 Applied Mathematics, 2010, 1, 153-158 doi:10.4236/am.2010.13020 Published Online September 2010 (http://www.SciRP.org/journal/am) Copyright © 2010 SciRes. AM Modified Efficient Families of Two and Three-Step Predictor-Corrector Iterative Methods for Solving Nonlinear Equations Sanjeev Kumar1, Vinay Kanwar2, Sukhjit Singh3 1Department of Applied Sciences, ICL Institute of Engineering and Technology, Sountli, India 2University Institute of Engineering and Technology, Panjab University, Chandigarh, India 3Department of Mathematics, Sant Longowal Institute of Engineering and Technology, Longowal, India E-mail: sanjeevbakshi1@gmail.com, vmithil@yahoo.co.in, sukhjit_d@yahoo.com Received June 9, 2010; revised July 13, 2010; accepted July 16, 2010 Abstract In this paper, we present and analyze modified families of predictor-corrector iterative methods for finding simple zeros of univariate nonlinear equations, permitting 0fx near the root. The main advantage of our methods is that they perform better and moreover, have the same efficiency indices as that of existing multipoint iterative methods. Furthermore, the convergence analysis of the new methods is discussed and several examples are given to illustrate their efficiency. Keywords: Nonlinear Equations, Iterative Methods, Multipoint Iterative Methods, Newton’s Method, Traub-Ostrowski’s Method, Predictor-Corrector Methods, Order of Convergence 1. Introduction One of the most important and challenging problems in computational mathematics is to compute approximate solutions of the nonlinear equation 0fx (1) Therefore, the design of iterative methods for solving the nonlinear equation is a very interesting and important task in numerical analysis. Assume that Equation (1) has a simple root r which is to be found and let 0x be our initial guess to this root. To solve this equation, one can use iterative methods such as Newton’s method [1,2] and its variants namely, Halley’s method [1-6], Chebyshev’s method [1-6], Chebyshev-Halley type methods  etc. The requirement of 0fx is an essential condition for the convergence of Newton’s method. The above- mentioned variants of Newton’s method have also two problems which restrict their practical applications rig-orously. The first problem is that these methods require the computation of second order derivative. The second problem is that like Newton’s method, these methods require the condition that 0fx in the vicinity of the root. For the first problem, Nedzhibov et al.  derived many families of multipoint iterative methods by discre-tizing the second order derivative involved in Cheby-shev-Halley type methods . We mention below only one root-finding technique (2.1) from , namely  1,2nnnnnnnnnn nfxzxfxfx fzxzf xfxfz, (2) where . For different specific values of , vari-ous multipoint iterative methods may result from (2). For 1 in (2), we get   12nnnnnnn nfxfx fzxxf xfxfz . (3) This is the famous Traub-Ostrowski’s formula [1,2,4,5, 7,8], which is an order four formula. This method re-quires one evaluation of the function and two evaluations of its derivative per iteration. Thus the efficiency index  of this method is equal to 34 1.587 which is bet-ter than the one of Newton’s method 221.414. Fur-thermore, Sharma and Guha  have developed a variant S. KUMAR ET AL. Copyright © 2010 SciRes. AM 154 of Traub-Ostrowski’s method (3) which is defined by   1,,2,2nnnnnnnnnn nnnnnnnn nfxzxfxfz fxyzf xfxfzfyfx afzxyfx fxafz  , (4) where a is a parameter. This family requires an additional evaluation of function fx at the point ite- rated by Traub-Ostrowski’s method (3), consequently, the local order of convergence is improved from four to six. For 0a, we obtain the method developed by Grau and Díaz-Barrero  defined by   1,,2,2nnnnnnnnnn nnnnnnn nfxzxfxfz fxyzf xfxfzfy fxxyfx fxfz   . (5) All these multipoint iterative methods are variants of Newton’s method. Therefore, they require sufficiently good initial approximation and fail miserably like New-ton’s method if at any stage of computation, the deriva-tive of the function is zero or very small in the vicinity of the root. Recently, Kanwar and Tomar [3,4] proposed an alter-native to the failure situation of Newton’s method and its various variants. They also derived modifications over the different families of Nedzhibov et al.  multipoint iterative methods. Unfortunately, the various families introduced by Kanwar and Tomar  produces only multipoint iterative methods of order three. Recently, Mir et al.  have proposed a new predic-tor-corrector method (designated as Simpson-Mamta method (SM)), which is defined by   2212,46,4/2nnnnnnnnnnnn nfxzxfxpfxf xfxxxfxfzxfz . (6) where p is chosen as a positive or negative sign so as to make the denominator largest in magnitude. This method is obtained by combining the quadratically convergent method due to Mamta et al.  and cubically conver-gent method due to Hasnov et al. . This method will not fail like existing methods if fx is very small or even zero in the vicinity of the root. This method re-quires one evaluation of the function and three evalua-tions of its derivative per iteration. Thus the efficiency index of this method is equal to 431.316 which is not better than the one of Newton’s method 221.414 or Traub-Ostrowski’s method 34 1.587. More recently, Gupta et al.  have developed a family of ellipse methods given by  1222nnnnnfxxxfxpfx , (7) where 0p and in which 0fx is permitted at some points in the vicinity of the root. The beauty of this method is that it converges quadratically and more-over, has the same error equation as Newton’s method. Therefore, this method is an efficient alternative to Newton’s method. In this paper, we present two families of predictor- corrector iterative methods based on quadratically con-vergent ellipse method (7), Nedihzbov et al. family (2) and the well-known Traub-Ostrowski’s Formula (3). 2. Development of Methods 2.1. Two-Step Iterative Method and its Order of Convergence Our aim is to develop a scheme that retains the order of convergence of Nedzhibov et al. family (3) and which can be used as an alternative to existing techniques or in cases where existing techniques are not successful. Thus we begin with the following predictor-corrector iterative scheme    2221222,,2nnnnnnnnnnnnnfxzxfx pfxfz fxxz fx fzfx pfx, (8) where the positive sign is taken if 0xr and the nega-tive sign is taken if 0xr. Geometrically, if slope of the curve 0fx at the point 00,xfx is nega-tive, then take positive sign otherwise, negative. It is interesting to note that by ignoring the term in p, pro-posed family (8) reduces to Nedzhibov et al. family (2). For 1 in (8), we get S. KUMAR ET AL. Copyright © 2010 SciRes. AM 155   1222 2nnnnnnnnnfxfx fzxxfxfzfx pfx , (9) This is the modification over the Formula (3) of Traub- Ostrowski [2,5,7], and is also an order four formula. This method requires same evaluation of the function and its derivative as Traub-Ostrowski’s method per iteration. Thus the efficiency index  of this method is equal to 34 1.587 which is better than the one of Newton’s method22 1.414 or SM method 431.316. More importantly, this method will not fail even if the deriva-tive of the function is small or even zero in the vicinity of the root. The asymptotic order of this method is presented in the following theorem. Theorem 1. Suppose fx is sufficiently differen-tiable function in the neighborhood of a simple root r and that 0x is close to r, then the iteration scheme (8) has 3rd and 4th order convergence for 1) 1, 2) 1, respectively. Proof: Since fx is sufficiently differentiable, ex-panding nfx and nfx about xr by Taylor’s expansion, we have 2345 62345nnnnnn nfx frececececeOe (10) and 2345234 512 345nnnnnnfxfrcecececeO e (11) where nnexr and 1, 2,3,...!kkfrckkfr Using Equations (10) and (11), we have  223345232 42322374nnnnn nnfx ececcecccceOefx  (12) Therefore,   222 2222233 2452232234211144814 63.22nnnn nnnnnnn nfx fxfx pfxfxfxp fxececc pecccccpeOe  (13) 2223324522322342114410146 3.22nn nnnfzf rceccp ecccccp eOe (14)   2223423 2321244 .nn nnnnfxfzf r ececccpeOe (15) Using Equations (12), (13) and (15), we obtain   2222232322242423231644228325 44.nnnnnnnfz cecccp efx fzccccp ccpeOe   (16) Using Equations (13)-(16) in Equation (8), we obtain,  22322 4512232217834914 4.2nn nnpececcc eOe     (17) S. KUMAR ET AL. Copyright © 2010 SciRes. AM 156 While for 1 in (18), we have 324512 2312.2nnnecccpeOe  (18) Thus Equation (18) establishes the maximum order of convergence equal to four, for iteration scheme (8). This completes the proof of the theorem. 2.2. Three-Step Iterative Method and its Order of Convergence On similar lines, we also propose a modification over the Formula (4) of Sharma and Guha . Mir and Zaman  have considered three-step quadrature based iterative methods with sixth, seventh and eight order of conver-gence for finding simple zeros of nonlinear equations. Milovannović and Cvetković  further presented modifications over three-step iterative methods consid-ered by Mir and Zaman . Also Rafiq et al.  have presented similar three-step iterative method based on Newton’s method with sixth-order convergence. All these modifications are targeted at increasing the local order of convergence with a view of increasing their ef-ficiency index. But all these methods are variants of Newton’s method and will not work if fx is very small or zero in the vicinity of the root. To overcome this problem, now we begin with the following predictor- corrector iterative scheme       2222221222,,2,nnnnnnnnnnnnnnnnnnnnnnfxzxfx pfxfz fxyz fx fzfx pfxfyfx afzxy fx bfzfx pfx  (19) where a and b are parameters to be determined from the following convergence theorem. Theorem 2. Let :fIR denote a real valued function defined on I, where I is a neighborhood of sim-ple root r of .fx Assume that fx is sufficiently differentiable function in I. Then the iteration scheme (19) defines a one-parameter (.., )iea family of sixth order convergence if 2ba and satisfies the following error equation: 22 2261223 237122 4124.n nne cccpaccpeOe (20) Proof: follows on the similar steps as given in the pre-vious theorem. The proposed scheme (19) is now given by      2222221222,,2,2nnnnnnnnnnnnnnnnnnnnnnfxzxfx pfxfz fxyz fx fzfx pfxfyfxafzxy fx a fzfx pfx (21) where a. Note that for 0,p we obtain method (4) obtained by Sharma and Guha  and for ,0,0pa , we obtain method (5) developed by Grau and Díaz- Barrero . 3. Numerical Results In this section, we shall present the numerical results obtained by employing the iterative methods namely Newton’s method (NM), Traub-Ostrowski’s method (3) (TOM), Simpson-Mamta method (6) (SM), modified Traub-Ostrowski’s method (9) (MTOM), method (4) for 1a (3M) and modified method (21) for 1a (3MM) respectively to solve nonlinear equations given in Table 1. The results are summarized in Table 2. We use 1510 as tolerance. Computations have been performed using Cin double precision arithmetic. Here all the formulas are tested for 1/2p. The fol-lowing stopping criteria are used for computer programs: 1) 1nnxx , 2) 1nfx. The behaviors of existing multipoint iterative schemes and proposed modifications can be compared by their corresponding correction factors. The correction factor (),()nnfxfx which appears in the existing multipoint itera-tive schemes is now modified by 222()() ()nnnfxfxpfx, where 0p. This is always well defined, even if ()0.nfx It is investigated that formulas (8) and (21) give very good approximation to the root when p is taken in between 01p. This is because that for small value of p, the ellipse will shrink in the vertical direction and extend along horizontal direction. This means that our next approximation will move faster to-wards the desired root. However, for 1p but not very large, the formulas work if the initial guess is very close S. KUMAR ET AL. Copyright © 2010 SciRes. AM 157 Table 1. Test problems. No Examples [a,b] Initial guess 0x Root (r) 1.1 1 6110x [1,3] 3.0 2.000000000000000 0.0 0.1 2 324100xx [0,2] 2.0 1.3652300134140969 0.0 3 cos 0xx [0,2] 2.0 0.7390851332151600 −2.0 −1.0 4 1tan 0x [-2,2] 2.0 0.000000000000000 1.8 5 324cos160xx x  [0.5,3] 3.0 1.000000000000000 2.0 2.5 2.8 6 2730 10xxe  [2,4] 3.5 3.000000000000000 −1.0 7 110xe [-1,3] 3.0 1.000000000000000 8 sin 0x [0,2] 1.5 0.000000000000000 Table 2. Comparison table. Number of iterations Number of functions evaluations Examples NM TOM SM MTOMM3 MM3 NM TOM SM MTOM M3 MM3 (3) (6) (9) (4) (21) (3) (6) (9) (4) (21) 58 25 6 3 D 5 116 75 24 9 - 20 1 8 4 5 4 3 3 16 12 20 12 12 12 F F 4 3 F 3 - - 16 9 - 12 9 4 4 2 8 2 18 12 16 6 32 8 2 4 2 3 2 2 2 8 6 12 6 8 8 3 2 3 2 2 2 6 6 12 6 8 8 3 3 2 3 2 2 2 6 6 12 6 8 8 D 5 5 3 8 3 - 15 20 9 32 12 5 3 3 3 3 2 10 9 12 9 12 8 4 D 5 5 3 8 3 - 15 20 9 32 12 5 2 3 2 2 2 10 6 12 6 8 8 5 6 3 4 2 3 2 12 9 16 6 12 8 D D D 2 D 2 - - - 6 - 8 D D D 6 D D - - - 18 - - 15 5 21 4 D D 30 15 84 12 - - 6 11 5 7 5 4 4 22 15 28 15 16 16 6 3 4 3 3 2 12 9 16 9 12 8 7 9 4 5 3 10 3 18 12 20 9 40 12 S. KUMAR ET AL. Copyright © 2010 SciRes. AM 158 to the required root. For larger value of p, the formulas do not work. This is perhaps due to the occurrence of numerical instability in the process of computation. Example 8. sin 0x. This equation has an infinite number of roots. New-ton’s method and Traub-Ostrowski’s method with initial 01.5x converge to 4 far away from the required root zero. Method (4) (3M) converges to 6. Our methods and SM method do not exhibit this type of be-havior and converge to the nearest root zero. 4. Conclusions The presented results indicate that the new proposed methods are more efficient and perform better than clas-sical existing methods. The computational results in Ta-ble 2 show that the modified Traub-Ostrowski’s method (MTOM) (9) requires a smaller number of function evaluations than Newton’s method (NM) and Traub- Ostrowski’s method (3) (TOM). The computational re-sults in Table 2 also show that modified method (21) (3MM) requires smaller number of function evaluations than method (4) (3M). On similar lines, we can also modify Mir and Zaman , Milovannović and Cvet- ković  three-step iterative methods. Now a reasona-bly close starting value 0x is not required for these methods to converge. This condition, however applies to practically all existing iterative methods for solving equations. Moreover, they have same efficiency indices as that of existing methods and do not fail if the deriva-tive of the function is either zero or very small in the vicinity of the root. Therefore, these techniques have a definite practical utility. 5. Acknowledgements We are grateful to the reviewer for the constructive re-marks and suggestions which enhanced our work. 6. References  A. M. Ostrowski, “Solution of Equations in Euclidean and Banach Space,” 3rd Edition, Academic Press, New York, 1973.  J. F. Traub, “Iterative Methods for the Solution of Equa-tions,” Prentice Hall, Englewood Cliffs, New Jersey, 1964.  V. Kanwar and S. K. Tomar, “Modified Families of New-ton, Halley and Chebyshev Methods,” Applied Mathe-matics and Computation, Vol. 192, No. 1, September 2007, pp. 20-26.  V. Kanwar and S. K. 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