IJDE International Journal of Differential Equations 1687-9651 1687-9643 Hindawi Publishing Corporation 495357 10.1155/2014/495357 495357 Research Article On Some Iterative Methods with Memory and High Efficiency Index for Solving Nonlinear Equations Eftekhari Tahereh Liu Fawang Faculty of Mathematics University of Sistan and Baluchestan Zahedan 987-98155 Iran usb.ac.ir 2014 642014 2014 25 12 2013 10 03 2014 06 04 2014 2014 Copyright © 2014 Tahereh Eftekhari. 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.

Based on iterative methods without memory of eighth-order convergence proposed by Thukral (2012), some iterative methods with memory and high efficiency index are presented. We show that the order of convergence is increased without any additional function evaluations. Numerical comparisons are made to show the performance of the presented methods.

1. Introduction

Multipoint iterative methods for solving nonlinear equations are of great practical importance since they overcome theoretical limits of one-point methods concerning the convergence order and computational efficiency. The main goal and motivation in constructing root-solvers is to achieve as high as possible convergence order consuming as small as possible function evaluations. Let f be a sufficiently smooth function of single variable in some neighborhood D of α , where α satisfies f ( α ) = 0 . Traub  considered the iterative function of order two: (1)    x n + 1 = x n - β f ( x n ) 2 f ( x n + β f ( x n ) ) - f ( x n ) , where β 0 is a real constant. The choice β = 1 produces the well-known Steffensen method . To improve the local order of convergence, many modified methods have been proposed in the open literature; see  and references therein. Thukral developed a scheme of optimal order of convergence eight , constructing a weight function as well in the following form: (2) ϕ 1 = ( 1 - f ( y n ) f ( w n ) ) - 1 , (3) ϕ 2 = 1 + f ( y n ) f ( w n ) + ( f ( y n ) f ( w n ) ) 2 , (4) ϕ 3 = f [ x n , w n ] f [ w n , y n ] , (5) w n = x n + β f ( x n ) , y n = x n - β f ( x n ) 2 f ( w n ) - f ( x n ) , z n = y n - ϕ k ( f ( y n ) f [ x n , y n ] ) , k = 1,2 , 3 , x n + 1 = z n - ( 1 - f ( z n ) f ( w n ) ) - 1 × ( 1 - f ( y n ) 3 f ( w n ) 2 f ( x n ) ) ( f [ x n , y n ] f ( z n ) f [ y n , z n ] f [ x n , z n ] ) , where β 0 is a real constant. These methods belong to the class of methods without memory. In this paper, we use optimal multipoint method without memory by Thukral as the base for constructing considerably faster methods employing information from the current and previous iteration without any additional evaluations of the function. Following Traub’s classification (see [1, pp. 8-9]), this class of root-finders is called methods with memory. Our main goal is to present some multipoint methods with high efficiency index to find the approximation of the root α of the nonlinear equation f ( x ) = 0 . The acceleration of convergence rate is attained by suitable variation of one free parameter in each iterative step. This self-accelerating parameter is calculated using information from the current and previous iteration by applying Newton’s interpolating polynomials. Since considerable acceleration of convergence is obtained without additional function evaluations, the computational efficiency of improved multipoint methods is significantly increased. The efficacy of the methods is tested on a number of numerical examples.

2. Multipoint Methods with Memory

In , Sharma et al. have presented three methods through the following forms of β n : (6) β n = - 1 f - ( α ) = - 1 N 3 ( x n ) , (7) β n = - 1 f - ( α ) = - 1 N - 3 ( x n ) , (8) β n = - 1 f - ( α ) = - 1 N 4 ( x n ) , for n = 1,2 , , where f - ( α ) denotes an approximation to f ( α ) , and (9) N 3 ( t ) = N 3 ( t ; x n , z n - 1 , y n - 1 , x n - 1 ) , N - 3 ( t ) = N - 3 ( t ; x n , z n - 1 , y n - 1 , w n - 1 ) are Newton’s interpolatory polynomials of the third degree, set through four available approximations: (10) ( x n , z n - 1 , y n - 1 , x n - 1 ) , ( x n , z n - 1 , y n - 1 , w n - 1 ) , respectively, and N 4 ( t ) = N 4 ( t ; x n , z n - 1 , y n - 1 , w n - 1 , x n - 1 ) is Newton’s interpolatory polynomial of the fourth degree, set through five available ( x n , z n - 1 , y n - 1 , w n - 1 , x n - 1 ) approximations. The derivatives are as follows: (11) N 3 ( x n ) = [ d d t N 3 ( t ) ] t = x n = [ d d t ( f ( x n ) + f [ x n , z n - 1 ] ( t - x n ) g lllmmllll + f [ x n , z n - 1 , y n - 1 ] ( t - x n ) ( t - z n - 1 ) g lllmmllll + f [ x n , z n - 1 , y n - 1 , x n - 1 ] glllmmllll × ( t - x n ) ( t - z n - 1 ) ( t - y n - 1 ) ) d d t ] t = x n = f [ x n , z n - 1 ] + f [ x n , z n - 1 , y n - 1 ] ( x n - z n - 1 ) + f [ x n , z n - 1 , y n - 1 , x n - 1 ] ( x n - z n - 1 ) ( x n - y n - 1 ) , N - 3 ( x n ) = [ d d t N - 3 ( t ) ] t = x n = [ d d t ( f ( x n ) + f [ x n , z n - 1 ] ( t - x n ) glllmmlll l + f [ x n , z n - 1 , y n - 1 ] ( t - x n ) ( t - z n - 1 ) glllmmlll l + f [ x n , z n - 1 , y n - 1 , w n - 1 ] g lllmmllll × ( t - x n ) ( t - z n - 1 ) ( t - y n - 1 ) ) d d t ] t = x n = f [ x n , z n - 1 ] + f [ x n , z n - 1 , y n - 1 ] ( x n - z n - 1 ) + f [ x n , z n - 1 , y n - 1 , w n - 1 ] ( x n - z n - 1 ) ( x n - y n - 1 ) , N 4 ( x n ) = [ d d t N 4 ( t ) ] t = x n = [ d d t ( f ( x n ) + f [ x n , z n - 1 ] ( t - x n ) glllmmlll l + f [ x n , z n - 1 , y n - 1 ] ( t - x n ) ( t - z n - 1 ) gllmmllll l + f [ x n , z n - 1 , y n - 1 , w n - 1 ] glllmmlll l × ( t - x n ) ( t - z n - 1 ) ( t - y n - 1 ) gllllmmll l + f [ x n , z n - 1 , y n - 1 , w n - 1 , x n - 1 ] ( t - x n ) glllmmlll l × ( t - z n - 1 ) ( t - y n - 1 ) ( t - w n - 1 ) ) d d t ] t = x n = f [ x n , z n - 1 ] + f [ x n , z n - 1 , y n - 1 ] ( x n - z n - 1 ) + f [ x n , z n - 1 , y n - 1 , w n - 1 ] ( x n - z n - 1 ) ( x n - y n - 1 ) + f [ x n , z n - 1 , y n - 1 , w n - 1 , x n - 1 ] × ( x n - z n - 1 ) ( x n - y n - 1 ) ( x n - w n - 1 ) . Now, we replace constant parameters β in the iterative formula (5) by the varying β n defined by (6), (7), and (8). Then, the multipoint methods with memory, following from (5), become (12) w n = x n + β n f ( x n ) , y n = x n - β n f ( x n ) 2 f ( w n ) - f ( x n ) , z n = y n - ϕ k ( f ( y n ) f [ x n , y n ] ) , k = 1,2 , 3 , x n + 1 = z n - ( 1 - f ( z n ) f ( w n ) ) - 1 × ( 1 - f ( y n ) 3 f ( w n ) 2 f ( x n ) ) ( f [ x n , y n ] f ( z n ) f [ y n , z n ] f [ x n , z n ] ) , where ϕ k for k = 1,2 , 3 is defined by (2), (3), and (4), respectively.

3. Convergence Theorems Theorem 1.

Let the function f ( x ) be sufficiently differentiable in a neighborhood of its simple zero α . If an initial approximation x 0 is sufficiently close to α and the parameter β n in (12) is recursively calculated by the forms given in (6)–(8), where ϕ k for k = 1 is defined by (2), then the R -order of convergence of the Steffensen-like method with memory (12) with the corresponding expressions (6)–(8) of β n is at least 10.7202, 11, and 11.2915, respectively.

Proof.

Let { x n } be a sequence of approximations generated by an iterative method with memory (12). If this sequence converges to the zero α of f with the R -order ( r ) of (12), then we write  (13) e n + 1 ~ D n , r e n r , e n = x n - α , where D n , r tends to the asymptotic error constant D r of (12) when n . Therefore, (14) e n + 1 ~ D n , r ( D n - 1 , r e n - 1 r ) r = D n , r D n - 1 , r r e n - 1 r 2 . Assume that the iterative sequences { w n } ,   { y n } , and { z n } have the R -orders q ,   p , and s , respectively; then, bearing in mind (13), we obtain (15) e ¯ n ~ D n , q e n q = D n , q ( D n - 1 , r e n - 1 r ) q = D n , q D n - 1 , r q e n - 1 r q , (16) e ~ n ~ D n , p e n p = D n , p ( D n - 1 , r e n - 1 r ) p = D n , p D n - 1 , r p e n - 1 r p , (17) e ^ n ~ D n , s e n s = D n , s ( D n - 1 , r e n - 1 r ) s = D n , s D n - 1 , r s e n - 1 r s . Let ϕ k for k = 1 be defined by (2). We now obtain the order of convergence of the methods with memory (12), where β n is calculated from (6). The error relations with the self-accelerating parameter β n for (12) are in what follows (where c i = f ( i ) ( α ) / ( i ! f ( α ) ) ,   i = 2,3 , ): (18) e ¯ n = w n - α ~ ( 1 + β n f ( α ) ) e n , (19) e ~ n = y n - α ~ c 2 ( 1 + β n f ( α ) ) e n 2 , (20) e ^ n = z n - α ~ c n , 4 ( 1 + β n f ( α ) ) 2 e n 4 , (21) e n + 1 = x n + 1 - α ~ c n , 8 ( 1 + β n f ( α ) ) 3 e n 8 . In order to find the error relation for (12), we need to find the expression for 1 + β n f ( α ) . Using a symbolic software such as M a t h e m a t i c a 8 with the use of (6), we attain that (22) 1 + β n f ( α ) ~ c 4 e n - 1 e ~ n - 1 e ^ n - 1 . According to (13), (16), (17), (19), and (22), we obtain (23) e ~ n ~ c 2 c 4 e n - 1 e ~ n - 1 e ^ n - 1 e n 2 ~ c 2 c 4 e n - 1 D n - 1 , p e n - 1 p D n - 1 , s e n - 1 s ( D n - 1 , r e n - 1 r ) 2 ~ c 2 c 4 D n - 1 , p D n - 1 , s D n - 1 , r 2 e n - 1 1 + p + s + 2 r . Similarly, by (13), (16), (17), (20), and (22), we can write (24) e ^ n ~ c n , 4 ( c 4 e n - 1 e ~ n - 1 e ^ n - 1 ) 2 e n 4 ~ c n , 4 ( c 4 e n - 1 D n - 1 , p e n - 1 p D n - 1 , s e n - 1 s ) 2 ( D n - 1 , r e n - 1 r ) 4 ~ c n , 4 c 4 2 e n - 1 2 D n - 1 , p 2 e n - 1 2 p D n - 1 , s 2 e n - 1 2 s D n - 1 , r 4 e n - 1 4 r ~ c n , 4 c 4 2 D n - 1 , p 2 D n - 1 , s 2 D n - 1 , r 4 e n - 1 2 + 2 p + 2 s + 4 r . Combining (13), (16), (17), (21), and (22) yields (25) e n + 1 ~ c n , 8 ( c 4 e n - 1 e ~ n - 1 e ^ n - 1 ) 3 e n 8 ~ c n , 8 ( c 4 e n - 1 D n - 1 , p e n - 1 p D n - 1 , s e n - 1 s ) 3 ( D n - 1 , r e n - 1 r ) 8 ~ c n , 8 c 4 3 e n - 1 3 D n - 1 , p 3 e n - 1 3 p D n - 1 , s 3 e n - 1 3 s D n - 1 , r 8 e n - 1 8 r ~ c n , 8 c 4 3 D n - 1 , p 3 D n - 1 , s 3 D n - 1 , r 8 e n - 1 3 + 3 p + 3 s + 8 r . Equating the exponents of the error e n - 1 in pairs of relations (19) and (23), (20) and (24), and then (21) and (25), we arrive at the following system of equations: (26) r p - 1 - p - s - 2 r = 0 , r s - 2 - 2 p - 2 s - 4 r = 0 , r 2 - 3 - 3 p - 3 s - 8 r = 0 . Positive solution of this system is p = ( 1 / 6 ) ( 7 + 109 ) ,   s = ( 1 / 3 ) ( 7 + 109 ) , and r = ( 1 / 2 ) ( 11 + 109 ) = 10.7202 . Therefore, the R -order of the methods with memory (12), when β n is calculated by (6), is at least 10.7202.

Now, using a symbolic software such as M a t h e m a t i c a 8 with the use of (7), we attain that (27) 1 + β n f ( α ) ~ c 4 e ¯ n - 1 e ~ n - 1 e ^ n - 1 . Combining (13), (15), (16), (17), (18), and (27), we obtain (28) e ¯ n ~ c 4 e ¯ n - 1 e ~ n - 1 e ^ n - 1 e n ~ c 4 D n - 1 , q e n - 1 q D n - 1 , p e n - 1 p D n - 1 , s e n - 1 s D n - 1 , r e n - 1 r ~ c 4 D n - 1 , q D n - 1 , p D n - 1 , s D n - 1 , r e n - 1 q + p + s + r . In the similar way, we find the following error relations: (29) e ~ n ~ c 2 c 4 e ¯ n - 1 e ~ n - 1 e ^ n - 1 e n 2 ~ c 2 c 4 D n - 1 , q e n - 1 q D n - 1 , p e n - 1 p D n - 1 , s e n - 1 s ( D n - 1 , r e n - 1 r ) 2 ~ c 2 c 4 D n - 1 , q D n - 1 , p D n - 1 , s D n - 1 , r 2 e n - 1 q + p + s + 2 r , (30) e ^ n ~ c n , 4 ( c 4 e ¯ n - 1 e ~ n - 1 e ^ n - 1 ) 2 e n 4 ~ c n , 4 ( c 4 D n - 1 , q e n - 1 q D n - 1 , p e n - 1 p D n - 1 , s e n - 1 s ) 2 ( D n - 1 , r e n - 1 r ) 4 ~ c n , 4 c 4 2 D n - 1 , q 2 e n - 1 2 q D n - 1 , p 2 e n - 1 2 p D n - 1 , s 2 e n - 1 2 s D n - 1 , r 4 e n - 1 4 r ~ c n , 4 c 4 2 D n - 1 , q 2 D n - 1 , p 2 D n - 1 , s 2 D n - 1 , r 4 e n - 1 2 q + 2 p + 2 s + 4 r , (31) e n + 1 ~ c n , 8 ( c 4 e ¯ n - 1 e ~ n - 1 e ^ n - 1 ) 3 e n 8 ~ c n , 8 ( c 4 D n - 1 , q e n - 1 q D n - 1 , p e n - 1 p D n - 1 , s e n - 1 s ) 3 ( D n - 1 , r e n - 1 r ) 8 ~ c n , 8 c 4 3 D n - 1 , q 3 e n - 1 3 q D n - 1 , p 3 e n - 1 3 p D n - 1 , s 3 e n - 1 3 s D n - 1 , r 8 e n - 1 8 r ~ c n , 8 c 4 3 D n - 1 , q 3 D n - 1 , p 3 D n - 1 , s 3 D n - 1 , r 8 e n - 1 3 q + 3 p + 3 s + 8 r . Comparing the exponents of e n - 1 on the right hand sides of (18) and (28), (19) and (29), (20) and (30), and then (21) and (31), we arrive at the following system of equations: (32) r q - q - p - s - r = 0 , r p - q - p - s - 2 r = 0 , r s - 2 q - 2 p - 2 s - 4 r = 0 , r 2 - 3 q - 3 p - 3 s - 8 r = 0 . Positive solution of this system is q = 2 ,   p = 3 ,   s = 6 , and r = 11 . Therefore, the R -order of the methods with memory (12), when β n is calculated by (7), is at least 11.

Using a symbolic software such as M a t h e m a t i c a 8 with the use of (8), we attain that (33) 1 + β n f ( α ) ~ c 5 e n - 1 e ¯ n - 1 e ~ n - 1 e ^ n - 1 . Using (33) and previously derived relations, we obtain the following error relations for the intermediate approximations: (34) e ¯ n ~ c 5 e n - 1 e ¯ n - 1 e ~ n - 1 e ^ n - 1 e n ~ c 5 e n - 1 D n - 1 , q e n - 1 q D n - 1 , p e n - 1 p D n - 1 , s e n - 1 s D n - 1 , r e n - 1 r ~ c 5 D n - 1 , q D n - 1 , p D n - 1 , s D n - 1 , r e n - 1 1 + q + p + s + r , (35) e ~ n ~ c 2 c 5 e n - 1 e ¯ n - 1 e ~ n - 1 e ^ n - 1 e n 2 ~ c 2 c 5 e n - 1 D n - 1 , q e n - 1 q D n - 1 , p e n - 1 p D n - 1 , s e n - 1 s ( D n - 1 , r e n - 1 r ) 2 ~ c 2 c 5 D n - 1 , q D n - 1 , p D n - 1 , s D n - 1 , r 2 e n - 1 1 + q + p + s + 2 r , (36) e ^ n ~ c n , 4 ( c 5 e n - 1 e ¯ n - 1 e ~ n - 1 e ^ n - 1 ) 2 e n 4 ~ c n , 4 ( c 5 e n - 1 D n - 1 , q e n - 1 q D n - 1 , p e n - 1 p D n - 1 , s e n - 1 s ) 2 ( D n - 1 , r e n - 1 r ) 4 ~ c n , 4 c 5 2 e n - 1 2 D n - 1 , q 2 e n - 1 2 q D n - 1 , p 2 e n - 1 2 p D n - 1 , s 2 e n - 1 2 s D n - 1 , r 4 e n - 1 4 r ~ c n , 4 c 5 2 D n - 1 , q 2 D n - 1 , p 2 D n - 1 , s 2 D n - 1 , r 4 e n - 1 2 + 2 q + 2 p + 2 s + 4 r . In the similar fashion, we find the final error relation (21) which is given by (37) e n + 1 ~ c n , 8 ( c 5 e n - 1 e ¯ n - 1 e ~ n - 1 e ^ n - 1 ) 3 e n 8 ~ c n , 8 ( c 5 e n - 1 D n - 1 , q e n - 1 q D n - 1 , p e n - 1 p D n - 1 , s e n - 1 s ) 3 × ( D n - 1 , r e n - 1 r ) 8 ~ c n , 8 c 5 3 e n - 1 3 D n - 1 , q 3 e n - 1 3 q D n - 1 , p 3 e n - 1 3 p D n - 1 , s 3 e n - 1 3 s D n - 1 , r 8 e n - 1 8 r ~ c n , 8 c 5 3 D n - 1 , q 3 D n - 1 , p 3 D n - 1 , s 3 D n - 1 , r 8 e n - 1 3 + 3 q + 3 p + 3 s + 8 r . Comparing the exponents of e n - 1 on the right hand sides of (18) and (34), (19) and (35), (20) and (36), and then (21) and (37), we arrive at the following system of equations: (38) r q - q - p - s - r - 1 = 0 , r p - q - p - s - 2 r - 1 = 0 , r s - 2 q - 2 p - 2 s - 4 r - 2 = 0 , r 2 - 3 q - 3 p - 3 s - 8 r - 3 = 0 . Positive solution of this system is q = ( 1 / 3 ) ( 1 + 2 7 ) ,   p = ( 2 / 3 ) ( 2 + 7 ) ,   s = ( 4 / 3 ) ( 2 + 7 ) , and r = 2 ( 3 + 7 ) = 11.2915 . Therefore, the R -order of the methods with memory (12), when β n is calculated by (8), is at least 11.2915.

Theorem 2.

Let the function f ( x ) be sufficiently differentiable in a neighborhood of its simple zero α . If an initial approximation x 0 is sufficiently close to α and the parameter β n in (12) is recursively calculated by the forms given in (6)–(8), where ϕ k for k = 2 is defined by (3), then the R -order of convergence of the Steffensen-like method with memory (12) with the corresponding expressions (6)–(8) of β n is at least 10, 10.2426, and 10.4721, respectively.

Proof.

The proof of this theorem is similar to the proof of Theorem 1; hence, it is omitted.

Theorem 3.

Let the function f ( x ) be sufficiently differentiable in a neighborhood of its simple zero α . If an initial approximation x 0 is sufficiently close to α and the parameter β n in (12) is recursively calculated by the forms given in (6)–(8), where ϕ k for k = 3 is defined by (4), then the R -order of convergence of the Steffensen-like method with memory (12) with the corresponding expressions (6)–(8) of β n is at least 10.7202, 11, and 11.2915, respectively.

Proof.

The proof of this theorem is similar to the proof of Theorem 1; hence, it is omitted.

4. Numerical Results

In this section, we demonstrate the convergence behavior of the methods with memory (12), where β n is calculated by one of formulas (6)–(8) and ϕ k for k = 1,2 , 3 is defined by (2)–(4), respectively. Numerical computations reported here have been carried out in a M a t h e m a t i c a 8.0 environment. Tables 1 and 2 show the difference of the root α and the approximation x n to α , where α is the exact root computed with 800 significant digits ( Digits : = 800 ). To check the theoretical order of convergence, we calculate the computational order of convergence r c using the following formula : (39) r c log | f ( x n ) / f ( x n - 1 ) | log | f ( x n - 1 ) / f ( x n - 2 ) | , taking into consideration the last three approximations in the iterative process. We use the following examples (selected from ): (40) f 1 ( x ) = e x 2 + x cos x - 1 sin π x + x log ( x sin x + 1 ) , α = 0 , x 0 = 0.6 , γ 0 = - 0.1 , f 2 ( x ) = log ( x 2 - 2 x + 2 ) + e x 2 - 5 x + 4 sin ( x - 1 ) , α = 1 , x 0 = 1.35 , γ 0 = - 0.1 . It is obvious from the tables that recursive calculation by the Newton interpolation (8) gives the best results.

Families of three-point methods with memory, for example, f 1 ( x ) .

Methods | x 1 - α | | x 2 - α | | x 3 - α | r c
ϕ 1
(12), (6) 0.665 e - 4 0.186 e - 43 0.374 e - 468 10.737
(12), (7) 0.665 e - 4 0.718 e - 44 0.592 e - 484 11.011
(12), (8) 0.665 e - 4 0.471 e - 44 0.321 e - 487 11.038
ϕ 2
(12), (6) 0.665 e - 4 0.468 e - 40 0.800 e - 402 10.007
(12), (7) 0.665 e - 4 0.231 e - 40 0.996 e - 406 10.021
(12), (8) 0.665 e - 4 0.276 e - 40 0.579 e - 405 10.023
ϕ 3
(12), (6) 0.665 e - 4 0.186 e - 43 0.374 e - 468 10.737
(12), (7) 0.665 e - 4 0.718 e - 44 0.592 e - 484 11.011
(12), (8) 0.665 e - 4 0.471 e - 44 0.321 e - 487 11.038

Families of three-point methods with memory, for example, f 2 ( x ) .

Methods | x 1 - α | | x 2 - α | | x 3 - α | r c
ϕ 1
(12), (6) 0.104 e - 3 0.127 e - 39 0.868 e - 425 10.725
(12), (7) 0.104 e - 3 0.116 e - 39 0.419 e - 436 11.027
(12), (8) 0.104 e - 3 0.261 e - 40 0.259 e - 443 11.010
ϕ 2
(12), (6) 0.106 e - 3 0.499 e - 36 0.142 e - 360 10.040
(12), (7) 0.106 e - 3 0.476 e - 36 0.507 e - 360 10.016
(12), (8) 0.106 e - 3 0.223 e - 36 0.254 e - 363 10.005
ϕ 3
(12), (6) 0.104 e - 3 0.127 e - 39 0.868 e - 425 10.725
(12), (7) 0.104 e - 3 0.116 e - 39 0.419 e - 436 11.027
(12), (8) 0.104 e - 3 0.261 e - 40 0.259 e - 443 11.010
5. Conclusion

In this paper, Newton’s interpolatory polynomials of the third and fourth degrees are applied for constructing considerably faster methods employing information from the current and previous iteration without any additional evaluations of the function. 11.291 5 1 / 4 = 1.83311 has been obtained as the highest possible computational efficiency index for the new methods with memory. The efficacy of the methods is tested on a number of numerical examples. The results show that this method is very useful to find an acceptable approximation of the exact solution of nonlinear equations.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The author is grateful to the referees for their comments and suggestions that helped to improve the paper.

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