- Email: [email protected]
New family of iterative methods for nonlinear equations
Applied Mathematics and Computation 190 (2007) 553–558 www.elsevier.com/locate/amc
New family of iterative methods for nonlinear equations Muhammad Aslam Noor Mathematics Department, COMSATS Institute of Information Technology, Sector-H-8/1, Islamabad 44000, Pakistan
Abstract In this paper, we suggest and analyze a new family of iterative methods for solving nonlinear equations using the system of coupled equations coupled with decomposition technique. Several numerical examples are given to illustrate the efficiency and performance of the new methods. These new iterative methods may be viewed as an extension and generalization of the existing methods for solving nonlinear equations. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Multi-step methods; Convergence; Decomposition methods; Numerical examples
1. Introduction In recent years, much attention has been given to develop several iterative methods for solving nonlinear equations, see [1–8] and the references therein. Abbasbandy [1] and Chun [3] have proposed and studied several one-step and two-step iterative methods with higher order convergence by using the decomposition technique. We again use this decomposition technique to generate a new family of iterative methods for solving nonlinear equations. One can easily show that the iterative methods considered by Abbasbandy [1] and Chun [3] are special cases of these proposed iterative methods. Our method can be considered as predictor–corrector type methods. Several numerical examples are given to illustrate the efficiency and the performance of the new iterative methods. Our results can be considered as an important improvement and refinement of the previous results. 2. Iterative methods Consider the nonlinear equation f ðxÞ ¼ 0:
ð1Þ
We assume that a is a simple root of (1) and c is an initial guess sufficiently close to a. We can rewrite the nonlinear equation (1) as a coupled system using the Taylor’s series
E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.01.045
554
M.A. Noor / Applied Mathematics and Computation 190 (2007) 553–558 2
ðx cÞ 00 f ðcÞ þ gðxÞ ¼ 0; 2 ðx cÞ2 00 f ðcÞ; gðxÞ ¼ f ðxÞ f ðcÞ f 0 ðcÞðx cÞ 2
f ðcÞ þ f 0 ðcÞðx cÞ þ
ð2Þ ð3Þ
where c is the initial approximation for a zero of (1). We can rewrite Eq. (3) in the following form: x¼c
f ðcÞ ðx cÞ2 f 00 ðcÞ gðxÞ : f 0 ðcÞ f 0 ðcÞ f 0 ðcÞ 2
ð4Þ
From the relation (3), it is clear that gðx0 Þ ¼ f ðx0 Þ:
ð5Þ
We remark that Eq. (5) plays a very important role in the derivation of new iterative method, see Chun [3, p. 1560]. It is worth mentioning that He [4] and Luo [8] have considered a very special case f ðx0 Þ ¼ 0: We also rectify this aspect in this paper. Also see Noor and Noor [7] for a different approach. We rewrite Eq. (4) in the following equivalent useful form: x ¼ c þ N ðxÞ;
ð6Þ
where c¼c
f ðcÞ f 0 ðcÞ
ð7Þ
and N ðxÞ ¼
ðx cÞ2 00 gðxÞ f ðcÞ 0 : f ðcÞ 2f 0 ðcÞ
ð8Þ
Here N(x) is a nonlinear operator. We now construct a new family of two-step iterative methods by using the decomposition method [2]. The main idea of this technique is to look for a solution of Eq. (6) having the series form 1 X xi : ð9Þ x¼ i¼0
The nonlinear operator N(x) can be decomposed as ! 1 1 X X N xi ¼ N ðxÞ ¼ Ai ; i¼0
ð10Þ
i¼0
where Ai are functions which are known as the Adomian polynomials depending on x0 ; x1 ; . . . ; given by the formulas " !# 1 X 1 dn i An ¼ N k xi ; n ¼ 0; 1; 2; . . . ð11Þ n! dkn i¼0 k¼0
First few Adomian polynomials are given by A0 ¼ N ðx0 Þ; 0
ð12Þ
A1 ¼ x1 N ðx0 Þ;
ð13Þ
1 A2 ¼ x2 N 0 ðx0 Þ þ x21 N 00 ðx0 Þ: 2
ð14Þ
M.A. Noor / Applied Mathematics and Computation 190 (2007) 553–558
555
It follows from (6), (7), (9), (10) and (11) that x x0 ¼ c ¼ c
f ðcÞ : f 0 ðcÞ
ð15Þ
This enables us to suggest the following iterative method, which is known as Newton method. Algorithm 2.1. For a given x0, compute the approximate solution xn+1 by the iterative scheme xnþ1 ¼ xn
f ðxn Þ ; f 0 ðxn Þ
n ¼ 0; 1; 2; . . . ; f 0 ðxn Þ 6¼ 0:
It is well known that Algorithm 2.1 has 2nd order convergence. From (5), (8) and (10), we have 2
x1 ¼ N ðx0 Þ ¼ ¼
ðx0 cÞ 00 gðx0 Þ f ðcÞ 0 0 f ðcÞ 2f ðcÞ
ð16Þ
ðx0 cÞ2 00 f ðx0 Þ f ðcÞ 0 : f ðcÞ 2f 0 ðcÞ
Again using (12), (15) and (16), we conclude that x c þ x1 ¼ x0 þ x1 ¼ x0 þ A0 ¼ x0 þ N ðx0 Þ ð17Þ
2
¼c
f ðcÞ ðx0 cÞ 00 f ðx0 Þ f ðcÞ 0 : f 0 ðcÞ f ðcÞ 2f 0 ðcÞ
Using this relation, we can suggest the following two-step iterative method for solving nonlinear equation (1). Algorithm 2.2. For a given x0, compute the approximate solution xn+1 by the iterative schemes: Predictor-step: y n ¼ xn
f ðxn Þ ; f 0 ðxn Þ
f 0 ðxn Þ 6¼ 0:
Corrector-step: xnþ1 ¼ xn
f ðxn Þ ðy n xn Þ2 00 f ðy Þ f ðxn Þ 0 n ; f 0 ðxn Þ f ðxn Þ 2f 0 ðxn Þ
n ¼ 0; 1; 2; . . .
Algorithm 2.2 can be written in the following form. Algorithm 2.3. For a given x0, compute the approximate solution xn+1 by the iterative scheme: f ðxn Þ 2 f x n 0 f ðxn Þ f ðxn Þ f ðxn Þ 00 f ðxn Þ ; n ¼ 0; 1; 2; . . . xnþ1 ¼ xn 0 f ðxn Þ 2f 03 ðxn Þ f 0 ðxn Þ We would like to emphasize the fact that Algorithms 2.2 and 2.3 are remarkably different that the methods of Abbasbandy [1] and Chun [3]. If f 00 ðxn Þ ¼ 0, then Algorithm 2.2 reduces to the following iterative method for solving nonlinear equation (1), which is mainly due to Chun [3]. Algorithm 2.4 [3]. For a given x0, compute the approximate solution xn+1 by the iterative schemes: Predictor-step: y n ¼ xn
f ðxn Þ ; f 0 ðxn Þ
f 0 ðxn Þ 6¼ 0:
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M.A. Noor / Applied Mathematics and Computation 190 (2007) 553–558
Corrector-step: xnþ1 ¼ xn
f ðxn Þ f ðy n Þ ; f 0 ðxn Þ f 0 ðxn Þ
n ¼ 0; 1; 2; . . .
If f 0 ðy n Þ ¼ 0; then Algorithm 2.2 collapses to the following iterative method. Algorithm 2.5. For a given x0, compute the approximate solution xn+1 by the iterative schemes: Predictor-step: y n ¼ xn
f ðxn Þ ; f 0 ðxn Þ
f 0 ðxn Þ 6¼ 0:
Corrector-step: 2
xnþ1 ¼ xn
f ðxn Þ ðy n xn Þ 00 f ðxn Þ; f 0 ðxn Þ 2f 0 ðxn Þ
n ¼ 0; 1; 2; . . .
Algorithm 2.5 can be considered a predictor–corrector method and appears to be a new one in this form. One can rewrite Algorithm 2.5 in the following equivalent form. Algorithm 2.6 [1]. For a given x0, find the approximate solution xn+1 by the iterative scheme: xnþ1 ¼ xn
f ðxn Þ f 2 ðxn Þ 00 f ðxn Þ; f 0 ðxn Þ 2f 03 ðxn Þ
n ¼ 0; 1; 2; . . .
which is known as the Houshoulder iterative method and was derived by Abbasbandy [1] using the Adomian decomposition method. This shows that Algorithms 2.2 and 2.3 include the methods of Chun [3] and Abbasbandy [1] as special cases. From (3) and (8), we have N 0 ðx0 Þ ¼ 1
f 0 ðx0 Þ ðx0 cÞ 00 0 f ðcÞ: f 0 ðcÞ f ðcÞ
ð18Þ
Combining (13), (16) and (18), we have x c þ x 1 þ x 2 ¼ x 0 þ A0 þ A1 2
¼c þ
2
f ðcÞ ðx0 cÞ f 00 ðcÞ f ðx0 Þ ðx0 cÞ f 0 ðx0 Þ 00 f ðx0 Þf 0 ðx0 Þ 2 0 þ f ðcÞ þ 2 0 0 f ðcÞ f ðcÞ f ðcÞ 1 2½f 0 ðcÞ ½f 0 ðcÞ2
ðx0 cÞf ðx0 Þf 00 ðcÞ ½f 0 ðcÞ2
3
þ
ðx0 cÞ ½f 00 ðcÞ 2½f 0 ðcÞ2
2
ð19Þ
:
Using this, we can suggest and analyze the following two-step iterative method for solving nonlinear equation (1). Algorithm 2.7. For a given x0, compute the approximate solution xn+1 by the iterative schemes: Predictor-steps: y n ¼ xn
f ðxn Þ ; f 0 ðxn Þ
f 0 ðxn Þ 6¼ 0:
ð20Þ
Corrector-step: xnþ1 ¼ xn þ
f ðxn Þ ðy n xn Þ2 f 00 ðxn Þ f ðy n Þ ðy n xn Þ2 f 0 ðy n Þ 00 2 þ f ðxn Þ 2 f 0 ðxn Þ f 0 ðxn Þ f 0 ðxn Þ 1 2½f 0 ðxn Þ
f ðy n Þf 0 ðy n Þ 2
½f 00 ðxn Þ
þ
ðy n xn Þf ðy n Þf 00 ðxn Þ ½f 0 ðxn Þ
2
þ
ðy n xn Þ3 ½f 00 ðxn Þ2 2½f 0 ðxn Þ
2
:
ð21Þ
M.A. Noor / Applied Mathematics and Computation 190 (2007) 553–558
557
Algorithm 2.7 is also called the predictor–corrector iterative method for solving nonlinear equations (1). Using the above arguments, one can show that methods developed in [1,3] are very special cases of Algorithm 2.7. Using the techniques and ideas of Chun [3] and Noor and Noor [7], one can study the convergence criteria of these new iterative methods. 3. Numerical examples We present some examples to illustrate the efficiency of the new developed three-step iterative methods in this paper. We compare the Newton method (NM), the method of Abbasbandy (AM), the method of Homeier Table 3.1 Examples and their comparison IT
xn
f(xn)
d
f1 ; x0 ¼ 1 NM AM HM CM NR2
7 5 4 5 6
1.4044916482153412260350868178 1.4044916482153412260350868178 1.4044916482153412260350868178 1.4044916482153412260350868178 1.4044916482153412260350868178
1.04e50 5.81e55 5.4e62 2.0e63 1.2e28
7.33e26 1.39e18 7.92e21 1.31e17 4.8e29
f2 ; x 0 ¼ 2 NM AM HM CM NR2
6 5 5 4 5
0.25753028543986076045536730494 0.25753028543986076045536730494 0.25753028543986076045536730494 0.25753028543986076045536730494 0.25753028543986076045536730494
2.93e55 1.0e63 0 1.0e63 6.0e24
9.1e28 1.45e26 9.33e43 9.46e29 1.5e16
f3 ; x0 ¼ 1:7 NM AM HM CM NR2
5 4 4 4 5
0.73908513321516064165537208767 0.73908513321516064165537208767 0.73908513321516064165537208767 0.73908513321516064165537208767 0.73908513321516064165537208767
2.03e32 7.14e47 5.02e59 0 1.4e15
2.34e16 8.6e16 9.64e20 1.86e53 8.4e16
f4 ; x0 ¼ 3:5 NM AM HM CM NR2
8 5 5 5 6
2 2 2 2 2
2.06e42 0 0 0 1.03e23
8.28e22 4.3e22 1.46e24 2.74e24 3.4e24
f5 ; x0 ¼ 1:5 NM AM HM CM NR2
7 5 4 5 7
2.1544346900318837217592935665 2.1544346900318837217592935665 2.1544346900318837217592935665 2.1544346900318837217592935665 2.1544346900318837217592935665
2.06e54 5.0e63 5.0e63 5.0e63 3.1e42
5.64e28 1.18e25 9.8e23 1.57e22 1.2e43
f6 ; x0 ¼ 2 NM AM HM CM NR2
9 6 6 6 6
1.2076478271309189270094167584 1.2076478271309189270094167584 1.2076478271309189270094167584 1.2076478271309189270094167584 1.2076478271309189270094167584
2.27e40 4.0e63 4.0e63 4.0e63 3.0e31
2.73e21 4.35e45 2.57e32 2.15e36 1.5e32
f7 ; x0 ¼ 3:5 NM AM HM CM NR2
13 7 8 8 8
1.52e47 4.33e48 2.0e62 2.0e62 2.1e29
4.2e25 2.25e17 2.43e33 2.12e23 1.6e28
3 3 3 3 3
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M.A. Noor / Applied Mathematics and Computation 190 (2007) 553–558
(HM), the method of Chun (CM) and Algorithm 2.7 (NR2), introduced in this present paper, see Table 3.1. We used e ¼ 1015 . The following stopping criteria is used for computer programs: ðiÞ jxnþ1 xn j < e;
ðiiÞ jf ðxnþ1 Þj < e:
The examples are the same as in Chun [3]. f1 ðxÞ ¼ sin2 x x2 þ 1; f2 ðxÞ ¼ x2 ex 3x þ 2; f3 ðxÞ ¼ cos x x; f4 ðxÞ ¼ ðx 1Þ3 1; f5 ðxÞ ¼ x3 10; 2
f6 ðxÞ ¼ xex sin2 x þ 3 cos x þ 5; 2 þ7x30
f7 ðxÞ ¼ ex
1:
As for the convergence criteria, it was required that the distance of two consecutive approximations d for the zero was less than 1015. Also displayed are the number of iterations to approximate the zero (IT), the approximate zero x and the value f ðx Þ: 4. Conclusion In this paper, we have suggested a family of two-step iterative methods for solving nonlinear equations by using the decomposition technique. Several examples are given to illustrate the efficiency of Algorithm 2.7. Using the technique and idea of this paper, one can suggest and analyze higher order multi-step iterative methods for solving nonlinear equations as well as system of nonlinear equations. This is the topic of further research. Acknowledgements This research is supported by the Higher Education Commission, Pakistan, through research Grant No: i-28/HEC/HRD/2005/90. The author would like to thank the referee for his/her several useful comments and suggestions. References [1] S. Abbasbandy, Improving Newton–Raphson method for nonlinear equations by modified Adomian decomposition method, Appl. Math. Comput. 145 (2003) 887–893. [2] G. Adomian, Nonlinear Stochastic Systems and Applications to Physics, Kluwer Academic Publishers, Dordrecht, 1989. [3] C. Chun, Iterative methods improving Newton’s method by the decomposition method, Comput. Math. Appl. 50 (2005) 1559–1568. [4] J.H. He, A new iterative method for solving algebraic equations, Appl. Math. Comput. 135 (2005) 81–84. [5] H.H. Homeier, On Newton-type methods with cubic convergence, J. Comput. Appl. Math. 176 (2005) 425–432. [6] M. Aslam Noor, Numerical Analysis and Optimization, Lecture Notes, COMSATS Institute of Information Technology, Islamabad, Pakistan, 2006. [7] M. Aslam Noor, K. Inayat Noor, Three-step iterative methods for nonlinear equations, Appl. Math. Comput. 183 (2006) 322–327. [8] X. Luo, A note on the new iteration for solving algebraic equations, Appl. Math. Comput. 171 (2005) 1177–1183.
New family of iterative methods for nonlinear equations Muhammad Aslam Noor Mathematics Department, COMSATS Institute of Information Technology, Sector-H-8/1, Islamabad 44000, Pakistan
Abstract In this paper, we suggest and analyze a new family of iterative methods for solving nonlinear equations using the system of coupled equations coupled with decomposition technique. Several numerical examples are given to illustrate the efficiency and performance of the new methods. These new iterative methods may be viewed as an extension and generalization of the existing methods for solving nonlinear equations. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Multi-step methods; Convergence; Decomposition methods; Numerical examples
1. Introduction In recent years, much attention has been given to develop several iterative methods for solving nonlinear equations, see [1–8] and the references therein. Abbasbandy [1] and Chun [3] have proposed and studied several one-step and two-step iterative methods with higher order convergence by using the decomposition technique. We again use this decomposition technique to generate a new family of iterative methods for solving nonlinear equations. One can easily show that the iterative methods considered by Abbasbandy [1] and Chun [3] are special cases of these proposed iterative methods. Our method can be considered as predictor–corrector type methods. Several numerical examples are given to illustrate the efficiency and the performance of the new iterative methods. Our results can be considered as an important improvement and refinement of the previous results. 2. Iterative methods Consider the nonlinear equation f ðxÞ ¼ 0:
ð1Þ
We assume that a is a simple root of (1) and c is an initial guess sufficiently close to a. We can rewrite the nonlinear equation (1) as a coupled system using the Taylor’s series
E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.01.045
554
M.A. Noor / Applied Mathematics and Computation 190 (2007) 553–558 2
ðx cÞ 00 f ðcÞ þ gðxÞ ¼ 0; 2 ðx cÞ2 00 f ðcÞ; gðxÞ ¼ f ðxÞ f ðcÞ f 0 ðcÞðx cÞ 2
f ðcÞ þ f 0 ðcÞðx cÞ þ
ð2Þ ð3Þ
where c is the initial approximation for a zero of (1). We can rewrite Eq. (3) in the following form: x¼c
f ðcÞ ðx cÞ2 f 00 ðcÞ gðxÞ : f 0 ðcÞ f 0 ðcÞ f 0 ðcÞ 2
ð4Þ
From the relation (3), it is clear that gðx0 Þ ¼ f ðx0 Þ:
ð5Þ
We remark that Eq. (5) plays a very important role in the derivation of new iterative method, see Chun [3, p. 1560]. It is worth mentioning that He [4] and Luo [8] have considered a very special case f ðx0 Þ ¼ 0: We also rectify this aspect in this paper. Also see Noor and Noor [7] for a different approach. We rewrite Eq. (4) in the following equivalent useful form: x ¼ c þ N ðxÞ;
ð6Þ
where c¼c
f ðcÞ f 0 ðcÞ
ð7Þ
and N ðxÞ ¼
ðx cÞ2 00 gðxÞ f ðcÞ 0 : f ðcÞ 2f 0 ðcÞ
ð8Þ
Here N(x) is a nonlinear operator. We now construct a new family of two-step iterative methods by using the decomposition method [2]. The main idea of this technique is to look for a solution of Eq. (6) having the series form 1 X xi : ð9Þ x¼ i¼0
The nonlinear operator N(x) can be decomposed as ! 1 1 X X N xi ¼ N ðxÞ ¼ Ai ; i¼0
ð10Þ
i¼0
where Ai are functions which are known as the Adomian polynomials depending on x0 ; x1 ; . . . ; given by the formulas " !# 1 X 1 dn i An ¼ N k xi ; n ¼ 0; 1; 2; . . . ð11Þ n! dkn i¼0 k¼0
First few Adomian polynomials are given by A0 ¼ N ðx0 Þ; 0
ð12Þ
A1 ¼ x1 N ðx0 Þ;
ð13Þ
1 A2 ¼ x2 N 0 ðx0 Þ þ x21 N 00 ðx0 Þ: 2
ð14Þ
M.A. Noor / Applied Mathematics and Computation 190 (2007) 553–558
555
It follows from (6), (7), (9), (10) and (11) that x x0 ¼ c ¼ c
f ðcÞ : f 0 ðcÞ
ð15Þ
This enables us to suggest the following iterative method, which is known as Newton method. Algorithm 2.1. For a given x0, compute the approximate solution xn+1 by the iterative scheme xnþ1 ¼ xn
f ðxn Þ ; f 0 ðxn Þ
n ¼ 0; 1; 2; . . . ; f 0 ðxn Þ 6¼ 0:
It is well known that Algorithm 2.1 has 2nd order convergence. From (5), (8) and (10), we have 2
x1 ¼ N ðx0 Þ ¼ ¼
ðx0 cÞ 00 gðx0 Þ f ðcÞ 0 0 f ðcÞ 2f ðcÞ
ð16Þ
ðx0 cÞ2 00 f ðx0 Þ f ðcÞ 0 : f ðcÞ 2f 0 ðcÞ
Again using (12), (15) and (16), we conclude that x c þ x1 ¼ x0 þ x1 ¼ x0 þ A0 ¼ x0 þ N ðx0 Þ ð17Þ
2
¼c
f ðcÞ ðx0 cÞ 00 f ðx0 Þ f ðcÞ 0 : f 0 ðcÞ f ðcÞ 2f 0 ðcÞ
Using this relation, we can suggest the following two-step iterative method for solving nonlinear equation (1). Algorithm 2.2. For a given x0, compute the approximate solution xn+1 by the iterative schemes: Predictor-step: y n ¼ xn
f ðxn Þ ; f 0 ðxn Þ
f 0 ðxn Þ 6¼ 0:
Corrector-step: xnþ1 ¼ xn
f ðxn Þ ðy n xn Þ2 00 f ðy Þ f ðxn Þ 0 n ; f 0 ðxn Þ f ðxn Þ 2f 0 ðxn Þ
n ¼ 0; 1; 2; . . .
Algorithm 2.2 can be written in the following form. Algorithm 2.3. For a given x0, compute the approximate solution xn+1 by the iterative scheme: f ðxn Þ 2 f x n 0 f ðxn Þ f ðxn Þ f ðxn Þ 00 f ðxn Þ ; n ¼ 0; 1; 2; . . . xnþ1 ¼ xn 0 f ðxn Þ 2f 03 ðxn Þ f 0 ðxn Þ We would like to emphasize the fact that Algorithms 2.2 and 2.3 are remarkably different that the methods of Abbasbandy [1] and Chun [3]. If f 00 ðxn Þ ¼ 0, then Algorithm 2.2 reduces to the following iterative method for solving nonlinear equation (1), which is mainly due to Chun [3]. Algorithm 2.4 [3]. For a given x0, compute the approximate solution xn+1 by the iterative schemes: Predictor-step: y n ¼ xn
f ðxn Þ ; f 0 ðxn Þ
f 0 ðxn Þ 6¼ 0:
556
M.A. Noor / Applied Mathematics and Computation 190 (2007) 553–558
Corrector-step: xnþ1 ¼ xn
f ðxn Þ f ðy n Þ ; f 0 ðxn Þ f 0 ðxn Þ
n ¼ 0; 1; 2; . . .
If f 0 ðy n Þ ¼ 0; then Algorithm 2.2 collapses to the following iterative method. Algorithm 2.5. For a given x0, compute the approximate solution xn+1 by the iterative schemes: Predictor-step: y n ¼ xn
f ðxn Þ ; f 0 ðxn Þ
f 0 ðxn Þ 6¼ 0:
Corrector-step: 2
xnþ1 ¼ xn
f ðxn Þ ðy n xn Þ 00 f ðxn Þ; f 0 ðxn Þ 2f 0 ðxn Þ
n ¼ 0; 1; 2; . . .
Algorithm 2.5 can be considered a predictor–corrector method and appears to be a new one in this form. One can rewrite Algorithm 2.5 in the following equivalent form. Algorithm 2.6 [1]. For a given x0, find the approximate solution xn+1 by the iterative scheme: xnþ1 ¼ xn
f ðxn Þ f 2 ðxn Þ 00 f ðxn Þ; f 0 ðxn Þ 2f 03 ðxn Þ
n ¼ 0; 1; 2; . . .
which is known as the Houshoulder iterative method and was derived by Abbasbandy [1] using the Adomian decomposition method. This shows that Algorithms 2.2 and 2.3 include the methods of Chun [3] and Abbasbandy [1] as special cases. From (3) and (8), we have N 0 ðx0 Þ ¼ 1
f 0 ðx0 Þ ðx0 cÞ 00 0 f ðcÞ: f 0 ðcÞ f ðcÞ
ð18Þ
Combining (13), (16) and (18), we have x c þ x 1 þ x 2 ¼ x 0 þ A0 þ A1 2
¼c þ
2
f ðcÞ ðx0 cÞ f 00 ðcÞ f ðx0 Þ ðx0 cÞ f 0 ðx0 Þ 00 f ðx0 Þf 0 ðx0 Þ 2 0 þ f ðcÞ þ 2 0 0 f ðcÞ f ðcÞ f ðcÞ 1 2½f 0 ðcÞ ½f 0 ðcÞ2
ðx0 cÞf ðx0 Þf 00 ðcÞ ½f 0 ðcÞ2
3
þ
ðx0 cÞ ½f 00 ðcÞ 2½f 0 ðcÞ2
2
ð19Þ
:
Using this, we can suggest and analyze the following two-step iterative method for solving nonlinear equation (1). Algorithm 2.7. For a given x0, compute the approximate solution xn+1 by the iterative schemes: Predictor-steps: y n ¼ xn
f ðxn Þ ; f 0 ðxn Þ
f 0 ðxn Þ 6¼ 0:
ð20Þ
Corrector-step: xnþ1 ¼ xn þ
f ðxn Þ ðy n xn Þ2 f 00 ðxn Þ f ðy n Þ ðy n xn Þ2 f 0 ðy n Þ 00 2 þ f ðxn Þ 2 f 0 ðxn Þ f 0 ðxn Þ f 0 ðxn Þ 1 2½f 0 ðxn Þ
f ðy n Þf 0 ðy n Þ 2
½f 00 ðxn Þ
þ
ðy n xn Þf ðy n Þf 00 ðxn Þ ½f 0 ðxn Þ
2
þ
ðy n xn Þ3 ½f 00 ðxn Þ2 2½f 0 ðxn Þ
2
:
ð21Þ
M.A. Noor / Applied Mathematics and Computation 190 (2007) 553–558
557
Algorithm 2.7 is also called the predictor–corrector iterative method for solving nonlinear equations (1). Using the above arguments, one can show that methods developed in [1,3] are very special cases of Algorithm 2.7. Using the techniques and ideas of Chun [3] and Noor and Noor [7], one can study the convergence criteria of these new iterative methods. 3. Numerical examples We present some examples to illustrate the efficiency of the new developed three-step iterative methods in this paper. We compare the Newton method (NM), the method of Abbasbandy (AM), the method of Homeier Table 3.1 Examples and their comparison IT
xn
f(xn)
d
f1 ; x0 ¼ 1 NM AM HM CM NR2
7 5 4 5 6
1.4044916482153412260350868178 1.4044916482153412260350868178 1.4044916482153412260350868178 1.4044916482153412260350868178 1.4044916482153412260350868178
1.04e50 5.81e55 5.4e62 2.0e63 1.2e28
7.33e26 1.39e18 7.92e21 1.31e17 4.8e29
f2 ; x 0 ¼ 2 NM AM HM CM NR2
6 5 5 4 5
0.25753028543986076045536730494 0.25753028543986076045536730494 0.25753028543986076045536730494 0.25753028543986076045536730494 0.25753028543986076045536730494
2.93e55 1.0e63 0 1.0e63 6.0e24
9.1e28 1.45e26 9.33e43 9.46e29 1.5e16
f3 ; x0 ¼ 1:7 NM AM HM CM NR2
5 4 4 4 5
0.73908513321516064165537208767 0.73908513321516064165537208767 0.73908513321516064165537208767 0.73908513321516064165537208767 0.73908513321516064165537208767
2.03e32 7.14e47 5.02e59 0 1.4e15
2.34e16 8.6e16 9.64e20 1.86e53 8.4e16
f4 ; x0 ¼ 3:5 NM AM HM CM NR2
8 5 5 5 6
2 2 2 2 2
2.06e42 0 0 0 1.03e23
8.28e22 4.3e22 1.46e24 2.74e24 3.4e24
f5 ; x0 ¼ 1:5 NM AM HM CM NR2
7 5 4 5 7
2.1544346900318837217592935665 2.1544346900318837217592935665 2.1544346900318837217592935665 2.1544346900318837217592935665 2.1544346900318837217592935665
2.06e54 5.0e63 5.0e63 5.0e63 3.1e42
5.64e28 1.18e25 9.8e23 1.57e22 1.2e43
f6 ; x0 ¼ 2 NM AM HM CM NR2
9 6 6 6 6
1.2076478271309189270094167584 1.2076478271309189270094167584 1.2076478271309189270094167584 1.2076478271309189270094167584 1.2076478271309189270094167584
2.27e40 4.0e63 4.0e63 4.0e63 3.0e31
2.73e21 4.35e45 2.57e32 2.15e36 1.5e32
f7 ; x0 ¼ 3:5 NM AM HM CM NR2
13 7 8 8 8
1.52e47 4.33e48 2.0e62 2.0e62 2.1e29
4.2e25 2.25e17 2.43e33 2.12e23 1.6e28
3 3 3 3 3
558
M.A. Noor / Applied Mathematics and Computation 190 (2007) 553–558
(HM), the method of Chun (CM) and Algorithm 2.7 (NR2), introduced in this present paper, see Table 3.1. We used e ¼ 1015 . The following stopping criteria is used for computer programs: ðiÞ jxnþ1 xn j < e;
ðiiÞ jf ðxnþ1 Þj < e:
The examples are the same as in Chun [3]. f1 ðxÞ ¼ sin2 x x2 þ 1; f2 ðxÞ ¼ x2 ex 3x þ 2; f3 ðxÞ ¼ cos x x; f4 ðxÞ ¼ ðx 1Þ3 1; f5 ðxÞ ¼ x3 10; 2
f6 ðxÞ ¼ xex sin2 x þ 3 cos x þ 5; 2 þ7x30
f7 ðxÞ ¼ ex
1:
As for the convergence criteria, it was required that the distance of two consecutive approximations d for the zero was less than 1015. Also displayed are the number of iterations to approximate the zero (IT), the approximate zero x and the value f ðx Þ: 4. Conclusion In this paper, we have suggested a family of two-step iterative methods for solving nonlinear equations by using the decomposition technique. Several examples are given to illustrate the efficiency of Algorithm 2.7. Using the technique and idea of this paper, one can suggest and analyze higher order multi-step iterative methods for solving nonlinear equations as well as system of nonlinear equations. This is the topic of further research. Acknowledgements This research is supported by the Higher Education Commission, Pakistan, through research Grant No: i-28/HEC/HRD/2005/90. The author would like to thank the referee for his/her several useful comments and suggestions. References [1] S. Abbasbandy, Improving Newton–Raphson method for nonlinear equations by modified Adomian decomposition method, Appl. Math. Comput. 145 (2003) 887–893. [2] G. Adomian, Nonlinear Stochastic Systems and Applications to Physics, Kluwer Academic Publishers, Dordrecht, 1989. [3] C. Chun, Iterative methods improving Newton’s method by the decomposition method, Comput. Math. Appl. 50 (2005) 1559–1568. [4] J.H. He, A new iterative method for solving algebraic equations, Appl. Math. Comput. 135 (2005) 81–84. [5] H.H. Homeier, On Newton-type methods with cubic convergence, J. Comput. Appl. Math. 176 (2005) 425–432. [6] M. Aslam Noor, Numerical Analysis and Optimization, Lecture Notes, COMSATS Institute of Information Technology, Islamabad, Pakistan, 2006. [7] M. Aslam Noor, K. Inayat Noor, Three-step iterative methods for nonlinear equations, Appl. Math. Comput. 183 (2006) 322–327. [8] X. Luo, A note on the new iteration for solving algebraic equations, Appl. Math. Comput. 171 (2005) 1177–1183.