New families of third-order iterative methods for finding multiple roots

Research Article New Families of Third-Order Iterative Methods for Finding Multiple Roots R. F. Lin,1H. M. Ren,2Z. Šmarda,3Q. B. Wu,4Y. Khan,4and J. L. Hu5 1Department of Mathematics, Taizhou University, Linhai, Zhejiang 317000, China 2College of Information and Engineering, Hangzhou Polytechnic, Hangzhou, Zhejiang 311402, China 3Department of Mathematics, Brno University of Technology, Brno, Czech Republic 4Department of Mathematics, Zhejiang University, Hangzhou, Zhejiang 310027, China 5Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou, Zhejiang 310018, China Correspondence should be addressed to Y. Khan; [email protected]m Received 5 February 2014; Revised 19 May 2014; Accepted 20 May 2014; Published 15 June 2014 Academic Editor: Alicia Cordero Copyright © 2014 R. F. Lin et al. This 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. Two families of third-order iterative methods for finding multiple roots of nonlinear equations are developed in this paper. Mild conditions are given to assure the cubic convergence of two iteration schemes (I) and (II). The presented families include many third-order methods for finding multiple roots, such as the known Dong’s methods and Neta’s method. Some new concrete iterative methods are provided. Each member of the two families requires two evaluations of the function and one of its first derivative per iteration. All these methods require the knowledge of the multiplicity. The obtained methods are also compared in their performance with various other iteration methods via numerical examples, and it is observed that these have better performance than the modified Newton method, and demonstrate at least equal performance to iterative methods of the same order. 1. Introduction Finding the roots of nonlinear equations is one of the most important problems in numerical analysis. In this study, we use iterative methods to find a multiple root 𝑥⋆of multiplicity 𝑚(𝑚>1);thatis,𝑓(𝑗)(𝑥⋆)=0,𝑗=0,1,...,𝑚−1,and 𝑓(𝑚)(𝑥⋆) =0, of a nonlinear equation 𝑓(𝑥)=0. ItisknownthatthemodifiedNewtonmethodformultiplerootsisgivenby 𝑥𝑛+1 =𝑥𝑛−𝑚𝑓(𝑥𝑛) 𝑓󸀠(𝑥𝑛),(1) which converges quadratically [1]. There exists a cubically convergent method for multiple roots, presented by Hansen and Patrick [2]. Consider 𝑥𝑛+1 =𝑥𝑛−(𝑓(𝑥𝑛)) ×(𝑚+1 2𝑚 𝑓󸀠(𝑥𝑛)−𝑓(𝑥𝑛)𝑓󸀠󸀠(𝑥𝑛) 2𝑓󸀠(𝑥𝑛))−1,(2) which is an extension of the classical Halley method of the third order. Another cubically convergent method for multiple roots is proposed by Traub [3]. Consider 𝑥𝑛+1 =𝑥𝑛−𝑚(3−𝑚) 2𝑓(𝑥𝑛) 𝑓󸀠(𝑥𝑛)−𝑚2 2𝑓(𝑥𝑛)2𝑓󸀠󸀠 (𝑥𝑛) 𝑓󸀠(𝑥𝑛)3,(3) which is an extension of the well-known Chebyshev method of the third order. In recent years, a lot of methods for multiple roots have been presented and analyzed, which require the knowledge of the multiplicity 𝑚;see[4–24] and references therein. Based on King’s fourth-order method (for simple roots) [25], Dong [4] has developed two third-order methods for Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2014, Article ID 812072, 9 pages http://dx.doi.org/10.1155/2014/812072 2Journal of Applied Mathematics multiple roots, requiring two evaluations of the function and one of its first derivative. Consider 𝑦𝑛=𝑥𝑛−√𝑚𝑓(𝑥𝑛) 𝑓󸀠(𝑥𝑛), 𝑥𝑛+1 =𝑦𝑛−𝑚(1− 1 √𝑚)(1−𝑚) 𝑓(𝑦𝑛) 𝑓󸀠(𝑥𝑛),(4) 𝑦𝑛=𝑥𝑛−𝑓(𝑥𝑛) 𝑓󸀠(𝑥𝑛), 𝑥𝑛+1 =𝑦𝑛−𝑓(𝑦𝑛) ((𝑚−1)/𝑚)𝑚−1𝑓(𝑥𝑛)−𝑓(𝑦𝑛)𝑓(𝑥𝑛) 𝑓󸀠(𝑥𝑛).(5) Using the same information, Victory Jr. and Neta [5]have developed a third method. Consider 𝑦𝑛=𝑥𝑛−𝑓(𝑥𝑛) 𝑓󸀠(𝑥𝑛), 𝑥𝑛+1 =𝑦𝑛−𝑓(𝑦𝑛) 𝑓󸀠(𝑥𝑛)𝑓(𝑥𝑛)+𝐴𝑓(𝑦𝑛) 𝑓(𝑥𝑛)+𝐵𝑓(𝑦𝑛),(6) where 𝐴=( 𝑚 𝑚−1)2𝑚 −( 𝑚 𝑚−1)𝑚+1, 𝐵=−(𝑚/(𝑚−1))𝑚(𝑚−2)(𝑚−1)+1 (𝑚−1)2.(7) Neta [9] has developed another third-order method requiring thesameinformation 𝑦𝑛=𝑥𝑛−𝛼𝑓(𝑥𝑛) 𝑓󸀠(𝑥𝑛), 𝑥𝑛+1 =𝑦𝑛−(𝛽−𝛼)𝑓(𝑥𝑛)+𝛾𝑓(𝑦𝑛) 𝑓󸀠(𝑥𝑛),(8) where 𝛼=1 2𝑚(𝑚+3) 𝑚+1 ,𝛽= 𝑚3+4𝑚2+9𝑚+2 (𝑚+3)2, 𝛾= 2𝑚+1 (𝑚2−1) (𝑚+3)2((𝑚−1)/(𝑚+1))𝑚.(9) Based on Halley’s method, Li et al. [15]haveproposedafamily of third methods using the same information. Consider 𝑦𝑛=𝑥𝑛−𝛼𝑓(𝑥𝑛) 𝑓󸀠(𝑥𝑛), 𝑥𝑛+1 =𝑥𝑛−𝑚𝛼2𝜇𝑚𝑓(𝑥𝑛) (𝑚−𝛼+𝛼2)𝜇𝑚𝑓(𝑥𝑛)−(𝑚−𝛼)𝑓(𝑦𝑛)𝑓(𝑥𝑛) 𝑓󸀠(𝑥𝑛), (10) where 𝛼is a real parameter and 𝛼 =0,𝑚,and𝜇=(𝑚−𝛼)/𝑚. Note also that, based on Traub’s method [2], Homeier [16] has suggested a family of third methods using the same information. Consider 𝑦𝑛=𝑥𝑛−𝛼𝑓(𝑥𝑛) 𝑓󸀠(𝑥𝑛), 𝑥𝑛+1 =𝑥𝑛−𝛽𝑓(𝑥𝑛) 𝑓󸀠(𝑥𝑛)−𝑓(𝑦𝑛) 𝛾𝑓󸀠(𝑥𝑛),(11) where 𝛼 =0,𝑚is a real parameter, 𝛽=(𝑚/𝛼2)(𝛼2+𝛼−𝑚), and 𝛾=(1/𝑚)(1−𝛼/𝑚)𝑚(𝛼2/(𝑚−𝛼)). In this paper, we propose two new families of third-order methods for multiple roots; each of the methods requires twofunction and one-derivative evaluation per iteration, respectively. The presented methods are obtained by investigating the following two iteration schemes: (I){ { { { { { { { { 𝑦𝑛=𝑥𝑛−𝛼𝑓(𝑥𝑛) 𝑓󸀠(𝑥𝑛), 𝑥𝑛+1 =𝑦𝑛−𝑎𝑓(𝑥𝑛)+𝑏𝑓(𝑦𝑛) 𝑐𝑓(𝑥𝑛)+𝑑𝑓(𝑦𝑛)𝑓(𝑦𝑛) 𝑓󸀠(𝑥𝑛),(12) (II){ { { { { { { { { 𝑦𝑛=𝑥𝑛−𝛼𝑓(𝑥𝑛) 𝑓󸀠(𝑥𝑛), 𝑥𝑛+1 =𝑦𝑛−𝑎𝑓(𝑥𝑛)+𝑏𝑓(𝑦𝑛) 𝑐𝑓(𝑥𝑛)+𝑑𝑓(𝑦𝑛)𝑓(𝑥𝑛) 𝑓󸀠(𝑥𝑛),(13) where 𝛼,𝑎,𝑏,𝑐,and𝑑are parameters to be determined. By specially choosing the parameters in (12)and(13), we get two new families of third-order methods, which include methods (4)–(6), (8), (10), and (11). In fact, the mild conditions to assurethecubicconvergenceof(I)-typeiteration(12) or (II)- type iteration (13) are given. Divided differences are adopted successfully in developing our methods, which will be useful in developing more new methods. Finally, we use some numerical examples to compare the presented methods with the modified Newton method and some known third-order methods. 2. Preliminaries We need the definitions of divided differences and their properties. Definition 1 (see [26]). The divided differences 𝑓[𝑎0,𝑎1, ...,𝑎𝑘]on 𝑘+1distinct points 𝑎0,𝑎1,...,𝑎𝑘of a function 𝑓(𝑥) are defined by 𝑓[𝑎0]=𝑓(𝑎0), 𝑓[𝑎0,𝑎1]=𝑓[𝑎0]−𝑓[𝑎1] 𝑎0−𝑎1, . . . 𝑓[𝑎0,𝑎1,...,𝑎𝑘]=𝑓[𝑎0,𝑎1,...,𝑎𝑘−1]−𝑓[𝑎1,𝑎2,...,𝑎𝑘] 𝑎0−𝑎𝑘. (14) Journal of Applied Mathematics 3 If the function 𝑓is sufficiently differentiable, then its divided differences 𝑓[𝑎0,𝑎1,...,𝑎𝑘]canbedefinedifsomeofthe arguments 𝑎𝑖coincide. For instance, if 𝑓(𝑥)has a derivative of the 𝑘th order at 𝑎0,thenitmakessensetodefine 𝑓[𝑎0,𝑎0,...,𝑎0 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 𝑘+1 ]=𝑓(𝑘) (𝑎0) 𝑘! .(15) Lemma 2 (see [26]). The divided differences 𝑓[𝑎0,𝑎1,...,𝑎𝑘] are symmetric functions of their arguments; that is, they are invariant to permutations of the 𝑎0,𝑎1,...,𝑎𝑘. Lemma 3 (see [26]). If the function 𝑓has (𝑘+1)st derivative, then, for every argument 𝑥, the following interpolation formula holds: 𝑓(𝑥)=𝑓[𝑎0]+𝑘 ∑ 𝑖=1𝑓[𝑎0,𝑎1,...,𝑎𝑖]𝑖−1 ∏ 𝑗=0 (𝑥−𝑎𝑗) +𝑓[𝑎0,𝑎1,...,𝑎𝑘,𝑥] 𝑘 ∏ 𝑖=0 (𝑥−𝑎𝑖). (16) Lemma 4. If the function 𝑓has a derivative of the (𝑚+1)th order, and 𝑥⋆isamultiplerootofmultiplicity𝑚,then,forevery argument 𝑥, the following formulae hold: 𝑓(𝑥)=𝑓[𝑥⋆,𝑥⋆,...,𝑥⋆ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 𝑚,𝑥](𝑥−𝑥⋆)𝑚,(17) 𝑓󸀠(𝑥)=𝑓[𝑥⋆,𝑥⋆,...,𝑥⋆ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 𝑚,𝑥,𝑥](𝑥−𝑥⋆)𝑚 +𝑚𝑓[𝑥⋆,𝑥⋆,...,𝑥⋆ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 𝑚,𝑥](𝑥−𝑥⋆)𝑚−1.(18) Proof. Applying Lemma 3 to the case of the multiple zero 𝑥⋆ of multiplicity 𝑚and using (15), we get (17). Differentiating both sides of (17)gives(18). 3. Development of New Families of Third-Order Methods Wewouldliketofindthefiveparameters𝛼,𝑎,𝑏,𝑐,and𝑑 in I-type iteration (12) and II-type iteration (13)soasto maximize its order of convergence to a root 𝑥⋆of multiplicity 𝑚,respectively.Let𝑒𝑛,𝑑𝑛be the errors at the 𝑛th step; that is, 𝑒𝑛=𝑥𝑛−𝑥⋆,𝑑 𝑛=𝑦𝑛−𝑥⋆.(19) Define functions 𝑔(𝑥)and ℎ(𝑥)as follows: 𝑔(𝑥)=𝑓[𝑥⋆,𝑥⋆,...,𝑥⋆ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟, 𝑚𝑥], ℎ(𝑥)=𝑓[𝑥⋆,𝑥⋆,...,𝑥⋆ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 𝑚,𝑥,𝑥]. (20) Write 𝑔𝑛=𝑔(𝑥𝑛), ℎ𝑛=ℎ(𝑥𝑛), 𝑔⋆=𝑔(𝑥⋆), ℎ⋆=ℎ(𝑥⋆). (21) In view of (17)and(18), we get the following: 𝑓(𝑥𝑛)=𝑔𝑛𝑒𝑚 𝑛,(22) 𝑓󸀠(𝑥𝑛)=ℎ𝑛𝑒𝑚 𝑛+𝑚𝑔𝑛𝑒𝑚−1 𝑛,(23) 𝑓(𝑦𝑛)=𝑔(𝑦𝑛)𝑑𝑚 𝑛.(24) Using the definitions of divided differences, we get the following: 𝑔(𝑦𝑛)=𝑔(𝑦𝑛)−𝑔(𝑥𝑛)+𝑔(𝑥𝑛) =𝑓[𝑥⋆,𝑥⋆,...,𝑥⋆ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 𝑚,𝑦𝑛,𝑥𝑛](𝑑𝑛−𝑒𝑛)+𝑔𝑛 =(𝑓[𝑥⋆,𝑥⋆,...,𝑥⋆ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 𝑚,𝑦𝑛,𝑥𝑛]−ℎ(𝑥𝑛)+ℎ(𝑥𝑛)) ×(𝑑𝑛−𝑒𝑛)+𝑔𝑛 =𝑝𝑛(𝑑𝑛−𝑒𝑛)2+ℎ𝑛(𝑑𝑛−𝑒𝑛)+𝑔𝑛,(25) where 𝑝𝑛=𝑓[𝑥⋆,𝑥⋆,...,𝑥⋆ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 𝑚,𝑦𝑛,𝑥𝑛,𝑥𝑛]. (26) In view of (22), (23), (12), and (13), we get in turn 𝑓(𝑥𝑛) 𝑓󸀠(𝑥𝑛)=𝑔𝑛𝑒𝑛 ℎ𝑛𝑒𝑛+𝑚𝑔𝑛,(27) 𝑑𝑛=𝑒𝑛−𝛼 𝑔𝑛𝑒𝑛 ℎ𝑛𝑒𝑛+𝑚𝑔𝑛=ℎ𝑛𝑒𝑛+(𝑚−𝛼)𝑔𝑛 ℎ𝑛𝑒𝑛+𝑚𝑔𝑛𝑒𝑛,(28) 𝑑𝑛−𝑒𝑛=− 𝛼𝑔𝑛𝑒𝑛 ℎ𝑛𝑒𝑛+𝑚𝑔𝑛.(29) Substituting (29)into(25)yields 𝑔(𝑦𝑛)=𝑝𝑛(− 𝛼𝑔𝑛𝑒𝑛 ℎ𝑛𝑒𝑛+𝑚𝑔𝑛)2+ℎ𝑛(− 𝛼𝑔𝑛𝑒𝑛 ℎ𝑛𝑒𝑛+𝑚𝑔𝑛)+𝑔𝑛 =𝑔𝑛 (ℎ𝑛𝑒𝑛+𝑚𝑔𝑛)2([𝛼2𝑝𝑛𝑔𝑛+(1−𝛼)ℎ2 𝑛]𝑒2 𝑛 +(2−𝛼)𝑚ℎ𝑛𝑔𝑛𝑒𝑛+𝑚2𝑔2 𝑛).(30) Substituting (30)and(28)into(24)leadsto 𝑓(𝑦𝑛)=𝑔𝑛[ℎ𝑛𝑒𝑛+(𝑚−𝛼)𝑔𝑛]𝑚 (ℎ𝑛𝑒𝑛+𝑚𝑔𝑛)𝑚+2 ×([𝛼2𝑝𝑛𝑔𝑛+(1−𝛼)ℎ2 𝑛]𝑒2 𝑛 +(2−𝛼)𝑚ℎ𝑛𝑔𝑛𝑒𝑛+𝑚2𝑔2 𝑛)𝑒𝑚 𝑛. (31) 4Journal of Applied Mathematics Write 𝐴𝑛=ℎ𝑛𝑒𝑛+𝑚𝑔𝑛, 𝐵𝑛=ℎ𝑛𝑒𝑛+(𝑚−𝛼)𝑔𝑛, 𝐶𝑛=[𝛼 2𝑝𝑛𝑔𝑛+(1−𝛼)ℎ2 𝑛]𝑒2 𝑛+(2−𝛼)𝑚ℎ𝑛𝑔𝑛𝑒𝑛+𝑚2𝑔2 𝑛. (32) Using (23), (24), and (32), we get 𝑓󸀠(𝑥𝑛)=𝐴𝑛𝑒𝑚−1 𝑛, 𝑓(𝑦𝑛)=𝑔𝑛𝐵𝑚 𝑛𝐶𝑛 𝐴𝑚+2 𝑛𝑒𝑚 𝑛.(33) Then, we can get the error equations as follows: 𝑒𝑛+1 =𝐵𝑛 𝐴𝑛𝑒𝑛−𝑎𝑔𝑛𝑒𝑚 𝑛+𝑏(𝑔𝑛𝐵𝑚 𝑛𝐶𝑛/𝐴𝑚+2 𝑛)𝑒𝑚 𝑛 𝑐𝑔𝑛𝑒𝑚 𝑛+𝑑(𝑔𝑛𝐵𝑚 𝑛𝐶𝑛/𝐴𝑚+2 𝑛)𝑒𝑚 𝑛 ×(𝑔𝑛𝐵𝑚 𝑛𝐶𝑛/𝐴𝑚+2 𝑛)𝑒𝑚 𝑛 𝐴𝑛𝑒𝑚−1 𝑛 =−([(𝑎𝐴𝑚+2 𝑛+𝑏𝐵𝑚 𝑛𝐶𝑛)𝑔𝑛𝐵𝑚−1 𝑛𝐶𝑛 −(𝑐𝐴𝑚+2 𝑛+𝑑𝐵𝑚 𝑛𝐶𝑛)𝐴𝑚+2 𝑛]𝐵𝑛𝑒𝑛) ×((𝑐𝐴𝑚+2 𝑛+𝑑𝐵𝑚 𝑛𝐶𝑛)𝐴𝑚+3 𝑛)−1, (34) for I-type iteration (12), and 𝑒𝑛+1 =𝐵𝑛 𝐴𝑛𝑒𝑛−𝑎𝑔𝑛𝑒𝑚 𝑛+𝑏(𝑔𝑛𝐵𝑚 𝑛𝐶𝑛/𝐴𝑚+2 𝑛)𝑒𝑚 𝑛 𝑐𝑔𝑛𝑒𝑚 𝑛+𝑑(𝑔𝑛𝐵𝑚 𝑛𝐶𝑛/𝐴𝑚+2 𝑛)𝑒𝑚 𝑛 ×𝑔𝑛𝑒𝑚 𝑛 𝐴𝑛𝑒𝑚−1 𝑛 =−([(𝑎𝐴𝑚+2 𝑛+𝑏𝐵𝑚 𝑛𝐶𝑛)𝑔𝑛−(𝑐𝐴𝑚+2 𝑛+𝑑𝐵𝑚 𝑛𝐶𝑛)𝐵𝑛]𝑒𝑛) ×((𝑐𝐴𝑚+2 𝑛+𝑑𝐵𝑚 𝑛𝐶𝑛)𝐴𝑛)−1,(35) for II-type iteration (13). In view of (34)and(35), the order of convergence for Itype or II-type iteration will arrive at three provided that Φ𝑛 (𝑐𝐴𝑚+2 𝑛+𝑑𝐵𝑚 𝑛𝐶𝑛)𝐴𝑚+3 𝑛=𝑂(𝑒2 𝑛),(36) or Ψ𝑛 (𝑐𝐴𝑚+2 𝑛+𝑑𝐵𝑚 𝑛𝐶𝑛)𝐴𝑛=𝑂(𝑒2 𝑛), (37) holds true, respectively. Here Φ𝑛=[(𝑎𝐴 𝑚+2 𝑛+𝑏𝐵𝑚 𝑛𝐶𝑛)𝑔𝑛𝐵𝑚−1 𝑛𝐶𝑛 −(𝑐𝐴𝑚+2 𝑛+𝑑𝐵𝑚 𝑛𝐶𝑛)𝐴𝑚+2 𝑛]𝐵𝑛, Ψ𝑛=(𝑎𝐴 𝑚+2 𝑛+𝑏𝐵𝑚 𝑛𝐶𝑛)𝑔𝑛−(𝑐𝐴𝑚+2 𝑛+𝑑𝐵𝑚 𝑛𝐶𝑛)𝐵𝑛. (38) Write 𝜆=𝑚−𝛼. (39) In view of (32), we can get, as 𝑛→∞, 𝐴𝑛󳨀→ 𝑚𝑔⋆=𝑚𝑓[𝑥⋆,𝑥⋆,...,𝑥⋆ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 𝑚+1 ]=𝑚𝑓(𝑚) (𝑥⋆) 𝑚! =0, 𝐵𝑛󳨀→ 𝜆𝑔⋆, 𝐶𝑛󳨀→ (𝑚𝑔⋆)2,(40) and then (𝑐𝐴𝑚+2 𝑛+𝑑𝐵𝑚 𝑛𝐶𝑛)𝐴𝑚+3 𝑛󳨀→ (𝑐𝑚𝑚+𝑑𝜆𝑚)𝑚𝑚+5𝑔2𝑚+5 ⋆, (𝑛󳨀→∞), (𝑐𝐴𝑚+2 𝑛+𝑑𝐵𝑚 𝑛𝐶𝑛)𝐴𝑛󳨀→ (𝑐𝑚𝑚+𝑑𝜆𝑚)𝑚3𝑔𝑚+3 ⋆, (𝑛󳨀→∞),(41) which show that, in order to assure the denominators in (36) and (37) are not equal to zero, we demand naturally that 𝑐𝑚𝑚+𝑑𝜆𝑚=0. (42) It is obvious that, under the condition (42), the error relations (36)and(37)areequivalentto Φ𝑛=𝑂(𝑒2 𝑛), (43) Ψ𝑛=𝑂(𝑒2 𝑛), (44) respectively. Next we will find conditions to assure (43)and(44). Note that the factor 𝐵𝑛of Φ𝑛plays an important role on the order of Φ𝑛. In fact, using the Taylor formula, we get from (32)the following: 𝐵𝑛=𝜆𝑔⋆+𝑂(𝑒𝑛).(45) Then, in the case 𝜆=0, to assure the relation (43)holdstrue, the following estimate is needed: Δ𝑛=(𝑎𝐴 𝑚+2 𝑛+𝑏𝐵𝑚 𝑛𝐶𝑛)𝑔𝑛𝐵𝑚−1 𝑛𝐶𝑛 −(𝑐𝐴𝑚+2 𝑛+𝑑𝐵𝑚 𝑛𝐶𝑛)𝐴𝑚+2 𝑛=𝑂(𝑒𝑛), (46) which demands Δ𝑛󳨀→ −𝑐(𝑚𝑔⋆)2𝑚+4 =0, (𝑛󳨀→∞);(47) that is, 𝑐=0. This is a contradiction to (42). Hence, in the case 𝜆=0,therelation(43) cannot be satisfied; that is, we cannot choose parameters so that the order of convergence of (I)-type iteration (12) arrives at three. Journal of Applied Mathematics 5 In what follows, we suppose 𝜆 =0.Inthiscase,𝐵𝑛→ 𝜆𝑔⋆=0(𝑛 → ∞),andthen(43)isequivalentto Δ𝑛=𝑂(𝑒2 𝑛). (48) In view of (32), and by a straight computation, we can get Δ𝑛=𝑢𝑛+V𝑛𝑒𝑛+𝑂(𝑒2 𝑛), (49) where 𝑢𝑛=(𝑚𝑚𝜆𝑚−1𝑎+𝜆2𝑚−1𝑏−𝑚2𝑚𝑐−𝑚𝑚𝜆𝑚𝑑)𝑚4𝑔2𝑚+4 𝑛, (50) V𝑛=(𝑚 𝑚𝜆𝑚−2 [𝜆2+4𝜆+𝑚(𝑚−1)]𝑎 +𝜆2𝑚−2 [2𝜆2+(4−2𝑚)𝜆+𝑚(2𝑚−1)]𝑏 −(2𝑚+4)𝑚2𝑚𝑐 −𝑚𝑚𝜆𝑚−1 (𝜆2+4𝜆+𝑚2)𝑑)𝑚3𝑔2𝑚+3 𝑛ℎ𝑛. (51) In view of (49) and in order to assure the relation (48)holds, we should choose parameters 𝜆,𝑎,𝑏,𝑐,and𝑑such that 𝑚𝑚𝜆𝑚−1𝑎+𝜆2𝑚−1𝑏−𝑚2𝑚𝑐−𝑚𝑚𝜆𝑚𝑑=0, 𝑚𝑚𝜆𝑚−2 [𝜆2+4𝜆+𝑚(𝑚−1)]𝑎 +𝜆2𝑚−2 [2𝜆2+(4−2𝑚)𝜆+𝑚(2𝑚−1)]𝑏 −(2𝑚+4)𝑚2𝑚𝑐−𝑚𝑚𝜆𝑚−1 (𝜆2+4𝜆+𝑚2)𝑑=0. (52) By a straight computation, we deduce that 𝑐=𝑚𝑚+1𝜆𝑚−1𝑎+[𝑚−(𝜆−𝑚)2]𝜆2𝑚−1𝑏 (𝜆−𝑚)2𝑚2𝑚 ,(53) 𝑑=[(𝜆−𝑚)2−𝑚]𝑚𝑚𝑎+[2(𝜆−𝑚)2−𝑚]𝜆𝑚𝑏 𝜆(𝜆−𝑚)2𝑚𝑚.(54) Substituting (53)and(54)intotheleftsideof(42), we get 𝑐𝑚𝑚+𝑑𝜆𝑚 =𝑚𝑚+1𝜆𝑚−1𝑎+[𝑚−(𝜆−𝑚)2]𝜆2𝑚−1𝑏 (𝜆−𝑚)2𝑚𝑚 +[(𝜆−𝑚)2−𝑚]𝑚𝑚𝜆𝑚−1𝑎+[2(𝜆−𝑚)2−𝑚]𝜆2𝑚−1𝑏 (𝜆−𝑚)2𝑚𝑚 =𝑚𝑚𝑎+𝜆𝑚𝑏 𝑚𝑚𝜆𝑚−1,(55) which shows the condition (42)isequivalentto 𝑎𝑚𝑚+𝑏(𝑚−𝛼)𝑚=0. (56) We summarize our development of new methods done so far in the following theorem. Theorem 5. Let 𝑥⋆∈𝐼be a multiple root of multiplicity 𝑚(𝑚>1)of a sufficiently differentiable function 𝑓:𝐼→R for an open interval 𝐼.If𝑥0is sufficiently close to 𝑥⋆,then the methods defined by (I)-type iteration (12)are cubically convergent for any parameters 𝛼,𝑎,𝑏,𝑐,and𝑑such that 𝛼 =0 and (53),(54),and(56)hold. Next, we turn to find the proper conditions to establish the relation (44). In view of (32), and by a straight computation, we can get Ψ𝑛=𝑟𝑛+𝑠𝑛𝑒𝑛+𝑂(𝑒2 𝑛), (57) where 𝑟𝑛=(𝑚𝑚𝑎+𝜆𝑚𝑏−𝑚𝑚𝜆𝑐−𝜆𝑚+1𝑑)𝑚2𝑔𝑚+3 𝑛, 𝑠𝑛=((𝑚+2)𝑚𝑚+1𝑎+[(2−𝛼)𝑚𝜆𝑚+𝑚3𝜆𝑚−1]𝑏 −[𝑚𝑚+2 +𝑚𝑚+1 (𝑚+2)𝜆]𝑐 −[(2−𝛼)𝑚𝜆𝑚+1 +𝑚2(𝑚+1)𝜆𝑚]𝑑)𝑔𝑚+2 𝑛ℎ𝑛.(58) In view of (57) and in order to assure the relation (44) holds, we should choose parameters 𝜆,𝑎,𝑏,𝑐,and𝑑such that 𝑚𝑚𝑎+𝜆𝑚𝑏−𝑚𝑚𝜆𝑐−𝜆𝑚+1𝑑=0, (𝑚+2)𝑚𝑚𝑎+[(2−𝛼)𝜆+𝑚2]𝜆𝑚−1𝑏 −[𝑚+(𝑚+2)𝜆]𝑚𝑚𝑐 −[(2−𝛼)𝜆+𝑚(𝑚+1)]𝜆𝑚𝑑=0. (59) By a straight computation, we deduce that 𝑐=[(𝜆−𝑚)2+𝑚]𝑚𝑚−1𝑎+𝜆𝑚𝑏 𝜆(𝜆−𝑚)2𝑚𝑚−1 , 𝑑=−𝑚𝑚+1𝑎+[(𝜆−𝑚)2−𝑚]𝜆𝑚𝑏 𝜆𝑚+1(𝜆−𝑚)2. (60) Substituting (60)intotheleftsideof(42), we get 𝑐𝑚𝑚+𝑑𝜆𝑚=𝑚𝑚𝑎+𝜆𝑚𝑏 𝜆,(61) which shows the condition (42)isalsoequivalenttothe condition given by (56). We can summarize the development of new methods involving (II)-type iteration (13)donesofarinthefollowing theorem. Theorem 6. Let 𝑥⋆∈𝐼be a multiple root of multiplicity 𝑚(𝑚>1)of a sufficiently differentiable function 𝑓:𝐼→R for an open interval 𝐼.If𝑥0is sufficiently close to 𝑥⋆,then the methods defined by (II)-type iteration (13)are cubically convergent for any parameters 𝛼,𝑎,𝑏,𝑐,and𝑑such that 𝛼 =0 and (60)and (56)hold. 6Journal of Applied Mathematics Choosing 𝛼=√𝑚,𝑎=1,and𝑏=0,wecandeducefrom (53), (54), and (56)that 𝑐=(𝑚−√𝑚)𝑚−1 𝑚𝑚,(62) 𝑑=0, (63) 𝑎𝑚𝑚+𝑏(𝑚−𝛼)𝑚=𝑚𝑚=0. (64) Using the parameters 𝛼,𝑎,𝑏,𝑐,and𝑑given above in (I)-type iteration (12), we can obtain Dong’s method (4), and its order of convergence arrives at three by Theorem 5. Choosing 𝛼=1,𝑎=1,and𝑏=0,wecandeducefrom (53), (54), and (56)that 𝑐=(𝑚−1)𝑚−1 𝑚𝑚−1 , 𝑑=−1, 𝑎𝑚𝑚+𝑏(𝑚−𝛼)𝑚=𝑚𝑚=0. (65) Using the parameters 𝛼,𝑎,𝑏,𝑐,and𝑑given above in (I)-type iteration (12),wecanobtainDong’smethod(5), and its order of convergence arrives at three by Theorem 5. Choosing 𝛼=1,𝑎=1,and𝑏 = (𝑚/(𝑚−1))2𝑚 − (𝑚/(𝑚−1))𝑚+1,wecandeducefrom(53), (54), and (56)that 𝑐=1, (66) 𝑑=−(𝑚/(𝑚−1))𝑚(𝑚−2)(𝑚−1)+1 (𝑚−1)2,(67) 𝑎𝑚𝑚+𝑏(𝑚−𝛼)𝑚=𝑚𝑚[1+( 𝑚 𝑚−1)𝑚−𝑚 𝑚−1] =0. (68) Using the parameters 𝛼,𝑎,𝑏,𝑐,and𝑑given above in (I)-type iteration (12), we can obtain Victory and Neta’s method (6), and its order of convergence arrives at three by Theorem 5. Let 𝛼=(1/2)(𝑚(𝑚+3)/(𝑚+1)),𝑎=(𝑚3+4𝑚2+9𝑚+ 2)/(𝑚+3)2− (1/2)(𝑚(𝑚+3)/(𝑚+1)),𝑏=2 𝑚+1(𝑚2− 1)/(𝑚+3)2((𝑚−1)/(𝑚+1))𝑚,𝑐=0,and𝑑=1.Usingthe definition of 𝜆,weget 𝜆=𝑚(𝑚−1) 2(𝑚+1).(69) We can verify that the parameters given above satisfy (52), andthustheyalsosatisfy(53)and(54)(as𝑐givenin(53)and 𝑑givenin(54)aresolvedfrom(52)). Furthermore, it is easy to verify that the condition (42)holds: 𝑐𝑚𝑚+𝑑𝜆𝑚=(𝑚(𝑚−1) 2(𝑚+1))𝑚=0. (70) This means that the condition (56)isalsotrue,since(56)is equivalent to (42). Using the parameters 𝛼,𝑎,𝑏,𝑐,and𝑑given in (I)-type iteration (12),wecanobtainNeta’smethod(8), and its order of convergence arrives at three by Theorem 5. We can verify that the family of methods (11)givenby Homeier [16] satisfies all conditions in Theorem 5.First,we can rewrite (11) as follows: 𝑦𝑛=𝑥𝑛−𝛼𝑓(𝑥𝑛) 𝑓󸀠(𝑥𝑛), 𝑥𝑛+1 =𝑦𝑛−(𝛽−𝛼)𝑓(𝑥𝑛)+(1/𝛾)𝑓(𝑦𝑛) 𝑓(𝑦𝑛)𝑓(𝑦𝑛) 𝑓󸀠(𝑥𝑛),(71) where 𝛼 =0,𝑚is a real parameter, 𝛽=(𝑚/𝛼2)(𝛼2+(𝛼−𝑚))= (𝑚/𝛼2)(𝛼2−𝜆),and𝛾 = (1/𝑚)(1−𝛼/𝑚)𝑚(𝛼2/(𝑚−𝛼)) = (𝜆𝑚−1𝛼2/𝑚𝑚+1).Choosing𝑎=𝛽−𝛼and 𝑏=1/𝛾,wecan deduce from (53), (54), and (56)that𝑐=0,𝑑=1,and 𝑐𝑚𝑚+𝑑𝜆𝑚=(𝑚−𝛼)𝑚=0. (72) This means that the condition (56)isalsotrue.Usingthe parameters 𝛼,𝑎,𝑏,𝑐,and𝑑given in (I)-type iteration (12), we can obtain Homeier’s family of methods (11), which has cubic convergence by Theorem 5. We can verify that the family of methods (10)givenby Homeier [16] satisfies all conditions in Theorem 6.First,we can rewrite (10) as follows: 𝑦𝑛=𝑥𝑛−𝛼𝑓(𝑥𝑛) 𝑓󸀠(𝑥𝑛),(73) 𝑥𝑛+1 =𝑦𝑛−((𝑚𝛼2−𝑚𝛼+𝛼2−𝛼3) ×𝜇𝑚𝑓(𝑥𝑛)+𝛼(𝑚−𝛼)𝑓(𝑦𝑛)) ×((𝑚−𝛼+𝛼2)𝜇𝑚𝑓(𝑥𝑛)−(𝑚−𝛼)𝑓(𝑦𝑛))−1 ×𝑓(𝑥𝑛) 𝑓󸀠(𝑥𝑛), (74) where 𝛼is a real parameter and 𝛼 =0,𝑚,and𝜇=(𝑚−𝛼)/𝑚. Let 𝑎=𝜆𝑚+1𝛼(𝛼−1)/𝑚𝑚,𝑏=𝛼𝜆,𝑐=(𝜆+𝛼2)𝜆𝑚/𝑚𝑚, and 𝑑=−𝜆. We can verify that the parameters given above satisfy (60)and(56). Using the parameters 𝛼,𝑎,𝑏,𝑐,and𝑑 givenin(II)-typeiteration(13), we can obtain Shengguo et al.’s family of methods (10), which has cubic convergence by Theorem 6. 4. Some Concrete Methods In this section, we give some concrete iterative forms of (I)- type iteration (12) and (II)-type iteration (13). Method 1. Choosing 𝛼=1,𝑎=0,and𝑏=1,weobtain from (63)and(64)that𝑐 = ((𝑚−1)/𝑚)2𝑚,𝑑=(2− 𝑚)(𝑚−1)𝑚−1/𝑚𝑚,and𝑎𝑚𝑚+𝑏(𝑚−𝛼)𝑚=(𝑚−1) 𝑚=0. Journal of Applied Mathematics 7 Using these parameters in (12), we get a new method. Consider 𝑦𝑛=𝑥𝑛−𝑓(𝑥𝑛) 𝑓󸀠(𝑥𝑛), 𝑥𝑛+1 =𝑦𝑛−(𝑓(𝑦𝑛))×((𝑚−1 𝑚)2𝑚𝑓(𝑥𝑛) +(2−𝑚)(𝑚−1)𝑚−1 𝑚𝑚𝑓(𝑦𝑛))−1 ×𝑓(𝑦𝑛) 𝑓󸀠(𝑥𝑛), (75) which has cubic convergence by Theorem 5. Method 2. Choosing 𝛼=1,𝑐=1,and𝑑=0,wecan obtain from (61)and(62)that𝑎 = (2−𝑚)𝑚𝑚/(𝑚−1)𝑚−1, 𝑏=𝑚2𝑚/(𝑚−1)2𝑚−2,and𝑐𝑚𝑚+𝑑(𝑚−𝛼)𝑚=𝑚𝑚=0.Using these parameters in (12), we get a new method. Consider 𝑦𝑛=𝑥𝑛−𝑓(𝑥𝑛) 𝑓󸀠(𝑥𝑛), 𝑥𝑛+1 =𝑦𝑛−((2−𝑚)𝑚𝑚 (𝑚−1)𝑚−1 𝑓(𝑥𝑛) +𝑚2𝑚 (𝑚−1)2𝑚−2 𝑓(𝑦𝑛)) ×(𝑓(𝑥𝑛))−1 ×𝑓(𝑦𝑛) 𝑓󸀠(𝑥𝑛), (76) which has cubic convergence by Theorem 5. Method 3.Choosing𝛼=1,𝑏=1,and𝑑=1,wecanobtain from (61)and(62)that𝑎=((𝑚−1)𝑚−1(2−𝑚)−𝑚𝑚)/𝑚𝑚, 𝑐 = ((𝑚−1)2𝑚−2 −𝑚 𝑚+1(𝑚−1)𝑚−1)/𝑚2𝑚,and𝑐𝑚𝑚+ 𝑑(𝑚−𝛼)𝑚=(𝑚−1) 𝑚−1[(𝑚−1)𝑚−1 −𝑚𝑚]/𝑚𝑚=0.Using these parameters in (12), we get a new method. Consider 𝑦𝑛=𝑥𝑛−𝑓(𝑥𝑛) 𝑓󸀠(𝑥𝑛), 𝑥𝑛+1 =𝑦𝑛−((𝑚−1)𝑚−1 (2−𝑚)−𝑚𝑚 𝑚𝑚𝑓(𝑥𝑛)+𝑓(𝑦𝑛)) ×((𝑚−1)2𝑚−2 −𝑚𝑚+1(𝑚−1)𝑚−1 𝑚2𝑚 ×𝑓(𝑥𝑛)+𝑓(𝑦𝑛))−1 ×𝑓(𝑦𝑛) 𝑓󸀠(𝑥𝑛), (77) which has cubic convergence by Theorem 5. Method 4.Choosing𝛼=1,𝑎=1,and𝑏=0,wecan obtain from (70)and(71)that𝑐 = (𝑚+1)/(𝑚−1),𝑑= −(𝑚/(𝑚−1))𝑚+1,and𝑎𝑚𝑚+𝑏(𝑚−𝛼)𝑚=𝑚 𝑚=0.Using these parameters in (13), we get a new method. Consider 𝑦𝑛=𝑥𝑛−𝑓(𝑥𝑛) 𝑓󸀠(𝑥𝑛), 𝑥𝑛+1 =𝑦𝑛−(𝑓(𝑥𝑛)) ×(𝑚+1 𝑚−1𝑓(𝑥𝑛)−( 𝑚 𝑚−1)𝑚+1𝑓(𝑦𝑛))−1 ×𝑓(𝑥𝑛) 𝑓󸀠(𝑥𝑛), (78) which has cubic convergence by Theorem 6. Method 5. Choosing 𝛼=1,𝑎=1,and𝑐=0,wecanobtain from (68)and(69)that𝑏=−(𝑚+1)𝑚 𝑚−1/(𝑚−1)𝑚,𝑑= −𝑚𝑚−1/(𝑚−1)𝑚+1,and𝑐𝑚𝑚+𝑑(𝑚−𝛼)𝑚=−𝑚 𝑚−1/(𝑚− 1) =0. Using these parameters in (13), we get a new method. Consider 𝑦𝑛=𝑥𝑛−𝑓(𝑥𝑛) 𝑓󸀠(𝑥𝑛), 𝑥𝑛+1 =𝑦𝑛+(𝑓(𝑥𝑛)−(𝑚+1)𝑚𝑚−1 (𝑚−1)𝑚𝑓(𝑦𝑛)) ×( 𝑚𝑚−1 (𝑚−1)𝑚+1 𝑓(𝑦𝑛))−1 ×𝑓(𝑥𝑛) 𝑓󸀠(𝑥𝑛), (79) which has cubic convergence by Theorem 6. Method 6. Choosing 𝛼=√𝑚,𝑎=1,and𝑏=0,we can obtain from (70)and(71)that𝑐 = 2/(𝑚−√𝑚),𝑑= −𝑚𝑚/(𝑚−√𝑚)(𝑚+1),and𝑎𝑚𝑚+𝑏(𝑚−𝛼)𝑚=𝑚𝑚=0.Using these parameters in (13), we get a new method. Consider 𝑦𝑛=𝑥𝑛−√𝑚𝑓(𝑥𝑛) 𝑓󸀠(𝑥𝑛), 𝑥𝑛+1 =𝑦𝑛−(𝑓(𝑥𝑛)) ×( 2 𝑚−√𝑚𝑓(𝑥𝑛)− 𝑚𝑚 (𝑚−√𝑚)𝑚+1 𝑓(𝑦𝑛))−1 ×𝑓(𝑥𝑛) 𝑓󸀠(𝑥𝑛), (80) which has cubic convergence by Theorem 6. 5. Numerical Examples We employ Method 1 (RM1), (75)–Method 6 (RM6), (80)to solve some nonlinear equations and compare them with the 8Journal of Applied Mathematics Table 1: Comparison of various iterative methods for the function 𝑓1(𝑥)under the same total number of function evaluations (TNFE) required by all methods. Method |𝑥𝑛−𝑥⋆||𝑓(𝑥 𝑛)| COC MNM 3.92𝑒−94 9.48𝑒−187 2.0000000 DM 4.83𝑒−117 1.44𝑒−232 3.0000000 VM 1.42𝑒−118 1.24𝑒−235 3.0000000 RM1 4.63𝑒−116 1.32𝑒−230 3.0000000 RM2 4.63𝑒−116 1.32𝑒−230 3.0000000 RM3 5.53𝑒−130 1.88𝑒−258 3.0000000 RM4 3.98𝑒−140 9.77𝑒−279 3.0000000 RM5 1.14𝑒−102 7.95𝑒−204 3.0000000 RM6 1.04𝑒−103 6.71𝑒−206 3.0000000 TNFE = 12. Table 2: Comparison of various iterative methods for the function 𝑓2(𝑥)under the same total number of function evaluations (TNFE) required by all methods. Method |𝑥𝑛−𝑥⋆||𝑓(𝑥 𝑛)| COC MNM 2.83𝑒−95 1.06𝑒−283 2.0000000 DM 1.88𝑒−109 3.11𝑒−326 3.0000000 VM 3.19𝑒−107 1.52𝑒−319 3.0000000 RM1 2.04𝑒−106 4.00𝑒−317 3.0000000 RM2 2.59𝑒−104 8.11𝑒−311 3.0000000 RM3 8.19𝑒−110 2.57𝑒−327 3.0000000 RM4 2.09𝑒−113 4.27𝑒−338 3.0000000 RM5 4.66𝑒−100 4.73𝑒−298 3.0000000 RM6 1.05𝑒−144 5.39𝑒−432 3.0000000 TNFE = 12. Table 3: Comparison of various iterative methods for the function 𝑓3(𝑥)under the same total number of function evaluations (TNFE) required by all methods. Method |𝑥𝑛−𝑥⋆||𝑓(𝑥 𝑛)| COC MNM 1.53𝑒−119 4.09𝑒−478 2.0000000 DM 5.39𝑒−167 6.27𝑒−668 3.0000000 VM 1.91𝑒−176 9.90𝑒−706 3.0000000 RM1 9.39𝑒−185 5.79𝑒−739 3.0000000 RM2 3.08𝑒−169 6.69𝑒−677 3.0000000 RM3 9.92𝑒−164 7.21𝑒−655 3.0000000 RM4 1.38𝑒−156 2.73𝑒−626 3.0000000 RM5 1.84𝑒−152 8.48𝑒−610 3.0000000 RM6 7.89𝑒−143 2.89𝑒−571 3.0000000 TNFE = 12. modifiedNewtonmethod(MNM),(1), Dong’s method (DM), (4), and Victory-Neta’s method (VM). Tables 1,2,and3show the difference of the root 𝑥⋆and the approximation 𝑥𝑛to 𝑥⋆for the function 𝑓1−𝑓3,respectively, where 𝑥⋆is the exact root computed with 650 significant digits and 𝑥𝑛iscalculatedbyusingthesametotalnumber of function evaluations (TNFE) for all methods. The absolute values of the function (|𝑓(𝑥𝑛)|)and the computational order of convergence (COC) are also shown in these tables. Here, COC is defined by [27] 𝜌≈ln 󵄨󵄨󵄨󵄨(𝑥𝑛+1 −𝑥⋆)/(𝑥𝑛−𝑥⋆)󵄨󵄨󵄨󵄨 ln 󵄨󵄨󵄨󵄨(𝑥𝑛−𝑥⋆)/(𝑥𝑛−1 −𝑥⋆)󵄨󵄨󵄨󵄨.(81) The following functions are used for the comparison: 𝑓1(𝑥)=(sin2𝑥−𝑥2+1)2, 𝑥0=1.45, 𝑥⋆≈1.4044916482153412260350868178, 𝑓2(𝑥)=(cos 𝑥−𝑥)3, 𝑥0=0.9, 𝑥⋆≈0.73908513321516064165531208767, 𝑓3(𝑥)=(ln (𝑥)+√𝑥−5)4, 𝑥0=8.0, 𝑥⋆≈8.3094326942315717953469556827. (82) As shown in Tables 1,2,and3, the presented methods in this contribution are preferable to the modified Newton method by numerical tests. Note also that the presented methods show at least equal performance as compared with some other methods of the same order. 6. Conclusion In this work, we obtained two families of third-order methods by using new techniques of divided differences for solving nonlinear equations with multiple roots. The proposed methods contain many known methods, such as Dong’s methods and Neta’s method. We conclude from the numerical examples that the proposed methods have at least equal performance as compared with the other iterative methods of the same order. Moreover, it was observed that these methods have better performance than the modified Newton method. We will continue our study to confirm if some fourth-order methods can be obtained using the same ideas in this contribution. Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper. Acknowledgments This work is supported by the National Basic Research 973 Program of China (no. 2011JB105001), National Natural Science Foundation of China (Grant no. 11371320), Zhejiang Natural Science Foundation (Grant no. LZ14A010002), the Foundation of Science and Technology Department (Grant no. 2013C31084) of Zhejiang Province, Scientific Research Journal of Applied Mathematics 9 Fund of Zhejiang Provincial Education Department (nos. Y201431077 and Y201329420), and the Grant FEKT-S-11-2921 of Faculty of Electrical Engineering and Communication, Brno University of Technology, Czech Republic. References [1] E. Schr¨ oder, “Ueber unendlich viele Algorithmen zur Aufl¨ osung der Gleichungen,” Mathematische Annalen,vol.2,no.2,pp.317– 365, 1870. [2] E. Hansen and M. Patrick, “A family of root finding methods,” Numerische Mathematik,vol.27,no.3,pp.257–269,1976. [3]J.F.Traub,Iterative Methods for the Solution of Equations, Chelsea, New York, NY, USA, 1977. [4] C. 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