Research Article On a Derivative-Free Variant of King s...
6
Research Article On a Derivative-Free Variant of King’s Family with Memory M. Sharifi, 1 S. Karimi Vanani, 1 F. Khaksar Haghani, 1 M. Arab, 1 and S. Shateyi 2 1 Department of Mathematics, Islamic Azad University, Shahrekord Branch, Shahrekord, Iran 2 Department of Mathematics and Applied Mathematics, University of Venda, ohoyandou 0950, South Africa Correspondence should be addressed to S. Shateyi; [email protected] Received 16 July 2014; Revised 29 August 2014; Accepted 1 September 2014 Academic Editor: Juan R. Torregrosa Copyright © 2015 M. Sharifi et al. 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. e aim of this paper is to construct a method with memory according to King’s family of methods without memory for nonlinear equations. It is proved that the proposed method possesses higher R-order of convergence using the same number of functional evaluations as King’s family. Numerical experiments are given to illustrate the performance of the constructed scheme. 1. Introduction Many problems arising in diverse disciplines of mathematical sciences can be described by a nonlinear equation of the following form (see, e.g., [1]): () = 0, (1) where : ⊆ R → R is a sufficiently differentiable function in a neighborhood of a simple zero of (1). If we are interested in approximating the root , we can do it by means of an iterative fixed-point method in the following form: +1 = ( ), ≥ 0, (2) provided that the starting point 0 is given. In this work, we are concerned with the fixed-point methods that generate sequences presumably convergent to the true solution of a given single smooth equation. ese schemes can be divided into one-point and multipoint schemes. We remark that the one-point methods can possess high order by using higher derivatives of the function, which is expensive from a computational point of view. On the other hand, the multipoint methods are allowing the user not to waste information that had already been used. is approach provides the construction of efficient iterative root-finding methods [2]. In such circumstance, special attention is devoted to multipoint methods with memory that use already computed information to considerably increase convergence rate with- out additional computational costs. is would be the focus of this paper. Traub in [2] proposed the following method with mem- ory (TM): = + ( ), =− 1 [ , −1 ] , = 0, 1, 2, . . . , +1 = − ( ) [ , ] , (3) with the order of convergence 1+ √ 2. e iterative methods with memory can improve the order of convergence of the without memory method without any additional functional calculations, and this results in a higher computational efficiency index. We remark that it is assumed that an initial approximation 0 close enough to the sought simple zero and 0 are given for iterative methods of type (3). Recently, authors in [3] designed an approach to make derivative-free families with low complexity out of optimal methods. In fact, they conjectured that every time that one applies the approximation of the derivative ( ) ≈ [ , ], with = + ( ) , on an optimal order 2, we will need ≥ for preserving the order of convergence. Hindawi Publishing Corporation e Scientific World Journal Volume 2015, Article ID 514075, 5 pages http://dx.doi.org/10.1155/2015/514075
Transcript of Research Article On a Derivative-Free Variant of King s...
Research ArticleOn a Derivative-Free Variant of Kingrsquos Family with Memory
M Sharifi1 S Karimi Vanani1 F Khaksar Haghani1 M Arab1 and S Shateyi2
1 Department of Mathematics Islamic Azad University Shahrekord Branch Shahrekord Iran2Department of Mathematics and Applied Mathematics University of Venda Thohoyandou 0950 South Africa
Correspondence should be addressed to S Shateyi stanfordshateyiunivenacza
Received 16 July 2014 Revised 29 August 2014 Accepted 1 September 2014
Academic Editor Juan R Torregrosa
Copyright copy 2015 M Sharifi et alThis is an open access article distributed under theCreativeCommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
The aim of this paper is to construct a method with memory according to Kingrsquos family of methods without memory for nonlinearequations It is proved that the proposed method possesses higher R-order of convergence using the same number of functionalevaluations as Kingrsquos family Numerical experiments are given to illustrate the performance of the constructed scheme
1 Introduction
Many problems arising in diverse disciplines of mathematicalsciences can be described by a nonlinear equation of thefollowing form (see eg [1])
119891 (119909) = 0 (1)
where 119891 119863 sube R rarr R is a sufficiently differentiablefunction in a neighborhood 119863 of a simple zero 120572 of (1) Ifwe are interested in approximating the root 120572 we can do itby means of an iterative fixed-point method in the followingform
119909119896+1 = 120595 (119909119896) 119896 ge 0 (2)
provided that the starting point 1199090 is givenIn this work we are concerned with the fixed-point
methods that generate sequences presumably convergentto the true solution of a given single smooth equationThese schemes can be divided into one-point and multipointschemes We remark that the one-point methods can possesshigh order by using higher derivatives of the function whichis expensive from a computational point of view On the otherhand the multipoint methods are allowing the user not towaste information that had already been usedThis approachprovides the construction of efficient iterative root-findingmethods [2]
In such circumstance special attention is devoted tomultipoint methods withmemory that use already computed
information to considerably increase convergence rate with-out additional computational costs This would be the focusof this paper
Traub in [2] proposed the following method with mem-ory (TM)
119908119896 = 119909119896 + 120573119896119891 (119909119896) 120573119896 = minus1
119891 [119909119896 119909119896minus1] 119896 = 0 1 2
119909119896+1 = 119909119896 minus119891 (119909119896)
119891 [119909119896 119908119896]
(3)
with the order of convergence 1 + radic2The iterative methods with memory can improve the
order of convergence of thewithoutmemorymethodwithoutany additional functional calculations and this results in ahigher computational efficiency index We remark that it isassumed that an initial approximation 1199090 close enough to thesought simple zero and 1205730 are given for iterative methods oftype (3)
Recently authors in [3] designed an approach to makederivative-free families with low complexity out of optimalmethods In fact they conjectured that every time thatone applies the approximation of the derivative 1198911015840(119909119899) asymp119891[119909119899 119908119899] with 119908119899 = 119909119899 + 120573119891(119909119899)
119897 on an optimal order 2119902we will need 119897 ge 119902 for preserving the order of convergence
Hindawi Publishing Corporatione Scientific World JournalVolume 2015 Article ID 514075 5 pageshttpdxdoiorg1011552015514075
2 The Scientific World Journal
For instance choosing the well-known optimal two-stepfamily of King (KM) [4]
119910119896 = 119909119896 minus119891 (119909119896)
1198911015840 (119909119896) 119896 = 0 1 2
119909119896+1 = 119910119896 minus119891 (119910119896)
1198911015840 (119909119896)
119891 (119909119896) + 120574119891 (119910119896)
119891 (119909119896) + (120574 minus 2) 119891 (119910119896) 120574 isin R
(4)
and the conjecture of Cordero-Torregrosa one may proposethe following method (DKM)
119910119896 = 119909119896 minus119891 (119909119896)
FD 119896 = 0 1 2
119909119896+1 = 119910119896 minus119891 (119910119896)
FD
119891 (119909119896) + 120574119891 (119910119896)
119891 (119909119896) + (120574 minus 2) 119891 (119910119896) 120574 isin R
(5)
wherein
FD =119891 (119909119896) minus 119891 (119908119896)
119909119896 minus 119908119896 119908119896 = 119909119896 + 120573119891 (119909119896)
2
120573 isin R 0
(6)
In this work we propose a two-stepmethodwithmemorypossessing a high efficiency index according to the well-known family of Kingrsquos methods (5)
Our inspiration andmotivation for constructing a higher-order method are linked in a direct manner with the fun-damental concept of numerical analysis that any numericalmethod should give as accurate as possible output results withminimal computational cost To state the matter differentlyit is necessary to pursue methods of higher computationalefficiency
For more background concerning this topic one mayrefer to [5 6]
The paper is organized as follows In Section 2 theaim of this paper is presented by contributing an iterativemethod with memory based on (5) for solving nonlinearequations The proposed scheme is an extension over (4)and has a simple structure with an increased computationalefficiency In Section 3 we compare the theoretical resultsby applying the definition of efficiency index and furthersupports are furnished whereas numerical reports are statedSome concluding remarks will be drawn in Section 4 to endthe paper
2 A New Method with Memory
In this section we propose the following iterative methodwith memory based on (5)
119910119896 = 119909119896 minus119891 (119909119896)
FD 119908119896 = 119909119896 + 120573119896119891 (119909119896)
2 119896 = 0 1 2
119909119896+1 = 119910119896 minus119891 (119910119896)
FD
119891 (119909119896) + 120574119891 (119910119896)
119891 (119909119896) + (120574 minus 2) 119891 (119910119896) 120574 = minus
1
2
(7)
wherein the self-accelerating parameter is 120573119896 The errorequation of (5) is (120574 = minus12)
119890119896+1 = minus1198882 (1198911015840(120572)2 1205731198882 + 1198883) 119890
4
119896+ 119874 (1198905
119896) (8)
where 119888119895 = (1119895)(119891(119895)(120572)1198911015840(120572)) We now must find a way so
as to vanish the asymptotic error constant 120578 = minus1198882(1198911015840(120572)21205731198882+
1198883)Toward this goal one can increase the 119877-order by consid-
ering the following substitution
120573 = minus1198883
1198911015840 (120572)2 1198882 (9)
Since the zero is not known relation (9) cannot be used inits exact form and we must approximate it recursively Thisbuilds a variant with memory for Kingrsquos family by using
120573119896 asymp minus1198883
1198911015840
(120572)2 1198882
(10)
where 119888119895 asymp 119888119895 Now if we consider 1198733(119905) to be Newtonrsquosinterpolation polynomial of third degree set through fouravailable approximations 119909119896 119909119896minus1 119910119896minus1 119908119896minus1 at the end ofeach cycle we can propose the following new method withmemory
120573119896 = minus1198731015840101584010158403(119909119896)
311987310158403(119909119896)2119873101584010158403(119909119896)
119910119896 = 119909119896 minus119891 (119909119896)
FD 119908119896 = 119909119896 + 120573119896119891 (119909119896)
2
119896 = 0 1 2
119909119896+1 = 119910119896 minus119891 (119910119896)
FD
119891 (119909119896) minus 12119891 (119910119896)
119891 (119909119896) minus 52119891 (119910119896)
(11)
Note that for example we have the following formulationfor the interpolating polynomial
11987310158403(119909119896) = [
119889
1198891199051198733(119905)]
119905=119909119896
= 119891 [119909119896 119909119896minus1] + 119891 [119909119896 119909119896minus1 119910119896minus1] (119909119896 minus 119909119896minus1)
+ 119891 [119909119896 119909119896minus1 119910119896minus1 119908119896minus1] (119909119896 minus 119909119896minus1) (119909119896 minus 119910119896minus1)
(12)
Acceleration in convergence for (11) is based on the use ofa variation of one free nonzero parameter in each iterativestep This parameter is calculated using information fromthe current and previous iteration(s) so that the developedmethod may be regarded as method with memory accordingto Traubrsquos classification [2]
We are at the time to write about the theoretical aspectsof our proposed solver (11)
The Scientific World Journal 3
Theorem 1 Let the function 119891(119909) be sufficiently differentiablein a neighborhood of its simple zero 120572 If an initial approxima-tion1199090 is sufficiently close to120572 then the119877-order of convergenceof the two-step method (11) with memory is at least 423607
Proof Let 119909119896 be a sequence of approximations generatedby an iterative method The error relations with the self-accelerating parameter 120573 = 120573119896 for (11) are in what follows
119890119896 = 119908119896 minus 120572 sim 1198881198961119890119896 (13)
119890119896 = 119910119896 minus 120572 sim 11988811989621198902
119896 (14)
119890119896+1 = 119909119896+1 minus 120572 sim 11988811989641198904
119896 (15)
Using a symbolic computations we attain that
minus1198882 (1198911015840(120572)2 1205731198882 + 1198883) sim 119890119896minus1 (16)
Substituting the value of minus1198882(1198911015840(120572)21205731198882 +1198883) from (16) in (15)
one may obtain
119890119896+1 sim 1198881198964119890119896minus11198904
119896 (17)
Note that in general we know that the error equation shouldread 119890119896+1 sim 119860119890
119901
119896 where119860 and 119901 are to be determined Hence
one has 119890119896 sim 119860119890119901
119896minus1 and subsequently
119890119896minus1 sim 119860minus11199011198901119901
119896 (18)
Thus it is easy to obtain
119890119901
119896sim 119860minus1119901119862119890
4+1119901
119896 (19)
wherein 119862 is a constant This results in
119901 = 4 +1
119901 (20)
with two solutions minus0236068 423607 Clearly the value for119901 = 423607 is acceptable and would be the convergence119877-order of method (11) with memory The proof is com-plete
The increase of 119877-order is attained without any (new)additional function calculations so that the novel methodwith memory possesses a high computational efficiencyindex This technique is an extension over scheme (5) toincrease the 119877-order from 4 to 423607
The accelerating method (11) is new simple and usefulproviding considerable improvement of convergence ratewithout any additional function evaluations in contrast to theoptimal two-step methods without memory
We also remark that an alternative form of our proposedmethod with memory could be deduced using backwardfinite difference formula at the beginning of the first substepand a minor modification in the accelerators that is to
say we have the following alternative method with memorypossessing 423607 as its 119877-order (APM) as well
120573119896 =1198731015840101584010158403(119909119896)
311987310158403(119909119896)2119873101584010158403(119909119896)
119910119896 = 119909119896 minus119891 (119909119896)
FD 119908119896 = 119909119896 minus 120573119896119891 (119909119896)
2
119909119896+1 = 119910119896 minus119891 (119910119896)
FD
119891 (119909119896) minus 12119891 (119910119896)
119891 (119909119896) minus 52119891 (119910119896)
(21)
Theorem 2 Let the function 119891(119909) be sufficiently differentiablein a neighborhood of its simple zero 120572 If an initial approxima-tion1199090 is sufficiently close to120572 then the119877-order of convergenceof the two-step method (21) with memory is at least 423607
Proof The proof of this theorem is similar toTheorem 1 It ishence omitted
3 Numerical Computations
Computational efficiency of different iterative methods withand without memory can be measured in a prosperousmanner by applying the definition of efficiency index For aniterative method with convergence (119877-)order 119903 that requires120579 functional evaluations the efficiency index (also namedcomputational efficiency) is calculated by Ostrowski-Traubrsquosformula [2]
119864 = 1199031120579 (22)
According to this we find
119864 (SM) asymp 14142 lt 119864 (3) asymp 15737 = 119864 (4) asymp 15874
= 119864 (5) asymp 15874 lt 119864 (11) asymp 16180(23)
where SM is the quadratically convergent method of Stef-fensen without memory [7]
It should be remarked that Dzunic in [8] designedan efficient one-step Steffensen-type method with memorypossessing (12)(3+radic17) 119877-order of convergence as follows
119908119896 = 119909119896 + 120573119896119891 (119909119896)
120573119896 = minus1
11987310158402(119909119896)
119901119896 = minus119873101584010158403(119908119896)
211987310158403(119908119896)
119909119896+1 = 119909119896 minus119891 (119909119896)
119891 [119909119896 119908119896] + 119901119896119891 (119908119896)
(24)
4 The Scientific World Journal
Table 1 Results of comparisons for Example 3 and to find 120572 = 2
Methods |119891(1199091)| |119891(119909
2)| |119891(119909
3)| |119891(119909
4)| coc
KM 18577 64890 72226 times 1010 32493 times 109 mdashOM 47484 00023129 13928 times 10minus16 18313 times 10minus69 400000DKM 053362 53207 times 10minus7 52711 times 10minus31 50774 times 10minus127 400000PM 053362 19202 times 10minus6 36106 times 10minus30 16392 times 10minus130 422928
Table 2 Results of comparisons for Example 4
Methods |119891(1199091)| |119891(1199092)| |119891(1199093)| |119891(1199094)| cocKM 20873 00095650 77971 times 10minus12 34597 times 10minus48 400000OM 081344 00010884 15476 times 10minus15 63280 times 10minus63 400000DKM 21909 0013379 29909 times 10minus11 75008 times 10minus46 400000PM 21909 00011772 70556 times 10minus16 84197 times 10minus68 423539APM 19861 000089226 23251 times 10minus16 75243 times 10minus70 423526
and Cordero et al in [9] presented a two-step biparametricSteffensen-type iterative method with memory possessingseventh 119877-order of convergence
119908119896 = 119909119896 + 120573119896119891 (119909119896) 120573119896 = minus1
11987310158403(119909119896)
119901119896 = minus119873101584010158404(119908119896)
211987310158404(119908119896)
119910119896 = 119909119896 minus119891 (119909119896)
119891 [119909119896 119908119896] + 119901119896119891 (119908119896)
119909119896+1 = 119910119896 minus119891 (119910119896)
119891 [119909119896 119910119896] + 119891 [119908119896 119909119896 119910119896] (119910119896 minus 119909119896)
(25)
Note that our main aim was to develop Kingrsquos family interms of efficiencies index and was not to achieve the highestpossible efficiency index
Although these methods possess higher computationalefficiency indices than our proposed method (11) we excludethem from numerical comparisons since our method is not aSteffensen-type method and it is a Newton-type method withmemory For more refer to [10]
Now we apply and compare the behavior of differentmethods for finding the simple zeros of some differentnonlinear test functions in the programming package Math-ematica [11] using multiple precision arithmetic to clearlyreveal the high 119877-order of PM and APM We comparemethods with the same number of functional evaluations percycle
We notice that by applying any root solver with localconvergence a special attention must be paid to the choiceof initial approximations If initial values are sufficientlyclose to the sought roots then the expected (theoretical)convergence speed is obtainable in practice otherwise theiterative methods show slower convergence especially at thebeginning of the iterative process
In this section the computational order of convergence(coc) has been computed by
coc =ln 1003816100381610038161003816119891 (119909119896) 119891 (119909119896minus1)
1003816100381610038161003816ln 1003816100381610038161003816119891 (119909119896minus1) 119891 (119909119896minus2)
1003816100381610038161003816 (26)
The calculated value coc estimates the theoretical order ofconvergence well when pathological behavior of the iter-ative method (ie slow convergence at the beginning ofthe implemented iterative method oscillating behavior ofapproximations etc) does not exist
Here the results of comparisons for the test functions aregiven by applying 1000 fixed floating point arithmetic usingthe stop termination |119891(119909119896)| le 10
minus100
Example 3 Weconsider the following nonlinear test functionin the interval119863 = [15 25]
119891 (119909) = (119909 minus 2 tan (119909)) (1199093 minus 8) (27)
using the initial approximation 1199090 = 17 The results areprovided in Table 1
In this section we have used 1205730 = 00001 wheneverrequired Furthermore for DKM we considered 120574 = minus12
Example 4 We compare the behavior of different methodsfor finding the complex solution of the following nonlinearequation
119892 (119909) = (minus1 + 2119868) +1
119909+ 119909 + sin (119909) (28)
using the initial approximation 1199090 = 1 minus 3119868 where 120572 =028860 sdot sdot sdot minus 124220 sdot sdot sdot 119868 The results for this test are givenin Table 2
It is evident from Tables 1 and 2 that approximations tothe roots possess great accuracy when the proposed methodwithmemory is applied Results of the fourth iterate in Tables1 and 2 are given only for demonstration of convergence speedof the tested methods and in most cases they are not requiredfor practical problems at present
The Scientific World Journal 5
We also incorporated and applied the developedmethodswith memory (11) and (21) for different test examples andobtained results with the same behavior as above Hencewe could mention that the theoretical results are upheld bynumerical experiments and thus the newmethod is goodwitha high computational efficiency index
4 Summary
In this paper we have proposed a new two-step Steffensen-type iterative method with memory for solving nonlinearscalar equations Using one self-correcting parameter calcu-lated by Newton interpolatory polynomial the 119877-order ofconvergence of the constructed method was increased from4 to 423607 without any additional calculations
The new method was compared in performance andcomputational efficiency with some existing methods bynumerical examples We have observed that the computa-tional efficiency index of the presentedmethod withmemoryis better than those of other existing two-step King-typemethods
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Authorsrsquo Contribution
The authors have made the same contribution All authorsread and approved the final paper
References
[1] A S Al-Fhaid S Shateyi M Z Ullah and F Soleymani ldquoAmatrix iteration for finding Drazin inverse with ninth-orderconvergencerdquo Abstract and Applied Analysis vol 2014 ArticleID 137486 7 pages 2014
[2] J F Traub Iterative Methods for the Solution of EquationsPrentice-Hall New York NY USA 1964
[3] A Cordero and J R Torregrosa ldquoLow-complexity root-findingiteration functions with no derivatives of any order of con-vergencerdquo Journal of Computational and Applied Mathematics2014
[4] R F King ldquoA family of fourth order methods for nonlinearequationsrdquo SIAM Journal on Numerical Analysis vol 10 pp876ndash879 1973
[5] N Huang and C Ma ldquoConvergence analysis and numericalstudy of a fixed-point iterative method for solving systems ofnonlinear equationsrdquo The Scientific World Journal vol 2014Article ID 789459 10 pages 2014
[6] J P Jaiswal ldquoSome class of third- and fourth-order iterativemethods for solving nonlinear equationsrdquo Journal of AppliedMathematics vol 2014 Article ID 817656 17 pages 2014
[7] J F Steffensen ldquoRemarks on iterationrdquo Skandinavisk Aktuari-etidskrift vol 16 pp 64ndash72 1933
[8] J Dzunic ldquoOn efficient two-parameter methods for solvingnonlinear equationsrdquo Numerical Algorithms vol 63 no 3 pp549ndash569 2013
[9] A Cordero T Lotfi P Bakhtiari and J R Torregrosa ldquoAnefficient two-parametric family with memory for nonlinearequationsrdquo Numerical Algorithms 2014
[10] X Wang and T Zhang ldquoA new family of Newton-type iterativemethods with and without memory for solving nonlinearequationsrdquo Calcolo vol 51 no 1 pp 1ndash15 2014
[11] S Wagon Mathematica in Action Springer New York NYUSA 3rd edition 2010
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2 The Scientific World Journal
For instance choosing the well-known optimal two-stepfamily of King (KM) [4]
119910119896 = 119909119896 minus119891 (119909119896)
1198911015840 (119909119896) 119896 = 0 1 2
119909119896+1 = 119910119896 minus119891 (119910119896)
1198911015840 (119909119896)
119891 (119909119896) + 120574119891 (119910119896)
119891 (119909119896) + (120574 minus 2) 119891 (119910119896) 120574 isin R
(4)
and the conjecture of Cordero-Torregrosa one may proposethe following method (DKM)
119910119896 = 119909119896 minus119891 (119909119896)
FD 119896 = 0 1 2
119909119896+1 = 119910119896 minus119891 (119910119896)
FD
119891 (119909119896) + 120574119891 (119910119896)
119891 (119909119896) + (120574 minus 2) 119891 (119910119896) 120574 isin R
(5)
wherein
FD =119891 (119909119896) minus 119891 (119908119896)
119909119896 minus 119908119896 119908119896 = 119909119896 + 120573119891 (119909119896)
2
120573 isin R 0
(6)
In this work we propose a two-stepmethodwithmemorypossessing a high efficiency index according to the well-known family of Kingrsquos methods (5)
Our inspiration andmotivation for constructing a higher-order method are linked in a direct manner with the fun-damental concept of numerical analysis that any numericalmethod should give as accurate as possible output results withminimal computational cost To state the matter differentlyit is necessary to pursue methods of higher computationalefficiency
For more background concerning this topic one mayrefer to [5 6]
The paper is organized as follows In Section 2 theaim of this paper is presented by contributing an iterativemethod with memory based on (5) for solving nonlinearequations The proposed scheme is an extension over (4)and has a simple structure with an increased computationalefficiency In Section 3 we compare the theoretical resultsby applying the definition of efficiency index and furthersupports are furnished whereas numerical reports are statedSome concluding remarks will be drawn in Section 4 to endthe paper
2 A New Method with Memory
In this section we propose the following iterative methodwith memory based on (5)
119910119896 = 119909119896 minus119891 (119909119896)
FD 119908119896 = 119909119896 + 120573119896119891 (119909119896)
2 119896 = 0 1 2
119909119896+1 = 119910119896 minus119891 (119910119896)
FD
119891 (119909119896) + 120574119891 (119910119896)
119891 (119909119896) + (120574 minus 2) 119891 (119910119896) 120574 = minus
1
2
(7)
wherein the self-accelerating parameter is 120573119896 The errorequation of (5) is (120574 = minus12)
119890119896+1 = minus1198882 (1198911015840(120572)2 1205731198882 + 1198883) 119890
4
119896+ 119874 (1198905
119896) (8)
where 119888119895 = (1119895)(119891(119895)(120572)1198911015840(120572)) We now must find a way so
as to vanish the asymptotic error constant 120578 = minus1198882(1198911015840(120572)21205731198882+
1198883)Toward this goal one can increase the 119877-order by consid-
ering the following substitution
120573 = minus1198883
1198911015840 (120572)2 1198882 (9)
Since the zero is not known relation (9) cannot be used inits exact form and we must approximate it recursively Thisbuilds a variant with memory for Kingrsquos family by using
120573119896 asymp minus1198883
1198911015840
(120572)2 1198882
(10)
where 119888119895 asymp 119888119895 Now if we consider 1198733(119905) to be Newtonrsquosinterpolation polynomial of third degree set through fouravailable approximations 119909119896 119909119896minus1 119910119896minus1 119908119896minus1 at the end ofeach cycle we can propose the following new method withmemory
120573119896 = minus1198731015840101584010158403(119909119896)
311987310158403(119909119896)2119873101584010158403(119909119896)
119910119896 = 119909119896 minus119891 (119909119896)
FD 119908119896 = 119909119896 + 120573119896119891 (119909119896)
2
119896 = 0 1 2
119909119896+1 = 119910119896 minus119891 (119910119896)
FD
119891 (119909119896) minus 12119891 (119910119896)
119891 (119909119896) minus 52119891 (119910119896)
(11)
Note that for example we have the following formulationfor the interpolating polynomial
11987310158403(119909119896) = [
119889
1198891199051198733(119905)]
119905=119909119896
= 119891 [119909119896 119909119896minus1] + 119891 [119909119896 119909119896minus1 119910119896minus1] (119909119896 minus 119909119896minus1)
+ 119891 [119909119896 119909119896minus1 119910119896minus1 119908119896minus1] (119909119896 minus 119909119896minus1) (119909119896 minus 119910119896minus1)
(12)
Acceleration in convergence for (11) is based on the use ofa variation of one free nonzero parameter in each iterativestep This parameter is calculated using information fromthe current and previous iteration(s) so that the developedmethod may be regarded as method with memory accordingto Traubrsquos classification [2]
We are at the time to write about the theoretical aspectsof our proposed solver (11)
The Scientific World Journal 3
Theorem 1 Let the function 119891(119909) be sufficiently differentiablein a neighborhood of its simple zero 120572 If an initial approxima-tion1199090 is sufficiently close to120572 then the119877-order of convergenceof the two-step method (11) with memory is at least 423607
Proof Let 119909119896 be a sequence of approximations generatedby an iterative method The error relations with the self-accelerating parameter 120573 = 120573119896 for (11) are in what follows
119890119896 = 119908119896 minus 120572 sim 1198881198961119890119896 (13)
119890119896 = 119910119896 minus 120572 sim 11988811989621198902
119896 (14)
119890119896+1 = 119909119896+1 minus 120572 sim 11988811989641198904
119896 (15)
Using a symbolic computations we attain that
minus1198882 (1198911015840(120572)2 1205731198882 + 1198883) sim 119890119896minus1 (16)
Substituting the value of minus1198882(1198911015840(120572)21205731198882 +1198883) from (16) in (15)
one may obtain
119890119896+1 sim 1198881198964119890119896minus11198904
119896 (17)
Note that in general we know that the error equation shouldread 119890119896+1 sim 119860119890
119901
119896 where119860 and 119901 are to be determined Hence
one has 119890119896 sim 119860119890119901
119896minus1 and subsequently
119890119896minus1 sim 119860minus11199011198901119901
119896 (18)
Thus it is easy to obtain
119890119901
119896sim 119860minus1119901119862119890
4+1119901
119896 (19)
wherein 119862 is a constant This results in
119901 = 4 +1
119901 (20)
with two solutions minus0236068 423607 Clearly the value for119901 = 423607 is acceptable and would be the convergence119877-order of method (11) with memory The proof is com-plete
The increase of 119877-order is attained without any (new)additional function calculations so that the novel methodwith memory possesses a high computational efficiencyindex This technique is an extension over scheme (5) toincrease the 119877-order from 4 to 423607
The accelerating method (11) is new simple and usefulproviding considerable improvement of convergence ratewithout any additional function evaluations in contrast to theoptimal two-step methods without memory
We also remark that an alternative form of our proposedmethod with memory could be deduced using backwardfinite difference formula at the beginning of the first substepand a minor modification in the accelerators that is to
say we have the following alternative method with memorypossessing 423607 as its 119877-order (APM) as well
120573119896 =1198731015840101584010158403(119909119896)
311987310158403(119909119896)2119873101584010158403(119909119896)
119910119896 = 119909119896 minus119891 (119909119896)
FD 119908119896 = 119909119896 minus 120573119896119891 (119909119896)
2
119909119896+1 = 119910119896 minus119891 (119910119896)
FD
119891 (119909119896) minus 12119891 (119910119896)
119891 (119909119896) minus 52119891 (119910119896)
(21)
Theorem 2 Let the function 119891(119909) be sufficiently differentiablein a neighborhood of its simple zero 120572 If an initial approxima-tion1199090 is sufficiently close to120572 then the119877-order of convergenceof the two-step method (21) with memory is at least 423607
Proof The proof of this theorem is similar toTheorem 1 It ishence omitted
3 Numerical Computations
Computational efficiency of different iterative methods withand without memory can be measured in a prosperousmanner by applying the definition of efficiency index For aniterative method with convergence (119877-)order 119903 that requires120579 functional evaluations the efficiency index (also namedcomputational efficiency) is calculated by Ostrowski-Traubrsquosformula [2]
119864 = 1199031120579 (22)
According to this we find
119864 (SM) asymp 14142 lt 119864 (3) asymp 15737 = 119864 (4) asymp 15874
= 119864 (5) asymp 15874 lt 119864 (11) asymp 16180(23)
where SM is the quadratically convergent method of Stef-fensen without memory [7]
It should be remarked that Dzunic in [8] designedan efficient one-step Steffensen-type method with memorypossessing (12)(3+radic17) 119877-order of convergence as follows
119908119896 = 119909119896 + 120573119896119891 (119909119896)
120573119896 = minus1
11987310158402(119909119896)
119901119896 = minus119873101584010158403(119908119896)
211987310158403(119908119896)
119909119896+1 = 119909119896 minus119891 (119909119896)
119891 [119909119896 119908119896] + 119901119896119891 (119908119896)
(24)
4 The Scientific World Journal
Table 1 Results of comparisons for Example 3 and to find 120572 = 2
Methods |119891(1199091)| |119891(119909
2)| |119891(119909
3)| |119891(119909
4)| coc
KM 18577 64890 72226 times 1010 32493 times 109 mdashOM 47484 00023129 13928 times 10minus16 18313 times 10minus69 400000DKM 053362 53207 times 10minus7 52711 times 10minus31 50774 times 10minus127 400000PM 053362 19202 times 10minus6 36106 times 10minus30 16392 times 10minus130 422928
Table 2 Results of comparisons for Example 4
Methods |119891(1199091)| |119891(1199092)| |119891(1199093)| |119891(1199094)| cocKM 20873 00095650 77971 times 10minus12 34597 times 10minus48 400000OM 081344 00010884 15476 times 10minus15 63280 times 10minus63 400000DKM 21909 0013379 29909 times 10minus11 75008 times 10minus46 400000PM 21909 00011772 70556 times 10minus16 84197 times 10minus68 423539APM 19861 000089226 23251 times 10minus16 75243 times 10minus70 423526
and Cordero et al in [9] presented a two-step biparametricSteffensen-type iterative method with memory possessingseventh 119877-order of convergence
119908119896 = 119909119896 + 120573119896119891 (119909119896) 120573119896 = minus1
11987310158403(119909119896)
119901119896 = minus119873101584010158404(119908119896)
211987310158404(119908119896)
119910119896 = 119909119896 minus119891 (119909119896)
119891 [119909119896 119908119896] + 119901119896119891 (119908119896)
119909119896+1 = 119910119896 minus119891 (119910119896)
119891 [119909119896 119910119896] + 119891 [119908119896 119909119896 119910119896] (119910119896 minus 119909119896)
(25)
Note that our main aim was to develop Kingrsquos family interms of efficiencies index and was not to achieve the highestpossible efficiency index
Although these methods possess higher computationalefficiency indices than our proposed method (11) we excludethem from numerical comparisons since our method is not aSteffensen-type method and it is a Newton-type method withmemory For more refer to [10]
Now we apply and compare the behavior of differentmethods for finding the simple zeros of some differentnonlinear test functions in the programming package Math-ematica [11] using multiple precision arithmetic to clearlyreveal the high 119877-order of PM and APM We comparemethods with the same number of functional evaluations percycle
We notice that by applying any root solver with localconvergence a special attention must be paid to the choiceof initial approximations If initial values are sufficientlyclose to the sought roots then the expected (theoretical)convergence speed is obtainable in practice otherwise theiterative methods show slower convergence especially at thebeginning of the iterative process
In this section the computational order of convergence(coc) has been computed by
coc =ln 1003816100381610038161003816119891 (119909119896) 119891 (119909119896minus1)
1003816100381610038161003816ln 1003816100381610038161003816119891 (119909119896minus1) 119891 (119909119896minus2)
1003816100381610038161003816 (26)
The calculated value coc estimates the theoretical order ofconvergence well when pathological behavior of the iter-ative method (ie slow convergence at the beginning ofthe implemented iterative method oscillating behavior ofapproximations etc) does not exist
Here the results of comparisons for the test functions aregiven by applying 1000 fixed floating point arithmetic usingthe stop termination |119891(119909119896)| le 10
minus100
Example 3 Weconsider the following nonlinear test functionin the interval119863 = [15 25]
119891 (119909) = (119909 minus 2 tan (119909)) (1199093 minus 8) (27)
using the initial approximation 1199090 = 17 The results areprovided in Table 1
In this section we have used 1205730 = 00001 wheneverrequired Furthermore for DKM we considered 120574 = minus12
Example 4 We compare the behavior of different methodsfor finding the complex solution of the following nonlinearequation
119892 (119909) = (minus1 + 2119868) +1
119909+ 119909 + sin (119909) (28)
using the initial approximation 1199090 = 1 minus 3119868 where 120572 =028860 sdot sdot sdot minus 124220 sdot sdot sdot 119868 The results for this test are givenin Table 2
It is evident from Tables 1 and 2 that approximations tothe roots possess great accuracy when the proposed methodwithmemory is applied Results of the fourth iterate in Tables1 and 2 are given only for demonstration of convergence speedof the tested methods and in most cases they are not requiredfor practical problems at present
The Scientific World Journal 5
We also incorporated and applied the developedmethodswith memory (11) and (21) for different test examples andobtained results with the same behavior as above Hencewe could mention that the theoretical results are upheld bynumerical experiments and thus the newmethod is goodwitha high computational efficiency index
4 Summary
In this paper we have proposed a new two-step Steffensen-type iterative method with memory for solving nonlinearscalar equations Using one self-correcting parameter calcu-lated by Newton interpolatory polynomial the 119877-order ofconvergence of the constructed method was increased from4 to 423607 without any additional calculations
The new method was compared in performance andcomputational efficiency with some existing methods bynumerical examples We have observed that the computa-tional efficiency index of the presentedmethod withmemoryis better than those of other existing two-step King-typemethods
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Authorsrsquo Contribution
The authors have made the same contribution All authorsread and approved the final paper
References
[1] A S Al-Fhaid S Shateyi M Z Ullah and F Soleymani ldquoAmatrix iteration for finding Drazin inverse with ninth-orderconvergencerdquo Abstract and Applied Analysis vol 2014 ArticleID 137486 7 pages 2014
[2] J F Traub Iterative Methods for the Solution of EquationsPrentice-Hall New York NY USA 1964
[3] A Cordero and J R Torregrosa ldquoLow-complexity root-findingiteration functions with no derivatives of any order of con-vergencerdquo Journal of Computational and Applied Mathematics2014
[4] R F King ldquoA family of fourth order methods for nonlinearequationsrdquo SIAM Journal on Numerical Analysis vol 10 pp876ndash879 1973
[5] N Huang and C Ma ldquoConvergence analysis and numericalstudy of a fixed-point iterative method for solving systems ofnonlinear equationsrdquo The Scientific World Journal vol 2014Article ID 789459 10 pages 2014
[6] J P Jaiswal ldquoSome class of third- and fourth-order iterativemethods for solving nonlinear equationsrdquo Journal of AppliedMathematics vol 2014 Article ID 817656 17 pages 2014
[7] J F Steffensen ldquoRemarks on iterationrdquo Skandinavisk Aktuari-etidskrift vol 16 pp 64ndash72 1933
[8] J Dzunic ldquoOn efficient two-parameter methods for solvingnonlinear equationsrdquo Numerical Algorithms vol 63 no 3 pp549ndash569 2013
[9] A Cordero T Lotfi P Bakhtiari and J R Torregrosa ldquoAnefficient two-parametric family with memory for nonlinearequationsrdquo Numerical Algorithms 2014
[10] X Wang and T Zhang ldquoA new family of Newton-type iterativemethods with and without memory for solving nonlinearequationsrdquo Calcolo vol 51 no 1 pp 1ndash15 2014
[11] S Wagon Mathematica in Action Springer New York NYUSA 3rd edition 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
The Scientific World Journal 3
Theorem 1 Let the function 119891(119909) be sufficiently differentiablein a neighborhood of its simple zero 120572 If an initial approxima-tion1199090 is sufficiently close to120572 then the119877-order of convergenceof the two-step method (11) with memory is at least 423607
Proof Let 119909119896 be a sequence of approximations generatedby an iterative method The error relations with the self-accelerating parameter 120573 = 120573119896 for (11) are in what follows
119890119896 = 119908119896 minus 120572 sim 1198881198961119890119896 (13)
119890119896 = 119910119896 minus 120572 sim 11988811989621198902
119896 (14)
119890119896+1 = 119909119896+1 minus 120572 sim 11988811989641198904
119896 (15)
Using a symbolic computations we attain that
minus1198882 (1198911015840(120572)2 1205731198882 + 1198883) sim 119890119896minus1 (16)
Substituting the value of minus1198882(1198911015840(120572)21205731198882 +1198883) from (16) in (15)
one may obtain
119890119896+1 sim 1198881198964119890119896minus11198904
119896 (17)
Note that in general we know that the error equation shouldread 119890119896+1 sim 119860119890
119901
119896 where119860 and 119901 are to be determined Hence
one has 119890119896 sim 119860119890119901
119896minus1 and subsequently
119890119896minus1 sim 119860minus11199011198901119901
119896 (18)
Thus it is easy to obtain
119890119901
119896sim 119860minus1119901119862119890
4+1119901
119896 (19)
wherein 119862 is a constant This results in
119901 = 4 +1
119901 (20)
with two solutions minus0236068 423607 Clearly the value for119901 = 423607 is acceptable and would be the convergence119877-order of method (11) with memory The proof is com-plete
The increase of 119877-order is attained without any (new)additional function calculations so that the novel methodwith memory possesses a high computational efficiencyindex This technique is an extension over scheme (5) toincrease the 119877-order from 4 to 423607
The accelerating method (11) is new simple and usefulproviding considerable improvement of convergence ratewithout any additional function evaluations in contrast to theoptimal two-step methods without memory
We also remark that an alternative form of our proposedmethod with memory could be deduced using backwardfinite difference formula at the beginning of the first substepand a minor modification in the accelerators that is to
say we have the following alternative method with memorypossessing 423607 as its 119877-order (APM) as well
120573119896 =1198731015840101584010158403(119909119896)
311987310158403(119909119896)2119873101584010158403(119909119896)
119910119896 = 119909119896 minus119891 (119909119896)
FD 119908119896 = 119909119896 minus 120573119896119891 (119909119896)
2
119909119896+1 = 119910119896 minus119891 (119910119896)
FD
119891 (119909119896) minus 12119891 (119910119896)
119891 (119909119896) minus 52119891 (119910119896)
(21)
Theorem 2 Let the function 119891(119909) be sufficiently differentiablein a neighborhood of its simple zero 120572 If an initial approxima-tion1199090 is sufficiently close to120572 then the119877-order of convergenceof the two-step method (21) with memory is at least 423607
Proof The proof of this theorem is similar toTheorem 1 It ishence omitted
3 Numerical Computations
Computational efficiency of different iterative methods withand without memory can be measured in a prosperousmanner by applying the definition of efficiency index For aniterative method with convergence (119877-)order 119903 that requires120579 functional evaluations the efficiency index (also namedcomputational efficiency) is calculated by Ostrowski-Traubrsquosformula [2]
119864 = 1199031120579 (22)
According to this we find
119864 (SM) asymp 14142 lt 119864 (3) asymp 15737 = 119864 (4) asymp 15874
= 119864 (5) asymp 15874 lt 119864 (11) asymp 16180(23)
where SM is the quadratically convergent method of Stef-fensen without memory [7]
It should be remarked that Dzunic in [8] designedan efficient one-step Steffensen-type method with memorypossessing (12)(3+radic17) 119877-order of convergence as follows
119908119896 = 119909119896 + 120573119896119891 (119909119896)
120573119896 = minus1
11987310158402(119909119896)
119901119896 = minus119873101584010158403(119908119896)
211987310158403(119908119896)
119909119896+1 = 119909119896 minus119891 (119909119896)
119891 [119909119896 119908119896] + 119901119896119891 (119908119896)
(24)
4 The Scientific World Journal
Table 1 Results of comparisons for Example 3 and to find 120572 = 2
Methods |119891(1199091)| |119891(119909
2)| |119891(119909
3)| |119891(119909
4)| coc
KM 18577 64890 72226 times 1010 32493 times 109 mdashOM 47484 00023129 13928 times 10minus16 18313 times 10minus69 400000DKM 053362 53207 times 10minus7 52711 times 10minus31 50774 times 10minus127 400000PM 053362 19202 times 10minus6 36106 times 10minus30 16392 times 10minus130 422928
Table 2 Results of comparisons for Example 4
Methods |119891(1199091)| |119891(1199092)| |119891(1199093)| |119891(1199094)| cocKM 20873 00095650 77971 times 10minus12 34597 times 10minus48 400000OM 081344 00010884 15476 times 10minus15 63280 times 10minus63 400000DKM 21909 0013379 29909 times 10minus11 75008 times 10minus46 400000PM 21909 00011772 70556 times 10minus16 84197 times 10minus68 423539APM 19861 000089226 23251 times 10minus16 75243 times 10minus70 423526
and Cordero et al in [9] presented a two-step biparametricSteffensen-type iterative method with memory possessingseventh 119877-order of convergence
119908119896 = 119909119896 + 120573119896119891 (119909119896) 120573119896 = minus1
11987310158403(119909119896)
119901119896 = minus119873101584010158404(119908119896)
211987310158404(119908119896)
119910119896 = 119909119896 minus119891 (119909119896)
119891 [119909119896 119908119896] + 119901119896119891 (119908119896)
119909119896+1 = 119910119896 minus119891 (119910119896)
119891 [119909119896 119910119896] + 119891 [119908119896 119909119896 119910119896] (119910119896 minus 119909119896)
(25)
Note that our main aim was to develop Kingrsquos family interms of efficiencies index and was not to achieve the highestpossible efficiency index
Although these methods possess higher computationalefficiency indices than our proposed method (11) we excludethem from numerical comparisons since our method is not aSteffensen-type method and it is a Newton-type method withmemory For more refer to [10]
Now we apply and compare the behavior of differentmethods for finding the simple zeros of some differentnonlinear test functions in the programming package Math-ematica [11] using multiple precision arithmetic to clearlyreveal the high 119877-order of PM and APM We comparemethods with the same number of functional evaluations percycle
We notice that by applying any root solver with localconvergence a special attention must be paid to the choiceof initial approximations If initial values are sufficientlyclose to the sought roots then the expected (theoretical)convergence speed is obtainable in practice otherwise theiterative methods show slower convergence especially at thebeginning of the iterative process
In this section the computational order of convergence(coc) has been computed by
coc =ln 1003816100381610038161003816119891 (119909119896) 119891 (119909119896minus1)
1003816100381610038161003816ln 1003816100381610038161003816119891 (119909119896minus1) 119891 (119909119896minus2)
1003816100381610038161003816 (26)
The calculated value coc estimates the theoretical order ofconvergence well when pathological behavior of the iter-ative method (ie slow convergence at the beginning ofthe implemented iterative method oscillating behavior ofapproximations etc) does not exist
Here the results of comparisons for the test functions aregiven by applying 1000 fixed floating point arithmetic usingthe stop termination |119891(119909119896)| le 10
minus100
Example 3 Weconsider the following nonlinear test functionin the interval119863 = [15 25]
119891 (119909) = (119909 minus 2 tan (119909)) (1199093 minus 8) (27)
using the initial approximation 1199090 = 17 The results areprovided in Table 1
In this section we have used 1205730 = 00001 wheneverrequired Furthermore for DKM we considered 120574 = minus12
Example 4 We compare the behavior of different methodsfor finding the complex solution of the following nonlinearequation
119892 (119909) = (minus1 + 2119868) +1
119909+ 119909 + sin (119909) (28)
using the initial approximation 1199090 = 1 minus 3119868 where 120572 =028860 sdot sdot sdot minus 124220 sdot sdot sdot 119868 The results for this test are givenin Table 2
It is evident from Tables 1 and 2 that approximations tothe roots possess great accuracy when the proposed methodwithmemory is applied Results of the fourth iterate in Tables1 and 2 are given only for demonstration of convergence speedof the tested methods and in most cases they are not requiredfor practical problems at present
The Scientific World Journal 5
We also incorporated and applied the developedmethodswith memory (11) and (21) for different test examples andobtained results with the same behavior as above Hencewe could mention that the theoretical results are upheld bynumerical experiments and thus the newmethod is goodwitha high computational efficiency index
4 Summary
In this paper we have proposed a new two-step Steffensen-type iterative method with memory for solving nonlinearscalar equations Using one self-correcting parameter calcu-lated by Newton interpolatory polynomial the 119877-order ofconvergence of the constructed method was increased from4 to 423607 without any additional calculations
The new method was compared in performance andcomputational efficiency with some existing methods bynumerical examples We have observed that the computa-tional efficiency index of the presentedmethod withmemoryis better than those of other existing two-step King-typemethods
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Authorsrsquo Contribution
The authors have made the same contribution All authorsread and approved the final paper
References
[1] A S Al-Fhaid S Shateyi M Z Ullah and F Soleymani ldquoAmatrix iteration for finding Drazin inverse with ninth-orderconvergencerdquo Abstract and Applied Analysis vol 2014 ArticleID 137486 7 pages 2014
[2] J F Traub Iterative Methods for the Solution of EquationsPrentice-Hall New York NY USA 1964
[3] A Cordero and J R Torregrosa ldquoLow-complexity root-findingiteration functions with no derivatives of any order of con-vergencerdquo Journal of Computational and Applied Mathematics2014
[4] R F King ldquoA family of fourth order methods for nonlinearequationsrdquo SIAM Journal on Numerical Analysis vol 10 pp876ndash879 1973
[5] N Huang and C Ma ldquoConvergence analysis and numericalstudy of a fixed-point iterative method for solving systems ofnonlinear equationsrdquo The Scientific World Journal vol 2014Article ID 789459 10 pages 2014
[6] J P Jaiswal ldquoSome class of third- and fourth-order iterativemethods for solving nonlinear equationsrdquo Journal of AppliedMathematics vol 2014 Article ID 817656 17 pages 2014
[7] J F Steffensen ldquoRemarks on iterationrdquo Skandinavisk Aktuari-etidskrift vol 16 pp 64ndash72 1933
[8] J Dzunic ldquoOn efficient two-parameter methods for solvingnonlinear equationsrdquo Numerical Algorithms vol 63 no 3 pp549ndash569 2013
[9] A Cordero T Lotfi P Bakhtiari and J R Torregrosa ldquoAnefficient two-parametric family with memory for nonlinearequationsrdquo Numerical Algorithms 2014
[10] X Wang and T Zhang ldquoA new family of Newton-type iterativemethods with and without memory for solving nonlinearequationsrdquo Calcolo vol 51 no 1 pp 1ndash15 2014
[11] S Wagon Mathematica in Action Springer New York NYUSA 3rd edition 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 The Scientific World Journal
Table 1 Results of comparisons for Example 3 and to find 120572 = 2
Methods |119891(1199091)| |119891(119909
2)| |119891(119909
3)| |119891(119909
4)| coc
KM 18577 64890 72226 times 1010 32493 times 109 mdashOM 47484 00023129 13928 times 10minus16 18313 times 10minus69 400000DKM 053362 53207 times 10minus7 52711 times 10minus31 50774 times 10minus127 400000PM 053362 19202 times 10minus6 36106 times 10minus30 16392 times 10minus130 422928
Table 2 Results of comparisons for Example 4
Methods |119891(1199091)| |119891(1199092)| |119891(1199093)| |119891(1199094)| cocKM 20873 00095650 77971 times 10minus12 34597 times 10minus48 400000OM 081344 00010884 15476 times 10minus15 63280 times 10minus63 400000DKM 21909 0013379 29909 times 10minus11 75008 times 10minus46 400000PM 21909 00011772 70556 times 10minus16 84197 times 10minus68 423539APM 19861 000089226 23251 times 10minus16 75243 times 10minus70 423526
and Cordero et al in [9] presented a two-step biparametricSteffensen-type iterative method with memory possessingseventh 119877-order of convergence
119908119896 = 119909119896 + 120573119896119891 (119909119896) 120573119896 = minus1
11987310158403(119909119896)
119901119896 = minus119873101584010158404(119908119896)
211987310158404(119908119896)
119910119896 = 119909119896 minus119891 (119909119896)
119891 [119909119896 119908119896] + 119901119896119891 (119908119896)
119909119896+1 = 119910119896 minus119891 (119910119896)
119891 [119909119896 119910119896] + 119891 [119908119896 119909119896 119910119896] (119910119896 minus 119909119896)
(25)
Note that our main aim was to develop Kingrsquos family interms of efficiencies index and was not to achieve the highestpossible efficiency index
Although these methods possess higher computationalefficiency indices than our proposed method (11) we excludethem from numerical comparisons since our method is not aSteffensen-type method and it is a Newton-type method withmemory For more refer to [10]
Now we apply and compare the behavior of differentmethods for finding the simple zeros of some differentnonlinear test functions in the programming package Math-ematica [11] using multiple precision arithmetic to clearlyreveal the high 119877-order of PM and APM We comparemethods with the same number of functional evaluations percycle
We notice that by applying any root solver with localconvergence a special attention must be paid to the choiceof initial approximations If initial values are sufficientlyclose to the sought roots then the expected (theoretical)convergence speed is obtainable in practice otherwise theiterative methods show slower convergence especially at thebeginning of the iterative process
In this section the computational order of convergence(coc) has been computed by
coc =ln 1003816100381610038161003816119891 (119909119896) 119891 (119909119896minus1)
1003816100381610038161003816ln 1003816100381610038161003816119891 (119909119896minus1) 119891 (119909119896minus2)
1003816100381610038161003816 (26)
The calculated value coc estimates the theoretical order ofconvergence well when pathological behavior of the iter-ative method (ie slow convergence at the beginning ofthe implemented iterative method oscillating behavior ofapproximations etc) does not exist
Here the results of comparisons for the test functions aregiven by applying 1000 fixed floating point arithmetic usingthe stop termination |119891(119909119896)| le 10
minus100
Example 3 Weconsider the following nonlinear test functionin the interval119863 = [15 25]
119891 (119909) = (119909 minus 2 tan (119909)) (1199093 minus 8) (27)
using the initial approximation 1199090 = 17 The results areprovided in Table 1
In this section we have used 1205730 = 00001 wheneverrequired Furthermore for DKM we considered 120574 = minus12
Example 4 We compare the behavior of different methodsfor finding the complex solution of the following nonlinearequation
119892 (119909) = (minus1 + 2119868) +1
119909+ 119909 + sin (119909) (28)
using the initial approximation 1199090 = 1 minus 3119868 where 120572 =028860 sdot sdot sdot minus 124220 sdot sdot sdot 119868 The results for this test are givenin Table 2
It is evident from Tables 1 and 2 that approximations tothe roots possess great accuracy when the proposed methodwithmemory is applied Results of the fourth iterate in Tables1 and 2 are given only for demonstration of convergence speedof the tested methods and in most cases they are not requiredfor practical problems at present
The Scientific World Journal 5
We also incorporated and applied the developedmethodswith memory (11) and (21) for different test examples andobtained results with the same behavior as above Hencewe could mention that the theoretical results are upheld bynumerical experiments and thus the newmethod is goodwitha high computational efficiency index
4 Summary
In this paper we have proposed a new two-step Steffensen-type iterative method with memory for solving nonlinearscalar equations Using one self-correcting parameter calcu-lated by Newton interpolatory polynomial the 119877-order ofconvergence of the constructed method was increased from4 to 423607 without any additional calculations
The new method was compared in performance andcomputational efficiency with some existing methods bynumerical examples We have observed that the computa-tional efficiency index of the presentedmethod withmemoryis better than those of other existing two-step King-typemethods
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Authorsrsquo Contribution
The authors have made the same contribution All authorsread and approved the final paper
References
[1] A S Al-Fhaid S Shateyi M Z Ullah and F Soleymani ldquoAmatrix iteration for finding Drazin inverse with ninth-orderconvergencerdquo Abstract and Applied Analysis vol 2014 ArticleID 137486 7 pages 2014
[2] J F Traub Iterative Methods for the Solution of EquationsPrentice-Hall New York NY USA 1964
[3] A Cordero and J R Torregrosa ldquoLow-complexity root-findingiteration functions with no derivatives of any order of con-vergencerdquo Journal of Computational and Applied Mathematics2014
[4] R F King ldquoA family of fourth order methods for nonlinearequationsrdquo SIAM Journal on Numerical Analysis vol 10 pp876ndash879 1973
[5] N Huang and C Ma ldquoConvergence analysis and numericalstudy of a fixed-point iterative method for solving systems ofnonlinear equationsrdquo The Scientific World Journal vol 2014Article ID 789459 10 pages 2014
[6] J P Jaiswal ldquoSome class of third- and fourth-order iterativemethods for solving nonlinear equationsrdquo Journal of AppliedMathematics vol 2014 Article ID 817656 17 pages 2014
[7] J F Steffensen ldquoRemarks on iterationrdquo Skandinavisk Aktuari-etidskrift vol 16 pp 64ndash72 1933
[8] J Dzunic ldquoOn efficient two-parameter methods for solvingnonlinear equationsrdquo Numerical Algorithms vol 63 no 3 pp549ndash569 2013
[9] A Cordero T Lotfi P Bakhtiari and J R Torregrosa ldquoAnefficient two-parametric family with memory for nonlinearequationsrdquo Numerical Algorithms 2014
[10] X Wang and T Zhang ldquoA new family of Newton-type iterativemethods with and without memory for solving nonlinearequationsrdquo Calcolo vol 51 no 1 pp 1ndash15 2014
[11] S Wagon Mathematica in Action Springer New York NYUSA 3rd edition 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
The Scientific World Journal 5
We also incorporated and applied the developedmethodswith memory (11) and (21) for different test examples andobtained results with the same behavior as above Hencewe could mention that the theoretical results are upheld bynumerical experiments and thus the newmethod is goodwitha high computational efficiency index
4 Summary
In this paper we have proposed a new two-step Steffensen-type iterative method with memory for solving nonlinearscalar equations Using one self-correcting parameter calcu-lated by Newton interpolatory polynomial the 119877-order ofconvergence of the constructed method was increased from4 to 423607 without any additional calculations
The new method was compared in performance andcomputational efficiency with some existing methods bynumerical examples We have observed that the computa-tional efficiency index of the presentedmethod withmemoryis better than those of other existing two-step King-typemethods
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Authorsrsquo Contribution
The authors have made the same contribution All authorsread and approved the final paper
References
[1] A S Al-Fhaid S Shateyi M Z Ullah and F Soleymani ldquoAmatrix iteration for finding Drazin inverse with ninth-orderconvergencerdquo Abstract and Applied Analysis vol 2014 ArticleID 137486 7 pages 2014
[2] J F Traub Iterative Methods for the Solution of EquationsPrentice-Hall New York NY USA 1964
[3] A Cordero and J R Torregrosa ldquoLow-complexity root-findingiteration functions with no derivatives of any order of con-vergencerdquo Journal of Computational and Applied Mathematics2014
[4] R F King ldquoA family of fourth order methods for nonlinearequationsrdquo SIAM Journal on Numerical Analysis vol 10 pp876ndash879 1973
[5] N Huang and C Ma ldquoConvergence analysis and numericalstudy of a fixed-point iterative method for solving systems ofnonlinear equationsrdquo The Scientific World Journal vol 2014Article ID 789459 10 pages 2014
[6] J P Jaiswal ldquoSome class of third- and fourth-order iterativemethods for solving nonlinear equationsrdquo Journal of AppliedMathematics vol 2014 Article ID 817656 17 pages 2014
[7] J F Steffensen ldquoRemarks on iterationrdquo Skandinavisk Aktuari-etidskrift vol 16 pp 64ndash72 1933
[8] J Dzunic ldquoOn efficient two-parameter methods for solvingnonlinear equationsrdquo Numerical Algorithms vol 63 no 3 pp549ndash569 2013
[9] A Cordero T Lotfi P Bakhtiari and J R Torregrosa ldquoAnefficient two-parametric family with memory for nonlinearequationsrdquo Numerical Algorithms 2014
[10] X Wang and T Zhang ldquoA new family of Newton-type iterativemethods with and without memory for solving nonlinearequationsrdquo Calcolo vol 51 no 1 pp 1ndash15 2014
[11] S Wagon Mathematica in Action Springer New York NYUSA 3rd edition 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
