 280 099119890 minus 171 043119890 minus 93 038119890 minus 48
1003816100381610038161003816119891 (119909119899)1003816100381610038161003816
035119890 minus 188 035119890 minus 280 027119890 minus 170 012119890 minus 92 011119890 minus 47
COC 6000000 6000000 5999998 5999951 2000000
TNFE 12 15 12 12 12
1198915 1199090= 1
1003816100381610038161003816119909119899minus 1205721003816100381610038161003816
021119890 minus 265 014119890 minus 204 031119890 minus 257 014119890 minus 133 054119890 minus 88
1003816100381610038161003816119891 (119909119899)1003816100381610038161003816
036119890 minus 265 024119890 minus 204 0515119890 minus 257 023119890 minus 133 091119890 minus 88
COC 6000000 6000000 6000000 6000024 2000000
TNFE 12 15 12 12 12
1198916 1199090= minus075
1003816100381610038161003816119909119899minus 1205721003816100381610038161003816
048119890 minus 109 011119890 minus 104 087119890 minus 74 018119890 minus 22 011119890 minus 12
1003816100381610038161003816119891 (119909119899)1003816100381610038161003816
015119890 minus 108 034119890 minus 104 026119890 minus 73 054119890 minus 22 034119890 minus 12
COC 5999904 5999991 5996977 5920939 1999927
TNFE 12 15 12 12 12
1198917 1199090= 32
1003816100381610038161003816119909119899minus 1205721003816100381610038161003816
014119890 minus 265 041119890 minus 199 061119890 minus 316 012119890 minus 168 021119890 minus 98
1003816100381610038161003816119891 (119909119899)1003816100381610038161003816
084119890 minus 266 024119890 minus 199 037119890 minus 316 072119890 minus 169 013119890 minus 98
COC 6000000 6000000 6000000 6000001 2000000
TNFE 12 15 12 12 12
Using (21) (24) (25) and (27) in the second step of (20) givesus
119911119899minus 120572 =
(1 + 1198881) 1198882
2
1198882
1
1198903
119899
+
minus (3 + 1198881(3 + 1198881)) 1198883
2+ 1198881(1 + 1198881) (3 + 119888
1) 11988821198883
1198883
1
1198904
119899
+ sdot sdot sdot + 119874 (1198907
119899)
(28)
This shows that at the end of the second step the methodis of third order convergence Therefore the third step is
introduced to achieve the higher orderThe Taylor expansionabout the simple root for 119891(119911
119899) is given as follows
119891 (119911119899) =
(1 + 1198881) 1198882
2
1198881
1198903
119899
+
minus (3 + 1198881(3 + 1198881)) 1198883
2+ 1198881(1 + 1198881) (3 + 119888
1) 11988821198883
1198882
1
1198904
119899
+ sdot sdot sdot + 119874 (1198907
119899)
(29)Using (27) and (29) in
119891 [119910119899 119911119899] =
119891 (119910119899) minus 119891 (119911
119899)
119910119899minus 119911119899
(30)
International Journal of Analysis 5
we get
119891 [119910119899 119911119899] = 1198881+
(1 + 1198881) 1198882
2
1198881
1198902
119899+ sdot sdot sdot + 119874 (119890
7
119899) (31)
Similarly
119891 [119909119899 119910119899] = 1198881+ 1198882119890119899
+ ((1 +
1
1198881
) 1198882
2+ 1198883) 1198902
119899+ sdot sdot sdot + 119874 (119890
7
119899)
119891 [119909119899 119911119899] = 1198881+ 1198882119890119899+ 11988831198902
119899+ sdot sdot sdot + 119874 (119890
7
119899)
(32)
Combining the above terms we have
1198911015840(119911119899) = 119891 [119910
119899 119911119899] minus 119891 [119909
119899 119910119899] + 119891 [119909
119899 119911119899]
= 1198881+
(1 + 1198881) 1198882(21198882
2minus 11988811198883)
1198882
1
1198903
119899+ sdot sdot sdot + 119874 (119890
7
119899)
(33)
Now dividing (33) by (29) and using the last step of (20) wehave
119890119899+1=
(1 + 1198881)2
1198883
2(1198882
2minus 11988811198883)
1198885
1
1198906
119899+ 119874 (119890
7
119899) (34)
This proves that our first proposed method defined by (20) isa sixth-order derivative-free algorithm and satisfies the aboveerror equation This completes the proof
Nowwe discuss the efficiency index of the newmethod byusing Definition 3 as 1199011119902 where 119901 is the order of the methodand 119902 is the number of function evaluations per iterationrequired of the method It is easy to know that the numberof function evaluations per iteration required by the methoddefined in algorithm 1 is four So the efficiency index is 614 =15651
4 Numerical Results
In this section we test the effectiveness of our new methodWe have used second-order method of Steffensen (SM) [2]sixth-order method of Cordero et al (CM) [3] six-ordermethod of Soleymani and Hosseinabadi (SHM) [4] andsix-order method of Yasmin and Iftikhar (YIM) [5] forcomparison with our method to find the simple root ofnonlinear equations The test functions of 119891(119909) are listed inTable 1
Numerical computations reported here have been carriedout in a 119872119886119905ℎ119890119898119886119905119894119888119886 80 environment Table 2 shows thedifference of the root 120572 and the approximation 119909
119899to 120572
where 120572 is the exact root computed with 800 significant digits(Digits = 800) The absolute values of the function (|119891(119909
119899)|)
the difference between the approximated root 119909119899and the
exact root 120572 the number of function evaluations (TNFE)and the computational order of convergence (COC) are alsoshown in Table 2 Here COC is defined in Definition 4
5 Conclusions
We have obtained a new Steffensen-type iterative method forsolving nonlinear equations The convergence order of thismethod is six and consists of four evaluations of the functionper iteration so it has an efficiency index equal to 614 =15651 Numerical examples also show that the numericalresults of our new method in equal iterations improve theresults of other existing methods
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The author is grateful to the referees for their comments andsuggestions that helped to improve the paper
References
[1] W Bi H Ren and Q Wu ldquoNew family of seventh-ordermethods for nonlinear equationsrdquo Applied Mathematics andComputation vol 203 no 1 pp 408ndash412 2008
[2] D Kincaid and W Cheney Numerical Analysis BrooksColePacific Grove Calif USA 2nd edition 1996
[3] A Cordero J L Hueso E Martınez and J R TorregrosaldquoSteffensen type methods for solving nonlinear equationsrdquoJournal of Computational andAppliedMathematics vol 236 no12 pp 3058ndash3064 2012
[4] F Soleymani and V Hosseinabadi ldquoNew third- and sixth-orderderivativefree techniques for nonlinear equationsrdquo Journal ofMathematics Research vol 3 no 2 pp 107ndash112 2011
[5] N Yasmin and S Iftikhar ldquoSome new Steffensen like three-stepmethods for solving nonlinear equationsrdquo International Journalof Pure and Applied Mathematics vol 82 pp 557ndash572 2013
[6] W Gautschi Numerical Analysis Birkhauser Boston MassUSA 1997
[7] S Weerakoon and T G I Fernando ldquoA variant of Newtonrsquosmethod with accelerated third-order convergencerdquo AppliedMathematics Letters vol 13 no 8 pp 87ndash93 2000
[8] M Basto V Semiao and F L Calheiros ldquoA new iterativemethod to compute nonlinear equationsrdquo Applied Mathematicsand Computation vol 173 no 1 pp 468ndash483 2006
[9] J Kou Y Li and XWang ldquoModified Halleyrsquos method free fromsecond derivativerdquo Applied Mathematics and Computation vol183 no 1 pp 704ndash708 2006
[10] M Dehghan and M Hajarian ldquoSome derivative free quadraticand cubic convergence iterative formulas for solving nonlinearequationsrdquoComputational andAppliedMathematics vol 29 no1 pp 19ndash30 2010
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
International Journal of Analysis 3
By combining the method (15) with Newtonrsquos method weobtain a new three-step iterative algorithm without memoryas follows
119910119899= 119909119899minus
119891 (119909119899)
1198911015840(119909119899)
119911119899= 119910119899+
119891 (119909119899) (1198911015840(119910119899) minus 1198911015840(119909119899))
21198911015840(119909119899) 1198911015840(119910119899)
119909119899+1= 119911119899minus
119891 (119911119899)
1198911015840(119911119899)
(16)
which is required of three evaluations of the first derivativeof the function To remedy these derivatives firstly weapproximate 1198911015840(119909
119899) by the divided difference of order one
1198911015840(119909119899) asymp 119891 [119908
119899 119909119899] =
119891 (119908119899) minus 119891 (119909
119899)
119908119899minus 119909119899
(17)
where 119908119899= 119909119899+ 119891(119909
119899)
Secondly approximate 1198911015840(119911119899) by the linear combination
of divided differences
1198911015840(119911119899) asymp 119891 [119910
119899 119911119899] minus 119891 [119909
119899 119910119899] + 119891 [119909
119899 119911119899] (18)
and for 1198911015840(119910119899) we use the following approximation [10]
1198911015840(119910119899) asymp
1198911015840(119909119899) (119891 (119909
119899) minus 119891 (119910
119899))
119891 (119909119899) + 119891 (119910
119899)
(19)
Thus our new three-step derivative-free iterative algorithmwithout memory is given as
119910119899= 119909119899minus
119891 (119909119899)
119891 [119909119899 119908119899]
119911119899= 119910119899minus
119891 (119909119899) 119891 (119910
119899)
119891 [119909119899 119908119899] (119891 (119909
119899) minus 119891 (119910
119899))
119909119899+1= 119911119899minus
119891 (119911119899)
119891 [119910119899 119911119899] minus 119891 [119909
119899 119910119899] + 119891 [119909
119899 119911119899]
(20)
where 119908119899= 119909119899+ 119891(119909
119899)
Theorem 5 demonstrates its convergence analysis
Theorem 5 Let 120572 isin 119868 be a simple root of a sufficientlydifferentiable function 119891 119868 sube R rarr R in an open interval 119868If the initial approximation 119909
0is sufficiently close to 120572 then the
derivative-freemethod defined by (20) has order of convergencesix
Proof Let 120572 be the simple root of 119891(119909) that is 119891(120572) = 01198911015840(120572) = 0 and the error equation is 119890
119899= 119909119899minus 120572
By Taylorrsquos expansion of 119891(119909119899) about 119909 = 120572 and putting
119891(120572) = 0 we have
119891 (119909119899) = 1198881119890119899+ 11988821198902
119899+ 11988831198903
119899+ 11988841198904
119899+ 11988851198905
119899+ 11988861198906
119899+ 119874 (119890
7
119899)
(21)
Table 1 The examples considered in this study
Test functions Zeros1198911(119909) = 119909
2minus 119890119909minus 3119909 + 2 120572 asymp 025753028543986079
1198912(119909) = 119909119890
1199092
minus sin2(119909) + 3 cos(119909) + 5 120572 asymp minus1207647827130919
1198913(119909) = sin(119909)119890119909 + Ln(1199092 + 1) 120572 asymp 0
1198914(119909) = 10119909119890
minus1199092
minus 1 120572 asymp 16796306104284499
1198915(119909) = cos(119909) minus 119909 120572 asymp 073908513321516064
1198916(119909) = 119890
minus1199092+119909+2minus 1 120572 asymp minus1000000000000000
1198917(119909) = Ln(1199092 + 119909 + 2) minus 119909 + 1 120572 asymp 41525907367571583
where
119888119896=
119891119896(120572)
119896
119896 = 1 2 3 (22)
Expanding the Taylor series of119891(119908119899) about the solution 120572 we
have
119891 (119908119899) =
infin
sum
119894=1
119888119894(119890119899+ 119891 (119909
119899))119894
(23)
substituting 119891(119909119899) given by (21) gives us
119891 (119908119899) = 1198881(1 + 1198881) 119890119899+ (1 + 119888
1(3 + 1198881)) 11988821198902
119899
+ (2 (1 + 1198881) 1198882
2+ 11988811198883+ (1 + 119888
1)3
1198883) 1198903
119899
+ sdot sdot sdot + 119874 (1198907
119899)
(24)
Note that to save the space we onlywrite someof the obtainedterms for the error equations and show the others by
Substituting (21) and (24) in the first step of (20) gives us
119910119899minus 120572 = 119909
119899minus 120572 minus
119891 (119909119899)
119891 [119908119899 119909119899]
= (1 +
1
1198881
) 11988821198902
119899
+
minus (2 + 1198881(2 + 1198881)) 1198882
2+ 1198881(1 + 1198881) (2 + 119888
1) 1198883
1198882
1
1198903
119899
+ sdot sdot sdot + 119874 (1198907
119899)
(25)
Using the Taylor expansion of 119891(119910119899) about the solution 120572 we
have
119891 (119910119899) =
infin
sum
119894=1
119888119894(119910119899minus 120572)119894
(26)
substituting (25) into the preceding equation we have
119891 (119910119899) = (1 + 119888
1) 11988821198902
119899
+ ((minus2 minus
2
1198881
minus 1198881) 1198882
2+ (1 + 119888
1) (2 + 119888
1) 1198883) 1198903
119899
+ sdot sdot sdot + 119874 (1198907
119899)
(27)
4 International Journal of Analysis
Table 2 Comparison of various iterative methods
New method (CM) (SHM) (YIM) (SM)1198911 1199090= 0
1003816100381610038161003816119909119899minus 1205721003816100381610038161003816
023119890 minus 291 046119890 minus 165 036119890 minus 267 045119890 minus 155 055119890 minus 80
1003816100381610038161003816119891 (119909119899)1003816100381610038161003816
088119890 minus 291 018119890 minus 164 014119890 minus 266 017119890 minus 154 021119890 minus 79
COC 6000000 6000006 6000000 5999992 2000000
TNFE 12 15 12 12 12
1198912 1199090= minus12
1003816100381610038161003816119909119899minus 1205721003816100381610038161003816
036119890 minus 345 034119890 minus 180 012119890 minus 157 067119890 minus 190 032119890 minus 40
1003816100381610038161003816119891 (119909119899)1003816100381610038161003816
073119890 minus 344 069119890 minus 179 024119890 minus 156 014119890 minus 188 065119890 minus 39
COC 6000000 5999994 6000004 6000000 2000000
TNFE 12 15 12 12 12
1198913 1199090= 01
1003816100381610038161003816119909119899minus 1205721003816100381610038161003816
043119890 minus 145 041119890 minus 158 052119890 minus 121 013119890 minus 88 071119890 minus 32
1003816100381610038161003816119891 (119909119899)1003816100381610038161003816
043119890 minus 145 041119890 minus 158 052119890 minus 121 013119890 minus 88 071119890 minus 32
COC 5999990 5999996 5999862 5999567 2000000
TNFE 12 15 12 12 12
1198914 1199090= 18
1003816100381610038161003816119909119899minus 1205721003816100381610038161003816
013119890 minus 188 012119890 minus 280 099119890 minus 171 043119890 minus 93 038119890 minus 48
1003816100381610038161003816119891 (119909119899)1003816100381610038161003816
035119890 minus 188 035119890 minus 280 027119890 minus 170 012119890 minus 92 011119890 minus 47
COC 6000000 6000000 5999998 5999951 2000000
TNFE 12 15 12 12 12
1198915 1199090= 1
1003816100381610038161003816119909119899minus 1205721003816100381610038161003816
021119890 minus 265 014119890 minus 204 031119890 minus 257 014119890 minus 133 054119890 minus 88
1003816100381610038161003816119891 (119909119899)1003816100381610038161003816
036119890 minus 265 024119890 minus 204 0515119890 minus 257 023119890 minus 133 091119890 minus 88
COC 6000000 6000000 6000000 6000024 2000000
TNFE 12 15 12 12 12
1198916 1199090= minus075
1003816100381610038161003816119909119899minus 1205721003816100381610038161003816
048119890 minus 109 011119890 minus 104 087119890 minus 74 018119890 minus 22 011119890 minus 12
1003816100381610038161003816119891 (119909119899)1003816100381610038161003816
015119890 minus 108 034119890 minus 104 026119890 minus 73 054119890 minus 22 034119890 minus 12
COC 5999904 5999991 5996977 5920939 1999927
TNFE 12 15 12 12 12
1198917 1199090= 32
1003816100381610038161003816119909119899minus 1205721003816100381610038161003816
014119890 minus 265 041119890 minus 199 061119890 minus 316 012119890 minus 168 021119890 minus 98
1003816100381610038161003816119891 (119909119899)1003816100381610038161003816
084119890 minus 266 024119890 minus 199 037119890 minus 316 072119890 minus 169 013119890 minus 98
COC 6000000 6000000 6000000 6000001 2000000
TNFE 12 15 12 12 12
Using (21) (24) (25) and (27) in the second step of (20) givesus
119911119899minus 120572 =
(1 + 1198881) 1198882
2
1198882
1
1198903
119899
+
minus (3 + 1198881(3 + 1198881)) 1198883
2+ 1198881(1 + 1198881) (3 + 119888
1) 11988821198883
1198883
1
1198904
119899
+ sdot sdot sdot + 119874 (1198907
119899)
(28)
This shows that at the end of the second step the methodis of third order convergence Therefore the third step is
introduced to achieve the higher orderThe Taylor expansionabout the simple root for 119891(119911
119899) is given as follows
119891 (119911119899) =
(1 + 1198881) 1198882
2
1198881
1198903
119899
+
minus (3 + 1198881(3 + 1198881)) 1198883
2+ 1198881(1 + 1198881) (3 + 119888
1) 11988821198883
1198882
1
1198904
119899
+ sdot sdot sdot + 119874 (1198907
119899)
(29)Using (27) and (29) in
119891 [119910119899 119911119899] =
119891 (119910119899) minus 119891 (119911
119899)
119910119899minus 119911119899
(30)
International Journal of Analysis 5
we get
119891 [119910119899 119911119899] = 1198881+
(1 + 1198881) 1198882
2
1198881
1198902
119899+ sdot sdot sdot + 119874 (119890
7
119899) (31)
Similarly
119891 [119909119899 119910119899] = 1198881+ 1198882119890119899
+ ((1 +
1
1198881
) 1198882
2+ 1198883) 1198902
119899+ sdot sdot sdot + 119874 (119890
7
119899)
119891 [119909119899 119911119899] = 1198881+ 1198882119890119899+ 11988831198902
119899+ sdot sdot sdot + 119874 (119890
7
119899)
(32)
Combining the above terms we have
1198911015840(119911119899) = 119891 [119910
119899 119911119899] minus 119891 [119909
119899 119910119899] + 119891 [119909
119899 119911119899]
= 1198881+
(1 + 1198881) 1198882(21198882
2minus 11988811198883)
1198882
1
1198903
119899+ sdot sdot sdot + 119874 (119890
7
119899)
(33)
Now dividing (33) by (29) and using the last step of (20) wehave
119890119899+1=
(1 + 1198881)2
1198883
2(1198882
2minus 11988811198883)
1198885
1
1198906
119899+ 119874 (119890
7
119899) (34)
This proves that our first proposed method defined by (20) isa sixth-order derivative-free algorithm and satisfies the aboveerror equation This completes the proof
Nowwe discuss the efficiency index of the newmethod byusing Definition 3 as 1199011119902 where 119901 is the order of the methodand 119902 is the number of function evaluations per iterationrequired of the method It is easy to know that the numberof function evaluations per iteration required by the methoddefined in algorithm 1 is four So the efficiency index is 614 =15651
4 Numerical Results
In this section we test the effectiveness of our new methodWe have used second-order method of Steffensen (SM) [2]sixth-order method of Cordero et al (CM) [3] six-ordermethod of Soleymani and Hosseinabadi (SHM) [4] andsix-order method of Yasmin and Iftikhar (YIM) [5] forcomparison with our method to find the simple root ofnonlinear equations The test functions of 119891(119909) are listed inTable 1
Numerical computations reported here have been carriedout in a 119872119886119905ℎ119890119898119886119905119894119888119886 80 environment Table 2 shows thedifference of the root 120572 and the approximation 119909
119899to 120572
where 120572 is the exact root computed with 800 significant digits(Digits = 800) The absolute values of the function (|119891(119909
119899)|)
the difference between the approximated root 119909119899and the
exact root 120572 the number of function evaluations (TNFE)and the computational order of convergence (COC) are alsoshown in Table 2 Here COC is defined in Definition 4
5 Conclusions
We have obtained a new Steffensen-type iterative method forsolving nonlinear equations The convergence order of thismethod is six and consists of four evaluations of the functionper iteration so it has an efficiency index equal to 614 =15651 Numerical examples also show that the numericalresults of our new method in equal iterations improve theresults of other existing methods
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The author is grateful to the referees for their comments andsuggestions that helped to improve the paper
References
[1] W Bi H Ren and Q Wu ldquoNew family of seventh-ordermethods for nonlinear equationsrdquo Applied Mathematics andComputation vol 203 no 1 pp 408ndash412 2008
[2] D Kincaid and W Cheney Numerical Analysis BrooksColePacific Grove Calif USA 2nd edition 1996
[3] A Cordero J L Hueso E Martınez and J R TorregrosaldquoSteffensen type methods for solving nonlinear equationsrdquoJournal of Computational andAppliedMathematics vol 236 no12 pp 3058ndash3064 2012
[4] F Soleymani and V Hosseinabadi ldquoNew third- and sixth-orderderivativefree techniques for nonlinear equationsrdquo Journal ofMathematics Research vol 3 no 2 pp 107ndash112 2011
[5] N Yasmin and S Iftikhar ldquoSome new Steffensen like three-stepmethods for solving nonlinear equationsrdquo International Journalof Pure and Applied Mathematics vol 82 pp 557ndash572 2013
[6] W Gautschi Numerical Analysis Birkhauser Boston MassUSA 1997
[7] S Weerakoon and T G I Fernando ldquoA variant of Newtonrsquosmethod with accelerated third-order convergencerdquo AppliedMathematics Letters vol 13 no 8 pp 87ndash93 2000
[8] M Basto V Semiao and F L Calheiros ldquoA new iterativemethod to compute nonlinear equationsrdquo Applied Mathematicsand Computation vol 173 no 1 pp 468ndash483 2006
[9] J Kou Y Li and XWang ldquoModified Halleyrsquos method free fromsecond derivativerdquo Applied Mathematics and Computation vol183 no 1 pp 704ndash708 2006
[10] M Dehghan and M Hajarian ldquoSome derivative free quadraticand cubic convergence iterative formulas for solving nonlinearequationsrdquoComputational andAppliedMathematics vol 29 no1 pp 19ndash30 2010
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
4 International Journal of Analysis
Table 2 Comparison of various iterative methods
New method (CM) (SHM) (YIM) (SM)1198911 1199090= 0
1003816100381610038161003816119909119899minus 1205721003816100381610038161003816
023119890 minus 291 046119890 minus 165 036119890 minus 267 045119890 minus 155 055119890 minus 80
1003816100381610038161003816119891 (119909119899)1003816100381610038161003816
088119890 minus 291 018119890 minus 164 014119890 minus 266 017119890 minus 154 021119890 minus 79
COC 6000000 6000006 6000000 5999992 2000000
TNFE 12 15 12 12 12
1198912 1199090= minus12
1003816100381610038161003816119909119899minus 1205721003816100381610038161003816
036119890 minus 345 034119890 minus 180 012119890 minus 157 067119890 minus 190 032119890 minus 40
1003816100381610038161003816119891 (119909119899)1003816100381610038161003816
073119890 minus 344 069119890 minus 179 024119890 minus 156 014119890 minus 188 065119890 minus 39
COC 6000000 5999994 6000004 6000000 2000000
TNFE 12 15 12 12 12
1198913 1199090= 01
1003816100381610038161003816119909119899minus 1205721003816100381610038161003816
043119890 minus 145 041119890 minus 158 052119890 minus 121 013119890 minus 88 071119890 minus 32
1003816100381610038161003816119891 (119909119899)1003816100381610038161003816
043119890 minus 145 041119890 minus 158 052119890 minus 121 013119890 minus 88 071119890 minus 32
COC 5999990 5999996 5999862 5999567 2000000
TNFE 12 15 12 12 12
1198914 1199090= 18
1003816100381610038161003816119909119899minus 1205721003816100381610038161003816
013119890 minus 188 012119890 minus 280 099119890 minus 171 043119890 minus 93 038119890 minus 48
1003816100381610038161003816119891 (119909119899)1003816100381610038161003816
035119890 minus 188 035119890 minus 280 027119890 minus 170 012119890 minus 92 011119890 minus 47
COC 6000000 6000000 5999998 5999951 2000000
TNFE 12 15 12 12 12
1198915 1199090= 1
1003816100381610038161003816119909119899minus 1205721003816100381610038161003816
021119890 minus 265 014119890 minus 204 031119890 minus 257 014119890 minus 133 054119890 minus 88
1003816100381610038161003816119891 (119909119899)1003816100381610038161003816
036119890 minus 265 024119890 minus 204 0515119890 minus 257 023119890 minus 133 091119890 minus 88
COC 6000000 6000000 6000000 6000024 2000000
TNFE 12 15 12 12 12
1198916 1199090= minus075
1003816100381610038161003816119909119899minus 1205721003816100381610038161003816
048119890 minus 109 011119890 minus 104 087119890 minus 74 018119890 minus 22 011119890 minus 12
1003816100381610038161003816119891 (119909119899)1003816100381610038161003816
015119890 minus 108 034119890 minus 104 026119890 minus 73 054119890 minus 22 034119890 minus 12
COC 5999904 5999991 5996977 5920939 1999927
TNFE 12 15 12 12 12
1198917 1199090= 32
1003816100381610038161003816119909119899minus 1205721003816100381610038161003816
014119890 minus 265 041119890 minus 199 061119890 minus 316 012119890 minus 168 021119890 minus 98
1003816100381610038161003816119891 (119909119899)1003816100381610038161003816
084119890 minus 266 024119890 minus 199 037119890 minus 316 072119890 minus 169 013119890 minus 98
COC 6000000 6000000 6000000 6000001 2000000
TNFE 12 15 12 12 12
Using (21) (24) (25) and (27) in the second step of (20) givesus
119911119899minus 120572 =
(1 + 1198881) 1198882
2
1198882
1
1198903
119899
+
minus (3 + 1198881(3 + 1198881)) 1198883
2+ 1198881(1 + 1198881) (3 + 119888
1) 11988821198883
1198883
1
1198904
119899
+ sdot sdot sdot + 119874 (1198907
119899)
(28)
This shows that at the end of the second step the methodis of third order convergence Therefore the third step is
introduced to achieve the higher orderThe Taylor expansionabout the simple root for 119891(119911
119899) is given as follows
119891 (119911119899) =
(1 + 1198881) 1198882
2
1198881
1198903
119899
+
minus (3 + 1198881(3 + 1198881)) 1198883
2+ 1198881(1 + 1198881) (3 + 119888
1) 11988821198883
1198882
1
1198904
119899
+ sdot sdot sdot + 119874 (1198907
119899)
(29)Using (27) and (29) in
119891 [119910119899 119911119899] =
119891 (119910119899) minus 119891 (119911
119899)
119910119899minus 119911119899
(30)
International Journal of Analysis 5
we get
119891 [119910119899 119911119899] = 1198881+
(1 + 1198881) 1198882
2
1198881
1198902
119899+ sdot sdot sdot + 119874 (119890
7
119899) (31)
Similarly
119891 [119909119899 119910119899] = 1198881+ 1198882119890119899
+ ((1 +
1
1198881
) 1198882
2+ 1198883) 1198902
119899+ sdot sdot sdot + 119874 (119890
7
119899)
119891 [119909119899 119911119899] = 1198881+ 1198882119890119899+ 11988831198902
119899+ sdot sdot sdot + 119874 (119890
7
119899)
(32)
Combining the above terms we have
1198911015840(119911119899) = 119891 [119910
119899 119911119899] minus 119891 [119909
119899 119910119899] + 119891 [119909
119899 119911119899]
= 1198881+
(1 + 1198881) 1198882(21198882
2minus 11988811198883)
1198882
1
1198903
119899+ sdot sdot sdot + 119874 (119890
7
119899)
(33)
Now dividing (33) by (29) and using the last step of (20) wehave
119890119899+1=
(1 + 1198881)2
1198883
2(1198882
2minus 11988811198883)
1198885
1
1198906
119899+ 119874 (119890
7
119899) (34)
This proves that our first proposed method defined by (20) isa sixth-order derivative-free algorithm and satisfies the aboveerror equation This completes the proof
Nowwe discuss the efficiency index of the newmethod byusing Definition 3 as 1199011119902 where 119901 is the order of the methodand 119902 is the number of function evaluations per iterationrequired of the method It is easy to know that the numberof function evaluations per iteration required by the methoddefined in algorithm 1 is four So the efficiency index is 614 =15651
4 Numerical Results
In this section we test the effectiveness of our new methodWe have used second-order method of Steffensen (SM) [2]sixth-order method of Cordero et al (CM) [3] six-ordermethod of Soleymani and Hosseinabadi (SHM) [4] andsix-order method of Yasmin and Iftikhar (YIM) [5] forcomparison with our method to find the simple root ofnonlinear equations The test functions of 119891(119909) are listed inTable 1
Numerical computations reported here have been carriedout in a 119872119886119905ℎ119890119898119886119905119894119888119886 80 environment Table 2 shows thedifference of the root 120572 and the approximation 119909
119899to 120572
where 120572 is the exact root computed with 800 significant digits(Digits = 800) The absolute values of the function (|119891(119909
119899)|)
the difference between the approximated root 119909119899and the
exact root 120572 the number of function evaluations (TNFE)and the computational order of convergence (COC) are alsoshown in Table 2 Here COC is defined in Definition 4
5 Conclusions
We have obtained a new Steffensen-type iterative method forsolving nonlinear equations The convergence order of thismethod is six and consists of four evaluations of the functionper iteration so it has an efficiency index equal to 614 =15651 Numerical examples also show that the numericalresults of our new method in equal iterations improve theresults of other existing methods
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The author is grateful to the referees for their comments andsuggestions that helped to improve the paper
References
[1] W Bi H Ren and Q Wu ldquoNew family of seventh-ordermethods for nonlinear equationsrdquo Applied Mathematics andComputation vol 203 no 1 pp 408ndash412 2008
[2] D Kincaid and W Cheney Numerical Analysis BrooksColePacific Grove Calif USA 2nd edition 1996
[3] A Cordero J L Hueso E Martınez and J R TorregrosaldquoSteffensen type methods for solving nonlinear equationsrdquoJournal of Computational andAppliedMathematics vol 236 no12 pp 3058ndash3064 2012
[4] F Soleymani and V Hosseinabadi ldquoNew third- and sixth-orderderivativefree techniques for nonlinear equationsrdquo Journal ofMathematics Research vol 3 no 2 pp 107ndash112 2011
[5] N Yasmin and S Iftikhar ldquoSome new Steffensen like three-stepmethods for solving nonlinear equationsrdquo International Journalof Pure and Applied Mathematics vol 82 pp 557ndash572 2013
[6] W Gautschi Numerical Analysis Birkhauser Boston MassUSA 1997
[7] S Weerakoon and T G I Fernando ldquoA variant of Newtonrsquosmethod with accelerated third-order convergencerdquo AppliedMathematics Letters vol 13 no 8 pp 87ndash93 2000
[8] M Basto V Semiao and F L Calheiros ldquoA new iterativemethod to compute nonlinear equationsrdquo Applied Mathematicsand Computation vol 173 no 1 pp 468ndash483 2006
[9] J Kou Y Li and XWang ldquoModified Halleyrsquos method free fromsecond derivativerdquo Applied Mathematics and Computation vol183 no 1 pp 704ndash708 2006
[10] M Dehghan and M Hajarian ldquoSome derivative free quadraticand cubic convergence iterative formulas for solving nonlinearequationsrdquoComputational andAppliedMathematics vol 29 no1 pp 19ndash30 2010
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
International Journal of Analysis 5
we get
119891 [119910119899 119911119899] = 1198881+
(1 + 1198881) 1198882
2
1198881
1198902
119899+ sdot sdot sdot + 119874 (119890
7
119899) (31)
Similarly
119891 [119909119899 119910119899] = 1198881+ 1198882119890119899
+ ((1 +
1
1198881
) 1198882
2+ 1198883) 1198902
119899+ sdot sdot sdot + 119874 (119890
7
119899)
119891 [119909119899 119911119899] = 1198881+ 1198882119890119899+ 11988831198902
119899+ sdot sdot sdot + 119874 (119890
7
119899)
(32)
Combining the above terms we have
1198911015840(119911119899) = 119891 [119910
119899 119911119899] minus 119891 [119909
119899 119910119899] + 119891 [119909
119899 119911119899]
= 1198881+
(1 + 1198881) 1198882(21198882
2minus 11988811198883)
1198882
1
1198903
119899+ sdot sdot sdot + 119874 (119890
7
119899)
(33)
Now dividing (33) by (29) and using the last step of (20) wehave
119890119899+1=
(1 + 1198881)2
1198883
2(1198882
2minus 11988811198883)
1198885
1
1198906
119899+ 119874 (119890
7
119899) (34)
This proves that our first proposed method defined by (20) isa sixth-order derivative-free algorithm and satisfies the aboveerror equation This completes the proof
Nowwe discuss the efficiency index of the newmethod byusing Definition 3 as 1199011119902 where 119901 is the order of the methodand 119902 is the number of function evaluations per iterationrequired of the method It is easy to know that the numberof function evaluations per iteration required by the methoddefined in algorithm 1 is four So the efficiency index is 614 =15651
4 Numerical Results
In this section we test the effectiveness of our new methodWe have used second-order method of Steffensen (SM) [2]sixth-order method of Cordero et al (CM) [3] six-ordermethod of Soleymani and Hosseinabadi (SHM) [4] andsix-order method of Yasmin and Iftikhar (YIM) [5] forcomparison with our method to find the simple root ofnonlinear equations The test functions of 119891(119909) are listed inTable 1
Numerical computations reported here have been carriedout in a 119872119886119905ℎ119890119898119886119905119894119888119886 80 environment Table 2 shows thedifference of the root 120572 and the approximation 119909
119899to 120572
where 120572 is the exact root computed with 800 significant digits(Digits = 800) The absolute values of the function (|119891(119909
119899)|)
the difference between the approximated root 119909119899and the
exact root 120572 the number of function evaluations (TNFE)and the computational order of convergence (COC) are alsoshown in Table 2 Here COC is defined in Definition 4
5 Conclusions
We have obtained a new Steffensen-type iterative method forsolving nonlinear equations The convergence order of thismethod is six and consists of four evaluations of the functionper iteration so it has an efficiency index equal to 614 =15651 Numerical examples also show that the numericalresults of our new method in equal iterations improve theresults of other existing methods
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgment
The author is grateful to the referees for their comments andsuggestions that helped to improve the paper
References
[1] W Bi H Ren and Q Wu ldquoNew family of seventh-ordermethods for nonlinear equationsrdquo Applied Mathematics andComputation vol 203 no 1 pp 408ndash412 2008
[2] D Kincaid and W Cheney Numerical Analysis BrooksColePacific Grove Calif USA 2nd edition 1996
[3] A Cordero J L Hueso E Martınez and J R TorregrosaldquoSteffensen type methods for solving nonlinear equationsrdquoJournal of Computational andAppliedMathematics vol 236 no12 pp 3058ndash3064 2012
[4] F Soleymani and V Hosseinabadi ldquoNew third- and sixth-orderderivativefree techniques for nonlinear equationsrdquo Journal ofMathematics Research vol 3 no 2 pp 107ndash112 2011
[5] N Yasmin and S Iftikhar ldquoSome new Steffensen like three-stepmethods for solving nonlinear equationsrdquo International Journalof Pure and Applied Mathematics vol 82 pp 557ndash572 2013
[6] W Gautschi Numerical Analysis Birkhauser Boston MassUSA 1997
[7] S Weerakoon and T G I Fernando ldquoA variant of Newtonrsquosmethod with accelerated third-order convergencerdquo AppliedMathematics Letters vol 13 no 8 pp 87ndash93 2000
[8] M Basto V Semiao and F L Calheiros ldquoA new iterativemethod to compute nonlinear equationsrdquo Applied Mathematicsand Computation vol 173 no 1 pp 468ndash483 2006
[9] J Kou Y Li and XWang ldquoModified Halleyrsquos method free fromsecond derivativerdquo Applied Mathematics and Computation vol183 no 1 pp 704ndash708 2006
[10] M Dehghan and M Hajarian ldquoSome derivative free quadraticand cubic convergence iterative formulas for solving nonlinearequationsrdquoComputational andAppliedMathematics vol 29 no1 pp 19ndash30 2010
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
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