Research Article A New Biparametric Family of Two-Point Optimal Fourth-Order Multiple-Root Finders

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1 Applied Matheatics, Article ID , 7 pages Research Article A New Biparaetric Faily of Two-Point Optial Fourth-Order Multiple-Root Finders Young Ik Ki and Young Hee Geu Departent of Applied Matheatics, Dankook University, Cheonan , Republic of Korea Correspondence should be addressed to Young Hee Geu; conpana@epal.co Received 21 February 2014; Revised 17 June 2014; Accepted 20 June 2014; Published 14 Septeber 2014 AcadeicEditor:JuanR.Torregrosa Copyright 2014 Y. I. Ki and Y. H. Geu. This is an open access article distributed under the Creative Coons Attribution License, which perits unrestricted use, distribution, and reproduction in any ediu, provided the original work is properly cited. We construct a biparaetric faily of fourth-order iterative ethods to copute ultiple roots of nonlinear equations. This ethod is verified to be optially convergent. Various nonlinear equations confir our proposed ethod with order of convergence of four and show that the coputed asyptotic error constant agrees with the theoretical one. 1. Introduction It is not surprising that odified Newton s ethod [1]inthe siple for x n+1 =x n f(x n) is ost widely used to find the approxiate ultiple root α of known ultiplicity for a given nonlinear equation f(x) = 0. Recall that nuerical schee (1) isaone-point optial ethod with quadratic convergence. In order to find nuerical solution for ultiple roots of nonlinear equations ore accurately, any researchers have ade enorous efforts in developing higher-order ethods with iproved convergence. In this paper, we extend odified Newton s ethod and propose two-point optial fourth-order ultiple-root finders by evaluating two derivatives and one function per iteration. The optiality will be pursued based on Kung- Traub s conjecture [2] in which the convergence order of any ultipoint ethod [3] without eory can reach at ost 2 k 1 for k evaluations of functions or derivatives. The contents of this paper consist of what follows. Described in Section 2 are previous studies on ultipleroot finders. Section 3 proposes a new biparaetric faily of two-point optial fourth-order ultiple-root finders. It (1) fullytreatsethoddevelopentandconvergenceanalysis. Derivation of the error equations for the proposed schees is an iportant task for ensuring convergence behavior. In Section4, a variety of nuerical exaples are presented forawideselectionoftestfunctions.itisiportantto copare the convergence behavior of the proposed schees with that of existing ethods. We confir that the proposed ethods well show the convergence behavior predicted by the developed theory. 2. Preliinary Review of Previous Studies A nuber of interesting fourth-order ultiple-root finders can be found in papers [4 16]. Aong these, we especially introduce five studies as follows. Shengguo et al. [14] introduced the following fourth-order ethod which needs evaluations of one function and two derivatives per iteration for x 0 chosen in a neighborhood of the sought zero α of f(x) with known ultiplicity 1: x n+1 =x n βf (x n )+φf (y n ) +δf (y n ) f(x n ), n=0,1,2,..., where y n =x n (2/( + 2))(f(x n )/), β= 2 /2, φ = (1/2)(( 2)/(/( + 2)) ),andδ = (/( + 2)). (2)

2 2 Applied Matheatics J. R. Shara and R. Shara [12]constructedthefollowing fourth-order schee with A = (1/8)( 3 4+8), B= (1/4)( 1)( + 2) 2 (/( + 2)) 2, C = (1/8)( + 2) 3 (/( + 2)) 2,andy n =x n (2/(+2))(f(x n )/): x n+1 =x n A f(x n) Bf(x n) f (y n ) C( f(x 2 n) f (y n ) ) ( f(x 1 n) ), n=0,1,2,... Li et al. [15] presented the fourth-order ethod with y n = x n (2/( + 2))(f(x n )/): x n+1 =x n ( 2 2 ) f(x n) f (x n ) +(/( + 2)) f (y n ). Zhou et al. [16] proposed the following fourth-order iterative schee with y n =x n (2/( + 2))(f(x n )/): x n+1 =x n 8 [3 ( +2 2 ) ( f 2 (y n ) ) 2 2 (+3) ( 2+ ) f (y n ) +( )] f(x n). Kanwar et al. [8] developed the fourth-order optial ultipoint iterative ethod for ultiple zeros: x n+1 =x n (f(x n )( 2 (3 2) {f (y n )} 2 (3) (4) (5) C 1 f (y n )+C 2 {} 2 )) (2(( 1)( +2 ) f (y n )) (f (y n ) ( +2 ) (+8) )) 1, (6) where y n = x n (2/( + 2))(f(x n )/), C 1 = (/( + 2)) ( ), and C 2 = (/( +2)) 2 ( ). 3. Method Developent and Convergence Analysis We first suppose that a function f:c C has a ultiple root α with integer ultiplicity 1 andisanalyticina sall neighborhood of α. Then a new iteration ethod free of second derivatives is proposed below to find an approxiate root α of ultiplicity,givenaninitialguessx 0 sufficiently close to α: where x n+1 =y n a f(x n) cf(x n) f (y n ) df(y n) f (y n ), y n =x n γ f(x n), n = 0, 1, 2,..., F(y n )=f(x n ) (y n x n ) λf (x n )f (y n ) +ρf (y n ), with a, c, d, γ, λ, and ρ are paraeters to be chosen for axial order of convergence [17, 18]. We establish a ain theore describing the convergence analysis regarding proposed schee (7) and find out how to select paraeters a, c,and d for optial fourth-order convergence. Definition 1 (error equation, asyptotic error constant, and order of convergence). Let x 0,x 1,x 2,...,x n,... be a sequence converging to α and let e n =x n αbe the nth iterate error. If there exist real nubers p R and b R {0}such that the following error equation holds (7) (8) e n+1 =be p n +O(ep+1 n ), (9) then b or b is called the asyptotic error constant and p is called the order of convergence [17, 18]. Theore 2. Let f : C C have a zero α with integer ultiplicity 1 andbeanalyticinasallneighborhood of α. Letκ = (/( + 2)) and θ j = f (+j) (α)/f () (α) for j N. Letx 0 be an initial guess chosen in a sufficiently sall neighborhood of α. Letλ, ρ R be two free constant paraeters. Let a= 3 ( 4κρ+(+2)(1+κρ))/8(+2)κρ, c=(+2){ 3 + ( + 2)κ( 3 + 2( + 2)κλ)ρ + ( + 2) 2 κ 2 ( 3+2κρ)ρ 2 (+2) 3 κ 3 ρ 3 }/16κλρ, d = (2+)(+(2+ )κρ) 3 /16κλρ, andγ = 2/(2 + ). Then iterative ethods (7) are optial and of order four and possess the following error equation: e n+1 =ψ 4 e 4 n +O(e5 n ), (10) where ψ 4 = (( (24κρ/(+(+ 2)κρ)))/3 2 ( + 1) 3 ( + 2))θ 3 1 (1/( + 1)2 ( + 2))θ 1 θ 2 + (/( + 2) 3 ( + 1)( + 3))θ 3. Proof. The optiality on convergence order of proposed schee (7) is clear in the sense of Kung-Traub due to three functional evaluations. Hence, it suffices to deterine the constant paraeters for fourth-order convergence.

3 Applied Matheatics 3 Applying the Taylor s series expansion about α,we get the following relations: f(x n ) = f() (α) e n! [1 + A 1e n +A 2 e 2 n +A 3e 3 n +A 4e 4 n +O(e5 n )], (11) = f() (α) ( 1)! e 1 n [1 + B 1 e n +B 2 e 2 n +B 3e 3 n +B 4e 4 n +O(e5 n )], (12) t 3 ( K 3 1 (2 3+2 ) (t 1) 3 +3A 1 K 2 1 (+1)(t 1)2 t 2 6K 1 (t 1) t (A 2 (+2) t 3 +K 2 ( 1)(1 t) +6t 2 (K 3 (1 t) +t(a 1 K 2 (+1)(t 1) +A 3 t 3 ))) e 3 n )+O(e4 n )}. (15) where A k =(!/(+k)!)θ k, B k = (( 1)!/( + k 1)!)θ k, and θ k =f (+k) (α)/f () (α) for k N. Dividing (11)by(12), we have f(x n ) = 1 [e n K 1 e 2 n K 2e 3 n K 3e 4 n +O(e5 n )], (13) Substituting (11) (15)into(7), we obtain the error equation: e n+1 =y n α K f =ψ 1 e n +ψ 2 e 2 n +ψ 3e 3 n +ψ 4e 4 n +O(e5 n ), (16) where K 1 = A 1 +B 1, K 2 = A 2 +A 1 B 1 B 2 1 +B 2, K 3 = A 3 +A 2 B 1 A 1 B 2 1 +B3 1 +A 1B 2 2B 1 B 2 +B 3. For algebraic convenience, we introduce a paraeter t defined by γ=(1 t);thatis,t=1 γ/to obtain where K f = a(f(x n )/) + c(f(x n )/f (y n )) + d(f(y n ) /f (y n )), ψ 1 = (a/) + t(1 ((c+ d)t /) (d(t 1)λ/(t + t ρ))), and coefficients ψ i, (i = 2, 3, 4) depend on the paraeters t, a, c, d, λ,andρ and the function f(x). Solving ψ 1 =0, ψ 2 =0for a and c,respectively,weget y n =x n γ f(x n) =e n +α γ f(x n) =α+te+k 1 (1 t) e 2 n +K 2 (1 t) e 3 n +K 3 (1 t) e 4 n +O(e5 n ). (14) a=t(1 (c+d) t d (t 1) λ t+t ρ ), c = (t (t + t ρ) 2 +d(t 1)(1+(t 1) +t) (t 2 +2t +1 ρ+t 2 ρ((t 1) λ+ρ))) (17) Evaluating f(y n ) fro (11) withe n being replaced by y n α in (14), we find f(y n )= f() (α) e! ((t 1)(1+(t 1)+t)(t+t ρ) 2 ) 1. We substitute a, c into ψ 3 and put ψ 3 =ψ 31 θ 2 1 +ψ 32θ 2.Solving ψ 3 =0independently of θ 1 and θ 2,thatis,solvingψ 31 =ψ 32 = 0 for d and t,weobtain {t +t 1 (A 1 t 2 +K 1 (1 t))e n d= (t 1 (2t 2 (2+t) + 2 (t 1)(1+2(t 1) t) (t 2 (K 2 1 ( 1) (t 1)2 2A 1 K 1 (+1)(t 1) t 2 +2t (A 2 t 3 +K 2 (1 t)))) e 2 n + (1 3t + 4t 3 )) (t + t ρ) 3 ) (2(t 1) 3 (1+(t 1) +t) 3 λρ) 1, t= +2. (18)

4 4 Applied Matheatics Table 1: Typical ethods with interesting paraeters (λ, ρ) and constants (a, c, d). Method (λ, ρ) (a,c,d) Y1 ( 10, 10) a= 3 { 40κ + ( + 2)(10κ + 1)} ( + 2)( + 10( + 2)κ)3, d= 80( + 2)κ 1600κ c= ( + 2){ ( + 2)κ + 500( + 2) 2 κ ( + 2) 2 (3 + 2)κ 3 } 1600κ Y2 ( + 2) ( 1, ( )κ ) (0, {( + 4)3 (+2) 3 ( )κ}, 2 ( + 4) 3 2( ) 2 2( ) ) 2 Y3 ( + 4) 3 ( ( + 2) 3 ( )κ, ( + 2) ( )κ ) (0, 0, ( + 2)3 κ 2( ) ) Y4 ( + ( + 2)κρ)3 ( 2( + 2) 2 κ 2 ρ(1 + κρ), 1 2 ) ( 3 (( + 2)(κ 2) 4κ),0, 1 8( + 2)κ 16 (+2)3 κ(κ + 2)) Substituting t = /(+2) into (17)and(18)withκ=(/(2+ )),wegetthefollowingrelations: a= 3 ( 4κρ + (+2)(1 + κρ)), 8 (+2) κρ c = ((+2) { 3 +(+2) κ ( (+2) κλ) ρ (16κλρ) 1, d= +(+2) 2 κ 2 ( 3 + 2κρ) ρ 2 (+2) 3 κ 3 ρ 3 }) (2+)( + (2+) κρ)3. 16κλρ (19) By the aid of sybolic coputation of Matheatica [19], we arrive at the relation below: e n+1 =ψ 4 e 4 n +O(e5 n ), (20) where ψ 4 = (( (24κρ/( + ( + 2)κρ)))/3 2 ( + 1) 3 ( + 2))θ 3 1 (1/( + 1)2 ( + 2))θ 1 θ 2 +(/(+2) 3 ( + 1)( + 3))θ 3. As a result, the proof is copleted. Reark 3. We observe that error equation (20)containsonly one free paraeter ρ, being independent of λ. Table 1 shows typically chosen paraeters λ and ρ and defines various ethods Y k, (k=1,2,3,4). 4. Nuerical Exaples and Conclusion We have perfored a variety of nuerical experients with Matheatica Version 5 [19] to confir the theory developed in Section 3. In these experients, we assign 300, via Matheatica coand $MinPrecision = 300, as the iniunuberofprecisiondigitstoachievethespecified sufficient accuracy. It is crucial to copute e n = x n α with high accuracy for desired nuerical results. When zero α is not exactly known, it is replaced by a highly accurate value α which has larger nuber of significant digits than the assigned iniu nuber of precision digits. To deal with nuerical results ore effectively, we first define α={ α, α, if α is exactly known, if α is not exactly known. (21) To properly display nuerical results, we need to define the nth coputational error e n =x n α for n = 0,1,2...We need further terinologies as defined below. Definition 4 (coputational asyptotic error constant and coputational convergence order). Assue that theoretical asyptotic error constant η = li n e n / e n 1 p and convergence order p 1 are known (usually via ain theore). Define q n = e n / e n 1 p as the coputational asyptoticerrorconstantandp n = log e n /η / log e n 1 as the coputational convergence order. Then we find that li n q n is equal or close to η, while li n p n is equal or close to p. If α has the sae accuracy of $MinPrecision as that of x n, then e n =x n α would be nearly zero and hence coputing e n+1 /e n 4 would unfavorably break down. Coputed values of x n areaccurateupto300significantdigits.forcurrent experients α is found to be accurate enough about up to 400 significant digits. To supply such α, asetoffollowing Matheatica coands are used: sol = FindRoot [f [x],{x,x 0 }, PrecisionGoal $MinPrecision, WorkingPrecision 2 $MinPrecision] ; α =sol[[1, 2]]. (22) Although the nuber of significant digits of x n and α is 300 and 400, respectively, theliitedpaperspace allowsus tolist both of the only up to 15 significant digits. We set the error bound ε to for x n α < ε. As a first exaple, we select a function f(x) = (x 2 + 7) 2 cscx log(x 2 +8)having a ultiple zero α = 7i with

5 Applied Matheatics 5 Table 2: Convergence behavior with f(x) = (x 2 +7) 2 csc x log(x 2 +8). (,λ,ρ)=(3,1/2, 1), α= 7i = 2.64i. n x n f(x n ) x n α e n+1 /e 4 n η log(e n /η)/ log(e n 1 ) i i i i i Table 3: Convergence behavior with f(x) = (x 2 +4x 1) 4 e x. (, λ, ρ) = (4, 10, 10), α = 2 5 = n x n f(x n ) x n α e n+1 /e 4 n η log(e n /η)/ log(e n 1 ) Table 4: Convergence behavior with f(x) = (e x2 +3x+7 1) 5 (x 2 +3x+7) 3. (, λ, ρ) = (8, 1/7, 3), α = ( 3 19i)/2. n x n f(x n ) x n α e n+1 /e 4 n η log(e n /η)/ log(e n 1 ) i i i i i i i = 1. Wechoosex 0 = 2.71i as an initial guess. We take another function f(x) = (x 2 +4x 1) 4 e x with a root α= 2 5. Weselectx 0 = 4.35 as an initial value. The order of convergence and the asyptotic error constant are clearly shown in Tables 2 and 3 revealing a good agreeent with the theory in Section 3. Taking another function f(x) = (e x2 +3x+7 1) 5 (x 2 +3x+7) 3 with a root α=( 3 19i)/2 with ultiplicity =8,weselectx 0 = i as an initialvalue.inthisexaple,wealsofindthattheorderof convergence is four and the coputational asyptotic error constant e n+1 /e 4 n well approaches the theoretical value η. The coputational convergence order and the coputational asyptotic error constant are certainly shown in Tables 2 4 reaching a good agreeent with the theory. It is certain that these ethods need one evaluation of the function f and two evaluations of the first derivative f. Additional test functions below are used to display the convergencebehaviorofproposedschee(7): f 1 (x) =( 1+3x+x 2 ) log (3 + x 2 ), α = , = 1, x 0 = 0.29, f 2 (x) =(x 2 5)log (x 2 4), α= 5, = 2, x 0 = 2.19, f 3 (x) =(x 2 25) 2 e x2 +1 cscx log (x 2 +3x 9), α= 5, =3, x 0 = 5, f 4 (x) =(x 7 x ) 4, α= 7, = 4, x 0 = 2.77, f 5 (x) = log 3 (x 2 1)(x 2 2) 2, α= 2, = 5, x 0 = 1.38, f 6 (x) =(x 2 +3) 4 (log (x 4 +x 2 5)) 2, α= 3i=1.73i, =6, x 0 = 1.68i, f 7 (x) = log (x 2 8)(e x2 +7x 30 1) 3 (x 3) 3, α = 3, = 7, x 0 = (23) In Table 5, we copare nuerical errors x n α of proposed ethods Y1 Y4 with those of existing optial fourthorder ultiple-root finders. Abbreviations S, J, L, Z, and K denote optial fourth-order ultiple-root finders obtained by Shengguo et al., Shara et al., Li et al., Zhou et al., and Kanwar et al., respectively. The least errors within the prescribed error bound are highlighted in boldface. Method Y2 shows best convergence

6 6 Applied Matheatics Table 5: Coparison of x n α for various ultiple-root finders. f(x) x 0 x n α S J L Z K Y1 Y2 Y3 Y4 f f f 3 5 f f f i f x 1 α 1.35e e e e e e e e e 9 x 2 α 1.63e e e e e e e e e 35 x 3 α 3.53e e e e e e e e e 139 x 4 α 0.e e e e e e e e e 300 x 1 α 1.83e e e e e e e 8 x 2 α 2.60e e e e e e e e e 33 x 3 α 1.05e e e e e e e e e 132 x 4 α 0.e e e e e e e e e 300 x 1 α x 2 α 2.73e e e e e e e e e 7 x 3 α 1.05e e e e e e e e e 27 x 4 α 2.35e e e e e e e e e 105 x 5 α 0.e e e e e e e e e 299 x 1 α 9.18e e e e e e e e e 7 x 2 α 2.93e e e e e e e e e 28 x 3 α 3.07e e e e e e e e e 112 x 4 α 0.e e e e e e e e e 299 x 1 α 1.04e e e e e e e e e 8 x 2 α 6.35e e e e e e e e e 30 x 3 α 8.75e e e e e e e e e 121 x 4 α 0.e e e e e e e e e 299 x 1 α 1.79e e e e e e e e e 3 x 2 α 1.36e e e e e e e e e 11 x 3 α 4.76e e e e e e e e e 42 x 4 α 6.94e e e e e e e e e 163 x 5 α 0.e e e e e e e e 299 x 1 α 1.380e e e e e e e e e 2 x 2 α 2.21e e e e e e e e e 7 x 3 α 4.23e e e e e e e e e 28 x 4 α 5.69e e e e e e e e e 109 x 5 α 0.e e e e e e e e e 299 Here, 1.35e 9 denotes

7 Applied Matheatics 7 for f 1, f 3, f 6, while ethod Y4 does show for f 2, f 4, f 5, f 7. It ust be kept in ind that such favorable perforance is shown only in the current nuerical experients with the particular choice of test functions. For a different choice of test functions and initial values, it is hardly expected that each of listed ethods would always give rise to better convergence behavior. One should be aware that no nuerical ethod exhibits better perforance for all the test functions as copared with other nuerical ethods. With the sae order of convergence, one should notice that the speed of local convergence is dependent on the function f(x), an initial value x 0,andaultiplezeroα itself. For efficiency check of ultipoint iterative ethods, we need to calculate the efficiency index [17] defined by EI = p 1/d,wherep is the order of convergence and d is the nuber of distinct functional or derivative evaluations per iteration. The proposed iterative ethods (7) haveei =4 1/ describing the optiality, which is the sae as that of any other listed ethod. This paper confirs optial fourthorder convergence and derives the correct error equation for proposed iterative ethods, using the weighted haronic ean of two derivatives to find approxiate ultiple zeros of nonlinear equations. Proposed optial fourth-order schees efficiently solve given probles without any difficulty, with a wide selection of free paraeters λ and ρ.infuturework,we will pursue higher-order optial ethods by extending the ethods developed here. Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper. Acknowledgents The authors would like to give special thanks to an anonyous referee for valuable suggestions and coents on iproveents of this paper. This research was supported by Basic Science Research Progra through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Project no ). References [6] Y. I. Ki and Y. H. Geu, A two-paraeter faily of fourthorder iterative ethods with optial convergence for ultiple zeros, Applied Matheatics, vol. 2013, ArticleID , 7 pages, [7] A. N. Johnson and B. Neta, High-order nonlinear solver for ultiple roots, Coputers & Matheatics with Applications, vol.55,no.9,pp ,2008. [8]V.Kanwar,S.Bhatia,andM.Kansal, Newoptialclassof higher-order ethods for ultiple roots, peritting = 0, Applied Matheatics and Coputation, vol. 222, pp , [9] X.Li,C.Mu,J.Ma,andL.Hou, Fifth-orderiterativeethod for finding ultiple roots of nonlinear equations, Nuerical Algoriths, vol. 57, no. 3, pp , [10] B. Neta, Extension of Murakai s high-order non-linear solver to ultiple roots, International Coputer Matheatics,vol.87,no.5,pp ,2010. [11] M. Petkovic, L. Petkovic, and J. Dzunic, Accelerating generators of iterative ethods for finding ultiple roots of nonlinear equations, Coputers & Matheatics with Applications,vol.59, no. 8, pp , [12] J. R. Shara and R. Shara, Modified Jarratt ethod for coputing ultiple roots, Applied Matheatics and Coputation, vol. 217, no. 2, pp , [13] J. R. Shara and R. Shara, Modified Chebyshev-Halley type ethod and its variants for coputing ultiple roots, Nuerical Algoriths,vol.61,no.4,pp ,2012. [14] L. Shengguo, L. Xiangke, and C. Lizhi, A new fourth-order iterative ethod for finding ultiple roots of nonlinear equations, Applied Matheatics and Coputation,vol.215,no.3,pp , [15] S.G.Li,L.Z.Cheng,andB.Neta, Soefourth-ordernonlinear solvers with closed forulae for ultiple roots, Coputers & Matheatics with Applications,vol.59,no.1,pp ,2010. [16] X. Zhou, X. Chen, and Y. Song, Constructing higher-order ethods for obtaining the ultiple roots of nonlinear equations, Coputational and Applied Matheatics, vol. 235, no. 14, pp , [17] J. F. Traub, Iterative Methods for the Solution of Equations, Prentice-Hall, New Jersey, NJ, USA, [18] B. Neta, Nuerical Methods for the Solution of Equations, Net- A-Sof, Monterey, Calif, USA, [19] S. Wolfra, The Matheatica Book, Cabridge University Press, Cabridge, UK, 4th edition, [1] L. B. Rall, Convergence of the Newton process to ultiple solutions, Nuerische Matheatik, vol. 9, pp , [2] H. T. Kung and J. F. Traub, Optial order of one-point and ultipoint iteration, the Association for Coputing Machinery,vol.21,pp ,1974. [3] P. Jarratt, Soe efficient fourth order ultipoint ethods for solving equations, BIT, vol. 9, pp , [4] C. Dong, A faily of ultipoint iterative functions for finding ultiple roots of equations, International Coputer Matheatics,vol.21,pp ,1987. [5] Y. H. Geu and Y. I. Ki, Cubic convergence of paraetercontrolled Newton-secant ethod for ultiple zeros, Journal of Coputational and Applied Matheatics, vol.233,no.4,pp , 2009.

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