MSE-Vol1-No1-_2015_-6-11-Nedzhibov 6 Mathematical and Software Engineering, Vol. 1, No. 1 (2015), 6-11 Varεpsilon Ltd, http://varepsilon.com Inverse Iterative Methods for Solving Nonlinear Equations Gyurhan Nedzhibov Faculty of Mathematics and Informatics, Shumen University, Bulgaria Abstract In his work we present an approach for obtaining new iterative methods for solving nonlinear equations. This approach can be applicable to arbitrary iterative process which is linearly or quadratically convergent. Analysis of convergence of the new methods demonstrates that the new method preserve the convergence conditions of primitive functions. Numerical examples are given to illustrate the efficiency and performance of presented methods. Subject Codes (PACS): 02.60.Cb Keywords: Nonlinear equations; Iterative methods; Order of convergence; Newton method. 1 Introduction The problem of solving nonlinear equations is one of the classical problems which arise in several branches of pure and applied mathematics. Including the fact that solution formulas do not in general exist, the finding new and efficient numerical algorithms for solving nonlinear equations always has been an attractive problem. In recent years, several numerical methods have been developed and analyzed under certain conditions. These methods have been constructed using different techniques such as Taylor series, quadrature formulas, acceleration techniques, decomposition method, and etc. (see, [1-9] and references therein). Our goal in this work is to suggest and explore a new approach of construction of iterative methods for solving nonlinear equations. The paper is organized as 7 follows. In Section 2 we consider some preliminary results and the new method. Analysis of convergence is provided in Section 3. Several sample iterative methods and numerical examples are explored in Section 4. 2 Main Results Let us consider the nonlinear equation (1.1) ,0)( =xf where RRUf →⊂: is a scalar function and U is an open interval. In this work, we consider iterative methods for finding a simple root α of f and assume that f is a 2C function in a neighborhood of α . An iterative scheme for solving the equation (1.1), generally takes the form (1.2) ),(1 kk xx ϕ=+ for 0≥k , where )( xϕ is a fixed function and Ux ∈0 is a given initial value of α . When studying the iterative methods, one of the most important aspects to consider is the convergence, and the order of convergence, respectively. Theorem 4 (Traub [1, Theorem 2.2]) Let ϕ be an iterative function such that ϕ and its derivatives )(,,, pϕϕϕ …′′′ are continuous in the neighborhood of a root α of a given function f . Then ϕ defines an iterative method of order p if and only if 0)(,0)()(',)( )()1( ≠==== − αϕαϕαϕααϕ pp… . Without loss of generality we assume that function ϕ in (1.2) has the form (1.3) )()( xgxx −=ϕ , where ),,,()( …ffxgxg ′= . Investigations in this work are inspired by our previous results in [6,7], where we suggest the Inverse Newton method (1.4) )( 2 1 kk k k xux x x + =+ , where ( ) )(' )( k k k xf xf xu = . which is a modification of the well known Newton iterative function (1.5) ( )kkk xuxx −=+1 . 8 Iterative function (1.4) is obtained as consequence of companion matrix method and similarity transformations between some companion matrices. The purpose of this paper is to generalize the approach presented in [7]. Let us consider an arbitrary, linearly or quadratically convergent iterative function (1.6) ),(1 kkk xgxx −=+ for 0≥k , then we call Inverse iterative function of (1.6) the following function (1.7) ,)( 2 1 kk k k xgx x x + =+ for 0≥k . Further we will show that iterative process (1.7) preserves the conditions of process (1.6) for nonzero roots. The following local convergence theorem is valid. Theorem 1.2 Let 0, ≠∈ αα U be a simple root of a sufficiently differentiable function RRUf →⊂: for an open interval U . Then if for 0x sufficiently close to α the iterative function (1.6) generates convergent sequence with order of convergence not higher than two, the iterative function (1.7) also generates convergent sequence with the same order of convergence. Proof: According to Theorem 1.1 for the function (1-3)-(1.6) are fulfilled (1.8) ααϕ =)( , i.e. 0)( =αg and )(1)( ααϕ g ′−=′ . Then (i) in the case of linear convergence of (1.6), the error equation is fulfilled (1.9) )()).(1()().( 22 1 nnnnn OgO εεαεεαϕε +′−=+′=+ , where αε −= nn x and 1)(1)(' <′−= ααϕ g ; (ii) in the case of quadratic convergence of (1.6), the following error equation is fulfilled (1.10) )(. 2 )( )(. 2 )( 3232 1 nnnnn O g O εεαεεαϕε + ′′ −=+ ′′ =+ . Let denote the function )( )( 2 xgx x x + =φ from the expression (1.7). It is not difficult to verify that ααφ =)( and (1.11) 2 2 ))(( ))(1( )( 2 )( xgx xgx xgx x xn + ′+ − + =′φ . 9 Then from (1.8) and (1.11) it follows that (1.12) )()(1)( αϕααφ ′=′−=′ g . Equations (1.9), (1.10) and (1.12) implies that the iterative process defined by (1.7) will have the same order of convergence as the process (1.6). Theorem is proved. 3 Sample iterative methods and numerical results In this we will consider some examples of well known iterative methods and their inverse version according approach (1.6)-(1.7). 3.1 Newton modification method (3.1) )( )( 0 1 xf xf xx kkk ′ −=+ , which is linearly convergent and the corresponding inverse method is (3.2) )()( )( 0 0 2 1 kk k k xfxfx xfx x +′ ′ =+ . 3.2 Consider the following quadratically convergent iterative function (3.3) ))(()( )(2 1 kkk k kk xfxfxf xf xx −− −=+ , and the inverse method function (3.4) )()))(()(( )))(()(( 2 2 1 kkkkk kkkk k xfxfxfxfx xfxfxfx x +−− −− =+ . Further we present some results of numerical experiments to illustrate the performance of the considered inverse iterative methods. We compare the Newton method (1.5) with the inverse method (1.4), Modified Newton methods (3.1) and (3.2), and the method s(3.3) and (3.4). We use the following stopping criteria (i) ε<−+ nn xx 1 , (ii) ε<)( nxf . We have used the fixed stopping criterion 1510 −=ε .When the stopping criterion is satisfied, 1+nx is taken as the exact root α computed. We have denoted: 0x - initial approx. It - the number of iterations to approximate the zero, nx - the approximate root, nn xx −+1 - the distance of two consecutive 10 aproximations , )( nxf the absolute value of f at nx , and COC- the computational order of convergence computed using the formula )/()(ln )/()(ln 211 11 −−− −+ −− −− = nnnn nnnn xxxx xxxx COC . We consider the following nonlinear equations as test problems. These equations can be found in many other papers on the subject, see for example [3,8,9]. 1) 1)1()( 3 1 −−= xxf ; 2) 1)(sin)( 22 2 +−= xxxf ; 3) 23)( 2 3 +−−= xexxf x ; The experimental results are included in the following Table 1: Table 1. Experimental results. Acknowledgements This work is partially supported by Shumen University under Grant No. RD-08-265/10.03.2015. References [1] J. F.Traub, Iterative methods for the Solution of Equations, Prentice Hall,Englewood Cliffs, New Jersey, (1964). 0);( xxfi IM It nx nn xx −+1 )( nxf COC (1.5) NM 6 2.0000 1.15e-14 0 2.00 (1.4) 6 2.0000 8.87e-11 0 2.00 (3.1) 29 2.0000 8.88e-16 1.33e-15 0.99 (3.2) 30 2.0000 4.44e-16 0 1.00 (3.3) 7 2.0000 1.79e-10 0 2.00 5.2);( 01 =xxf (3.4) 7 2.0000 7.79e-15 0 1.99 (1.5) NM 5 1.4044 2.04e-12 4.44 e-16 2.00 (1.4) 6 1.4044 2.75e-14 3.33 e-16 1.99 (3.1) 21 1.4044 4.44e-16 3.33e-16 0.94 (3.2) 20 1.4044 1.55e-15 4.44 e-16 0.93 (3.3) 7 1.4044 7.46e-14 3.33 e-16 2.00 2.1);( 02 =xxf (3.4) 78 1.4044 6.97e-12 3.33 e-16 2.00 (1.5) NM 5 0.257 3.36e-14 0 2.00 (1.4) 8 0.257 3.29e-11 0 1.99 (3.1) 10 0.257 4.38e-15 0 0.99 (3.2) 13 0.257 8.93e-15 8.88e-16 0.99 (3.3) 12 0.257 1.00e-11 0 2.00 5.1);( 03 =xxf (3.4) 15 0.257 1.97e-14 0 1.99 11 [2] G.H.Nedzhibov, V.I.Hasanov, M.G.Petkov, On Some Families of Multi-point Iterative methods for Solving Nonlinear Equations, Numer. Algor., 42, (2006), 127-136. [3] S. Weerakoon and T. G. I. Fernando, “A variant of Newton’s method with accelerated third-order convergence,” Applied Mathematics Letters, vol. 13, no. 8, pp. 87–93, 2000. [4] G.H. Nedzhibov, An acceleration of iterative methods for solving nonlinear equations, Applied Mathematics and Computation, Sep2005, Vol. 168, Issue 1, 320–332 (2004). [5] M.A.Noor, K.I. Noor, E. Al-Said and M. Waseem, Some new iterative methods for nonlinear equations, Math. Prob. Eng. (2010), Article ID, 98943: 12. [6] G.H.Nedzhibov, Similarity transformations between some companion matrices, Application of Mathematics in Engineering and Economics (AMEE14), AIP Conf. Proc., 1631, pp. 375-382, (2014). [7] G.H.Nedzhibov, On two modifications of Weierstrass-Dochev iterative method for solving polynomial equations, MATHTECH 2014, Proceedings of the international conference, Volume 1, pp. 84-90, (2014). [8] C. Chun, Construction of Newton-like iteration methods for solving nonlinear equations, Numerical Mathematics 104, (2006), 297-315. [9] M. Aslam Noor and K. Inayat Noor, “Some iterative schemes for nonlinear equations,” Applied Mathematics and Computation, vol. 183, no. 2, pp.774–779, 2006. Copyright © 2015 Gyurhan Nedzhibov. 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.