We present a new third order convergence iterative method for solving multiple roots of nonlinear equation, which requires one function evaluation and two evaluation of first derivative of function per step. Our present method free from second derivative function. Error term is proved to possess a third order method. Numerical experiments exhibit that our method gives the smallest error of bound per iteration and it is highly accurate as compared to other existing iterative methods

Ismail, Fudziah

Jamaludin, Nur Alif Akid

Nik Long, Nik Mohd Asri

Salimi, Mehdi

Universiti Putra Malaysia Institutional Repository

Malaysian Journal of Mathematical Sciences 13(S) April: 51–59 (2019)Special Issue: 3rd International Conference on Mathematical Sciences and Statistics(ICMSS2018)MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCESJournal homepage: http://einspem.upm.edu.my/journalThird Order Convergence Iterative Method forMultiple Roots of Nonlinear EquationJamaludin, N. A. A.1, Nik Long, N.M.A. ∗1,2, Salimi, M.3, andIsmail, F.1,21Department of Mathematics, Faculty of Science, Universiti PutraMalaysia, Malaysia2Institute for Mathematical Research, Universiti Putra Malaysia,Malaysia3Center for Dynamics and Institute for Analysis, Department ofMathematics, Technische UniversitÃt Dresden, GermanyE-mail: nmasri@upm.edu.my∗Corresponding authorABSTRACTWe present a new third order convergence iterative method for solvingmultiple roots of nonlinear equation, which requires one function eval-uation and two evaluation of first derivative of function per step. Ourpresent method free from second derivative function. Error term is provedto possess a third order method. Numerical experiments exhibit that ourmethod gives the smallest error of bound per iteration and it is highlyaccurate as compared to other existing iterative methods.Keywords: Multi-point iterative methods, Multiple roots, Nonlinearequations, Order of convergence, Root-finding.Jamaludin, N. A. A., Nik Long, N.M.A., Salimi, M. & Ismail, F.1. IntroductionSolving nonlinear equations become the subject of interest to many re-searchers (see Chun et al. (2009), Dong (1982, 1987), Ferrara et al. (2015),Homeier (2009), Neta (2008), Victory Jr and Neta (1983), Zhou et al. (2013)).The well known modified Newton’s method Schroeder and Stewart (1998) iswritten asxk+1 = xk −m f(xk)f ′(xk), (1)which is of second order convergence. Victory and Neta Victory Jr and Neta(1983) developed two-point third order iterative method for solving multipleroots which is given byyk = xk − f(xk)f ′(xk),xk+1 = yk − f(yk)f ′(xk)· f(xk) +Af(yk)f(xk) +Bf(yk),(2)where A = µ2m − µm+1, B = −µm(m− 2)(m− 1) + 1(m− 1)2 and µ =mm− 1 . Equa-tion (2) requires two evaluations of function, f and one evaluation of firstderivative of function, f′. Osada (1994) proposed the iterative method forfinding the multiple roots of nonlinear equation, written asxk+1 = xk − 12m(m+ 1)f(xk)f ′(xk)+12(m− 1)2 f′(xk)f ′′(xk). (3)This iterative method requires one evaluation of function, f , two evalua-tions of f ′ and one evaluation of second derivative, f ′′. By combining Osada’smethod (3) and Euler-Chebyshev’s method Traub (1977), Chun et al. Chunet al. (2009) produced the new iterative method for multiple zeros given byxk+1 = xk−m((2γ − 1)m+ 3− 2γ)2f(xk)f ′(xk)+γ(m− 1)22f ′(xk)f ′′(xk)−m2(1− γ)2f(xk)2f ′′(xk)(f ′(xk))3(4)where,A = µ2m − µm+1, B = −µm(m− 2)(m− 1) + 1(m− 1)2 , µ =mm− 1 , γ = −1 or12.Neta Neta (2008) developed the new iterative method for multiple zeroswith the same number of function evaluations as in (2), written as yk = xk −f(xk)f ′(xk),xk+1 = yk − f(xk)f ′(xk)(β + γf(yk)f ′(xk)),(5)52 Malaysian Journal of Mathematical SciencesThird Order Convergence Iterative Method for Multiple Roots of Nonlinear Equationwhere,β = m− m(m− θ)θ2, γ =m(m− θ)ρθ2, ρ =(m− θm)mand θ ∈ R.In this paper we develop the iterative method which is free from the evalu-ation of second derivative of functions.2. Construction of MethodThe well known Chebyshev’s method for computing the simple iterativemethod is given by Petkovic et al. (2012)xk+1 = xk − f(xk)f ′(xk)− f′′(xk)2f(xk)(f(xk)f ′(xk))2. (6)Equation (6) requires one evaluation of function, one evaluation of firstderivative function and one evaluation of second derivative function. Let theNewton-type iterative method bexk+1 = xk − 2mm+ 2f(xk)f ′(xk). (7)Expand f ′(xk+1) in (7) about x = xk yields,f ′′(xk) ∼= (m+ 2)f′(xk) [f ′(xk)− f ′(yk)]2mf(xk). (8)Subtituting (8) into (6) givesyk = xk − 2mm+ 2f(xk)f ′(xk),xk+1 = yk − (m+ 2)f(xk)4m[f ′(xk)− f ′(yk)(f ′(xk))2].(9)We introduced the free disposable parameters, α and β, such that (9) becomesyk = xk − 2mm+ 2f(xk)f ′(xk),xk+1 = yk − (m+ 2)f(xk)4m[αf ′(xk)− βf ′(yk)(f ′(xk))2].(10)Malaysian Journal of Mathematical Sciences 53Jamaludin, N. A. A., Nik Long, N.M.A., Salimi, M. & Ismail, F.3. Convergence AnalysisTheorem 1. Let x∗ ∈ D be a multiple roots of a sufficiently smooth functionf : D ⊆ R → R defined on an open interval D with the multiplicity m > 1,which includes x0 as an initial approximation of x∗. Then, the iterative methodsdefined by (10) has third order convergence when α =m3 (m(m+ 4) + 8)(m+ 2)2 ,β = m−m+4 (m+ 2)m−1 ,(11)with the error termen+1 =2c21e3nm3+O(e4n).Proof. Let en := xn−x∗, en,y := yn−x∗, ci := m!(m+ i)!fm+i(x∗)fm(x∗), c0 = 1, p =m + 1, q = m + 2, r = m − 1. Since f(x∗) = 0, Taylor expansion of f at x∗yieldsf(xn) = emn(1 + c1en + c2e2n + c3e3n)+O(e4n), (12)andf ′(xn) = er(m+ enpc1 + e2n(q)c2 + e3n((m+ 3)c3 +O(e4n))), (13)henceen,y = yn − x∗ = menq+2c1e2nm(q)+(−2(p)c21 + 4mc2) e3nm2(q)+O(e4n), (14)For f(yn) we havef(yn) = emn,y(1 + c1en,y + c2e2n,y + c3e3n,y)+O(e4n,y). (15)Substituting (12)-(15) into (10) gives the error term asen+1 = D1en +D2e2n +D3e3n +O(e4n), (16)whereD1 =mm+ 2+−m(m+ 2)α+mm(m+ 2)−m+2β4m3, (17)D2 =q−p(m2qm(8m2 + q2α)−mmq2(4 +mq)β)c14m5, (18)54 Malaysian Journal of Mathematical SciencesThird Order Convergence Iterative Method for Multiple Roots of Nonlinear EquationD3 =14m7q−p(H + Jmmq2β), (19)H = −m3qm (8m2 + q2α) (pc21 − 2mc2) (20)andJ =((8 +mp(4 +mq))c21 − 2m2(4 +mq)c2). (21)To obtain the convergence order of three, it is necessary to choose Di = 0(i = 1, 2) which yieldα =m3 (m(m+ 4) + 8)(m+ 2)2 ,andβ = m−m+4 (m+ 2)m−1 ,and the error terms becomeen+1 =2c21e3nm3+O(e4n),which complete the proof.4. Numerical AnalysisTo analyse the performance of the proposed method, we conduct the nu-merical comparison by using different type of test functions listed in Table 1.We utilized the software package Mathematica 11 with 200 significant digitmulti-precision and compute the error bound, (EB) the computational order ofconvergence (COC) see Weerakoon and Fernando (2000) and the approximatedcomputational order of convergence (ACOC)as in Grau-Sánchez et al. (2010),which are defined respectively as EB = |xk − x∗|,COC ≈ ln |(xk+1 − α)/(xk − α)|ln |(xk − α)/(xk−1 − α)| (22)andACOC ≈ ln |(xk+1 − xk)/(xk − xk−1)|ln |(xk − xk−1)/(xk−1 − xk−2)| . (23)The following are the list of existing third order iterative methods for mul-tiple roots :Malaysian Journal of Mathematical Sciences 55Jamaludin, N. A. A., Nik Long, N.M.A., Salimi, M. & Ismail, F.1. Dong’s method (DM) Dong (1987), given byyk = xk − f(xk)f ′(xk),xk+1 = yk − f(xk)(mm− 1)m+1f ′(yk) +m−m2 − 1(m− 1)2 f′(xk). (24)2. Chun et al.’s method (CBN) Chun et al. (2009), given in (4) with thevalue of γ = −1.3. Hommier’s method (HM) Homeier (2009), given byyk = xk − f(xk)f ′(xk),xk+1 = xk −m2(mm+ 1)m−1f(xk)f ′(xk)+m(m− 1) f(xk)f ′(xk).(25)4. Zhou et al.’s method (ZCS) Zhou et al. (2013), given byyk = xk − f(xk)f ′(xk),xk+1 = xk +m(m− 2) f(xk)f ′(xk)−m(m− 1)(mm− 1)mf(yk)f ′(xk).(26)5. Ferrara et al.’s method (FSS) Ferrara et al. (2015), given byyk = xk − f(xk)f ′(xk),xk+1 = xk − θf(xk)θf(xk)− f(yk)f(xk)f ′(xk),(27)where θ =(−1 +mm)−1+m.Table 2 shows that from test functions f1-f5 the absolute error of our PMmethod gives smallest value as compared to other. The value of COC andACOC proved that our method posses as third order convergence method.Thus, our method is converge faster and give more accurate numerical resultsas compared to other existing iterative methods.56 Malaysian Journal of Mathematical SciencesThird Order Convergence Iterative Method for Multiple Roots of Nonlinear EquationTable 1: List of test functionsTest functions Roots Multiplicityfn x∗ mf1(x) = (sin2 x+ x)5 0 5f2(x) = (ln (1 + x2) + ex2−3x sinx)6 0 6f3(x) = (x3 + ln (1 + x))7 0 7f4(x) = (x6 − 8)2 ln (x6 − 7)√2 3f5(x) = (ln(x3 − x+ 1) + 4 sinx− 1)10 1 10Table 2: Error, COC and ACOC of test functionsMethods PM DM CBN HM ZCS FSSf1, x0 = 0.1|x1 − x∗| 0.455e−3 0.420e−3 0.113e−2 0.115e−2 0.131e−2 0.740e−3|x2 − x∗| 0.379e−10 0.314e−10 0.215e−8 0.242e−8 0.424e−8 0.364e−9|x3 − x∗| 0.217e−31 0.132e−31 0.150e−25 0.227e−25 0.145e−24 0.434e−28COC 3.0000 3.0000 3.0000 3.0000 3.0000 3.0000ACOC 3.0001 3.0000 2.9998 2.9997 2.9996 2.9999f2, x0 = 0.3|x1 − x∗| 0.114e−1 0.479e−1 0.623e−1 0.503e−1 0.544e−1 0.564e−1|x2 − x∗| 0.205e−5 0.116e−3 0.665e−3 0.478e−3 0.711e−3 0.178e−4|x3 − x∗| 0.115e−16 0.223e−11 0.320e−9 0.300e−9 0.116e−8 0.437e−14COC 3.0000 3.0001 3.0011 3.0001 3.0002 3.0000ACOC 3.0040 2.9490 3.2118 3.0735 3.0804 2.7447f3, x0 = 0.2|x1 − x∗| 0.644e−3 0.781e−2 0.104e−1 0.797e−2 0.881e−2 0.925e−2|x2 − x∗| 0.189e−10 0.376e−6 0.967e−6 0.283e−6 0.0.418e−6 0.702e−6|x3 − x∗| 0.486e−33 0.425e−19 0.830e−18 0.134−19 0.0.482e−19 0.316e−18COC 3.0000 3.0000 3.0000 3.0000 3.0000 3.0000ACOC 2.9995 2.9986 2.9934 2.9936 2.9921 2.9972f4, x0 = 1.5|x1 − x∗| 0.956e−3 0.221e−2 0.422e−2 0.299e−2 0.329e−2 0.329e−2|x2 − x∗| 0.944e−8 0.329e−6 0.592e−5 0.631e−6 0.915e−6 0.163e−5|x3 − x∗| 0.848−23 0.970e−18 0.129e−13 0.551e−17 0.184e−16 0.171e−15COC 3.0000 3.0000 3.0000 3.0000 3.0000 3.0000ACOC 3.0059 3.0134 3.0339 3.0088 3.0085 3.0207f5, x0 = 1.2|x1 − x∗| 0.552e−4 0.146e−2 0.182e−2 0.152e−2 0.165e−2 0.164e−2|x2 − x∗| 0.337e−15 0.688e−9 0.171e−8 0.832e−9 0.0.116e−8 0.109e−8|x3 − x∗| 0.763e−49 0.719e−28 0.141e−26 0.137e−27 0.404e−27 0.322e−27COC 3.0000 3.0000 3.0000 3.0000 3.0000 3.0000ACOC 3.0001 2.9999 2.9998 2.9998 2.9998 2.9999Malaysian Journal of Mathematical Sciences 57Jamaludin, N. A. A., Nik Long, N.M.A., Salimi, M. & Ismail, F.5. ConclusionIn this paper, we developed a new third order convergence iterative methodto obtain multiple roots for nonlinear equation. The new proposed methodrequires one function evaluation and two evaluation of the first derivatives.From the numerical, the new method shows rapid convergence compared toother existing methods in the scientific literature.ReferencesChun, C., Bae, H. J., and Neta, B. (2009). New families of nonlinear third-order solvers for finding multiple roots. Computers and Mathematics withApplications, 57(9):1574–1582.Dong, C. (1982). A basic theorem of constructing an iterative formula of thehigher order for computing multiple roots of an equation. Mathematics Nu-merical Sinica, 11:445–450.Dong, C. (1987). A family of multipoint iterative functions for finding multipleroots of equations. International Journal of Computer Mathematics, 21(3-4):363–367.Ferrara, M., Sharifi, S., and Salimi, M. (2015). Computing multiple zeros byusing a parameter in Newton-Secant method. SeMA Journal, pages 1–9.Grau-Sánchez, M., Noguera, M., and Gutiérrez, J. M. (2010). On some compu-tational orders of convergence. Applied Mathematics Letters, 23(4):472–478.Homeier, H. H. H. (2009). On Newton-type methods for multiple roots withcubic convergence. Journal of Computational and Applied Mathematics,231(1):249–254.Neta, B. (2008). New third order nonlinear solvers for multiple roots. AppliedMathematics and Computations, 202(1):162–170.Osada, N. (1994). An optimal multiple root-finding method of order three.Journal of Computational and Applied Mathematics, 51(1):131–133.Petkovic, M., Neta, B., Petkovic, L., and Dzunic, J. (2012). Multipoint methodsfor solving nonlinear equations. Academic Press, London.Schroeder, E. and Stewart, G. W. (1998). On infinitely many algorithms forsolving equations. Instititue for Advanced Computer Studies, University ofMaryland, College Park,, UMIACS-TR-92–121.58 Malaysian Journal of Mathematical SciencesThird Order Convergence Iterative Method for Multiple Roots of Nonlinear EquationTraub, J. F. (1977). Iterative methods for the solution of equations. New York.Victory Jr, H. D. and Neta, B. (1983). A higher order method for multiple ze-ros of nonlinear functions. International Journal of Computer Mathematics,12(3-4):329–335.Weerakoon, S. and Fernando, T. G. I. (2000). A variant of Newton’s methodwith accelerated third-order convergence. Applied Mathematics Letters,13(8):87–93.Zhou, X., Chen, X., and Song, Y. (2013). Families of third and fourth ordermethods for multiple roots of nonlinear equations. Applied Mathematics andComputation, 219(11):6030–6038.Malaysian Journal of Mathematical Sciences 59

Third order convergence iterative method for multiple roots of nonlinear equation

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Third order convergence iterative method for multiple roots of nonlinear equation

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