SOME MULTI-STEP ITERATIVE METHODS FOR SOLVING NONLINEAR EQUATIONS

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1 Open J. Math. Sci., Vol. 1(017, No. 1, pp ISSN Website: SOME MULTI-STEP ITERATIVE METHODS FOR SOLVING NONLINEAR EQUATIONS MUHAMMAD SAQIB 1, MUHAMMAD IQBAL Abstract. In this paper, we suggest and analyze two new algorithm of fourth and fifth order convergence. We rewrite nonlinear equation as an equivalent coupled system and then use modified decomposition technique to develop our algorithms. Convergence analysis of newly introduced algorithms has been discussed. To see efficiency and performance of these algorithms, we have made comparison of these algorithms with some well known algorithms existing in literature. AMS Mathematics Subject Classification : 39B1, 34A34. Key words and phrases : Iterative methods; nonlinear system; Convergence. 1. Introduction To find the root of nonlinear equation of the form f(x = 0, is the oldest and basic problem in numerical analysis. Newton s method is one of oldest and most powerful formula to approximate the root of nonlinear equations. It has second order convergence. Many modifications in Newton s method have been made to increase its convergence order using various techniques. Recently, many iterative methods with higher order convergence have been established using different techniques like Taylor s series, Adomain decomposition, homotopy, modified homotopy, decomposition, modified decomposition, interpolation, quadrature rules and their many modifications. see [1,, 3, 4, 5, 6, 7, 8, 9, 10, 11] and references therein. Chun [3] introduced some multi-step iterative methods using Adomain decomposition. These method requires higher order derivatives. Later on, Noor [4, 5] established some multi-step iterative methods that do not require higher order derivative using a different technique. Received 08 July 017. Revised 10 September Corresponding Author c 017 Muhammad Saqib, Muhammad Iqbal. 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. 5

2 6 M. Saqib, M. Iqbal In this paper, some numerical methods based on decomposition technique are proposed for solving algebraic nonlinear equations. For this purpose, we write nonlinear equations as an equivalent system of equations and use technique introduced by Gejji and Jafari [8]. Recently, several iterative methods have been suggested by writing nonlinear equations as an equivalent system of equations and then using different techniques. In section 3, we give detailed proof regarding convergence of our newly established iterative methods. In the last section, numerical results are given to make comparison of these algorithm with some classical methods. Consider the nonlinear equation. Iterative methods f(x = 0; x R, (1 assume α is a simple root of (1 and γ is an initial guess sufficiently close to α. We can write (1 as following coupled system; f(γ + (x γf ( x + γ + g(x = 0, ( g(x = f(x f(γ (x γf ( x + γ. (3 From Eq. (, we have Let where and x = γ f(γ g(x. x = c + N(x, (4 c = γ f(γ (5, N(x = g(x (6. Now, we construct a sequence of higher-order iterative methods by applying modified decomposition technique introduced by Gejji and Jafari [8]. This technique consists a solution of equation (3 that can be written in the form of infinite series: x = x i, (7 i=0 and using Gejji and Jafari [8] technique, we decompose the nonlinear function N as; i i 1 N( x i = N(x 0 {N( x j N( x j }. (8 i=0 i=1 j=0 j=0

3 Some multi-step iterative methods for solving nonlinear equations 7 Thus from Eqs. (4, (7 and (8 we have i i 1 x i = x 0 + N(x 0 + {N( x j N( x j }. i=0 So we have iteration scheme as i=1 j=0 j=0 When x 0 = c x 1 = N(x 0 x = N(x 0 + x 1 N(x 0... x n+1 = N(x 0 + x x n N(x 0 + x x n 1, n = 1,,.. x x 0 = c = γ f(γ. From above relation, we can write the algorithm as follows; Algorithm.1: For any initial value x 0, we compute the approximation solution x n+1, by the iteration scheme; y n = x n f(x n f (x n x n+1 = x n f(x n. This algorithm has third order convergence and has been established by Frontini and Sormani [1]. When x x 0 + x 1 = γ f(γ g(x 0. From Eq. (3, we see that g(x 0 = f(x 0, by substituting in above, we get x = γ f(γ f(x 0 f ( x0+γ,

4 8 M. Saqib, M. Iqbal Using this, we suggest the following three step iterative methods for solving Eq. (1 as follows; Algorithm.: For any initial value x 0, we compute the approximation solution x n+1, by the iteration scheme; When Predictor Steps; y n = x n f(x n f (x n z n = x n f(x n Corrector Step; x n+1 = z n f(z n f ( xn+zn x x 0 + x 1 + x = x 0 + N(x 0 + x 1 = x 0 g(x 0 + x 1 f ( x0+x1+γ = x 0 f(x 0 + x 1 + f(x 0 f ( x0+x1+γ From this relation, we formulate four step iterative method for solving Eq. (1 as follows; Algorithm.3: For any initial value x 0, we compute the approximation solution x n+1, by the iteration scheme; Predictor Steps; y n = x n f(x n f (x n z n = x n f(x n u n = z n f(z n f ( xn+zn Corrector Step; x n+1 = z n f(u n + f(z n f ( xn+un 3. Convergence Analysis In this section, we discuss in detail the convergence analysis of Algorithm. and.3 established above. Theorem 3.1. For an open interval I R let f : I R and α I be simple zero of f(x = 0. If f is differentiable and x 0 is sufficiently close to α then threestep iterative method defined by Algorithm. has fourth order convergence.

5 Some multi-step iterative methods for solving nonlinear equations 9 Proof. Expanding f(x by Taylor s series about α, we have f(x n = f (α(e n + c e n + c 3 e 3 n + c 4 e 4 n + c 5 e 5 n +... (9 where c k = 1 f (k (α k!, and e f (α n = x n α. f (x n = f (α(1 + c e n + 3c 3 e n + 4c 4 e 3 n + 5c 5 e 4 n + 6c 6 e 5 n +... (10 From above equations,we get y n = x n f(x n f (x n y n = α + c e n + (c 3 c e 3 n + (3c 4 7c c 3 + 4c 3 e 4 n + (4c 5 10c c 4 6c 3 + 0c 3 c 8c 4 e 5 n +... (11 Now, expanding by Taylor s series about α, f ( x n + y n = f (α{1 + c e n + (c c 3e n + ( 7 c c 3 c c 4e 3 n Dividing Eq. (3.1 by Eq. (3.4 Now + (3c c 3c + 3 c c 4 + 4c c 5e 4 n +...} (1 f(x n = e n + ( c c 3e 3 n + ( 15 4 c c 3 + 3c c 4e 4 n + ( c c 3c c c 4 6c c 5e 5 n +... (13 z n = x n f(x n Putting values in above, we get z n = α + (c 1 4 c 3e 3 n + ( 15 4 c c 3 3c 3 1 c 4e 4 n + ( c 3 7 c 3c + c c 4 6c c 5e 5 n +... (14 Expanding f(z n and f ( xn+zn by Taylor s series about α, f(z n = f (α{(c 1 4 c 3e 3 n + ( 15 4 c c 3 3c 3 1 c 4e 4 n + ( c 3 7 c 3c + c c 4 6c c 5e 5 n +...} (15 f ( x n + z n = f (α{1 + c e n c 3e n + (c c c c 4e 3 n + ( 1 4 c 3c 3c 3 1 c c c c 3e 4 n +...} (16

6 30 M. Saqib, M. Iqbal Dividing (15 by Eq. (16, we have f(z n f ( xn+zn = (c 1 4 c 3e 3 n + (4c c 3 4c 3 1 c 4e 4 n + ( 7 8 c c 3c + 5 c c c c 5e 5 n +... Corrector step of Algorithm. is x n+1 = z n f(z n f ( xn+zn By substitution and simplification, we have x n+1 = α + (4c c 3 4c 3 1 c 4e 4 n + ( 7 8 c c 3c + 5 c c c c 5e 5 n +... Hence Algorithm. has fourth order convergence. (17 Theorem 3.. Let f : I R R for an open interval I and α I be simple zero of f(x = 0. If f is differentiable and x 0 is sufficiently close to α then three-step iterative method defined by Algorithm.3 has fifth order convergence and satisfies the error equation x n+1 = α + ( 1 4 c 3c + c 4 e 5 n + O(e 6 n, and e n = x n α. Proof. From Eq. (17, we have u n = α + (4c c 3 4c 3 1 c 4e 4 n + ( 7 8 c c 3c + 5 c c c c 5e 5 n +..., Expanding f(u n and f ( xn+un by Taylor s series about α, f(u n = f (α{(4c c 3 4c 3 1 c 4e 4 n+( 7 8 c c 3c + 5 c c 4 +10c c 5e 5 n+...}, (18 f ( x n + u n = f (α{1+c e n c 3e n+ 1 c 4e 3 n+(c c 3c c 4e 4 n+...}. (19 Now x n+1 = z n f(u n + f(z n, f ( xn+un by substituting values and simplifying, we have x n+1 = α + ( 1 4 c 3c + c 4 e 5 n + O(e 6 n. Hence Algorithm.3 has fifth order convergence.

7 Some multi-step iterative methods for solving nonlinear equations Applications In this section, we present some numerical examples to examine the validity and efficieny of our newly devolped algorithms namely, Algorithm. and Algorithm.3. We also provide comparison of these algorithms with Newton s method(nm, Abbasbandy s method (AM, Householder s method(hhm, Halley s method (HM, Chun s method (CM [3] and Noor s method(nr[4]. We use ɛ = We use the following stopping criteria; (i x n x n 1 < (ii f(x n < We consider the following nonlinear scalar equations for comparison: f 1 (x = sin x x + 1 = 0 f (x = x e x 3x + = 0 f 3 (x = cos x x = 0 f 4 (x = (x = 0 f 5 (x = x 3 10 = 0 f 6 (x = e x +7x 30 1 = 0. Comparison Table Examples Iterations x n f(x n f 1, x 0 = 1 NM e 50 AM e 54 HM e 63 CM e 64 HHM e 5 NR e 6 Alg e 5 Alg e 44 f, x 0 = NM e 55 AM e 63 HM CM e 6 HHM e 5 NR e 3 Alg e 69 Alg e 9

8 3 M. Saqib, M. Iqbal Examples Iterations x n f(x n f 3, x 0 = 1.7 NM e 31 AM e 47 HM e 58 CM HHM e 17 NR e 18 Alg e 50 Alg e 100 f 4, x 0 = 1.5 NM e 4 AM 5 0 HM 5 0 CM 5 0 HHM e NR e 4 Alg e 44 Alg e 7 f 5, x 0 = 1.5 N.M e 54 A.M e 63 H.M e 6 C.M e 64 H.H.M e 7 NR.M e 4 Alg e 7 Alg e 50 f 6, x 0 = 3.5 N.M e 48 A.M e 48 H.M e 6 C.M e 6 H.H.M e 5 NR.M e 06 Alg e 30 Alg e Conclusion We have estalished two new algorithms of fourth and fifth order convergence by using modified decomposition technique to coupled system. We have discussed convergence analysis of these newly established algorithms. Computational comparison of these algorithms has been made with some well known methods for solving nonlinear equations. From numerical table, we see that our

9 Some multi-step iterative methods for solving nonlinear equations 33 new methods converge fastly to the true solution of equations and are comparable with some classical methods. Competing Interests The author(s do not have any competing interests in the manuscript. References 1. Frontini, M., & Sormani, E. (003. Some variant of Newtons method with third-order convergence. Applied Mathematics and Computation, 140(, Adomain G. (1989. Nonlinear Atochastic Systems and Applications to Physics, Kluwer Academy Publishers. Dordrecht. 3. Chun, C. (005. Iterative methods improving newton s method by the decomposition method. Computers & Mathematics with Applications, 50 (10, Noor, M. A., & Noor, K. I. (006. Some iterative schemes for nonlinear equations. Applied Mathematics and Computation, 183 (, Noor, M. A., Noor, K. I., Mohyud-Din, S. T., & Shabbir, A. (006. An iterative method with cubic convergence for nonlinear equations. Applied Mathematics and Computation, 183 (, Babolian, E., & Biazar, J. (00. On the order of convergence of Adomian method. Applied Mathematics and Computation, 130 (, Weerakoon, S., & Fernando, T. G. I. (000. A variant of Newton s method with accelerated third-order convergence. Applied Mathematics Letters, 13 (8, Daftardar-Gejji, V., & Jafari, H. (006. An iterative method for solving nonlinear functional equations. Journal of Mathematical Analysis and Applications, 316(, Saqib, M., Iqbal, M., Ali, S., & Ismaeel, T. (015. New Fourth and Fifth-Order Iterative Methods for Solving Nonlinear Equations. Applied Mathematics, 6 (08, Frontini, M., & Sormani, E. (003. Modified Newton s method with third-order convergence and multiple roots. Journal of computational and applied mathematics, 156(, Hasanov, V. I., Ivanov, I. G., & Nedzhibov, G. (00. A new modification of Newtons method. Appl. Math. Eng, 7, Muhammad Saqib Department of Mathematics, Lahore Leads University, Lahore-54590, Pakistan. saqib70@yahoo.com Muhammad Iqbal Department of Mathematics, Lahore Leads University, Lahore-54590, Pakistan. iqbal66dn@yahoo.com

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