Iterative Methods for Single Variable Equations

Transcription

1 International Journal of Mathematical Analysis Vol 0, 06, no 6, HII Ltd, wwwm-hikaricom Iterative Methods for Single Variable Equations Shin Min Kang Department of Mathematics and RINS Gyeongsang National University Jinju 588, Korea Arif Rafiq Department of Mathematics and Statistics Virtual University of Pakistan Lahore 54000, Pakistan Faisal Ali Center for Advanced Studies in Pure and Applied Mathematics Bahauddin Zakariya University Multan 60800, Pakistan Young Chel Kwun Department of Mathematics Dong-A University Busan 4935, Korea Corresponding author Copyright c 06 Shin Min Kang et al 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 Abstract In this paper, we establish new iterative methods for the solution of scalar equations by using the decomposition technique given in [7] (M A Noor and K I Noor, Some iterative schemes for nonlinear equations, Appl Math Comput, 83 (006,

2 80 Shin Min Kang et al Mathematics Subject Classification: 65H05 Keywords: Iterative method, nonlinear equation, order of convergence, multiple root Introduction In science and engineering, one of the most important problems is to find the solution of nonlinear equations Numerous iterative methods involving various techniques have been developes to find the approximate roots of nonlinear equations, see [, 3 5, 7, 0,, 3] and references there in These methods can be categorized as one-step, two-step and three-step methods Chun [5] has proposed some one-step and two-step iterative methods with higher-order convergence by using the decomposition technique of Adomian [] Several other iterative methods have also been established for finding the simple zero of nonlinear equations The leading equations in many problems related to science and technology have roots with multiplicity greater than one The first contribution in dealing with such equations was made by Schröder [9] who developed the following modified Newton s method: x n+ = x n m f (x n f (x n, f (x n 0 If m =, then the equation is the classical Newton s formula During the last two decades, much attention has been devoted by various researchers for solving nonlinear equation with multiple roots Chun et al [6], Chun and Neta [8], Homeier [], Osada [8] have developed some techniques to find the multiple roots of nonlinear equations In the recent years, the researchers have made significant and interesting contribution in this field [9, 5, 6] In the present paper, we explore new iterative methods using the decomposition technique due to Noor and Noor [7] for solving scalar equations with roots of multiplicity m Consider the nonlinear equation f(x = 0 ( It is well known that if α is a root with multiplicity m, then it is also a root of f (x = 0 with multiplicity m, of f (x = 0 with multiplicity m

3 Iterative methods for single variable equations 8 and so on Hence if initial guess x 0 is sufficiently close to α, the expressions x 0 m f(x 0 f (x 0, x 0 (m f (x 0 f (x 0, x 0 (m f (x 0 f (x 0, ( will have the same value Remark The generalized Newton s formula x n+ = x n f(x n f (x n (3 gives a quadratic convergence when the equation f(x = 0 has a pair of double roots in the neighbourhood of x 0 It may be noted that for the double root α near to x 0, f(α = 0 = f (α New iterative methods We can rewrite the nonlinear equation ( as a coupled system: f(γ + (x γ f (γ + f (x + g(x = 0, ( g(x = f(x f(γ (x γ f (γ + f (x, ( where γ is the initial approximation for a zero of ( By taking g(x = 0, we can rewrite ( in the following form: x = γ f(γ f (γ (x γf (x f (γ = c + N(x, (3 where and c = γ f(γ f (γ (4 N(x = (x γ f (x f (γ (5

4 8 Shin Min Kang et al Here N(x is a nonlinear operator As in [7], the solution of (3 has the series form, x = x i (6 i=0 The nonlinear operator N(x can be decomposed as it has been shown in [7] Also the series i=0 x i converges absolutely and uniformly to a unique solution of equation (3 if the nonlinear operator ( { ( i } N(x = N x i = N(x 0 + N x j (7 i=0 is a contraction Combining (3, (6 and (7, we have { ( i x i = c + N(x 0 + N x j } (8 i=0 i= Thus we have the following iterative scheme: Then and and x 0 = c, x = N(x 0, x = N(x 0 + x, i= j=0 j=0 x n+ = N(x 0 + x + + x n, n =,, x + x + + x n+ = N(x 0 + N(x 0 + x + + N(x 0 + x + + x n, n =,, x = c + From (4, (5 and (9, we have (9 x i (0 i= x 0 = c = γ f(γ f (γ, ( x = N(x 0 = (x 0 γ f (x 0 f (γ = f(γf (x 0 f (γ (

5 Iterative methods for single variable equations 83 It follows from (4, (9, and (0 that x x 0 = c = γ f(γ f (γ (3 This enables us to suggest the following method for solving the nonlinear equation ( Algorithm For the given x 0 compute the approximate solution x n+ by the iterative schemes: x n+ = x n f(x n f (x n, f (x n 0, n = 0,,,, which is known as the generalized Newton s formula and is quadratically convergent Again by using (4, (5, (9 and (0, we conclude that x x 0 + x = c + x = x 0 + N(x 0 = γ f(γ f (γ + f(γf (x 0 f (γ (4 Using (4, we can suggest the following two-step iterative method for solving nonlinear equation ( Algorithm For the given x 0 compute the approximate solution x n+ by the iterative schemes: y n = x n f (x n f (x n, f (x n 0, n = 0,,,, x n+ = y n + f(x nf (y n (f (x n, which is introduced by Kang et al [4] Again, using (5, ( and (, we can calculate N(x 0 + x = (x 0 + x γ f (x 0 + x f (γ = f(γ ( f (x 0 f (x 0 + x f (γ f (γ f (γ (5

6 84 Shin Min Kang et al From (0, (, (, and (5, we get x x 0 + x + x = c + N(x 0 + N(x 0 + x = γ f(γ f (γ + f(γf (x 0 f (γ ( γ f(γ f (γ + f(γf (x 0 f (x 0 + x γ f (γ f (γ (6 Using (6, we can suggest and analyze the following three-step iterative method for solving nonlinear equation ( Algorithm 3 For a given x 0, compute the approximate solution by the iterative schemes: y n = x n f (x n f (x n, f (x n 0, n = 0,,,, z n = f(x nf (y n (f (x n, x n+ = x n f (y n + z n f (x n 3 Convergence analysis ( + (y n + z n f (y n + z n f (x n In this section, the convergence analysis of Algorithm 3 is given Theorem 3 Assume that the function f : D R R for an open interval D has a multiple root α D of multiplicity Let f(x be sufficiently smooth in the neighborhood of the root α Then the order of convergence of the methods defined by Algorithm 3 is Proof Let α be a root of f(x of multiplicity Then by expanding f(x n, (f (x n, in Taylor s series about α, we obtain f(x n = c e n + c 3 e 3 n + c 4 e 4 n + O ( e 5 n, (3 f (x n = c e n + 3c 3 e n + 4c 4 e c 5 e 4 n + O ( e 5 n, (3 (f (x n = 4c e n + c c 3 e 3 n + ( 6c c 4 + 9c 3 e O ( e 5 n, (33 where e n = x n α and c k = f (k (α, k =, 3, k! Using (3 and (3, we have f(x n f (x n = e n ( c 3 e n + c 4 4 c ( + 3 c c 3 c c 4 c 6 c c 3 3 c 3 c 3 c e 3 n e 4 n + O ( e 5 n (34

7 Iterative methods for single variable equations 85 Thus y n = x n f(x n f (x n = α + ( c 3 e c4 n + 3 c 3 e 3 c c 4 c n ( 3 c c 3 c c 3 3 e 4 c c 8 c 3 n + O ( e 5 n Expanding f (y n by Taylor s series about α, we get ( f (y n = c 3 e 3 5c3 c 4 n + 5 c 3 3 e 4 n c c 4 + ( 9c3 c 5 c 4c 4c 3 c c Now we use (3, (33 and (36 to compute z n as follows z n = f(x nf (y n (f (x n = ( 5 e 3 n + c 3 c c 3 c 4 c 3 8 c 4 3 c 3 c 3 3 c 3 Using (35 and (37, we have y n + z n = α + c ( 3 e c4 n + 5 c c ( 3c5 + 7 c 3 c 4 + 6c3 3 c c c 3 Thus (35 + 6c 4 e 5 n + O ( (36 e 6 n c e 4 n + O ( (37 e 5 n c 3 c e 3 n e 4 n + O ( (38 e 5 n Expanding f (y n + z n by Taylor s series about α, we obtain ( f (y n + z n = c 3 e 3 5c3 c 4 n + 5 c 3 3 e 4 c c 4 c n ( 9c3 c c c 4 3c 4c 3 e 5 c 8c 3 c c c n + O ( (39 e 6 n f (y n + z n f (x n = + c 3 c ( 9 e n + c 3 c 5 c ( 5 c 3 c 4 c c4 3 8c c 4 c c 3 3 c 3 e 3 n 5c 4c 3 4c 3 e 4 n + O ( (30 e 5 n Now using (34, (35, (39 and (30, we have the error term as e n+ = c ( 3 e n + 5 c 3 + 4c 4 e 3 n + O ( e c c n 4 c

8 86 Shin Min Kang et al The last equation shows that the convergence order of Algorithm 3 is This completes the proof 4 Numerical examples In this section, we present some numerical examples to illustrate the validity and efficiency of the methods proposed by us In the following Tables -, we compare our method (Algorithm 3 ( with classical Newton s method (, generalized Newton s method (G, Chun et al [6, Equations (35 and (36] (CBN and (CBN and Kang et al [4] (Algorithm ( All the computations for above mentioned methods are performed using software Maple and ε = 0 0 as tolerance and also the following criteria is used for estimating the zero: (i δ = x n+ x n < ε, (ii f (x n < ε, (iii Maximum numbers of iterations = 500 Here N denotes the number of iterations Table Comparison for f (x = (x f(x x Method N x [k] f(x n δ G e 0 000e e e 0 f (x 97 CBN e e 0 CBN e e e e e 0 300e 0 Table Comparison for f (x = arctan x f(x x Method N x [k] f(x n δ G e 0 84e 95e 05 00e 0 f (x 00 CBN e 00e 0 CBN e 00e e 0 456e 0 00e 0 00e 0 Table 3 Comparison for f 3 (x = ( e x 4x f(x x Method N x [k] f(x n δ G e 0 6e 3 40e e 04 f 3 (x 039 CBN e 7 69e 05 CBN e 78e e 643e 0 63e 04 3e 03

9 Iterative methods for single variable equations 87 Table 4 Comparison for f 4 (x = x 3 x x + f(x x Method N x [k] f(x n δ G e 0 93e 6 49e 05 0e 04 f 4 (x 03 CBN e 300e 0 CBN e 300e e 4 05e 434e 04 85e 04 Table 5 Comparison for f 5 (x = (sin x cos x f(x x Method N x [k] f(x n δ G e 0 8e 3 08e 05 08e 04 f 5 (x 07 CBN e 6 07e 04 CBN e 6 03e e 5e 8 48e e 04 Table 6 Comparison for f 6 (x = (x + ln x + sin x f(x x Method N x [k] f(x n δ G e 0 37e 4 543e 06 49e 04 f 6 (x 06 CBN e 9 8e 05 CBN e 7 34e e 4e 0 8e 04 6e 03 Table 7 Comparison for f 7 (x = x 3 3x + f(x x Method N x [k] f(x n δ G e 0 959e 5 49e 05 58e 04 f 7 (x 06 CBN e 30 94e 05 CBN e 600e e 3 39e 4e 03 3e 03 Table 8 Comparison for f 8 (x = (x e cos x f(x x Method N x [k] f(x n δ G e 0 39e 3 95e 06 97e 04 f 8 (x 3 CBN e 6 709e 05 CBN e 8 404e e 447e 0 8e e 03

10 88 Shin Min Kang et al Table 9 Comparison for f 9 (x = (x tan x f(x x Method N x [k] f(x n δ G e 0 53e 0 833e 03 67e 0 f 9 (x 005 CBN e e 0 CBN e 44e e 0 343e 0 44e 0 8e 0 Table 0 Comparison for f 0 (x = (x sin x f(x x Method N x [k] f(x n δ G e 0 4e 0 00e 0 00e 0 f 0 (x 006 CBN e 53e 0 CBN e 93e e 0 56e 0 74e 0 4e 0 Table Comparison for f (x = (x + sin x 5 4 f(x x Method N x [k] f(x n δ G e 0 87e 80e 05 70e 03 f (x 037 CBN e 37e 04 CBN e 33 30e e 0 7e 4 368e 03 58e 04 Table Comparison for f (x = (x + 7x 30 f(x x Method N x [k] f(x n δ G e 0 6e 7 84e e 05 f (x 303 CBN e 300e 0 CBN e 5 300e e 5 886e 4 37e 04 73e 04 5 Conclusions In the present work, we have proposed two iterative methods ( and ( with convergence order for finding the multiple roots of nonlinear equations The numerical results presented in the tables given in the previous section reveal that our iterative methods not only perform better that Newton s method but are even comparable with the methods developed by Chun et

11 Iterative methods for single variable equations 89 al [6] (CBN and (CBN with convergence order 3 The idea and technique employed in this paper can be developed to higher-order multi-step iterative methods for solving nonlinear equations having roots with multiplicity greater than one Acknowledgments This work was supported by the Dong-A University research fund References [] S Abbasbandy, Improving Newton-Raphson method for nonlinear equations modified Adomian decomposition method, Appl Math Comput, 45 (003, [] G Adomian, Nonlinear Stochastic Systems and Applications to Physics, Kluwer Academic Publishers, Dordrecht, 989 [3] E Babolian and J Biazar, On the order of convergence of Adomian method, Appl Math Comput, 30 (00, [4] E Babolian and J Biazar, Solution of nonlinear equations by modified Adomian decomposition method, Appl Math Comput, 3 (00, [5] C Chun, Iterative methods improving Newton s method by the decomposition method, Comput Math Appl, 50 (005, [6] C Chun, H J Bae and B Neta, New families of nonlinear third-order solvers for finding multiple roots, Comput Math Appl, 57 (009, [7] C Chun and Y Ham, A one-parameter fourth-order family of iterative methods for nonlinear equations, Appl Math Comput, 89 (007, [8] C Chun and B Neta, A third-order modification of Newton s method for multiple roots, Appl Math Comput, (009, [9] C Chun and B Neta, Basin of attraction for Zhou-Chen-Song fourth order family of methods for multiple roots, Math Comput Simulation, 09 (05,

12 90 Shin Min Kang et al [0] V Daftardar-Gejji and H Jafari, An iterative method for solving nonlinear functional equations, J Math Anal Appl, 36 (006, [] V I Hasanov, I G Ivanov and G Nedzhibov, A new modification of Newton s method, Appl Math Engin Econom, (Sozopol, 00, 78 86, Heron Press, Sofia, 00 [] H H H Homeier, On Newton-type methods for multiple roots with cubic convergence, J Comput Appl Math, 3 (009, [3] E Isaacson and H B Keller, Analysis of Numerical Methods, John Wiley & Sons, Inc, New York, 966 [4] S M Kang, F Ali and A Rafiq, Iterative methods for solving scalar equations, J Nonlinear Sci Appl, 9 (06, [5] B Neta and C Chun, On a family of Laguerre methods to find multiple roots of nonlinear equations, Appl Math Comput, 9 (03, [6] B Neta, C Chun and M Scott, On a development of iterative methods for multiple roots, Appl Math Comput, 4 (03, [7] M A Noor and K I Noor, Some iterative schemes for nonlinear equations, Appl Math Comput, 83 (006, [8] N Osada, An optimal multiple root-finding method of order three, J Comput Appl Math, 5 (994, [9] E Schröder, Ueber unendlich viele Algorithmen zur Auflösung der Gleichungen, Math Ann, (870, Received: January 7, 06; Published February 8, 06