A Family of Optimal Multipoint Root-Finding Methods Based on the Interpolating Polynomials

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1 Applied Mathematical Sciences, Vol. 8, 2014, no. 35, HIKARI Ltd, A Family of Optimal Multipoint Root-Finding Methods Based on the Interpolating Polynomials Beong In Yun Department of Statistics and Computer Science Kunsan National University, Gunsan, Republic of Korea Copyright c 2014 Beong In Yun. 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. Abstract A family of simple derivative-free multipoint iterative methods, based on the interpolating polynomials, for solving nonlinear equations is presented. It is shown that the presented n-point iterative method has the convergence order 2 n 1 with n function evaluations per iteration. It is an optimal iterative method in the sense of the Kung-Traub s conjecture. Numerical examples are included to support the result of theoretical convergence analysis and demonstrate efficiency of the proposed method. Mathematics Subject Classification: 65H05 Keywords: Nonlinear equation, multiploint method, optimal iterative method, interpolating polynomial 1 Introduction For finding roots of nonlinear equations multipoint iterative methods without memory are most often used since they have a definite advantage in both order of convergence and computational efficiency [1 3, 5 9, 11]. General investigation of the multipoint and one-point methods can be found in the literature [10]. Moreover, Kung and Traub [4] conjectured that the order of

2 1724 Beong In Yun convergence of a multipoint method with n function evaluations per iteration can not exceed d n = 2 n 1 which is called an optimal order. On the other hand, the computational efficiency of an iterative method of order d n, requiring n function evaluations, is defined by d 1/n n [10]. Therefore, the computational efficiency of an iterative method having the optimal order d n is 2 (n 1)/n. In this paper we propose a new family of derivative-free multipoint methods based on the interpolating polynomials. We prove that the presented n-point methods, requiring n function evaluations, have the optimal convergence order 2 n 1 and thus have the optimal computational efficiency 2 (n 1)/n. Numerical results for some examples show the consistency of the presented method with the theoretical convergence analysis. 2 A new family of derivative free multipoint methods We suppose that the equation f(x) = 0 has a simple root α in an interval (a, b). For an integer q 1 and for a k-th iterate x k near the root α we propose a family of n-point multipoint methods as follows. ψ 0,k = x k ψ 1,k = x k + f(x k ) q ψ 2,k = ψ 1,k f (ψ 1,k) P 1 (ψ 1,k ) ψ 3,k = ψ 2,k f (ψ 2,k) P 2 (ψ 2,k ). ψ n,k = ψ n 1,k f (ψ n 1,k) P n 1 (ψ n 1,k ) = x k+1, (1) where n 2 and P λ (t) is an interpolating polynomial of degree 1 λ n 1 for f at the points ψ 0,k, ψ 1,k,, ψ λ,k. Without loss of generality, we may write P λ (t) by the divided differences as P λ (t) = λ i 1 f [ψ 0,k, ψ 1,k,, ψ i,k ] (t ψ j,k ) (2) i=0 j=0 satisfying P λ (ψ j,k ) = f (ψ j,k ), j = 0, 1,, λ. For example, 2- and 3-point iterative methods can be respectively expressed

3 A family of optimal multipoint root-finding methods 1725 by (ψ 1,k = ) s k = x k + f(x k ) q (ψ 2,k = ) x k+1 = s k f (s k) P 1 (s k ) with P 1 (t) = f (x k ) + f [x k, s k ] (t x k ) and (ψ 1,k = ) s k = x k + f(x k ) q (ψ 2,k = ) y k = s k f (s k) P 1 (s k ) (ψ 3,k = ) x k+1 = y k f (y k) P 2 (y k ) (3) (4) with P 2 (t) = P 1 (t) + f [x k, s k, y k ] (t x k )(t s k ). Theorem 2.1 Assume that f is a real analytic function on an interval (a, b) containing a simple zero α with f (x) 0 for all x (a, b), and let x 0 be sufficiently close to α. Then, for any integer q 1, the family of n-point iterative methods in (1) has the order of convergence 2 n 1 with n function evaluations. In other words, for ɛ k = x k α, it follows that ( ) ɛ k+1 = ψ n,k α = O ɛ d n k, d n = 2 n 1 for every integer n 2. Proof: We will prove by induction. For the initial value of n = 2 x k+1 = ψ 2,k = ψ 1,k f (ψ 1,k) P 1 (ψ 1,k ). Therein f (ψ 1,k ) satisfies f (ψ 1,k ) = f (α) (ψ 1,k α) + O ( (ψ 1,k α) 2) = f (α) (ψ 1,k α) + O ( ) ɛ 2 k since ψ 1,k α = x k α + f(x k ) q = ɛ k + O (ɛ q k ). In addition, P 1 (ψ 1,k ) satisfies P 1 (ψ 1,k ) = f [x k, x k + f(x k ) q ] = f (x k ) + O (f(x k ) q ) = f (α) + O (ɛ k ). These results imply that, for sufficiently small ɛ k, ψ 2,k = ψ 1,k (ψ 1,k α) + O ( ɛk) 2.

4 1726 Beong In Yun Thus we have ɛ k+1 = ψ 2,k α = O (ɛ 2 k ), that is, the 2-point iterative method is of order d 2 = 2. Suppose that for some integer n 3 the n-point iterative method satisfies ψ n,k α = O ( ɛ dn k ), dn = 2 n 1. Then we consider the (n + 1)-point iterative method such as ψ n+1,k = ψ n,k f (ψ n,k) P n (ψ n,k ). Similarly to the case of n = 2, we can see that and Therefore f (ψ n,k ) = f (α) (ψ n,k α) + O ( (ψ n,k α) 2) P n (ψ n,k ) = f [x k, x k + f(x k ) q ] + O (ɛ k ) = f (x k ) + O (ɛ k ) = f (α) + O (ɛ k ). ψ n+1,k = ψ n,k (ψ n,k α) + O ( (ψ n,k α) 2) = α + O ( (ψ n,k α) 2) and, by the above assumption, we have ψ n+1,k α = O ( (ψ n,k α) 2) = O ( ɛ 2dn k ) ) = O (ɛ d n+1 k. This completes the proof. 3 Examples We consider three equations f i (x) = 0, i = 1, 2, 3, having a simple root α on a given interval for each function f i (x) defined as f 1 (x) = (3x 2 1)e x x + 1, 0 < x < 1 (α = 1 3 ) f 2 (x) = (x + 2) log(3 x), 1 < x < 3.5 (α = 2) f 3 (x) = cos ( πx 2 /4 ) + x 2 2, 0 < x < 1.5 (α = 2). (5) For these examples numerical results of the presented multipoint methods are given in Table 1. Therein, dn,k denotes the computational order of convergence defined as d n,k = log (x k α)/(x k 1 α) log (x k 1 α)/(x k 2 α), k 2

5 A family of optimal multipoint root-finding methods 1727 for the k-th iterate x k obtained by the presented n-point multipoint method. In addition, each initial approximation x 0 was chosen by the following formula, with N = 4, given in the literature [12]. x 0 = x + F (a) h 2 N 1 j=1 F ( x + (2j N)h/2), (6) where F (x) = Sign(f(x)), h = (b a)/n and x = (a + b)/2 for a given interval (a, b) containing the simple root α. One can see that the results of the presented multipoint methods are consistent with the results of theoretical convergence analysis in Theorem 1. Table 1. Numerical results of the presented n-point iterative methods (1) with q = 1. f 1 (x) = 0 f 2 (x) = 0 f 3 (x) = 0 k x k α dn,k x k α dn,k x k α dn,k n = 3 n = 4 n = 5 n = On the other hand, we may surmise that convergence of the iterates x k s becomes better as the exponent q in (1) increases since the approximation of P 1(ψ 1,k ) to the derivative f (x k ) is accurate for large q as long as f(x k ) is sufficiently small. To identify this surmise we include Figure 1 which shows behavior of errors of the approximation of the second iterates x 2 s obtained by the proposed 4-point method, requiring 4 function evaluations, with respect to the exponents q = 1, 2, 3, 4, 8. Furthermore, it is compared with the Sharma s

6 1728 Beong In Yun method of order eight [8] as follows. w k = x k f(x k) f (x k ) f(x k ) f(w k ) z k = w k f(x k ) 2f(w k ) f (x k ) x k+1 = z k W (µ k ) f[x k, w k ]f(z k ) f[x k, z k ]f[w k, z k ], (7) where µ k = f(z k )/f(x k ) and W is a weight function defined as W (t) = 1+t+t 2. As a result, one can find that the presented method with q 2 provides better approximate iterates than the Sharma s method. Acknowledgements. This research was supported by Basic Science Research program through the National Research Foundation of Korea(NRF- 2013R1A1 A4A ). References [1] W. Bi, H. Ren, Q. Wu, Three-step iterative methods with eighth-order convergence for solving nonlinear equations, J. Comput. Appl. Math. 225 (2009), [2] J. Džunić, M.S. Petković, L.D. Petković, Three-point methods with and without memory for solving nonlinear equations, Appl. Math. Comput. 218 (2012), [3] J. Kou, A. Wang, Y. Li, Some eighth-order root-finding three-step methods, Commun. Nonlin. Sci. Numer. Simul. 15 (2010), [4] H.T. Kung, J.F. Traub, Optimal order of one-point and multipoint iteration, J. ACM 21 (1974), [5] X. Li, C. Mu, J. Ma, C. Wang, Sixteenth-order method for nonlinear equations, Appl. Math. Comput. 215 (2010), [6] L. Liu, X. Wang, Eighth-order methods of high efficiency index for solving nonlinear equations, Appl. Math. Comput. 215 (2010), [7] M.S. Petković, On a general class of multipoint root-finding methods of high computational efficiency, SIAM J. Numer. Anal. 47 (2010),

7 A family of optimal multipoint root-finding methods 1729 [8] J.R. Sharma, R. Sharma, A new family of modified Ostrowski s methods with accelerated eighth order convergence, Numer. Algor. 54 (2010), [9] R. Thukral, A new eighth-order iterative method for solving nonlinear equations, Appl. Math. Comput. 217 (2010), [10] J.F. Traub, Iterative Methods for the Solution of Equations, Prentice- Hall, New Jersey, [11] X. Wang, L. Liu, New eighth-order iterative methods for solving nonlinear equations, J. Comput. Appl. Math. 234 (2010), [12] B.I. Yun, A non-iterative method for solving non-linear equations, Appl. Math. Comput. 198 (2008), Received: January 11, 2014

8 1730 Beong In Yun (q) Figure 1. Errors x 2 α of the second iterates x 2 obtained by the presented 4- point method with degrees q = 1, 2, 3, 4, 8 for the example f 1 (x) = 0. The dotted line indicates the error of the second iterate from Sharma s method.