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New classes of iterative methods for nonlinear equations
Applied Mathematics and Computation 191 (2007) 128–131 www.elsevier.com/locate/amc
New classes of iterative methods for nonlinear equations Muhammad Aslam Noor Mathematics Department, COMSATS Institute of Information Technology, Islamabad, Pakistan
Abstract In this paper, we suggest and analyze some new iterative methods for solving nonlinear equations f ðxÞ ¼ 0 by using the variational iteration technique. These new methods includes the Householder and its variant forms as special cases. We also give several examples to illustrate the efficiency of these methods. Comparison with other similar methods is also given. These new methods can be considered as an alternative to the Newton method. This technique can be used to suggest a wide class of new iterative methods for solving system of nonlinear equations. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Variational iteration; Iterative method; Convergence; Newton method; Taylor series; Examples
1. Introduction Iterative methods for finding the approximate solutions of the nonlinear equation f ðxÞ ¼ 0 are being developed using several different techniques including Taylor series, quadrature formulas, homotopy and decomposition techniques, see [1–9,11] and the references therein. In this paper, we use the variational iteration technique to suggest and analyze some new iterative methods for solving the nonlinear equations, the origin of which can be traced back to Inokuti et al. [4]. However it is He [2] who realized the potential of this method for solving a wide class of both linear and nonlinear problems arising in various branches of pure and applied sciences. See also Noor and Mohyud-Din [10] and the references therein. Essentially using the idea and technique of He [2], we suggest and analyze a class of iterative methods for solving the nonlinear equations. We show that the new methods include Newton and Householder iterative methods and their variant forms as special cases. Several examples are given to illustrate the efficiency and performance of these new methods and their comparison with other iterative methods. These new methods can be considered as alternative to the Newton method. 2. Iterative methods Consider the nonlinear equation of the type f ðxÞ ¼ 0: E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.02.098
ð1Þ
M.A. Noor / Applied Mathematics and Computation 191 (2007) 128–131
129
For simplicity, we assume that a is a simple root and c is an initial guess sufficiently close to a. For the sake of complete and to give the idea, we consider the approximate solution xn of (1) such that f ðxn Þ 6¼ 0: Let gðxÞ be an arbitrary function and k be a parameter, which is usually called the Lagrange multiplier and can be identified by the optimality condition. Consider the following iterative relation: xnþ1 ¼ xn þ kgðxn Þf ðxn Þ:
ð2Þ
Using the optimality criteria, from (2), we have k¼
1 : g0 ðxn Þf ðxn Þ þ gðxn Þf 0 ðxn Þ
ð3Þ
From (2) and (3), we have xnþ1 ¼ xn
g0 ðx
gðxn Þf ðxn Þ : 0 n Þf ðxn Þ þ gðxn Þf ðxn Þ
ð4Þ
Eq. (4) is the main recurrence relation for the iterative methods. We will use (4) to deduce several iterative methods for solving nonlinear equations and this is the main motivation of this paper. We now consider some special of the auxiliary functions g and f. I. Let gðxn Þ ¼ eaxn . Then g0 ðxn Þ ¼ aeaxn and consequently from (4), we obtain the following iterative method. Algorithm 2.1 [2]. For a given x0, find the approximate solution xnþ1 by the iterative scheme xnþ1 ¼ xn
f ðxn Þ : f 0 ðxn Þ af ðxn Þ
Algorithm 2.1 has been obtained by Noor by using a different technique. See also Noor [5]. For a = 0, Algorithm 2.3 reduces to the well known Newton Method. Algorithm 2.2. For a given x0, find the approximate solution xn by the iterative scheme xnþ1 ¼ xn
f ðxn Þ : f 0 ðxn Þ
Algorithm 2.2 is wella known Newton method and has a quadratic convergence, see [1,3,10] for more details. II. Let gðxn Þ ¼ ef 0 ðxn Þ . Then, from (4), we have the following iterative methods for solving the nonlinear Eq. (1). Algorithm 2.3. For a given x0, find the approximate solution xnþ1 by the iterative scheme 2
xnþ1 ¼ xn
f ðxn Þ½f 0 ðxn Þ 3
½f 0 ðxn Þ af ðxn Þf 00 ðxn Þ
:
If a ¼ 12 f 0 ðxn Þ; then Algorithm 2.3 reduce to the following iterative method. Algorithm 2.4. For a given x0, find the approximate solution xnþ1 by the iterative scheme 2
xnþ1 ¼ xn
2f ðxn Þ½f 0 ðxn Þ
2½f 0 ðxn Þ2 f ðxn Þf 00 ðxn Þ
;
which is well known Halley method and has cubic convergence, see [3,5–9] and the references therein. III. Let gðxn Þ ¼ e linear Eq. (1).
af ðx Þ n
f 0 ðx nÞ
. Then, from (4), we can deduce the following iterative method for solving the non-
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M.A. Noor / Applied Mathematics and Computation 191 (2007) 128–131
Algorithm 2.5. For a given x0, find the approximate solution xnþ1 by the iterative scheme f ðxn Þ
xnþ1 ¼ xn
: h i2 f 0 ðxn Þ af ðxn Þ þ a ff0ðxðxnnÞÞ f 00 ðxn Þ h i2 If we neglect the term ff0ðxðxnnÞÞ in the above algorithm, then Algorithm 2.5 reduces to Algorithm 2.1. In a similar way, we can suggest the following iterative method. Algorithm 2.6. For a given x0, find the approximate solution xnþ1 by the iterative scheme xnþ1 ¼ xn
f ðxn Þf 0 ðxn Þ 2
½f 0 ðxn Þ af ðxn Þf 0 ðxn Þ f ðxn Þf 00 ðxn Þ
:
We would like to pint out that Halley method and its variant forms can be deduced from these Algorithms with suitable and appropriate choice of the parametera. This is main innovative and novel aim of the present technique. In fact, using the Eq. (4), one can obtain a wide class of iterative methods for solving nonlinear equations. In view of this fact, the results proved in this paper may stimulate and motivate future research in this direction. Using essentially the techniques and ideas of Noor and Noor [7–9] and Noor [5], one can study the convergence criteria of Algorithms 2.1, 2.3, 2.5 and 2.6. 3. Numerical results We now present some examples to illustrate the efficiency of the new developed two-step iterative methods, see Table 1. We compare the Newton method (NM), Algorithms 2.1, 2.3, 2.5, 2.6, introduced in this present paper. We use 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 [5–9].
Table 1 Examples and comparison of various iterative schemes f ðxn Þ
xn
NM (Algorithm 2.2) a = 1
Algorithm 2.1 a = 1
Algorithm 2.3 a = 1
Algorithm 2.5 a = 1
Algorithm 2.6 a = 1
f1 f2 f3 f4 f5 f6 f7
1 0.5 1 2.1 1.5 1.5 3.1
6 4 4 4 6 6 6
4 5 5 3 5 6 6
5 5 4 4 6 8 11
5 5 5 3 6 6 6
6 5 5 3 6 5 8
Table 2 Examples and comparison of various iterative schemes f ðxn Þ
xn
NM (Algorithm 2.2) a=1
Algorithm 2.1 a = 0.5
Algorithm 2.3 a = 0.5
Algorithm 2.5 a = 0.5
Algorithm 2.6 a = 0.5
f1 f2 f3 f4 f5 f6 f7
1 0.5 1 2.1 1.5 1.5 3.1
6 4 4 4 6 6 6
5 5 4 4 5 6 6
5 5 4 6 5 8 11
5 5 4 4 5 6 6
6 5 5 5 6 6 8
M.A. Noor / Applied Mathematics and Computation 191 (2007) 128–131
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f1 ðxÞ ¼ sin2 x x2 þ 1; f2 ðxÞ ¼ x2 ex 3x þ 2; f3 ðxÞ ¼ cos x x; f5 ðxÞ ¼ x3 10; f7 ðxÞ ¼ ex
2 þ7x30
3
f4 ðxÞ ¼ ðx 1Þ 1; 2 f6 ðxÞ ¼ xex sin2 x þ 3 cos x þ 5;
1:
From Tables 1 and 2, we see that our methods are compatible with the method of Newton Method. In view of this fact, these new methods can be viewed as a significant improvement of the previously known methods and can be considered as alternative method to that of Newton and its variant forms. Acknowledgement This research is supported by the Higher Education Commission, Pakistan, through research Grant No: I-28/HEC/HRD/2005/90. References [1] Richard L. Burden, J. Douglas Faires, Numerical Analysis, PWS publishing company, Bostan, USA, 2001. [2] J.H. He, Variational iteration method-some recent results and new interpretations, J. Comput. Appl. Math. (2006), doi:10.1016/ j.cam.2006.07.009. [3] A.S. Householder, The Numerical Treatment of a Single Nonlinear Equation, McGraw-Hill, New York, 1970. [4] M. Inokuti, H. Sekine, T. Mura, General use of the Lagrange multiplier in nonlinear mathematical physics, in: S. Nemat-Nasser (Ed.), Variational Methods in the Mechanics of Solids, Pergamon Press, New York, 1978, pp. 156–162. [5] M. Aslam Noor, Numerical Analysis and Optimization, Lecture Notes, Mathematics Department, COMSATS Institute of Information Technology, Islamabad, Pakistan, 2006. [6] Khalida Inayat Noor, M. Aslam Noor, Predictor-corrector Halley method for nonlinear equations, Appl. Math. Comput. (2006), doi:10.1016/j.amc.2006.11.023. [7] K. Inayat Noor, M. Aslam Noor, Iterative methods with fourth-order convergence for nonlinear equations, Appl. Math. Comput. (2007), doi:10.1016/j.amc.2006.11.080. [8] K. Inayat Noor, M. Aslam Noor, Shaher Momani, Modified Householder iterative methods for nonlinear equations, Appl. Math. Comput. (2007), doi:10.1016/j.amc,2007.02.036. [9] M. Aslam Noor, K. Inayat Noor, Iterative schemes for solving nonlinear equations, Appl. Math. Comput. 183 (2006) 774–779. [10] M. Aslam Noor, Syed T. Mohyud-Din, Variational iteration techniques for solving higher-order boundary value problems, Appl. Math. Comput. (2006), doi:10.1016/j.amc.2006.12.071. [11] J.F. Traub, Iterative Methods for Solution of Equations, Prentice-Hall, Englewood Cliffs, NJ, 1964.
New classes of iterative methods for nonlinear equations Muhammad Aslam Noor Mathematics Department, COMSATS Institute of Information Technology, Islamabad, Pakistan
Abstract In this paper, we suggest and analyze some new iterative methods for solving nonlinear equations f ðxÞ ¼ 0 by using the variational iteration technique. These new methods includes the Householder and its variant forms as special cases. We also give several examples to illustrate the efficiency of these methods. Comparison with other similar methods is also given. These new methods can be considered as an alternative to the Newton method. This technique can be used to suggest a wide class of new iterative methods for solving system of nonlinear equations. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Variational iteration; Iterative method; Convergence; Newton method; Taylor series; Examples
1. Introduction Iterative methods for finding the approximate solutions of the nonlinear equation f ðxÞ ¼ 0 are being developed using several different techniques including Taylor series, quadrature formulas, homotopy and decomposition techniques, see [1–9,11] and the references therein. In this paper, we use the variational iteration technique to suggest and analyze some new iterative methods for solving the nonlinear equations, the origin of which can be traced back to Inokuti et al. [4]. However it is He [2] who realized the potential of this method for solving a wide class of both linear and nonlinear problems arising in various branches of pure and applied sciences. See also Noor and Mohyud-Din [10] and the references therein. Essentially using the idea and technique of He [2], we suggest and analyze a class of iterative methods for solving the nonlinear equations. We show that the new methods include Newton and Householder iterative methods and their variant forms as special cases. Several examples are given to illustrate the efficiency and performance of these new methods and their comparison with other iterative methods. These new methods can be considered as alternative to the Newton method. 2. Iterative methods Consider the nonlinear equation of the type f ðxÞ ¼ 0: E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.02.098
ð1Þ
M.A. Noor / Applied Mathematics and Computation 191 (2007) 128–131
129
For simplicity, we assume that a is a simple root and c is an initial guess sufficiently close to a. For the sake of complete and to give the idea, we consider the approximate solution xn of (1) such that f ðxn Þ 6¼ 0: Let gðxÞ be an arbitrary function and k be a parameter, which is usually called the Lagrange multiplier and can be identified by the optimality condition. Consider the following iterative relation: xnþ1 ¼ xn þ kgðxn Þf ðxn Þ:
ð2Þ
Using the optimality criteria, from (2), we have k¼
1 : g0 ðxn Þf ðxn Þ þ gðxn Þf 0 ðxn Þ
ð3Þ
From (2) and (3), we have xnþ1 ¼ xn
g0 ðx
gðxn Þf ðxn Þ : 0 n Þf ðxn Þ þ gðxn Þf ðxn Þ
ð4Þ
Eq. (4) is the main recurrence relation for the iterative methods. We will use (4) to deduce several iterative methods for solving nonlinear equations and this is the main motivation of this paper. We now consider some special of the auxiliary functions g and f. I. Let gðxn Þ ¼ eaxn . Then g0 ðxn Þ ¼ aeaxn and consequently from (4), we obtain the following iterative method. Algorithm 2.1 [2]. For a given x0, find the approximate solution xnþ1 by the iterative scheme xnþ1 ¼ xn
f ðxn Þ : f 0 ðxn Þ af ðxn Þ
Algorithm 2.1 has been obtained by Noor by using a different technique. See also Noor [5]. For a = 0, Algorithm 2.3 reduces to the well known Newton Method. Algorithm 2.2. For a given x0, find the approximate solution xn by the iterative scheme xnþ1 ¼ xn
f ðxn Þ : f 0 ðxn Þ
Algorithm 2.2 is wella known Newton method and has a quadratic convergence, see [1,3,10] for more details. II. Let gðxn Þ ¼ ef 0 ðxn Þ . Then, from (4), we have the following iterative methods for solving the nonlinear Eq. (1). Algorithm 2.3. For a given x0, find the approximate solution xnþ1 by the iterative scheme 2
xnþ1 ¼ xn
f ðxn Þ½f 0 ðxn Þ 3
½f 0 ðxn Þ af ðxn Þf 00 ðxn Þ
:
If a ¼ 12 f 0 ðxn Þ; then Algorithm 2.3 reduce to the following iterative method. Algorithm 2.4. For a given x0, find the approximate solution xnþ1 by the iterative scheme 2
xnþ1 ¼ xn
2f ðxn Þ½f 0 ðxn Þ
2½f 0 ðxn Þ2 f ðxn Þf 00 ðxn Þ
;
which is well known Halley method and has cubic convergence, see [3,5–9] and the references therein. III. Let gðxn Þ ¼ e linear Eq. (1).
af ðx Þ n
f 0 ðx nÞ
. Then, from (4), we can deduce the following iterative method for solving the non-
130
M.A. Noor / Applied Mathematics and Computation 191 (2007) 128–131
Algorithm 2.5. For a given x0, find the approximate solution xnþ1 by the iterative scheme f ðxn Þ
xnþ1 ¼ xn
: h i2 f 0 ðxn Þ af ðxn Þ þ a ff0ðxðxnnÞÞ f 00 ðxn Þ h i2 If we neglect the term ff0ðxðxnnÞÞ in the above algorithm, then Algorithm 2.5 reduces to Algorithm 2.1. In a similar way, we can suggest the following iterative method. Algorithm 2.6. For a given x0, find the approximate solution xnþ1 by the iterative scheme xnþ1 ¼ xn
f ðxn Þf 0 ðxn Þ 2
½f 0 ðxn Þ af ðxn Þf 0 ðxn Þ f ðxn Þf 00 ðxn Þ
:
We would like to pint out that Halley method and its variant forms can be deduced from these Algorithms with suitable and appropriate choice of the parametera. This is main innovative and novel aim of the present technique. In fact, using the Eq. (4), one can obtain a wide class of iterative methods for solving nonlinear equations. In view of this fact, the results proved in this paper may stimulate and motivate future research in this direction. Using essentially the techniques and ideas of Noor and Noor [7–9] and Noor [5], one can study the convergence criteria of Algorithms 2.1, 2.3, 2.5 and 2.6. 3. Numerical results We now present some examples to illustrate the efficiency of the new developed two-step iterative methods, see Table 1. We compare the Newton method (NM), Algorithms 2.1, 2.3, 2.5, 2.6, introduced in this present paper. We use 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 [5–9].
Table 1 Examples and comparison of various iterative schemes f ðxn Þ
xn
NM (Algorithm 2.2) a = 1
Algorithm 2.1 a = 1
Algorithm 2.3 a = 1
Algorithm 2.5 a = 1
Algorithm 2.6 a = 1
f1 f2 f3 f4 f5 f6 f7
1 0.5 1 2.1 1.5 1.5 3.1
6 4 4 4 6 6 6
4 5 5 3 5 6 6
5 5 4 4 6 8 11
5 5 5 3 6 6 6
6 5 5 3 6 5 8
Table 2 Examples and comparison of various iterative schemes f ðxn Þ
xn
NM (Algorithm 2.2) a=1
Algorithm 2.1 a = 0.5
Algorithm 2.3 a = 0.5
Algorithm 2.5 a = 0.5
Algorithm 2.6 a = 0.5
f1 f2 f3 f4 f5 f6 f7
1 0.5 1 2.1 1.5 1.5 3.1
6 4 4 4 6 6 6
5 5 4 4 5 6 6
5 5 4 6 5 8 11
5 5 4 4 5 6 6
6 5 5 5 6 6 8
M.A. Noor / Applied Mathematics and Computation 191 (2007) 128–131
131
f1 ðxÞ ¼ sin2 x x2 þ 1; f2 ðxÞ ¼ x2 ex 3x þ 2; f3 ðxÞ ¼ cos x x; f5 ðxÞ ¼ x3 10; f7 ðxÞ ¼ ex
2 þ7x30
3
f4 ðxÞ ¼ ðx 1Þ 1; 2 f6 ðxÞ ¼ xex sin2 x þ 3 cos x þ 5;
1:
From Tables 1 and 2, we see that our methods are compatible with the method of Newton Method. In view of this fact, these new methods can be viewed as a significant improvement of the previously known methods and can be considered as alternative method to that of Newton and its variant forms. Acknowledgement This research is supported by the Higher Education Commission, Pakistan, through research Grant No: I-28/HEC/HRD/2005/90. References [1] Richard L. Burden, J. Douglas Faires, Numerical Analysis, PWS publishing company, Bostan, USA, 2001. [2] J.H. He, Variational iteration method-some recent results and new interpretations, J. Comput. Appl. Math. (2006), doi:10.1016/ j.cam.2006.07.009. [3] A.S. Householder, The Numerical Treatment of a Single Nonlinear Equation, McGraw-Hill, New York, 1970. [4] M. Inokuti, H. Sekine, T. Mura, General use of the Lagrange multiplier in nonlinear mathematical physics, in: S. Nemat-Nasser (Ed.), Variational Methods in the Mechanics of Solids, Pergamon Press, New York, 1978, pp. 156–162. [5] M. Aslam Noor, Numerical Analysis and Optimization, Lecture Notes, Mathematics Department, COMSATS Institute of Information Technology, Islamabad, Pakistan, 2006. [6] Khalida Inayat Noor, M. Aslam Noor, Predictor-corrector Halley method for nonlinear equations, Appl. Math. Comput. (2006), doi:10.1016/j.amc.2006.11.023. [7] K. Inayat Noor, M. Aslam Noor, Iterative methods with fourth-order convergence for nonlinear equations, Appl. Math. Comput. (2007), doi:10.1016/j.amc.2006.11.080. [8] K. Inayat Noor, M. Aslam Noor, Shaher Momani, Modified Householder iterative methods for nonlinear equations, Appl. Math. Comput. (2007), doi:10.1016/j.amc,2007.02.036. [9] M. Aslam Noor, K. Inayat Noor, Iterative schemes for solving nonlinear equations, Appl. Math. Comput. 183 (2006) 774–779. [10] M. Aslam Noor, Syed T. Mohyud-Din, Variational iteration techniques for solving higher-order boundary value problems, Appl. Math. Comput. (2006), doi:10.1016/j.amc.2006.12.071. [11] J.F. Traub, Iterative Methods for Solution of Equations, Prentice-Hall, Englewood Cliffs, NJ, 1964.