It is attempted to derive an optimal class of methods without memory from Ozban’s method [A. Y. Ozban, Some New Variants of Newton’s Method, Appl. Math. Lett. 17 (2004) 677-682]. To this end, we try to introduce a weight function in the second step of the method and to find some suitable conditions, so that the modified method is optimal in the sense of Kung and Traub’s conjecture. Also, convergence analysis along with numerical implementations are included to verify both theoretical and practical aspects of the proposed optimal class of methods without memory. © 2020 Forum-Editrice Universitaria Udinese SRL. All rights reserved

Agarwal, Praveen

Attary, Maryam

English

Kırşehir Ahi Evran University Institutional Repository

ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS – N. 43–2020 (523–530) 523On developing an optimal Jarratt-like class for solvingnonlinear equationsMaryam Attary∗Department of MathematicsKaraj BranchIslamic Azad UniversityKarajIranmaryam.attari@kiau.ac.irPraveen AgarwalDepartment of MathematicsANAND International College of EngineeringJaipur-303012IndiaandDepartment of MathematicsAhi Evran University40100 KircontractiblesehirTurkeyAbstract. It is attempted to derive an optimal class of methods without memory fromOzban’s method [A. Y. Ozban, Some New Variants of Newton’s Method, Appl. Math.Lett. 17 (2004) 677-682]. To this end, we try to introduce a weight function in thesecond step of the method and to find some suitable conditions, so that the modifiedmethod is optimal in the sense of Kung and Traub’s conjecture. Also, convergenceanalysis along with numerical implementations are included to verify both theoreticaland practical aspects of the proposed optimal class of methods without memory.Keywords: nonlinear equations, Kung and Traub’s conjecture, iterative method,optimal method, convergence analysis.1. IntroductionThe main objective of this work is to derive an optimal class of methods withoutmemory for approximating a simple root of a nonlinear equation. For this pur-pose, we consider a non-optimal method without memory developed by Ozbanin [8]. Although this method is one of the most cited works in the literature,it is not optimal in the sense of Kung and Traub’s conjecture. Based on thisconjecture, any two-step method without memory is optimal if it has conver-gence order four using three functional evaluations per iteration [4, 12], while∗. Corresponding author524 MARYAM ATTARY and PRAVEEN AGARWALthe pointed method uses three functional evaluations per iteration and has con-vergence order three, see Theorem 4.1 in [8].There are so many two-step optimal methods without memory which werecall some of them here. To the best of our knowledge, there are three generalkinds of optimal methods without memory: Jarratt-, Ostrowski- and Steffensen-type methods. Jarratt’s method [2] is given by:(1.1)yn = xn −23f(xn)f ′(xn),xn+1 = xn −3f ′(yn) + f′(xn)6f ′(yn)− 2f ′(xn)f(xn)f ′(xn),where its error equation is en+1 = (c32− c2c3+ c49 c)e4n+O(e5n), with ck =f (k)(α)k!f ′(α) ,k = 2, 3, ..., and α is a simple zero of f(x) = 0, i.e., f ′(α) ̸= 0 = f(α). Jarratt’smethod (1.1) uses three functional evaluations per iteration and has convergenceorder four so it is optimal. In other words, it uses functional evaluation of itsderivation in two points, say f ′(xn) and f′(yn), and one functional evaluationof the given function, says f(xn), in each iteration. Such methods in which, oneuses two evaluations of the derivatives of the given functions and one evaluationof the given function are called Jarratt-type methods. Soleymani et al. [10]suggested the following optimization of Jarratt-type method:(1.2)yn = xn −23f(xn)f ′(xn),xn+1 = xn − (1 + (f(xn)f ′(xn))3)(2− 74s+34s2)2f(xn)f ′(xn) + f ′(yn).Another optimal method of this type is considered by Lotfi [5](1.3)yn = xn −23f(xn)f ′(xn),xn+1 = xn − (2−74s+34s2)2f(xn)f ′(xn) + f ′(yn).Some other optimal Jarratt-type methods and different anomalies in a Jarrattfamily can be found in the literature [5, 10]. Similar to Jarratt-type methods,there is another set of methods in which they use derivative of the functionin each iteration. However, these kinds use two function evaluations and onederivative evaluation, say f(xn), f(yn) and f′(xn). We call these kinds of itera-tive methods Ostrowski-type methods. Indeed, Ostrowski’s method is given by[3](1.4)yn = xn −f(xn)f ′(xn),xn+1 = yn −f(xn)f(xn)− 2f(yn)f(yn)f ′(xn),ON DEVELOPING AN OPTIMAL JARRATT-LIKE CLASS ... 525with the following error equation en+1 = (c32 − c2c3)e4n +O(e5n).It is worth noting that Ostrowski’s method (1.4) is a special case, b=0, ofKing’s family [3] which is defined as follows(1.5)yn = xn −f(xn)f ′(xn),xn+1 = yn −f(xn) + bf(yn)f(xn) + (b− 2)f(yn)f(yn)f ′(xn).Also, we consider the first two-step iterative method by Kung and Traub [4] asfollows yn = xn −f(xn)f ′(xn),xn+1 = yn −f(xn)2f(yn)f ′(xn)(f(xn)− f(yn))2.There is another kind of the Ostrowski-type method which can be obtained viaHermit-interpolation as follows(1.6)yn = xn −f(xn)f ′(xn),xn+1 = yn −f(yn)f [xn, yn] + f [yn, xn, xn](yn − xn).Finally, there is another kind of optimal two-step methods without memoryin which, one does not use derivatives. We call them Steffensen-type method.In what follows, we recall two of them. First, we consider the first two-stepderivative-free version of Kung and Traub [4] given by(1.7)yn = xn −f(xn)f [xn, ωn], ωn = xn + f(xn),xn+1 = yn −f(yn)f [xn, yn]f(ωn)(f(ωn)− f(yn)).Bi et al. [9] and Zheng et al. [14] simultaneously derived the following methodbased on Newton interpolation(1.8)yn = xn −f(xn)f [xn, ωn], ωn = xn + γf(xn),xn+1 = yn −f(yn)f [xn, yn] + f [yn, xn, ωn](yn − xn).All of the mentioned methods can be considered as a special case of the optimalclass of two-step methods without memory. Detailed description, convergenceand analysis of these methods may be found in [1, 3, 5, 6, 7, 12] and referencestherein .526 MARYAM ATTARY and PRAVEEN AGARWALThis work is organized as follows: Section 2 is devoted to extracting optimalmethod from non-optimal method by Ozban [8]. Furthermore, we discuss theconvergence analysis of the developed method in this section, and also someconcrete functions are given based on the developed method. Section 3 repre-sents numerical implimentations and comparisons. Finally, Section 4 concludesthis work. For some given methods in this work, we append their Mathematicacodes, too.2. Method and resultIn this section, we deal with developing a new optimal class of Jarratt-typemethods to approximate a simple zero of f(x) = 0. Also, we discuss a theoret-ical aspect of the developed class, namely convergence analysis. We recall thefollowing method by Ozban [8](2.1)yn = xn −f(xn)f ′(xn),xn+1 = xn −(f ′(xn) + f′(yn))2f ′(yn)f(xn)f ′(xn), (n = 0, 1, ...).Theorem 4.1. in [8] considers the error analysis of this method. The follow-ing self-explanatory Mathematica code decodes and deciphers the same resultsquickly. We introduce the following abbreviations used in this program.ck = f(k)(α)/(k!f′(α)), e = xn − α, ey1 = yn − α, ey = xn+1 − α, f[e] =f(e), f1a = f′(α).Program 1. Mathematica code:f [e] = f1a(e+ c2e2 + c3e3 + c4e4);ey1 = e− Series[f [e]f ′[e] , {e, 0, 3}]//Simplify;ey = e− Series[f [e](f ′[e]+f ′[ey1])2f ′[e]f ′[ey1] , {e, 0, 3}]//SimplifyOut[ey] = c3e32+ O[e]4Remark 1. As can be seen, this method is not optimal based on Kung andTraub’s conjecture. It uses three functional evaluations per iteration while ithas convergence order three. Here, our aim is to modify method (2.1) in such away that it becomes optimal. More details are given in what follows.Let us consider the following changes to (2.1). The first step of Ozban’smethod, namely Newton’s method, is exchanged with the first step of Jarratt’smethod, namely weighted Newton’s method. Then in the second step of Ozban’sON DEVELOPING AN OPTIMAL JARRATT-LIKE CLASS ... 527method, we introduce a weight function, say h(t), as follows(2.2)yn = xn −23f(xn)f ′(xn),xn+1 = xn − h(tn)f(xn)f ′(xn)(f ′(xn) + f′(yn))2f ′(yn),where tn =f ′(yn)f ′(xn). Now, it is tried to optimize this new method. To this end, weimpose some conditions on h(t) so that we achieve an optimal class of Jarratt-type methods. Instead of using pencil-paper method to discuss the mentionedaim, we prefer to use the Mathematica approach. We think this technique hasseveral advantages: it is fast, it saves space of the paper, and it avoids involvingtedious and cumbersome calculations with using many terms of Taylor’s series.We reuse the symbols introduced before in giving the error equation for themethod (2.1), also the rest of the abbreviations used are introduced as followsh = h(0), h1 = h′(0), h2 = h′′(0).Program 2. Mathematica code:f [e] = f1a(e+ c2e2 + c3e3 + c4e4);ey1 = e− Series[2∗f [e]3f ′[e] , {e, 0, 8}];t = f′[ey1]f ′[e] ;h[t] = h+ h1t+ h2t22 ;ey = e− f[e](f′[e]+f′[ey1])∗h[t]2f’[e]f′[ey1]//FullSimplify;a1 = Coefficient[ey, e]//Simplifya2 = Coefficient[ey, e2]//Simplifya3 = Coefficient[ey, e3]//Simplifya4 = Coefficient[ey, e4]//SimplifyOut[a1]=1-h− h1− h22Out[a2] = 16(2 h+10h1+ 9h2)c2Out[a3] = 19(−(2 h+42h1+ 49h2)c22 + 3 (2 h+10h1+ 9h2)c3)Out[a4] = 154((20 h+684h1+ 978h2)c32 − 3(22 h+294h1+ 347h2)c2c3+ (58 h+266h1+ 237h2)c4))To obtain an optimal class, the coefficients of e, e2, and e3 in the errorequation of the new class (2.1) need to vanish. By solving the above system ofequations simultaneously, equations Out[a1], Out[a2] and Out[a3], the desiredresults are obtained and we have{{h → 74, h1 → −54, h2 → 1}}. In other words,to provide the fourth order of convergence of the proposed method, it is neces-sary to choose h = 74 , h1 =−54 and h2 = 1. Therefore, we have established thefollowing theorem about the convergence order of the optimal class (2.2).Theorem 1. Let α be a simple zero of f(x) = 0 and function h(t) is cho-528 MARYAM ATTARY and PRAVEEN AGARWALsen so that the conditions h(0) = 74 , h′(0) = −54 , and h′′(0) = 1 hold. If aninitial approximation is sufficiently close to α, then the equation (2.2) has theorder of convergence four with the following error equation(2.3) en+1 = (7927c32 − c2c3 +c49)e4n +O(e5n).The function h(t) can take many forms satisfying the conditions of Theorem 1,examples of which are:h1(t) =74− 54t+12t2, h2(t) =147+2049t+44343t2, h3(t) =74− 12t− 34t21 + t.Accordingly, we can consider the following optimal method as a typical ex-ample of our proposed class(2.4)yn = xn −23f(xn)f ′(xn),xn+1 = xn − (74− 54tn +12t2n)f(xn)f ′(xn)(f ′(xn) + f′(yn))2f ′(yn).3. Numerical implimentationsTo verify the applicability of the derived method (2.2) of the optimal class ofJarratt-type methods we give two examples. Also, we report the results of theother methods given in this work for comparison. The implementations wereran in Mathematica. In Tables 1 and 2, the values of the computational orderof convergence are computed by the following approximate formula ( see Weer-akoon and Fernando [13])coc =ln(|xn+1 − α|/|xn − α|)ln(|xn − α|/|xn−1 − α|),where |xn − α| denots absolute errors of approximations and a(−b) meansa× 10−b.Example 1. Consider the following nonlinear equationf(x) = e2+x−x2 − cos(1 + x) + x3 + 1, α = −1,with the initial approximation x0 = −0.7.Example 2. Consider the following nonlinear equationf(x) = ln(1 + x2) + e−3x+x2sin(x), α = 0,with the initial approximation x0 = 0.35.We have reported the obtained numerical results in Table 1 and 2. These resultsON DEVELOPING AN OPTIMAL JARRATT-LIKE CLASS ... 529Table 1: Numerical results of Example 1 in the first three iterationsAbsolute ErrorTwo-point methods |x1 − α| |x2 − α| |x3 − α| cocNew Method (2.4) 0.2189(−3) 0.1566(−15) 0.4104(−64) 4Method (1.4) 0.4557(−3) 0.2790(−14) 0.3925(−59) 4Method (1.7) 0.4357(−1) 0.1170(−5) 0.5534(−23) 4Method (1.1) 0.6543(−3) 0.1411(−13) 0.3056(−56) 4Method (1.2) 0.6113(−2) 0.1490(−8) 0.5245(−35) 4Table 2: Numerical results of Example 2 in the first three iterationsAbsolute ErrorTwo-point methods |x1 − α| |x2 − α| |x3 − α| cocNew Method (2.4) 0.1965(−2) 0.2035(−9) 0.2326(−37) 4Method (1.4) 0.5733(−2) 0.2999(−8) 0.2158(−33) 4Method (1.7) 0.8517(−2) 0.1282(−6) 0.5757(−26) 4Method (1.3) 0.1990(−2) 0.3071(−9) 0.1734(−36) 4Method (1.2) 0.1948(−1) 0.3038(−5) 0.1747(−20) 4confirm the theoretical prediction, which has been proved in the previous section.Moreover, it can be concluded that the proposed method (2.4) generates slightlybetter results in comparison with the other numerical methods mentioned in thispaper.4. ConclusionIn this research, a new optimal fourth order method based on Ozban’s methodhas been developed for solving simple roots of nonlinear equations. The pre-sented method has the convergence order four. It supports the Kung and Traub’sconjecture requiring only three function evaluations per iteration and it has theefficiency index 41/3 ≈ 1.587, which is better than Ozban’s method 31/3 ≈ 1.390( for the definition of efficiency index see [11]).References[1] A. Corderoa, J. M. Gutierrezb, A. A. Magrenan, J. R. Torregrosa, Stabilityanalysis of a parametric family of iterative methods for solving nonlinearmodels, Appl. Math. Comput., 285 (2016), 26-40.[2] P. Jarratt, Some efficient fourth order multiple methods for solving equa-tions, BIT, 9 (1969), 119-124.530 MARYAM ATTARY and PRAVEEN AGARWAL[3] R. King, A family of fourth order methods for nonlinear equations, SIAMJ. Numer. Anal., 10 (1973), 876-879.[4] H. T. Kung, J. F. Traub, Optimal order of one-point and multipoint itera-tion, J. ACM, 21 (1974), 643-651.[5] T. Lotfi, A new optimal method of fourth-order convergence for solvingnonlinear equations, Int. J. Indust. Math., 6 (2008).[6] T. Lotfi, P. Assari, New three- and four-parametric iterative with memorymethods with efficiency index near 2, Appl. Math. Comput., 270 (2015),1004-1010.[7] T. Lotfi, P. Assari, A new two step class of methods with memory for solv-ing nonlinear equations with high efficiency index, International Journal ofMathematical Modelling and Computations, 4 (2014), 277-288.[8] A. Y. Ozban, Some new variants of Newton’s method, Appl. Math. Lett.,17 (2004), 677-682.[9] H. Ren, Q. Wu, W. Bi, A class of two-step Steffensen type methods withfourth-order convergence, Appl. Math. Comput., 209 (2009), 206-210.[10] F. Soleymani, SK. Khattri, S. Karimi Nanani, Two new classes of optimalJarratt-type fourth-order methods, Appl. Math. Lett., 25 (2005), 847-853.[11] F. Soleymani, T. Lotfi, P. Bakhtiari, A multi-step class of iterative methodsfor nonlinear systems, Optim. Lett., 8 (2004), 1001-1015.[12] J. F. Traub, Iterative methods for solution of equations, Prentice-Hall, En-glewood Cliff, New Jersey, 1964.[13] S. Weerakoon, T. G. I. Fernando, A variant of Newton’s method with ac-celerated third-order convergence, Appl. Math. Lett., 13 (2000), 87-93.[14] Q. Zheng, J. Li, F. Huang, An optimal Steffensen-type family for solvingnonlinear equations, Appl. Math. Comput., 217 (2011), 9592-9597.Accepted: 2.12.2018

On developing an optimal Jarratt-like class for solving nonlinear equations

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On developing an optimal Jarratt-like class for solving nonlinear equations

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