Original article Biomath 2 (2013), 1305155, 1–7 B f Volume ░, Number ░, 20░░ BIOMATH ISSN 1314-684X Editor–in–Chief: Roumen Anguelov B f BIOMATH h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / Biomath Forum On some multipoint methods arising from optimal in the sense of Kung–Traub algorithms Nikolay Kyurkchiev Faculty of Mathematics and Informatics Plovdiv University Plovdiv, Bulgaria Institute of Mathematics and Informatics Bulgarian Academy of Sciences Sofia, Bulgaria Email: nkyurk@uni-plovdiv.bg Anton Iliev Faculty of Mathematics and Informatics Plovdiv University Plovdiv, Bulgaria Institute of Mathematics and Informatics Bulgarian Academy of Sciences Sofia, Bulgaria Email: aii@uni-plovdiv.bg Received: 20 April 2013, accepted: 15 May 2013, published: 29 June 2013 Abstract—In this paper we will examine self-accelerating in terms of convergence speed and the corresponding index of efficiency in the sense of Ostrowski–Traub of certain standard and most commonly used in practice multipoint iterative methods using several initial approximations for numerical solution of nonlinear equations due to optimal in the sense of the Kung–Traub algorithm of order 4, 8 and 16. Some hypothetical iterative procedures generated by algorithms from order of convergence 32 and 64 are also studied (the receipt and publication of which is a matter of time, having in mind the increased interest in such optimal algorithms). The corresponding model theorems for their convergence speed and efficiency index have been formulated and proved. Keywords-solving nonlinear equations; order of conver- gence; optimal algorithm; efficiency index I. INTRODUCTION One of the most basic problems in scientific and engineering applications is to find the solution of a nonlinear equation f(x) = 0. (1) In literature, it is known that the computational effi- ciency of a method is measured by the concept of the efficiency index p 1 n , where p is the order of convergence and n is the whole number of functional evaluations per iteration. Subsequently, the maximum efficiency index for Newton’s iteration with two functional evaluations is 2 1 2 ≈ 1.414 [30]. According to the conjecture of Kung and Traub [13], the maximum convergence order of a scheme (without memory) including n evaluations per step is 2n−1. By taking into account the optimality concept, many authors have tried to build iterative procedures of optimal order of convergence p = 4, p = 8, p = 16. The recent results of M. Petkovic [20] and M. Petkovic and L. Petkovic [22], Bi, Wu and Ren [2], Geum and Kim [7], Thurkal and Petkovic [29], Wang and Liu [31], Kou, Wang and Sun [12], Chun and Neta [3], Soleymani and Soleymani [24], Soleymani [25], Bi, Ren and Wu [1], Sargolzaei and Soleymani [26], Soleymani and Mousavi [27], Soleymani and Sharifi [28], Igna- tova, Kyurkchiev and Iliev [10], M. Petkovic, Neta, L. Petkovic and Dzunic [21] are presented for optimal multipoint methods for solving nonlinear equations. For other results see Dzunic and M. Petkovic [6]. M. Petkovic [20] gives a useful detailed review about computational efficiency of many methods in the sense of Kung–Traub hypothesis. For other nontrivial methods for solving nonlinear equations see, Kyurkchiev and Iliev [14] and Iliev and Citation: N Kyurkchiev, A Iliev, On some multipoint methods arising from optimal in the sense of Kung–Traub algorithms, Biomath 2 (2013), 1305155, http://dx.doi.org/10.11145/j.biomath.2013.05.155 Page 1 of 7 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2013.05.155 N Kyurkchiev at al., On some multipoint methods arising from optimal in the sense of Kung–Traub algorithms Kyurkchiev [11]. In many natural science tasks, from purely physical considerations, the user of numerical algorithms for solving nonlinear equation (1) knows a set of initial approximations x01,x 0 2, . . . ,x 0 k for the root ξ of equation (1). As an example, regula falsi methods and modifications of Euler–Chebyshev method and Halley method with a lower order of convergence use two or three initial approximations for the root ξ. In [16], refined conditions of convergence for the difference analogue of Halley method (using three initial approximations) for solving nonlinear equation are given (see, also [32]). An efficient modification of a finite–difference ana- logue of Halley method is proposed in [9]. Naturally arises the task of designing and testing multipoint variants of the classical procedures in the light of the achievements over the past five years important theoretical results related to obtaining optimal in the sense of Kung–Traub algorithms. In this sense the task of detailed refinement of the self-accelerating multipoint methods using several initial approximations become very actual. II. MAIN RESULTS A. Optimal algorithm in the sense of Kung–Traub with order of convergence p = 4 We consider the following nonstationary iterative scheme based on the 4-point iteration function in combi- nation with an optimal algorithm in the sense of Kung– Traub with order of convergence p = 4: x2n+1 = ϕ1(x2n,x2n−1,x2n−2,x2n−3), x2n+2 = ϕ2(x2n+1). (2) It is known that for the error �i = xi − ξ, i = −3,−2,−1,0,1,2 . . . ; [30] is valid �2n+1 ∼ C1(ξ)�2n�2n−1�2n−2�2n−3, (3) �2n+2 ∼ C2(ξ)�42n+1. (4) Let K9 = max{|C1(ξ))|, |C2(ξ)|} , d2n−1 = K 1 3 9 |�2n−1|, d2n = K 1 3 9 |�2n|, and let 0 < d < 1, and x−3, x−2, x−1 and x0 be chosen so that the following inequalities d−3 = K 1 3 9 |x−3 − ξ| ≤ d < 1, d−2 = K 1 3 9 |x−2 − ξ| ≤ d < 1, d−1 = K 1 3 9 |x−1 − ξ| ≤ d < 1, d0 = K 1 3 9 |x0 − ξ| ≤ d < 1 hold true. From (3) and (4), we have d2n+1 = K 1 3 9 |�2n+1| ≤ K 1 3 9 K9|�2n||�2n−1||�2n−2||�2n−3| = K 1 3 9 |�2n−1|K 1 3 9 |�2n|K 1 3 9 |�2n−2|K 1 3 9 |�2n−3| = d2nd2n−1d2n−2d2n−3, d2n+2 = K 1 3 9 |�2n+2| ≤ K 1 3 9 K9� 4 2n+1 = ( K 1 3 9 �2n+1 )4 = d42n+1. (5) Evidently, from (5), we find d1 ≤ d4, d2 ≤ d16, d3 ≤ d22, d4 ≤ d88, d5 ≤ d130, d6 ≤ d520, d7 ≤ d760, d8 ≤ d3040, d9 ≤ d4450, d10 ≤ d17800. Our results concerning the order of convergence gener- ated by (2) are summarized in the following theorem. Theorem A. Assume that the initial approximations x0, x−1, x−2, x−3 are chosen so that d−3 ≤ d, d−2 ≤ d, d−1 ≤ d < 1 and d0 ≤ d < 1. Then for the error of the sequences {x2n+1}∞n=0 and {x2n+2} ∞ n=0 determined by (2), we have d2n−1 ≤ dτ2n−1, d2n ≤ dτ2n, (6) where τm+4 = 5τm+2 + 5τm, m = 1,2, . . . (7) and the order of convergence of the iteration (2) is τ = 5 + 3 √ 5 2 . Proof. It is well known that the recursion: γi+1 = n∑ j=1 Ajγi−j+1, i = n−1,n−2, . . . , Biomath 2 (2013), 1305155, http://dx.doi.org/10.11145/j.biomath.2013.05.155 Page 2 of 7 http://dx.doi.org/10.11145/j.biomath.2013.05.155 N Kyurkchiev at al., On some multipoint methods arising from optimal in the sense of Kung–Traub algorithms (for any initial conditions) corresponds to the character- istic polynomial: ρn = n∑ j=1 Ajρ n−j. In our case, for the recursion τm+4 = 5τm+2 + 5τm, the characteristic polynomial is of the type ρ2 −5ρ−5 = 0. (8) Equation (8) has the roots: ρ1 = 5 + 3 √ 5 2 , ρ2 = 5−3 √ 5 2 . From the general iterative theory [30], (see, also [8]) it follows that the order of convergence of the iteration procedure, defined by (2) is given by the only real root of equation (8) with magnitude greater than 1. On the other hand, |�2n+1| ≤ K −1 3 9 d2n+1, |�2n+2| ≤ K −1 3 9 d2n+2, and consequently we can conclude that the order of convergence of iteration (2) is τ = 5 + 3 √ 5 2 ≈ 5.8541... Thus, the theorem is proven. B. Optimal algorithm in the sense of Kung–Traub with order of convergence p = 8 We consider the following nonstationary iterative scheme based on the 4-point iteration function in combi- nation with an optimal algorithm in the sense of Kung– Traub with order of convergence p = 8: x2n+1 = ϕ1(x2n,x2n−1,x2n−2,x2n−3), x2n+2 = ϕ3(x2n+1). (9) For the error �i = xi − ξ, i = −3,−2,−1,0,1,2 . . . ; [30], [23] is valid �2n+1 ∼ C1(ξ)�2n�2n−1�2n−2�2n−3, (10) �2n+2 ∼ C3(ξ)�82n+1. (11) Let K10 = max{|C1(ξ))|, |C3(ξ)|} , d2n−1 = K 3 17 10 |�2n−1|, d2n = K 7 17 10 |�2n|, and let d > 0, and x−3, x−2, x−1 and x0 be chosen so that the following inequalities d−3 = K 3 17 10 |x−3 − ξ| ≤ d < 1, d−2 = K 7 17 10 |x−2 − ξ| ≤ d < 1, d−1 = K 3 17 10 |x−1 − ξ| ≤ d < 1, d0 = K 7 17 10 |x0 − ξ| ≤ d < 1 hold true. From (10) and (11), we have d2n+1 = K 3 17 10 |�2n+1| ≤ K 3 17 10 K10|�2n||�2n−1||�2n−2||�2n−3| = K 3 17 10 K 3 17 + 7 17 + 7 17 10 |�2n||�2n−1||�2n−2||�2n−3| = d2nd2n−1d2n−2d2n−3, d2n+2 = K 7 17 10 |�2n+2| ≤ K 7 17 10 K10� 8 2n+1 = ( K 3 17 10 �2n+1 )8 = d82n+1. (12) From (12), we find d1 ≤ d4, d2 ≤ d32, d3 ≤ d38, d4 ≤ d304, d5 ≤ d378, d6 ≤ d3024, d7 ≤ d3744, d8 ≤ d29952, d9 ≤ d37098, d10 ≤ d296784. Our results concerning the order of convergence gener- ated by (9) are summarized in the following theorem. Theorem B. Assume that the initial approximations x0, x−1, x−2, x−3 are chosen so that d−3 ≤ d, d−2 ≤ d, d−1 ≤ d < 1 and d0 ≤ d < 1. Then for the error of the sequences {x2n+1}∞n=0 and {x2n+2} ∞ n=0 determined by (9), we have d2n−1 ≤ dτ2n−1, d2n ≤ dτ2n, (13) where τm+4 = 9τm+2 + 9τm, m = 1,2, . . . (14) and the order of convergence of the iteration (9) is τ = 3 ( 3 + √ 13 ) 2 . Proof. In our case, for the recursion τm+4 = 9τm+2 + 9τm, Biomath 2 (2013), 1305155, http://dx.doi.org/10.11145/j.biomath.2013.05.155 Page 3 of 7 http://dx.doi.org/10.11145/j.biomath.2013.05.155 N Kyurkchiev at al., On some multipoint methods arising from optimal in the sense of Kung–Traub algorithms the characteristic polynomial is of the type ρ2 −9ρ−9 = 0. (15) Equation (15) has the roots: ρ1 = 3 ( 3 + √ 13 ) 2 , 3 ( 3− √ 13 ) 2 . From the general iterative theory it follows that the order of convergence of the iteration procedure, defined by (9) is given by the only real root of equation (15) with magnitude greater than 1. On the other hand, |�2n+1| ≤ K − 3 17 10 d2n+1, |�2n+2| ≤ K − 7 17 10 d2n+2, and consequently we can conclude that the order of convergence of iteration (9) is τ = 3 ( 3 + √ 13 ) 2 ≈ 9.9083... Thus, the theorem is proven. C. Optimal algorithm in the sense of Kung–Traub with order of convergence p = 16 We consider the following nonstationary iterative scheme based on the 4-point iteration function in combi- nation with an optimal algorithm in the sense of Kung– Traub with order of convergence p = 16: x2n+1 = ϕ1(x2n,x2n−1,x2n−2,x2n−3), x2n+2 = ϕ4(x2n+1). (16) It is known that for the error �i = xi − ξ, i = −3,−2,−1,0,1,2 . . . ; [30] is valid �2n+1 ∼ C1(ξ)�2n�2n−1�2n−2�2n−3, (17) �2n+2 ∼ C4(ξ)�162n+1. (18) Let K11 = max{|C1(ξ))|, |C4(ξ)|} , d2n−1 = K 1 11 11 |�2n−1|, d2n = K 5 11 11 |�2n|, and let d > 0, and x−3, x−2, x−1 and x0 be chosen so that the following inequalities d−3 = K 1 11 11 |x−3 − ξ| ≤ d < 1, d−2 = K 5 11 11 |x−2 − ξ| ≤ d < 1, d−1 = K 1 11 11 |x−1 − ξ| ≤ d < 1, d0 = K 5 11 11 |x0 − ξ| ≤ d < 1 hold true. From (17) and (18), we have d2n+1 = K 1 11 11 |�2n+1| ≤ K 1 11 11 K11|�2n||�2n−1||�2n−2||�2n−3| = K 1 11 11 K 1 11 + 1 11 + 1 11 11 |�2n|�2n−1||�2n−2|�2n−3| = d2nd2n−1d2n−2d2n−3, d2n+2 = K 5 11 11 |�2n+2| ≤ K 5 11 11 K11� 16 2n+1 = ( K 1 11 11 �2n+1 )16 = d162n+1. (19) From (19), we find d1 ≤ d4, d2 ≤ d64, d3 ≤ d70, d4 ≤ d1120, d5 ≤ d1258, d6 ≤ d20128, d7 ≤ d22576, d8 ≤ d361216, d9 ≤ d405178, d10 ≤ d6482848. Our results concerning the order of convergence gener- ated by (19) are summarized in the following theorem. Theorem C. Assume that the initial approximations x0, x−1, x−2, x−3 are chosen so that d−3 ≤ d, d−2 ≤ d, d−1 ≤ d < 1 and d0 ≤ d < 1. Then for the error of the sequences {x2n+1}∞n=0 and {x2n+2} ∞ n=0 determined by (16), we have d2n−1 ≤ dτ2n−1, d2n ≤ dτ2n, (20) where τm+4 = 17τm+2 + 17τm, m = 1,2, . . . (21) and the order of convergence of the iteration (16) is τ = ( 17 + √ 357 ) 2 . Proof. In our case, for the recursion τm+4 = 17τm+2 + 17τm, the characteristic polynomial is of the type ρ2 −17ρ−17 = 0. (22) Equation (22) has the roots: ρ1 = ( 17 + √ 357 ) 2 , ( 17− √ 357 ) 2 . From the general iterative theory it follows that the order of convergence of the iteration procedure, defined by Biomath 2 (2013), 1305155, http://dx.doi.org/10.11145/j.biomath.2013.05.155 Page 4 of 7 http://dx.doi.org/10.11145/j.biomath.2013.05.155 N Kyurkchiev at al., On some multipoint methods arising from optimal in the sense of Kung–Traub algorithms (16) is given by the only real root of equation (22) with magnitude greater than 1. On the other hand, |�2n+1| ≤ K − 1 11 11 d2n+1, |�2n+2| ≤ K − 5 11 11 d2n+2, and consequently we can conclude that the order of convergence of iteration (16) is τ = ( 17 + √ 357 ) 2 ≈ 17.9472... Thus, the theorem is proven. D. Optimal algorithm in the sense of Kung–Traub with order of convergence p = 32 We consider the following nonstationary iterative scheme based on the 4-point iteration function in combi- nation with an optimal algorithm in the sense of Kung– Traub with order of convergence p = 32: x2n+1 = ϕ1(x2n,x2n−1,x2n−2,x2n−3), x2n+2 = ϕ5(x2n+1). (23) For the error �i = xi − ξ, i = −3,−2,−1,0,1,2 . . . ; [30] is valid �2n+1 ∼ C1(ξ)�2n�2n−1�2n−2�2n−3, (24) �2n+2 ∼ C5(ξ)�162n+1. (25) Let K12 = max{|C1(ξ))|, |C5(ξ)|} , d2n−1 = K 3 65 12 |�2n−1|, d2n = K 31 65 12 |�2n|, and let d > 0, and x−3, x−2, x−1 and x0 be chosen so that the following inequalities d−3 = K 3 65 12 |x−3 − ξ| ≤ d < 1, d−2 = K 31 65 12 |x−2 − ξ| ≤ d < 1, d−1 = K 3 65 12 |x−1 − ξ| ≤ d < 1, d0 = K 31 65 12 |x0 − ξ| ≤ d < 1 hold true. From (24) and (25), we have d2n+1 = K 3 65 12 |�2n+1| ≤ K 1 12 12 K12|�2n||�2n−1||�2n−2||�2n−3| = K 3 65 12 K 3 65 + 31 65 + 31 65 12 |�2n|�2n−1||�2n−2|�2n−3| = d2nd2n−1d2n−2d2n−3, d2n+2 = K 31 65 12 |�2n+2| ≤ K 31 65 12 K12� 32 2n+1 = ( K 3 65 12 �2n+1 )32 = d322n+1. (26) Evidently, from (26), we find d1 ≤ d4, d2 ≤ d128, d3 ≤ d134, d4 ≤ d4288, d5 ≤ d4554, d6 ≤ d145728, d7 ≤ d154704, d8 ≤ d4950528, d9 ≤ d5255514, d10 ≤ d168176448. Our results concerning the order of convergence gener- ated by (23) are summarized in the following theorem. Theorem D. Assume that the initial approximations x0, x−1, x−2, x−3 are chosen so that d−3 ≤ d, d−2 ≤ d, d−1 ≤ d < 1 and d0 ≤ d < 1. Then for the error of the sequences {x2n+1}∞n=0 and {x2n+2} ∞ n=0 determined by (23), we have d2n−1 ≤ dτ2n−1, d2n ≤ dτ2n, (27) where τm+4 = 33τm+2 + 33τm, m = 1,2, . . . (28) and the order of convergence of the iteration (23) is τ = ( 33 + √ 1221 ) 2 . Proof. In our case, for the recursion τm+4 = 33τm+2 + 33τm, characteristic polynomial is of the type ρ2 −33ρ−33 = 0. (29) Equation (29) has the roots: ρ1 = ( 33 + √ 1221 ) 2 , ( 33− √ 1221 ) 2 . From the general iterative theory it follows that the order of convergence of the iteration procedure, defined by (23) is given by the only real root of equation (22) with magnitude greater than 1. On the other hand, |�2n+1| ≤ K − 3 65 12 d2n+1, |�2n+2| ≤ K −31 65 12 d2n+2, Biomath 2 (2013), 1305155, http://dx.doi.org/10.11145/j.biomath.2013.05.155 Page 5 of 7 http://dx.doi.org/10.11145/j.biomath.2013.05.155 N Kyurkchiev at al., On some multipoint methods arising from optimal in the sense of Kung–Traub algorithms and consequently we can conclude that the order of convergence of iteration (23) is τ = ( 33 + √ 1221 ) 2 ≈ 33.9714... Thus, the theorem is proven. III. CONCLUSION If in the literature in terms of optimal Kung–Traub algorithm of order 64 appeared, we have shown that the acceleration in the light of our considerations is: τ = ( 65 + √ 4485 ) 2 ≈ 65.9851... Intensively working scientific groups in this branch of numerical analysis should directed theirs efforts to make new interval numerical algorithms which are based on recently arised schemes which are optimal in the sense of Kung–Traub. For methodical construction of numerical algorithms with result verification see Markov [17], [18] and his coauthors [4], [5], [19], [15]. ACKNOWLEDGMENT The work presented here is dedicated to the 70th anniversary of Prof. Dr. Svetoslav Markov. This article is partially supported by project NI13 FMI-002 of the Department for Scientific Research, Paisii Hilendarski University of Plovdiv. REFERENCES [1] W. Bi, H. Ren and Q. Wu, Three-step iterative methods with eight-order convergence for solving nonlinear equations, J. Com- put. Appl. Math. 225 (2009) 105–112. http://dx.doi.org/10.1016/j.cam.2008.07.004 [2] W. Bi, Q. Wu and H. Ren, A new family of eight-order iterative methods for solving nonlinear equations, Appl. Math. Comput. 214 (2009) 236–245. http://dx.doi.org/10.1016/j.amc.2009.03.077 [3] C. Chun and B. Neta, Certain improvements of Newton’s method with fourth-order convergence, Appl. Math. Comput. 215 (2009) 821–824. http://dx.doi.org/10.1016/j.amc.2009.06.007 [4] N. Dimitrova and S. Markov, Interval methods of Newton type for nonlinear equations, Pliska (Studia math. bulg.) 5 (1983) 105– 117. [5] N. Dimitrova and S. Markov, A Validated Newton Type Method for Nonlinear Equations, Interval Computations 2 (1994) 27–51. [6] J. Dzunic and M. Petkovic, On generalized biparametric multi- point root finding methods with memory, J. Comput. Appl. Math. 255 (2014) 362–375. http://dx.doi.org/10.1016/j.cam.2013.05.013 [7] Y. H. Geum and Y. I. Lim, A multiparameter family of three-step eight-order iterative methods locating a simple root, Appl. Math. Comput. 215 (2010) 3375–3382. http://dx.doi.org/10.1016/j.amc.2009.10.030 [8] A. Householder, The Numerical Treatment of a Single Nonlinear Equations, McGraw-Hill Book Company, 1970. [9] V. Hristov, A. Iliev and N. Kyurkchiev, A note on the conver- gence of nonstationary finite-difference analogues, Comp. Math. and Math. Phys. 45 (2005) 194–201. [10] B. Ignatova, N. Kyurkchiev and A. Iliev, Multipoint algorithms arising from optimal in the sense of Kung–Traub iterative pro- cedures for numerical solution of nonlinear equations, General Mathematics Notes 6 (2011) 45–79. [11] A. Iliev and N. Kyurkchiev, Nontrivial Methods in Numerical Analysis: Selected Topics in Numerical Analysis, LAP LAM- BERT Academic Publishing, 2010. [12] J. Kou, X. Wang and S. Sun, Some new root-finding meth- ods with eight-order convergence, Bull. Math. Soc. Sci. Math. Roumanie 53 (2010) 133–143. [13] H. Kung and J. Traub, Optimal order of one point and multi- point iteration, ACM 21 (1974) 643–651. http://dx.doi.org/10.1145/321850.321860 [14] N. Kyurkchiev and A. Iliev, A note on the constructing of non- stationary methods for solving nonlinear equations with raised speed of convergence, Serdica J. Computing 3 (2009) 47–74. [15] N. Kyurkchiev and S. Markov, Two interval methods for al- gebraic equations with real roots, Pliska (Studia math. bulg.) 5 (1983) 118–131. [16] X. Liangzang and M. Xiangjiand, Convergence of an iteration method without derivative, Numer. Math. J. Chinese Univ. Engl. Ser. 11 (2002) 113–120. [17] S. Markov, Extended Interval Arithmetic Involving Infinite Intervals, Mathematica Balkanica, New Series 3 (1992) 269–304. [18] S. Markov, Iterative method for algebraic solution to interval equations, Applied Numerical Mathematics 30 (1999) 225–239. http://dx.doi.org/10.1016/S0168-9274(98)00112-3 [19] S. Markov and N. Kyurkchiev, A method for solving algebraic equations, ZAMM 89 (1989) T106–T107. [20] M. Petkovic, On optimal multipoint methods for solving non- linear equations, Novi Sad J. Math. 39 (2009) 123–130. [21] M. Petkovic, B. Neta, L. Petkovic and J. Dzunic, Multipoint methods for solving nonlinear equations, Academic press, 2012. [22] M. Petkovic and L. Petkovic, Families of optimal multipoint methods for solving nonlinear equations: A Survey, Appl. Anal. Discrete Math. 4 (2010) 1–22. http://dx.doi.org/10.2298/AADM100217015P [23] J. R. Sharma and R. Sharma, A new family of modified Ostrowski’s methods with accelerated eight order convergence, Numer. Algor. 54 (2010) 445–458. http://dx.doi.org/10.1007/s11075-009-9345-5 [24] F. Soleimani and F. Soleymani, Computing simple roots by a sixth-order iterative method, Int. J. of Pure and Appl. Math. 69 (2011) 41–48. [25] F. Soleymani, Regarding the Accuracy of Optimal Eight-Order Methods, Math. and Comput. Modelling 53 (2011) 1351–1357. http://dx.doi.org/10.1016/j.mcm.2010.12.032 [26] P. Sargolzaei and F. Soleymani, Accurate fourteenth-order meth- ods for solving nonlinear equations, J. Numerical Algorithms (2011). http://dx.doi.org/10.1007/s11075-011-9467-4 [27] F. Soleymani and B. Mousavi, A novel computational technique for finding simple roots of nonlinear equations, Int. J. of Math. Analysis 5 (2011) 1813–1819. [28] F. Soleymani and M. Sharifi, On a general efficient class of four-step root-finding methods, Int. J. of Math. and Comp. in Simulation 5 (2011) 181–189. Biomath 2 (2013), 1305155, http://dx.doi.org/10.11145/j.biomath.2013.05.155 Page 6 of 7 http://dx.doi.org/10.1016/j.cam.2008.07.004 http://dx.doi.org/10.1016/j.amc.2009.03.077 http://dx.doi.org/10.1016/j.amc.2009.06.007 http://dx.doi.org/10.1016/j.cam.2013.05.013 http://dx.doi.org/10.1016/j.amc.2009.10.030 http://dx.doi.org/10.1145/321850.321860 http://dx.doi.org/10.1016/S0168-9274(98)00112-3 http://dx.doi.org/10.2298/AADM100217015P http://dx.doi.org/10.1007/s11075-009-9345-5 http://dx.doi.org/10.1016/j.mcm.2010.12.032 http://dx.doi.org/10.1007/s11075-011-9467-4 http://dx.doi.org/10.11145/j.biomath.2013.05.155 N Kyurkchiev at al., On some multipoint methods arising from optimal in the sense of Kung–Traub algorithms [29] R. Thurkal and M. Petkovic, A family of three-point methods of optimal order for solving nonlinear equations, J. Comput. Appl. Math. 233 (2010) 2278–2284. http://dx.doi.org/10.1016/j.cam.2009.10.012 [30] J. Traub, Iterative Methods for Solution of Equations, Prentice- Hall, Englewood Cliffs, N.J., 1964. [31] X. Wang and L. Liu, New eight-order iterative methods for solving nonlinear equations, J. Comput. Appl. Math. 234 (2010) 1611–1620. http://dx.doi.org/10.1016/j.cam.2010.03.002 [32] W. Xinghua, Z. Shiming and H. Danfu, Convergence on Euler’s series, the iteration of Euler’s and Halley’s families, Acta Math. Sinica 33 (1990) 721–738. Biomath 2 (2013), 1305155, http://dx.doi.org/10.11145/j.biomath.2013.05.155 Page 7 of 7 http://dx.doi.org/10.1016/j.cam.2009.10.012 http://dx.doi.org/10.1016/j.cam.2010.03.002 http://dx.doi.org/10.11145/j.biomath.2013.05.155 Introduction Main Results Optimal algorithm in the sense of Kung–Traub with order of convergence p=4 Optimal algorithm in the sense of Kung–Traub with order of convergence p=8 Optimal algorithm in the sense of Kung–Traub with order of convergence p=16 Optimal algorithm in the sense of Kung–Traub with order of convergence p=32 Conclusion References