Four-Point Optimal Sixteenth-Order Iterative Method for Solving Nonlinear Equations

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1 Four-Point Optimal Sixteenth-Order Iterative Method for Solving Nonlinear Equations Malik Zaka Ullah a, A.S. Al-Fhaid a, Fayyaz Ahmad b, a Department of Mathematics, King Abdulaziz University, Jeddah 21589, Kingdom of Saudi Arabia b Dept. de Fisica i Enginyeria Nuclear, Universitat Politecnica de Catalunya, Barcelona 836, Spain Abstract In this research article, we present sixteenth-order iterative method for solving nonlinear equations. The Iterative method has optimal order of convergence sixteen in the sense of Kung-Traub conjecture [1], It means iterative scheme uses five function evaluations to achieve 16(= ) order of convergence. The proposed iterative method utilize one derivative evaluation and weight functions. Numerical experiments are made with comparison to some existing methods to demonstrate the convergence and validation of iterative method. Keywords: Non-linear equations, Iterative methods, Optimal order of convergence, Weight function 1. Introduction In order of construct sixteenth-order convergent iterative method for solving nonlinear equations, we need fourth and eight optimal-order iterative methods as a part of sixteenth-order iterative scheme. Many authors have been developed the optimal eighth-order iterative methods namely Bi-Ren-Wu[2], Bi-Wu-Ren[3], Guem-Kim[4], Liu-Wang[5] and Wang-Liu[6]. For the proposed iterative method, we developed new optimal fourth and eighth order iterative methods to construct optimal sixteenth-order iterative scheme. Babajee-Thukral [7] suggested 4-point sixteenth-order king family of iterative methods for solving nonlinear equations (BT): 1 + βt 1 f (y n ) z n = y n 1 + (β 2)t 1 w n = z n (θ + θ 1 + θ 2 + θ 3 ) f (y (1) n) x n+1 = w n (θ + θ 1 + θ 2 + θ 3 + θ 4 + θ 5 + θ 6 + θ 7 ) f (w n) t 1 = f (y n) f (x n ), t 2 = f (z n) f (x n ), t 3 = f (z n) f (y n ), t 4 = f (w n) f (x n ), t 5 = f (w n) f (z n ), t 6 = f (w n) f (y n ), 1 + βt 1 + 3/2βt1 2 θ = 1, θ 1 = 1, θ 1 + (β 2)t 1 + (3/2β 1)t1 2 2 = t 3, θ 3 = 4t 2, θ 4 = t 5 + t 1 t 2, θ 5 = 2t 1 t 5 + 4(1 β)t 3 1 t 3 + 2t 2 t 3, θ 6 = 2t 6 + (7β 2 47/2β + 14)t 3 t1 4 + (2β 3)t2 2 + (5 2β)t 5t1 2 t3 3, θ 7 = 8t 4 + ( 12β + 2β )t 5 t 3 1 4t3 3 t 1 + ( 2β β 22)t 2 3 t3 1 + ( 1β /2β 2 15β + 46)t 2 t 4 1. Corresponding author addresses: mzhussain@kau.edu.sa ( Malik Zaka Ullah ), aalfhaid@hotmail.com,aalfhaid@kau.edu.sa (A.S. Al-Fhaid), fayyaz.ahmad@upc.edu ( Fayyaz Ahmad ) April 11, 213

2 In 211, Geum-Kim [8] proposed a family of optimal sixteenth-order multipoint methods (GK2): f? ((x n ), z n = K f (u n ) f (y n) s n = z n H f (u n, v n, w n ) f (z n) x n+1 = s n W f (u n, v n, w n, t n ) f (s n) (2) u n = f (y n) f (x n ), v n = f (z n) f (y n ), w n = f (z n) f (x n ), t n = f (s n) f (z n ), K f (u n ) = 1 + βu n + ( 9 + 5/2β)u 2 n 1 + (β 2)u n + ( 4 + β/2)u 2 n, H f = 1 + 2u n + (2 + σ)w n 1 v n + σw n, W f = 1 + 2u n 1 v n 2w n t n + G(u n, v n, w n ), one of the choice for G(u n, v n, w n ) along with β = 24/11 and σ = 2: G(u n, v n, w n ) = 6u 3 nv n 244/11u 4 nw n + 6w 2 n + u n (2v 2 n + 4v 3 n + w n 2w 2 n). In the same year, Geum-Kim[9] presented a biparametric family of optimally convergent sixteenth-order multipoint methods(gk1): f? ((x n ), z n = K f (u n ) f (y n) s n = z n H f (u n, v n, w n ) f (z n) x n+1 = s n W f (u n, v n, w n, t n ) f (s n) u n = f (y n) f (x n ), v n = f (z n) f (y n ), w n = f (z n) f (x n ), t n = f (s n) f (z n ), 1 + βu n + ( 9 + 5/2β)u 2 n K f (u n ) =, H 1 + (β 2)u n + ( 4 + β/2)u 2 f = 1 + 2u n + (2 + σ)w n, n 1 v n + σw n 1 + 2u n + (2 + σ)v n w n W f = + G(u n, w n ), 1 v n 2w n t n + 2(1 + σ)v n w n one of the choice for G(u n, w n ) along with β = 2 and σ = 2: G(u n, w n ) = 1/2? u n w n (6 + 12u n + (24 11β)u 2 n + u 3 nφ 1 + 4σ)? + φ 2 w 2 n, φ 1 = (11β 2 66β + 136), φ 2 = (2u n (σ 2 2σ 9) 4σ 6). (3) 2

3 2. Four-point sixteenth-order iterative method and convergence analysis The proposed sixteenth-order iterative method is describe as follows (MA): z n = y n 1 + 2t 1 t1 2 f (y n ) 1 6t1 2 1 t 1 + t 3 f (z n ) w n = z n 1 3t 1 + 2t 3 t 2 x n+1 = w n (q 1 + q 2 + q 3 + q 4 + q 5 + q 6 + q 7 ) f (w n) (4) t 1 = f (y n) f (x n ), t 2 = f (z n) f (y n ), t 3 = f (z n) f (x n ), t 4 = f (w n) f (x n ), t 5 = f (w n) f (y n ), t 6 = f (w n) f (z n ), q 1 = q 5 = 1 1 2(t 1 + t t3 1 + t4 1 + t5 1 + t6 1 + t7 1 ), q 2 = 4t /4t 3, q 3 = t 2 1 t 2 2t 3 2, q 4 = 8t 4 1 t 4 + 2t 5 1 t 5 + t 6 1 t 6, 15t 1 t 3, q 6 = 54t2 1 t /15t 3 1 t1 2t, q 7 = 7t 2 t 3 + 2t 1 t 6 + 6t 6 t t 3t t 6t t2 2 t t1 4 t 3. 3 Theorem 1. Let f : D?? be a sufficiently differentiable function, and α D is a simple root of f (x) =, for an open interval D. If x is chosen sufficiently close to α, then the iterative scheme (4) converges to α and shows an order of convergence at least equal to sixteen. Proof. Let error at step n be denoted by e n = x n α and c 1 = f? (α) and c k = 1 k! computer assisted proof in Figure 1, 2 and got the following error equation: f (k) (α) f?, k = 2, 3,. We provided (α) e n+1 = c 4 c 3 c 2 2 (c 5c 3 c c 4c 2 c 2 3 2c4 3 51c3 3 c c2 3 c c 3c c8 2 3c 4c 3 c c 4 c 5 2 )e16 n + O(e 17 n ) (5) 3. Numerical Results A set of five nonlinear equations is used for numerical computations in Table 1. Three iterations are performed to calculate the absolute error ( x n α ) and computational order of convergence. Table 2 and Table 3 show absolute error and computational order of convergence respectively. Functions Roots f 1 (x) = e x sin(x) + log(1 + x 2 ) α = f 2 (x) = (x 2)(x ( 1) + x + 1)e x 1 α = 2 f 3 (x) = sin(x) 2 x α = f 4 (x) = e x cos(x) α = f 5 (x) = x 3 + log(x) α = Table 1: Set of six nonlinear functions 3

4 (1) (2) Figure 1: Maple program assisted proof part1. 4

5 (3) Figure 2: Maple program assisted proof part2. 5

6 ( f n (x),x ) MA BT GK1 GK2 f 1, e e e e-523 f 2, e-121 divergent 4.79e e-317 f 3, e e e e-121 f 4, 1/6 2.8e e e e-893 f 5, e e e e-563 Table 2: Numerical comparison of absolute error x n α, number of iterations =4 ( f n (x),x ) MA BT GK1 GK2 f f f f f Table 3: Computational order of convergence 4. Conclusion we have shown the validity of our proposed iterative scheme by comparing it with other existing optimal sixteenorder iterative methods. The numerical results show the performance of iterative scheme is competitive as compare to other methods. The computational order of convergence also verify the claimed sixteenth-order. Acknowledgements The third author is supported for this research under Spanish MEC grants AYA References [1] H. Kung and J. F. Traub. Optimal order of one-point and multipoint iteration. J. Assoc. Comput. Math., 21:643:651, [2] W. Bi, H. Ren, Q. Wu, Three-step iterative methods with eighth-order convergence for solving nonlinear equations, Journal of Computational and Applied Mathematics, 225 (29), pp [3] W. Bi, Q. Wu, H. Ren, A new family of eighth-order iterative methods for solving nonlinear equations, Applied Mathematics and Computation, 214 (29), pp [4] Y.H. Geum, Y.I. Kim, A multi-parameter family of three-step eighth-order iterative methods locating a simple root, Applied Mathematics and Computation, 215 (21), pp [5] L. Liu, X. Wang, Eighth-order methods with high efficiency index for solving nonlinear equations, Applied Mathematics and Computation, 215 (21), pp [6] X. Wang, L. Liu, Modified Ostrowskis method with eighth-order convergence and high efficiency index, Applied Mathematics Letters, 23 (21), pp [7] Diyashvir Kreetee Rajiv Babajee and Rajinder Thukral, On a 4-Point Sixteenth-Order King Family of Iterative Methods for Solving Nonlinear Equations, International Journal of Mathematics and Mathematical Sciences, Volume 212 (212), Article ID , 13 pages, doi:1.1155/212/ [8] Young Hee Geum, Young Ik Kim, A family of optimal sixteenth-order multipoint methods with a linear fraction plus a trivariate polynomial as the fourth-step weighting function,journal of Computers & Mathematics with Applications Volume 61, Issue 11, June 211, Pages [9] Young Hee Geum, Young Ik Kim, A biparametric family of optimally convergent sixteenth-order multipoint methods with their fourth-step weighting function as a sum of a rational and a generic two-variable function, Journal of Computational and Applied Mathematics, Volume 235, Issue 1, 15 March 211, Pages