In the paper a new two-step iterative method has been suggested for estimating nonlinear problems by using quadrature formula. We have also proved that the proposed two-step iterative method has third-order of convergence. For the duration of study, it has been detected that the third-order iterative Method faster than existing cubic Methods such as variant of Newton Raphson Method and Halley Method. Numerous numerical illustrations are specified to demonstrate the eﬃciency and the performance of the new two-step method. Henceforth, the third-order iterative Method has been considered as an imperative enhancement and reﬁnement to find the root of nonlinear equations. Keyword: Non-linear equations, cubic methods, order of convergence, error and accuracy

Qureshi, Umair Khalid

English

International Institute for Science, Technology and Education (IISTE): E-Journals

Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.8, No.5, 2018  64 A New Accelerated Third-Order Two-Step Iterative Method for Solving Nonlinear Equations  Umair Khalid Qureshi Department of Basic Science & Related Studies, Mehran University of Engineering and technology, Pakistan  Email: khalidumair531@gmail.com Abstract In the paper a new two-step iterative method has been suggested for estimating nonlinear problems by using quadrature formula. We have also proved that the proposed two-step iterative method has third-order of convergence. For the duration of study, it has been detected that the third-order iterative Method faster than existing cubic Methods such as variant of Newton Raphson Method and Halley Method. Numerous numerical illustrations are specified to demonstrate the efficiency and the performance of the new two-step method. Henceforth, the third-order iterative Method has been considered as an imperative enhancement and refinement to find the root of nonlinear equations. Keyword: Non-linear equations, cubic methods, order of convergence, error and accuracy.  1. INTRODUCTION For determining nonlinear problems is an imperious area of research in numerical analysis at it had interested applications in numerous field of pure and applied science have been deliberate in the general framework of the nonlinear problems [1-3], for instance                                                                                        𝑓(𝑥) = 0                                                                                             (1) To overview these difficulties, countless numerical techniques had planned and examined for estimating nonlinear problems. Such as most useful numerical techniques include Newton Raphson Method [4-5]. Newton Raphson Method are fast converging numerical techniques but are not reliable because keeping a kind of pitfall. However, it is most convenient and useful numerical techniques. In recent years, in literature several modifications had been done by using this technique for finding a single root of a nonlinear equation [6-8]. Furthermore, by combine Newton Raphson Method in literature, which has given a method equation with better accuracy as well as iteration perspective for solving non-linear [9-11]. Correspondingly, Numerous numerical have been illustrated and specified to demonstrate the efficiency and the performance of the new two-step method. These numerical methods have been constructed by using different techniques such as Taylor series, homotopy perturbation method and its variant forms, quadrature formula, variation iteration method and decomposition method [12–15]. Respectively, in this paper a Modified cubic iterated method has been suggested. Which is combination of classical Newton Raphson Method and quadrature formula. We have proven that two-step method is third-order convergence and present the comparison of new method with cubic methods such as Variant of Newton Raphson Method [16] 𝑦𝑛 = 𝑥𝑛 –𝑓(𝑥𝑛)𝑓`(𝑥𝑛) 𝑥𝑛+1 = 𝑥𝑛 −2𝑓(𝑥𝑛)𝑓`(𝑥𝑛) + 𝑓`(𝑦𝑛)  and Halley Method [17]   𝑥𝑛+1 = 𝑥𝑛 −2𝑓(𝑥𝑛)𝑓`(𝑥𝑛)2𝑓`2(𝑥𝑛) + 𝑓(𝑥𝑛)𝑓``(𝑥𝑛)  From the assessment of existing cubic methods to the new third-order iterated method is that the proposed is fast converging and more efficient to approaching the root.   2. ITERATIVE METHOD The proposed third-order iterated method is developed by using the quadrature formula and the fundamental theorem of calculus, such as Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.8, No.5, 2018  65               𝑓(𝑥) = 𝑓(𝑟) +ℎ3[𝑓`(𝑟) + 4𝑓` ((𝑥 + 𝑟)2) + 𝑓`(𝑥)]                                                  (2) Where, the Simson 1/3𝑟𝑑 rule with n=2, then h become ℎ =(𝑥 − 𝑟)2 Now, h substitute in (2), then (2) become               𝑓(𝑥) = 𝑓(𝑟) +(𝑥 − 𝑟)6[𝑓`(𝑟) + 4𝑓` ((𝑥 + 𝑟)2) + 𝑓`(𝑥)]                                          (3) Here `r` is an initial guess sufficiently close to root `x` and 𝑓(𝑥) = 0, then (3) become               𝑓(𝑟) +(𝑥 − 𝑟)6[𝑓`(𝑟) + 4𝑓` ((𝑥 + 𝑟)2) + 𝑓`(𝑥)] = 0                                               (4) or                        𝑥 = 𝑟 −6𝑓(𝑟)𝑓`(𝑟) + 4𝑓` ((𝑥 + 𝑟)2 ) + 𝑓`(𝑥)                                                              (5) For using this relation, we can recommend that the following two-step iterative method as,                        𝑥𝑛+1 = 𝑦𝑛 −6𝑓(𝑦𝑛)𝑓`(𝑦𝑛) + 4𝑓` (𝑥𝑛 + 𝑦𝑛2 ) + 𝑓`(𝑥𝑛)                                               (6) 𝑓𝑜𝑟 𝑛 = 0,1,2 …  Where 𝑦𝑛 = 𝑥𝑛 –𝑓(𝑥𝑛) 𝑓`(𝑥𝑛)    The relationship (6) is a new iterative method  3. CONVERGENCE ANALYSIS The following section will be shows that the New Developed Method is cubically Convergence.  Proof  Let `a` be a simple zero of 𝑓. Then, by expanding 𝑓(𝑥𝑛) and 𝑓`(𝑥𝑛) in Taylor’s Series about `a`, we have  𝑓(𝑥𝑛) = 𝑓`(𝑎)(𝑒𝑛 + 𝑐2𝑒2𝑛 + 𝑐3𝑒3𝑛 + 𝑜(𝑒4𝑛))            − − − (𝑖)        𝑓`(𝑥𝑛) = 𝑓`(𝑎)(1 + 2𝑐2𝑒𝑛 + 3𝑐3𝑒2𝑛 + 4𝑐4𝑒3𝑛 + 𝑜(𝑒4𝑛))            − − − (𝑖𝑖) By using 𝑐𝑘 =𝑓𝑘(𝑎)𝑘!𝑓𝑘−1(𝑎), k=2,3, 4, … and 𝑒𝑛 = 𝑥𝑛 − a From (𝑖) and (𝑖𝑖), we have             𝑓(𝑥𝑛)𝑓`(𝑥𝑛)= 𝑒𝑛 − 𝑐2𝑒2𝑛 + 2(𝑐22  − 𝑐3)𝑒3𝑛                            − − − (𝑖𝑖𝑖) From  (𝑖𝑖𝑖), we get                     𝑦𝑛 = 𝑐2𝑒2𝑛 − 2(𝑐22  − 𝑐3)𝑒3𝑛                                     − − − (𝑖𝑣) expanding 𝑓(𝑦𝑛) 𝑓`(𝑦𝑛)  𝑎𝑛𝑑 𝑓` (𝑥𝑛+𝑦𝑛2) in Taylor’s Series about `a`, by using (iv), we have  𝑓(𝑦𝑛) = 𝑓`(a)[𝑐2𝑒2𝑛 + 2(𝑐3 − 𝑐22)𝑒3𝑛]                                   Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.8, No.5, 2018  66 Or  𝑓`(𝑦𝑛) = 𝑓`(a)[1 + 2𝑐22𝑒2𝑛 + 4(𝑐2𝑐3 − 𝑐32)𝑒3𝑛]                   − − − (𝑣) and                      𝑓` (𝑥𝑛 + 𝑦𝑛2) = 𝑓`(a)[1 + 𝑐2𝑒𝑛 + (𝑐22 +34𝑐3) 𝑒2𝑛 + (72𝑐2𝑐3 − 2𝑐32) 𝑒3𝑛]                   − − − (𝑣𝑖) From(𝑖𝑖), (𝑣), and (𝑣𝑖), we have 𝑓`(𝑦𝑛) + 4𝑓` (𝑥𝑛 + 𝑦𝑛2) + 𝑓`(𝑥𝑛)= 𝑓`(a) [6 + 6𝑐2𝑒𝑛 + (6𝑐22 + 6𝑐3)𝑒2𝑛 + 6 (3𝑐2𝑐3 −13𝑐32) 𝑒3𝑛]      − − − (𝑣𝑖𝑖) From (𝑣) and (𝑣𝑖𝑖), we obtain              6𝑓(𝑦𝑛)𝑓`(𝑦𝑛) + 4𝑓` (𝑥𝑛 + 𝑦𝑛2 ) + 𝑓`(𝑥𝑛)=6𝑓`(a)[𝑐2𝑒2𝑛 + 2(𝑐3 − 𝑐22)𝑒3𝑛] 6𝑓`(a) [1 + 𝑐2𝑒𝑛 + (𝑐22 + 𝑐3)𝑒2𝑛 + (3𝑐2𝑐3 −13 𝑐32) 𝑒3𝑛] 6𝑓(𝑦𝑛)𝑓`(𝑦𝑛) + 4𝑓` (𝑥𝑛 + 𝑦𝑛2 ) + 𝑓`(𝑥𝑛)= 𝑐2𝑒2𝑛 − (3𝑐22 − 2𝑐3)𝑒3𝑛            − − − (𝑣𝑖𝑖𝑖) From (𝑖𝑣) and (𝑣𝑖𝑖𝑖) substitute in (6), we get 𝑒𝑛+1 = 𝑐2𝑒2𝑛 − 2(𝑐22  − 𝑐3)𝑒3𝑛 − 𝑐2𝑒2𝑛 + (3𝑐22 − 2𝑐3)𝑒3𝑛   𝑒𝑛+1 = (3𝑐22 − 2𝑐3 − 2𝑐22 + 2𝑐3)𝑒3𝑛                                      𝑒𝑛+1 = 𝑐22𝑒3𝑛 + 𝑜(𝑒4𝑛)                                         − − − (𝑖𝑥) Error equation (𝑖𝑥) shows that the proposed method (6) is cubic order of convergence.  4. RESULTS AND DISCUSSIONS In this section, the developed third-order iterative method is pragmatic on few examples of nonlinear equations and developed method compared with the variant of Newton Raphson Method and Halley Method. From the numerical result of table-1, it has been detected that the cubic iterative method is reducing the iterations number which is less than the number of iteration of the existing cubic methods and likewise accuracy side. Mathematical package such as C++ and EXCEL have used to justify the proposed Iterative Method. From the results and comparison of proposed cubic iterative method with the cubic methods that the proposed cubic iterative method is execution healthier than prevailing Techniques.   Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.8, No.5, 2018  67 Table-1 FUNCTION METHODS ITERATION X A E   Sinx-x+1 X=2 Halley Method Variant Newton Method Two-Step Method 2 3 2  1.93456  4.33922e-5 1.19209e-7 2.68221e-5 2x-lnx-7 X=4 Halley Method Variant Newton Method Two-Step Method 3 2 2  4.21991  3.10421e-4 1.19209e-5 6.19888e-6 xex–2 X=1 Halley Method Variant Newton Method Two-Step Method 2 8 2  0.852605 7.42078e-4 2.98023e-7 1.52845e-2 x3 -9x+1 X=0 Halley Method Variant Newton Method Two-Step Method 2 4 2  0.111264  1.53050e-4 1.49012e-7 1.56462e-7 Cosx-x3 X=1 Halley Method Variant Newton Method Two-Step Method 2 6 2  0.865474 1.28573e-3 1.78814e-7 1.76102e-3        5. CONCLUSION In this study, it is obvious that two-step method has been presented. The proposed method has third-order of convergence, which is performing better than the third-order iterated methods such as variant of Newton Raphson Method and Halley Method. Through the Table-1, it is concluding from the fallouts and assessment of suggested third-order method is that the proposed iterated method is loftier than Newton Raphson Method from precision viewpoint as well as iterative interpretation. Henceforth, the new third-order iterated method is a 02468101 2 3Figure-1: Comparative illustration A E % of sinx-x+1 05101520251 2 3Figure-2: Comparative Illustration A E % of 2x-lnx-7 Mathematical Theory and Modeling                                                                                                                                                  www.iiste.org ISSN 2224-5804 (Paper)    ISSN 2225-0522 (Online) Vol.8, No.5, 2018  68 supercilious and a good achievement to find root of nonlinear equations in one variable with the comparison of existing Cubic Methods.  REFERENCE Akram, S. and Q. U. Ann., 2015.  Newton Raphson Method, International Journal of Scientific & Engineering Research, Volume 6. Allame M., and N. Azad, 2012. On Modified Newton Method for Solving a Nonlinear Algebraic Equations by Mid-Point, World Applied Sciences Journal 17 (12): 1546-1548, ISSN 1818-4952 IDOSI Publications. Allame M., and N. Azad, 2012. 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A New Accelerated Third-Order Two-Step Iterative Method for Solving Nonlinear Equations

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A New Accelerated Third-Order Two-Step Iterative Method for Solving Nonlinear Equations

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