Applied Mathematics and Computation 219 (2012) 3565–3574

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate/amc

New iterative methods for solving nonlinear equationby using homotopy perturbation method

Muhammad Aslam Noor a,b, Waseem Asghar Khan a,⇑a Department of Mathematics, COMSATS Institute of Information Technology, Park Road, Chak Shahzad, Islamabad, Pakistanb Mathematics Department, College of Science, King Saud University, Reyal, Saudi Arabia

a r t i c l e i n f o a b s t r a c t

Keywords:Nonlinear equationsSeries solutionModified homotopy perturbation methodConvergence criteriaNumerical examples

0096-3003/$ - see front matter � 2012 Elsevier Inchttp://dx.doi.org/10.1016/j.amc.2012.09.040

⇑ Corresponding author.E-mail addresses: nooraslam@hotmail.com (M.A.

In this paper, we suggest and analyze a new class of iterative methods for solving nonlinearequations by using the homotopy perturbation method. Convergence of their method isalso considered. Here we also discuss the efficiency index and computational order of con-vergence of new methods. Several numerical examples are given to illustrate the efficiencyand performance of these new methods. These new iterative methods may be viewed as anextension and generalization of the existing methods for solving nonlinear equations.

� 2012 Elsevier Inc. All rights reserved.

1. Introduction

In recent years, several iterative type methods have been developed by using the Taylor series, decomposition, quadratureformulae and perturbation technique for solving nonlinear equations see [1–19] and the references therein. Nonlinear equa-tions are difficult to solve, especially analytically. Perturbation techniques [1,2,15] are widely applied in science and engi-neering, and do great contribution to help us understand many nonlinear phenomena. However, it is well known thatperturbation methods are strongly dependent upon small/large physical parameters, and therefore are valid in principle onlyfor weakly nonlinear problems. The so-called non-perturbation techniques, Adomian’s decomposition method [4,5,10],perturbation methods [7,8,17] and so on, are formally independent of small/large physical parameters.

But, all of these traditional non-perturbation methods cannot ensure the convergence of solution series: they are in factonly valid for weakly nonlinear problems, too.

In this paper, we introduce a modified homotopy perturbation method by combining the homotopy analysis method andhomotopy perturbation technique. This new homotopy perturbation method is quite flexible and allows us to choose theauxiliary operator and the auxiliary parameter arbitrarily. This is one of the advantages of this method. This new modifiedhomotopy perturbation technique may be a starting point for a wide range of further applications for solving linear and non-linear problems arising in various branches of pure and applied sciences. In recent years in order to solve f(x) = 0, manyresearchers have used various numerical methods. He [6] has suggested an iterative method for solving the nonlinear equa-tions by rewriting the given nonlinear equation as a system of coupled equations. This technique has been used by Chun [4],Noor [10,15], Noor and Noor [12] and Noor et al. [16] to suggest some higher order convergent iterative methods for solvingnonlinear equations.

It has been shown that these new iterative methods include a wide class of known and new iterative methods as specialcases. Here we also discuss the efficiency index and computational order of convergence of new methods. Several examplesare given to illustrate the efficiency and performance of these new methods. We also compare these new methods with otherrecent methods of the same convergence order.

. All rights reserved.

Noor), waseemasg@gmail.com (W.A. Khan).

3566 M.A. Noor, W.A. Khan / Applied Mathematics and Computation 219 (2012) 3565–3574

2. Iterative methods

We consider iterative methods to find a simple root of a nonlinear equation

f ðxÞ ¼ 0: x 2 R: ð1Þ

We assume that a is simple root of the Eq. (1) and c is an initial guess sufficiently close to a. Using the technique and idea ofHe [6], we can rewrite the nonlinear Eq. (1) as a coupled system using the Taylor’s series:

f ðcÞ þ ðx� cÞf 0ðcÞ þ ðx� cÞ2

2!f 00ðcÞ þ gðxÞ ¼ 0; ð2Þ

gðxÞ ¼ f ðxÞ � f ðcÞ � ðx� cÞ1!

f 0ðcÞ � ðx� cÞ2

2!f 00ðcÞ; ð3Þ

where c is the initial approximation for a zero of Eq. (1).We can rewrite Eq. (3) in the following form

x ¼ c� f ðcÞf 0ðcÞ �

ðx� cÞ2

2!

f 00ðcÞf 0ðcÞ �

gðxÞf 0ðcÞ : ð4Þ

From relation Eq. (3), it is clear that

f ðx0Þ ¼ gðx0Þ: ð5Þ

We remark that Eq. (5) plays a very important role in the derivation of the iterative method [4]. It is worth mentioning thatHe [6] and Luo [9] have considered a very special case f(x0) = 0. In this case, from Eq. (5), we note that g(x0) = 0. We also rec-tify this aspect in this paper. Also see Noor [10] for a different approach.

We rewrite Eq. (4) in the following equivalent useful form as:

x ¼ c þ �hNðxÞ; ð6Þ

where

c ¼ c� f ðcÞf 0ðcÞ ; ð7Þ

and

NðxÞ ¼ � ðx� cÞ2

2!

f 00ðcÞf 0ðcÞ �

gðxÞf 0ðcÞ : ð8Þ

Here N(x) is a nonlinear operator. We use the homotopy perturbation technique to develop a class of new iterative methodsfor solving nonlinear equations.

Following the homotopy perturbation technique [2], we define the modified homotopy H(v, p, T) : (R � [0, 1]) � R ? Ras:

Hðv ;p; TÞ ¼ ð1� pÞ½LðvÞ � Lðx0Þ� þ p½LðvÞ � c � �hNðvÞ� � �hp2ð1� pÞT ¼ 0; ð9Þ

or

Hðv ;p; TÞ ¼ LðvÞ � Lðx0Þ þ pLðx0Þ þ p½�c � �hNðvÞ� � �hp2ð1� pÞT ¼ 0; ð10Þ

where p 2 [0, 1] is an embedding parameter, �h – 0 an auxiliary parameter, x0 an initial approximation and T an arbitrary aux-iliary operator. We would like to emphasize that one has great freedom to select the initial guess, the auxiliary parameter �hand the auxiliary operator T.

Clearly from Eqs. (9) and (10), we have

Hðv ; 0; TÞ ¼ LðvÞ � Lðx0Þ ¼ v � c ¼ 0; ð11Þ

and

Hðv ; 1; TÞ ¼ LðvÞ � c � �hNðxÞ ¼ v � c � �hNðxÞ ¼ 0: ð12Þ

The embedding parameter p monotonically increases from zero to unity as the trivial problem H(x; 0, T) = x � c = 0, is con-tinuously deformed to the original problem H(v, 1, T) = v � c��hN(v) = 0. The changing process of p from zero to unity is calleddeformation. L(v) � L(x0) and L(v) � c��hN(x) are homotopic. The basic assumption is that the solution of Eqs. (9) and (10) canbe expressed as a power series in p:

v ¼ x0 þXþ1k¼1

pkxk: ð13Þ

M.A. Noor, W.A. Khan / Applied Mathematics and Computation 219 (2012) 3565–3574 3567

The approximate solution of Eq. (6) can be readily obtained:

x ¼ limp!1

v ¼ x0 þXþ1k¼1

xk: ð14Þ

For the application of homotopy perturbation method to Eq. (1), we can write Eq. (10) as follows by expanding N(x) into aTaylor series around x0.

v � c � p�h Nðx0Þ þ ðv � x0ÞN0ðx0Þ þ ðv � x0Þ2N00ðx0Þ

2!þ :::

� �� �hp2ð1� pÞT ¼ 0: ð15Þ

Substitution of Eq. (13) into Eq. (15), we have

x0 þ x1p1 þ x2p2 þ :::� c � p�h Nðx0Þ þ ðx1p1 þ x2p2 þ :::ÞN0ðx0Þ þ ðx1p1 þ x2p2 þ :::Þ2 N00ðx0Þ2!

þ :::� �

� �hp2ð1� pÞT ¼ 0:

ð16Þ

By equating the coefficients of the identical power of p, we have

p0 : x0 � c ¼ 0; ð17Þ

p1 : x1 ¼ �hNðx0Þ; ð18Þ

p2 : x2 ¼ �hx1N0ðx0Þ þ �hT; ð19Þ

p3 : x3 ¼ �h x2N0ðx0Þ þ x21

N00ðx0Þ2!

� �� �hT: ð20Þ

From Eqs. (7) and (17), we have

x0 � c ¼ 0; ð21Þ

x1 ¼ �hNðx0Þ; ð22Þ

x2 ¼ �hx1N0ðx0Þ þ �hT; ð23Þ

x3 ¼ �h x2N0ðx0Þ þ x21

N00ðx0Þ2!

� �� �hT: ð24Þ

From Eqs. (3), (5), and (8), we have

Nðx0Þ ¼ �ðx0 � cÞ2

2!

f 00ðcÞf 0ðcÞ �

gðx0Þf 0ðcÞ ¼ �

ðx0 � cÞ2

2!

f 00ðcÞf 0ðcÞ �

f ðx0Þf 0ðcÞ ; ð25Þ

N0ðx0Þ ¼ 1� f 0ðx0Þf 0ðcÞ ; ð26Þ

N00ðx0Þ ¼ 1� f 00ðx0Þf 0ðcÞ : ð27Þ

Note that x is approximated by

x ¼ limn!1

Xn ¼ x0 þ x1 þ x2 þ :::þ xn: ð28Þ

To find the auxiliary operator, we may take either x2 = 0 or x3 = 0. We remark that for x2 = 0, we obtain the same iterativemethods for solving nonlinear equation as derived in Noor [10]. We consider the case x2 = 0 here. In this case, we have

x1 ¼ �hNðx0Þ;

x3 ¼�h2

x21N00ðx0Þ � �hT ¼ �h

2x2

1N00ðx0Þ þ �hx1N0ðx0Þ:

For n = 0,

x � x0 ¼ c ¼ c� f ðcÞf 0ðcÞ :

This formulation allows us the following iterative method.

3568 M.A. Noor, W.A. Khan / Applied Mathematics and Computation 219 (2012) 3565–3574

Algorithm 2.1. For a given x0, compute approximates solution xn+1 by the iterative scheme.

xnþ1 ¼ xn �f ðxnÞf 0ðxnÞ

;

which is the well-known Newton method and has quadratic convergence. Chun [4] obtained Algorithm 2.1 using theAdomian decomposition technique.

For n = 1

x � x0 þ x1 ¼ c þ �hNðx0Þ ¼ c� f ðcÞf 0ðcÞ � �h

ðx0 � cÞ2

2!

f 00ðcÞf 0ðcÞ þ

f ðx0Þf 0ðcÞ

!:

This formulation allows us the following new iterative method.

Algorithm 2.2. For a given x0, compute approximates solution xn+1 by the iterative schemes.

yn ¼ xn �f ðxnÞf 0ðxnÞ

;

xnþ1 ¼ xn �f ðxnÞf 0ðxnÞ

� �hðyn � xnÞ2

2!

f 00ðxnÞf 0ðxnÞ

þ f ðynÞf 0ðxnÞ

!:

For �h = 1/2, we get the new well-known iterative Newton method.For n = 3

x � x0 þ x1 þ x3 ¼ c þ �hNðx0Þ þ �hx1N0ðx0Þ þ�h2

x21N00ðx0Þ

¼ c� f ðcÞf 0ðcÞ � �h

ðx0 � cÞ2

2!

f 00ðcÞf 0ðcÞ � �h

f ðx0Þf 0ðcÞ � �h2 ðx0 � cÞ2

2!

f 00ðcÞf 0ðcÞ þ

f ðx0Þf 0ðcÞ

!1� f 0ðx0Þ

f 0ðcÞ

� �� �h3

2ðx0 � cÞ2

2!

f 00ðcÞf 0ðcÞ þ

f ðx0Þf 0ðcÞ

!2

� f 00ðx0Þf 0ðcÞ :

This formulation allows us the following new iterative method.

Algorithm 2.3. For a given x0, compute approximates solution xn+1 by the iterative schemes.

yn ¼ xn �f ðxnÞf 0ðxnÞ

;

xnþ1 ¼ xn �f ðxnÞf 0ðxnÞ

� �hðyn � xnÞ2

2!

f 00ðxnÞf 0ðxnÞ

� �hf ðynÞf 0ðxnÞ

� �h2 ðyn � xnÞ2

2!

f 00ðxnÞf 0ðxnÞ

þ f ðynÞf 0ðxnÞ

!1� f 0ðynÞ

f 0ðxnÞ

� �

� �h3

2ðyn � xnÞ2

2!

f 00ðxnÞf 0ðxnÞ

þ f ðynÞf 0ðxnÞ

!2f 00ðynÞf 0ðxnÞ

:

We would like to mention that for �h = 1, Algorithm 2.3 reduces to the following.

Algorithm 2.4. For a given x0, compute approximates solution xn+1 by the iterative schemes.

yn ¼ xn �f ðxnÞf 0ðxnÞ

;

xnþ1 ¼ xn �f ðxnÞf 0ðxnÞ

� ðyn � xnÞ2

2f 00ðxnÞf 0ðxnÞ

� f ðynÞf 0ðxnÞ

� ðyn � xnÞ2

2!

f 00ðxnÞf 0ðxnÞ

þ f ðynÞf 0ðxnÞ

!1� f 0ðynÞ

f 0ðxnÞ

� �� ðyn � xnÞ2

2!

f 00ðxnÞf 0ðxnÞ

þ f ðynÞf 0ðxnÞ

!2

� f 00ðynÞf 0ðxnÞ

:

For f 00ðynÞ ¼ 0; Algorithm 2.4 reduces to the following new third-order iterative method for solving nonlinear Eq. (1).

Algorithm 2.5. For a given x0, compute approximates solution xn+1 by the iterative schemes.

yn ¼ xn �f ðxnÞf 0ðxnÞ

;

M.A. Noor, W.A. Khan / Applied Mathematics and Computation 219 (2012) 3565–3574 3569

xnþ1 ¼ xn �f ðxnÞf 0ðxnÞ

� ðyn � xnÞ2

2!

f 00ðxnÞf 0ðxnÞ

� f ðynÞf 0ðxnÞ

� ðyn � xnÞ2

2!

f 00ðxnÞf 0ðxnÞ

þ f ðynÞf 0ðxnÞ

!1� f 0ðynÞ

f 0ðxnÞ

� �:

For f 00ðxnÞ ¼ 0; Algorithm 2.4 reduces the following iterative method for solving nonlinear Eq. (1).

Algorithm 2.6. For a given x0, calculate approximates solution xn+1 by the iterative schemes.

yn ¼ xn �f ðxnÞf 0ðxnÞ

;

xnþ1 ¼ xn �f ðxnÞf 0ðxnÞ

� f ðynÞf 0ðxnÞ

þ � f ðynÞf 0ðxnÞ

� �1� f 0ðynÞ

f 0ðxnÞ

� �� 1

2f ðynÞf 0ðxnÞ

� �2 f 00ðynÞf 0ðxnÞ

:

Algorithm 2.6 is exactly the well-known iterative Newton method, which is mainly due to Noor [15].In order to implement this method, one has to find the second derivative of this function, which may create some

problems. To overcome this drawback, several authors have developed involving only the first derivative. In order toovercome drawback, one usually uses the finite difference approximation of the second derivative. This idea is veryimportant and plays a significant part in developing some iterative methods free from second derivatives. To be moreprecise, we consider

f 00ðynÞ ¼f 0ðynÞ � f 0ðxnÞ

yn � xn: ð29Þ

Using Eq. (29) in Algorithm 2.6, and reduced the following new iterative method for solving nonlinear Eq. (1).

Algorithm 2.7. For a given x0, compute approximates solution xn+1 by the iterative schemes.

yn ¼ xn �f ðxnÞf 0ðxnÞ

; ð30Þ

xnþ1 ¼ xn �f ðxnÞf 0ðxnÞ

� 2f ðynÞf 0ðxnÞ

þ f ðynÞf 0ðynÞf 02 ðxnÞ

þ f 0ðynÞ � f 0ðxnÞ2f ðxnÞ

f ðynÞf 0ðxnÞ

� �2

: ð31Þ

This shows that Algorithm 2.3 is quite general and includes several known and new iterative methods for solving nonlinearequations.

It should be mentioned that the methods defined by Algorithm 2.2 and Algorithm 2.5 are of third-order requiring twofunctional evaluation, one of its first derivative and one of its second derivative evaluations per iteration. If we consider thedefinition of efficiency index [18] as p1/m, where p is the order of the method and m is the number of function evaluations periteration required by method, then above methods have efficiency index to 31/4 = 1.3161. Similarly, Algorithm 2.6 andAlgorithm 2.7 are of fourth-order. The efficiency index of Algorithm 2.6 is 41/5 = 1.3195 and Algorithm 2.7 are41/4 = 1.4143. The efficiency index of Algorithm 2.7 is same as of Newton’s method.

Essentially, using the technique of Noor [10], Chun [4] and Noor and Noor [13,14], we consider the convergence criteria ofAlgorithm 2.4 and this is the main motivation of our next result.

3. Convergence criteria

Now we consider the convergence criteria of Algorithm 2.7. In similar way, we can discuss the convergence of otherAlgorithms.

Theorem 1. Let a 2 D be a simple zero of sufficiently differentiable function f:D � R ? R for an open interval D. And x0 isinitial choice, then Algorithm 2.7 has fourth-order convergences.

Proof. If a is the root and en be the error at nth iteration, than en = xn - a, using Taylor’s expansion, we have

f ðxnÞ ¼ f 0ðxnÞen þ12!

f 00ðxnÞe2n þ

13!

f 000ðxnÞe3n þ

14!

f ðivÞðxnÞe4n þ

15!

f ðvÞðxnÞe5n þ

16!

f ðviÞðxnÞe6n þ Oðe7

nÞ;

f ðxnÞ ¼ f 0ðaÞ en þ c2e2n þ c3e3

n þ c4e4n þ c5e5

n þ Oðe6nÞ

� �; ð32Þ

f 0ðxnÞ ¼ f 0ðaÞ 1þ 2c2en þ 3c3e2n þ 4c4e3

n þ 5c5e4n þ 6c6e5

n þ Oðe6nÞ

� �; ð33Þ

3570 M.A. Noor, W.A. Khan / Applied Mathematics and Computation 219 (2012) 3565–3574

where

ck ¼1k!

f ðkÞðaÞf 0ðaÞ ; k ¼ 2;3; . . . ; let en ¼ xn � a; and:

From Eqs. (32) and (33), we have

f ðxnÞf 0ðxnÞ

¼ en � c2e2n � ð2c3 � 2c2

2Þe3n � ð3c4 � 7c2c3 þ 4c3

2Þe4n þ ð�6c2

3 þ 20c3c22 � 10c2c4 þ 4c5 � 8c4

2Þe5n þ Oðe6

nÞ: ð34Þ

From Eqs. (30) and (34), we have

yn ¼ aþ c2e2n þ ð2c3 � 2c2

2Þe3n þ ð3c4 � 7c2c3 þ 4c3

2Þe4n þ Oðe5

nÞ; ð35Þ

f ðynÞ ¼ f 0ðaÞ c2e2n þ 2ðc3 � c2

2Þe3n þ ð3c4 � 7c2c3 þ 5c3

2Þe4n þ Oðe5

n� �

ð36Þ

and,

f 0ðynÞ ¼ f 0ðaÞ 1þ c22e2

n þ 4ðc2c3 � c32Þe3

n þ ð8c42 þ 6c2c4 � 11c2

2c3Þe4n þ Oðe5

n� �

: ð37Þ

Using Eqs. (32)–(37) in Algorithm 2.7, we have

xnþ1 ¼ aþ c32e4

n þ Oðe5nÞ:

Thus, we have

enþ1 ¼ c32e4

n þ Oðe5nÞ;

which shows that Algorithm 2.7 has fourth-order convergence.

4. Numerical examples

We present some examples to illustrate the efficiency of the new developed two-step iterative methods (Tables 1–7). Wecompare the Newton Method (NM), methods of Chun (CM1) and (CM2) [4], method of Noor (NR1) [15], Algorithm 2.2 (NW1),

Table 1Approximate solution of Example 1.

Methods ITnx ( )nf x δ COC

NM 6 3.98235e-43 2.21790e-22 2

CM1 4 -1.40000e-58 9.34581e-25 3.98

CM2 5 0 1.81711e-39 3

NR1 4 0 1.36796e-28 4

NW1 4 -1.52137e-53 1.63687e-18 3

NW2 4 -1.40000e-58 1.01789e-25 3

NW3 4 0 3.03653e-28 4

Table 2Approximate solution of Example 2.

Table 3Approximate solution of Example.3.

NW2 4 0.2575302854398607604553673 2.00000e-59 4.55706e-20 3

NW3 4 0.2575302854398607604553673

0 1.19088e-30 3.99

M.A. Noor, W.A. Khan / Applied Mathematics and Computation 219 (2012) 3565–3574 3571

Algorithm 2.5 (NW2) and Algorithm 2.7 (NW3) introduced in this present paper. All computations have been done by usingthe Maple 11 package with 25 digit floating point arithmetic. We accept an approximate solution rather than the exact root,depending on the precision (e) of the computer. We use the following stopping criteria for computer programs:

Table 4Approximate solution of Example.4.

Methods ITnx ( )nf x δ COC

NM 5 0.7390851332151606416553121 -2.03197e-32 2.34491e-16 1.99

CM1 4 0.7390851332151606416553121 -1.00000e-60 1.86747e-53 4

CM2 4 0.7390851332151606416553121 1.00000e-60 1.74996e-24 2.99

NR1 4 0.7390851332151606416553121 -1.00000e-60 2.34005e-55 4

NW1 4 0.7390851332151606416553121 -2.27853e-52 1.18270e-17 2.99

NW2 4 0.7390851332151606416553121 -4.04369e-53 6.64633e-18 3

NW3 4 0.7390851332151606416553121

1.00000e-60 1.57697e-54 4

Table 5Approximate solution of Example 5.

Methods IT

nx ( )nf x δ COC

NM 7 2 5.03100e-56 1.29484e-28 2

CM1 4 2 0 6.95860e-20 3.98

CM2 5 2 0 1.50483e-28 3

NR1 4 2 0 8.03070e-21 3.98

NW1 5 2 0 5.17377e-40 3

NW2 5 2 0 1.90133e-36 2.99

NW3 4 2 0 6.56889e-21 3.98

3572 M.A. Noor, W.A. Khan / Applied Mathematics and Computation 219 (2012) 3565–3574

(i) |xn+1 � xn| < e,(ii) |f(xn+1)| < e,

and so, when the stopping criterion is satisfied, xn+1 is taken as the exact root a computed. For numerical illustrations wehave used the fixed stopping criterion e = 10�15.

Table 6Approximate solution of example.6.

Methods IT

nx ( )nf x δ COC

NM 5 2.1544346900318837217592936 3.29189e-35 2.25681e-18 2

CM1 4 2.1544346900318837217592936 1.00000e-58 5.22400e-56 4

CM2 4 2.1544346900318837217592936 1.00000e-58 2.15498e-26 3

NR1 4 2.1544346900318837217592936 1.00000e-58 2.50000e-58 4.01

NW1 4 2.1544346900318837217592936 1.00000e-58 1.79623e-31 3

NW2 4 2.1544346900318837217592936 1.00000e-58 1.56606e-32 3

NW3 4 2.1544346900318837217592936

1.00000e-58 2.90000e-58 4.01

Table 7Approximate solution of Example 7.

Methods IT

nx ( )nf x δ COC

NM 9 3 1.37562e-53 4.01112e-28 2

CM1 6 3 0 5.02650e-52 4

CM2 6 3 9.99684e-48 2.07148e-17 2.99

NR1 6 3 0 6.22300e-56 3.99

NW1 6 3 0 5.08126e-39 3

NW2 6 3 -2.00000e-58 3.30462e-28 3

NW3 6 3 0 2.10000e-57 4

M.A. Noor, W.A. Khan / Applied Mathematics and Computation 219 (2012) 3565–3574 3573

As for the convergence criteria, it was required that the distance of two consecutive approximations d for the root was lessthan e = 10�15. Also displayed are the number of iterations to approximate the zero (IT), the approximate root xn, the value

f(xn) and the computational order of convergence (COC) can be approximated using the formula, COC � ln jxnþ1�xn�1 j=jxn�xn�1 j½ �ln jxn�xn�1 j=jxn�xn�2 j½ � . All

examples are same in [4].

3574 M.A. Noor, W.A. Khan / Applied Mathematics and Computation 219 (2012) 3565–3574

Example 1. Consider the equation Table 1

f1ðxÞ ¼ x3 þ 4x2 � 10; x0 ¼ 1:

Example 2. Consider the equation Table 2

f2ðxÞ ¼ sin2 x� x2 þ 1; x0 ¼ 1:3:

Example 3. Consider the equation Table 3

f3ðxÞ ¼ x2 � ex � 3xþ 2; x0 ¼ 2:

Example 4. Consider the equation Table 4

f4ðxÞ ¼ cos x� x; x0 ¼ 1:7:

Example 5. Consider the equation Table 5

f5ðxÞ ¼ ðx� 1Þ3 � 1; x0 ¼ 2:5:

Example 6. Consider the equation Table 6

f6ðxÞ ¼ x3 � 10; x0 ¼ 2:

Example 7. Consider the equation Table 7

f7ðxÞ ¼ ex2þ7x�30 � 1; x0 ¼ 3:2:

5. Conclusions

In this paper, we have suggested new fourth-order iterative methods for solving nonlinear equations by using the mod-ified homotopy perturbation method [15]. Here we also discuss the efficiency index and computational order of convergenceof new methods. Several examples are given to illustrate the efficiency of Algorithms 2.2, 2.5, 2.7. Using the idea of this pa-per, one can suggest and analyze higher-order multi-step iterative methods for solving nonlinear equations.

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