Applications of Mathematics and Computer Engineering MASOUD ALLAME1, NAFISEH AZAD2 Mathematics Department 1. Islamic Azad University, Khorasgan branch 2. Islamic Azad University, Khorasgan branch, young researchers club Arghavaneh St. Korasgan, Isfahan IRAN 1. masoudallame@yahoo.com 2. Nafiseh_1007@yahoo.com The present paper illustrates an iterative numerical method to solve nonlinear equation f(x) = 0, especially those containing the partial and nonparties involvement of transcendental term. We proposed a new iteration method by Taylor expansion to order 3 and we show by numerical results that this new iteration method is faster than Newton – Raphson method, hybrid iteration method. The new hybrid iteration method and present iteration needs lesser number of functional evaluations. Algebraic equation, Nonlinear equation, Transcendental equation, Taylor expansion, Newton Method, Hybrid method, New hybrid Method. evaluations. Then those of recently proposed iterative methods like hybrid iteration method [11], new hybrid iteration method [2] and Newton's method. Here, the principle of present iteration method is given in section 2. The result and discussion are present in section 3 and conclusion in the last section. Many of the complex problems in Science and Engineering contains the function of nonlinear and Transcendental nature in the equation of the from f(x)=0 Numerical iterative methods like Newton's method [12] are often used to obtain the approximate solution of such problems because it is not always possible to obtain its exact solution by usual algebraic process. There are many methods develops on the improvement of quadratic ally convergent Newton's Method so as to get a superior convergence order than it. Earlier, many investigations [1,6,8,9] have made to explain the root of nonlinear algebraic and transcendental equation. For the same purpose the variants of the Newton's formulas have been discussed by Babajee and Dauhoo [5], whereas other [3,4] suggests the multiCstep iterative method for it. MultiCstep iterative methods have multiple step process to follow the computation rout of each step which is generally cumbersome to deal with. The usual expectation from a mathematical method of such type is to obtain the fast result. Present work found these expectations by converting the given nonlinear problems to a well defined numerical iterative scheme through the use of Taylor's theorem; thereby obtaining a well efficient, highly convergent method that not only works faster than conventional methods but also takes lesser iteration step and lesser number of functional ISBN: 978-960-474-270-7 Consider the following nonlinear algebraic equation: 0 (1) Let α be the root of this equation in the open interval , in which the function is continuous and well defined f' , f'' , f'''. Following the basic assumption of Abbasbandy and maheshweri [1,11] and also others(See [7,13]), we also take Taylor expansion of f(x) one step ahead of Newton, ! ! (2) Where x is the nCth approximation to the root of Eq (1). Since, α is the root of Eq.(2) , so ! ! 65 ! Applications of Mathematics and Computer Engineering We start x0=2. (3) ! " Using Eq.(1) , we can write 0 ! Formula n Newton 5 Hybrid 10 New hybrid 6 Present 4 ! (4) Considering four terms of Eq.(4) , the value of the root of Eq.(1) can be obtained if let , ! 0 ! ! 4. Consider the following equation [2,13], tan 1 0 We start x0=2. (5) ! . Which can solve by Solve # !!! !! ! ! " (6) Formula n 11 Hybrid 6 New hybrid 7 Present 4 Newton 3 Following examples illustrate the result obtained by present method to solve nonlinear equations. At the same time, the results of Newton's method and some other recently proposed methods are presented to compare the efficiency and accuracy of the method by MATLAB software and show the following tables: Formula n 5 Hybrid 9 New hybrid 5 Present 2 Newton % Formula n 7 Hybrid 4 New hybrid 6 Present 2 Newton . Consider the following equation [2,13], sin 0.5 0 We start x0=1.6. 2.23606898849978980 C2.23606797749978980 2.236067977499789800 2.236067977499789800 | | 8.8818EC12 8.8818EC12 8.8818EC12 8.8818EC12 7. Consider the following equation [2,13], 4 2 10 0 We start x0=1.5. results obtained by different method for 0.5 0. | | 1.895494267033980 0.000000000 0.000000000000000 0.000000000 1.895494267033999 0.000000000 1.895494267033980 0.000000000 & Formula n 5 Hybrid 6 New hybrid 4 Present 2 Newton ". Consider the following equation [2,13], ISBN: 978-960-474-270-7 2.09455148151852778 2.09455148151852700 2.09455148154232650 2.09455148154232650 | | 8.8818EC12 3.55271EC11 8.8818EC12 8.8818EC12 6. Consider the following equation [2,13], 5 0 We start x0=1. Comparison of the results obtained by different method for # 0. solving Formula n | | Newton 6.5 E – 17 5 0.77288295914921012 Hybrid 6.5 E – 17 6 0.77288295914921012 New hybrid 0.772882959149210113 1.62630325 E C19 6 Present iteration 3 0.772882959149210120 0.0 1.2 | | CC 0.0(<EC18) 0.0(<EC18) 0.0(<EC18) $ Starting by initial point $ 1 , we will obtain the solution x=0.7729 after few iterations. The results obtained by Newton iteration, hybrid iteration [13], new hybrid iteration [2], and present iteration method are shown in table 1. ln Failure 0.0(<EC18) 0.0(<EC18) 0.0(<EC18) 5. Consider the following equation [2,13], 2 5 0 We start x0=2. ! Consider the following equation [2,13] # 0 Comparison of the solving sin Formula n Newton 5 Hybrid 6 New hybrid 5 Present iteration 4 1.88808675302834340 1.88808675302834340 1.88808675302834340 1.88808675302834360 | | 2.2204EC12 2.2204EC12 2.2204EC12 2.2204EC12 1.365230013414096850 1.365230013893928000 1.36523001344889900 1.36523003414096900 | | 0.0(<EC17) 7.923642EC9 5.565379EC9 0.0(<EC17) 8. Reconsider the following equation of Example 7: 0 66 Applications of Mathematics and Computer Engineering 2 4 10 0 The results obtained by Newton iteration, Hybrid iteration method [13], New hybrid iteration [2] and present iteration method on taking the starting condition of Example 7 are shown in table 8. [1] Abbasbandy, S. Improving NewtonCRaphson method for nonlinear equations by modified Adomian decomposition method,, Vol. 145, 2003, pp. 887 – 893. [2] Nasr AlCDin Ide, A new Hybrid iteration method for solving algebraic equations, ,Vol. 195, 2008, pp. 772C774. ' Method n No. of . functional Newton iteration 5 10 Hybrid iteration 6 24 New hybrid iteration 4 12 Present iteration 8 2 [3] Muhammad Aslam Noor, Khalida Inayat Noor, ThreeCstep iterative methods for nonlinear equations, Vol. 183, 2006, pp. 322C327. [4] Muhammad Aslam Noor, Khalida Inayat Noor, Syed Tauseef MohyudC Din, Asim Shabbir, An iterative method with cubic convergence for nonlinear equations, , Vol. 183, 2006, pp. 1249C1255. [5] D.K.R. Babajee, M.Z. Dauhoo, An analysis of the properties of the variants of Newton's method with third order convergense, , Vol. 183, No 1, 2006, pp. 659 – 684. [6] Golbabai, M. Javibi, NewtonClike iterative methods for solving system of nonClinear equations, Vol. 192, 2007, pp. 546C551. [7] Jinhai Chen, Weiguo Li, On new exponential quadratically convergent iterative formulae,, ,Vol. 180, 2006, pp. 242C246. [8] Jisheng kou, Yitian Li, Xiuhua Wang, On modified Newton methods with cubic convergence, Vol. 176 , 2006, pp. 123C 127. [9] Jisheng Kou, Yitian Li, Xiuhua Wang, A uniparametric ChebyshevCtype method free from second derivatives, Vol. 179, 2006, pp. 296C300. [10] Jisheng Kou, Yitian Li, Xiuhua Wang, A modification of Newton method With thirdCorder convergence, , Vol. 181, 2006, pp. 1106C1111. [11] Amit kumar Maheshwari, A fourth order iterative method for solving nonlinear eqations, , Vol.211, 2009, pp. 383C391. [12] Avram Sidi, Unified treatment of regular falai, Newton–Raphson, Secent, and Steffensen methods for nonlinear equations, ! " 2006, pp. 1C13. [13] XingCGuo Luo, A note on the new iterative method for solving algebraic equation, , Vol. 171, No 2, 2005, pp. 1177C 1183. 9. Reconsider the following equation of Example 5: 2 5 0 The results obtained by Newton iteration, Hybrid iteration method [13], New hybrid iteration [2] and present iteration method on taking the starting condition of Example 5 are shown in table 9. ( Method n No. of . functional Newton iteration 5 10 Hybrid iteration 14 56 New hybrid iteration 5 15 Present iteration 8 2 The main observations are as follows: I. present method takes lesser number of iteration than the others compared here. II. Examples show that the present method requires lesser number of functional evaluations, as compared to other methods. Now, one question can ask, In your present method, you use "solve" in software, so it can use for basic function and then find the root it? We mast note that we use "solve" in our method for a polynomial diger 3, and we know that it always have a real root. # ) In the paper, we proposed a new iteration method by Taylor expansion to order 3 to solve nonlinear equations f(x) =0. , that this new iteration method is faster than Newton – Raphson method, hybrid iteration method, new hybrid iteration method and present iteration needs lesser number of functional evaluations. ISBN: 978-960-474-270-7 67