A new sixth-order scheme for nonlinear equations

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1 Calhoun: The NPS Institutional Archive DSpace Repository Faculty and Researchers Faculty and Researchers Collection 202 A new sixth-order scheme for nonlinear equations Chun, Changbum Downloaded from NPS Archive: Calhoun

2 Applied Mathematics Letters ( ) Contents lists available at SciVerse ScienceDirect Applied Mathematics Letters journal homepage: A new sixth-order scheme for nonlinear equations Changbum Chun a,, Beny Neta b, a Department of Mathematics, Sungkyunkwan University, Suwon , Republic of Korea b Naval Postgraduate School, Department of Applied Mathematics, Monterey, CA 93943, United States a r t i c l e i n f o a b s t r a c t Article history: Received 0 February 20 Received in revised form 3 August 20 Accepted 4 August 20 Keywords: Newton s method Iterative methods Nonlinear equations Order of convergence Root-finding methods In this paper we present a new efficient sixth-order scheme for nonlinear equations. The method is compared to several members of the family of methods developed by Neta (979) [B. Neta, A sixth-order family of methods for nonlinear equations, Int. J. Comput. Math. 7 (979) 57 6]. It is shown that the new method is an improvement over this well known scheme. Published by Elsevier Ltd. Introduction Solving nonlinear equations is one of the most important problems in numerical analysis. To solve nonlinear equations, iterative methods such as Newton s method are usually used. Throughout this paper we consider iterative methods to find a simple root ξ, i.e., f (ξ) = 0 and f (ξ) 0, of a nonlinear equation f (x) = 0, where f : I R R for an open interval I. Newton s method for the calculation of ξ is probably the most widely used iterative scheme defined by x n+ = x n f (x n). () It is well known (see e.g. []) that this method is quadratically convergent. Some modifications of Newton s method to achieve higher order and better efficiency have been suggested and analyzed in the literature. See e.g. the books by Ostrowski [2], Traub [] and Neta [3]. See also more recent results by Kim [4] who discussed a wide collection of sixth-order methods, Soleymani [5] and Khattri and Argyros [6]. This last paper gives a family of three step methods free from derivatives. Most of the methods improve the order of convergence and computational efficiency of Newton s method with an additional evaluation of the function or its derivatives. To be more precise, we define informational efficiency E by E = p d where p is the order of the method and d is the number of function- (and derivative-) evaluations per step. We also mention another measure, the efficiency index I I = p /d. Corresponding author. Tel.: ; fax: addresses: cbchun@skku.edu (C. Chun), bneta@nps.edu, byneta@gmail.com (B. Neta). Tel.: ; fax: /$ see front matter. Published by Elsevier Ltd doi:0.06/j.aml

3 2 C. Chun, B. Neta / Applied Mathematics Letters ( ) Here we compare the sixth-order family of methods [7] given by w n = x n f (x n), z n = w n f (w n) f (x n ) + βf (w n ) f (x n ) + (β 2)f (w n ), x n+ = z n f (w n) f (x n ) f (w n ) + γ f (z n ) f (x n ) 3f (w n ) + γ f (z n ), (2) to our new sixth-order scheme. Both of these methods have the same efficiency index 6 /4.565 which is better than that of Newton s method. This method has an error term ϵ n+ = c 2 c 3 c3 (2β + )c 2 2 ϵ 6 + n O(ϵ7 n ), (3) where ϵ n = x n ξ and c i = f (i) (ξ) i!f (ξ), i. Note that γ does not appear in the error constant and therefore we can take γ = 0. The first two substeps constitute King s fourth-order scheme [8]. For the parameter β, we note that if β = 0 then the first two steps are Ostrowski s fourthorder method [2]. If we choose β = then the factor in the second and third substeps is identical, and thus we can save on computation. The choice β = /2 minimizes the error term, ϵ n+ = c 2 3 c 2ϵ 6 n + O(ϵ7 n ). In the numerical experiments section we will use these three parameters for comparison. 2. Development of method and convergence analysis We suggest replacing the first two substeps in (2) by the fourth-order method due to Kung and Traub [9] (4) w n = x n f (x n), x n+ = w n f (w n) and then consider the method w n = x n f (x n), z n = w n f (w n) x n+ = z n f (z n) f (w n) f (x n ) f (w n) f (x n ) 2, 2, f (w n) f (x n ) f (z n) f (x n ) 2. (5) (6) For the method defined by (6), we have the following analysis of convergence. Theorem 2.. Let ξ I be a simple zero of a sufficiently differentiable function f : I R for an open interval I. Let ϵ n = x n ξ. Then the new method defined by (6) is of sixth-order. The error at the (n + ) st step, ϵ n+, satisfies the relation ϵ n+ = 5c 3 c c5 + 2 c 2c 2 3 ϵ 6 + n O(ϵ7 n ), (7) where c i i =, 2, 3 are given by (4). Proof. Let ϵ n = x n ξ, u n = w n ξ and v n = z n ξ. Using the Taylor expansion of f (x) around x = ξ and taking f (ξ) = 0 into account, we get f (x n ) = f (ξ) ϵ n + c 2 ϵ 2 n + c 3ϵ 3 n + c 4ϵ 4 n + O(ϵ5 n ), (8) = f (ξ) + 2c 2 ϵ n + 3c 3 ϵ 2 n + 4c 4ϵ 3 n + O(ϵ4 n ). (9)

4 C. Chun, B. Neta / Applied Mathematics Letters ( ) 3 Dividing (8) by (9) gives u n = ϵ n f (x n) = c 2ϵ 2 n ( 2c 3 + 2c 2 2 )ϵ3 n ( 3c 4 + 7c 2 c 3 4c 3 2 )ϵ4 n (0c 2 c 4 + 6c c 3c c4 2 )ϵ5 + n O(ϵ6 n ), (0) so that, after elementary calculation, f (w n ) = f (ξ)[u n + c 2 u 2 n + c 3u 3 n + c 4u 4 n + O(u5 n )] = f (ξ)[c 2 ϵ 2 + n (2c 3 2c 2 2 )ϵ3 + n (3c 4 7c 2 c 3 + 5c 3 2 )ϵ4 + n ( 2c4 2 6c2 3 0c 2c c 3 c 2 2 )ϵ5 + n O(ϵ6 n )]. () Using (8) (), we find v n = u n f (w n) [ f (w n )/f (x n )] 2 = (2c 3 2 c 2c 3 )ϵ 4 n + ( 2c2 3 0c4 2 2c 2c 4 + 4c 3 c 2 2 )ϵ5 n + (2c 4c 2 2 7c 4c c 2 c c5 2 72c 3c 3 2 )ϵ6 n + ( 00c 4 c c 4c 2 c 3 88c 2 3 c c 3c 4 2 6c c3 3 74c6 2 )ϵ7 + n O(ϵ8 n ), (2) so that, after elementary calculation, f (z n ) = f (ξ)[v n + c 2 v 2 n + c 3v 3 n + O(v4 n )] = f (ξ) ( c 2 c 3 + 2c 3 2 )ϵ4 n + ( 0c4 2 2c 2c 4 2c c 3c 2 2 )ϵ5 n + (3c c 2c c 4c 2 2 7c 4c 3 72c 3 c 3 2 )ϵ6 n + O(ϵ7 n ). (3) By doing simple calculations with () (3) we obtain ϵ n+ = v n f (z n) [ f (w n )/f (x n ) f (z n )/f (x n )] 2 = ( 5c 3 c c5 + 2 c 2c 2 3 )ϵ6 + n O(ϵ7 n ), (4) which means that the method defined by (6) is at least sixth-order. This completes the proof. Remark. Kung and Traub [9] have conjectured that one can get an optimal eighth-order scheme using the same information as our scheme. That is for three function- and one derivative-evaluation one can get an eighth-order method. Therefore our scheme is not optimal, but the computational cost of our additional step is not as high as the optimal eighth-order obtained by interpolation. 3. Numerical examples In this section we present some numerical experiments using our new method and compare these results to the three members of Neta s family of schemes. All computations were done using MAPLE using 28 digit floating point arithmetics (Digits := 28). We accept an approximate solution rather than the exact root, depending on the precision (ϵ) of the computer. We use the following stopping criteria for computer programs: (i) x n+ x n < ϵ, (ii) f (x n+ ) < ϵ, and so, when the stopping criterion is satisfied, x n+ is taken as the exact root ξ computed. For numerical illustrations in this section we used the fixed stopping criterion ϵ = We used the following 23 test functions, some are taken from [0] and some from []. Test function x 0 x f (x) = x 3 + 4x f 2 (x) = sin 2 (x) x f 3 (x) = (x ) f 4 (x) = x f 5 (x) = xe x2 sin 2 (x) + 3 cos(x) f 6 (x) = e x2 +7x f 7 (x) = sin(x) x f 8 (x) = x 5 + x

5 4 C. Chun, B. Neta / Applied Mathematics Letters ( ) f 9 (x) = x x f 0 (x) = e x + x f (x) = ln(x) + x f 2 (x) = x 3 x f 3 (x) = x 2 e x 3x f 4 (x) = arctan(x) f 5 (x) = e x sin(x) + ln( + x 2 ).0 0 f 6 (x) = ln(x 2 + x + 2) x f 7 (x) = e x2 +x f 8 (x) = x 5 + x 4 + 4x f 9 (x) = x f 20 (x) = x f 2 (x) = 2 + x 2 sin π f 22 (x) = cos 2 x f 23 (x) = x 4 + sin π x x ln(x2 + 2x + 2) x 2 π x 2 Table Comparison of sixth-order iterative schemes. f N0 N Nh New f IT f (x ) 6e 27 6e 27 6e 27 6e 27 f 2 IT f (x ) e 27 e 27 e 27 e 27 f 3 IT f 4 IT f 5 IT f (x ) e 26.e 26.2e 26.e 26 f 6 IT div 6 9 f (x ) f 7 IT f (x ) 2e 28 2e 28 2e 28 2e 28 f 8 IT div div 7 5 f (x ) 0 0 f 9 IT div div div 4 f (x ) 0 f 0 IT div div div 7 f (x ) 0 f IT 5 div div 4 f (x ) e 27 e 27 f 2 IT f (x ) e 27 e 27 e 27 e 27 f 3 IT f (x ) e 27 e 27 e 27 e 27 f 4 IT f 5 IT f 6 IT (continued on next page)

6 C. Chun, B. Neta / Applied Mathematics Letters ( ) 5 Table (continued) f N0 N Nh New f 7 IT f 8 IT div f (x ) e 26 e 26 e 26 f 9 IT f 20 IT 6 div 4 4 f (x ) 5e 28 5e 28 e 27 f 2 IT f 22 IT f (x ) 7e 28 7e 28 7e 28 7e 28 f 23 IT f (x ) 5e 27 5e 27 5e 27 5e 27 In Table we presented the results for N0 (the case for β = 0), N (the case for β = ), Nh (the case for β = /2) and our new scheme. The number of iterations IT is given along with the value of the function at the last iteration f (x ). Notice that out of 23 cases our method diverged only in one case but for the three members of Neta s family we found divergence in 3 5 cases. In cases the methods gave the same answer with the same number of iterations. The β = /2 and the new method were superior in 5 cases whereas the other methods were superior in 3 or 4 cases. Therefore we can conclude that the new method is competitive with Neta s family of sixth-order schemes. Acknowledgments The first author s research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology ( ). References [] J.F. Traub, Iterative Methods for the Solution of Equations, Chelsea Publishing Company, New York, 977. [2] A.M. Ostrowski, Solution of Equations and Systems of Equations, Academic Press, New York, London, 966. [3] B. Neta, Numerical methods for the solution of equations, net-a-sof, Monterey, 983. [4] Y.I. Kim, A new two-step biparametric family of sixth-order iterative methods free from second derivatives for solving nonlinear algebraic equations, Appl. Math. Comput. 25 (200) [5] F. Soleymani, Regarding the accuracy of optimal eighth-order methods, Math. Comput. Modelling 53 (20) [6] S.K. Khattri, I.K. Argyros, Sixth order derivative free family of iterative methods, Appl. Math. Comput. 27 (20) [7] B. Neta, A sixth order family of methods for nonlinear equations, Int. J. Comput. Math. 7 (979) [8] R.F. King, A family of fourth order methods for nonlinear equations, SIAM J. Numer. Anal. 0 (973) [9] H.T. Kung, J.F. Traub, Optimal order of one-point and multipoint iterations, J. Assoc. Comput. Mach. 2 (974) [0] S. Weerakoon, G.I. Fernando, A variant of Newton s method with accelerated third-order convergence, Appl. Math. Lett. 7 (2000) [] B. Neta, Several new schemes for solving equations, Int. J. Comput. Math. 23 (987)