- Email: [email protected]
Sixth-order variants of Chebyshev–Halley methods for solving non-linear equations
Applied Mathematics and Computation 190 (2007) 1839–1843 www.elsevier.com/locate/amc
Sixth-order variants of Chebyshev–Halley methods for solving non-linear equations Jisheng Kou
a,*
, Xiuhua Wang
b
a
b
Department of Mathematics, Shanghai University, Shanghai 200444, China Department of Mathematics, Xiaogan University, Xiaogan 432100, Hubei, China
Abstract In this paper, we present a family of new variants of Chebyshev–Halley methods with sixth-order convergence. Compared with Chebyshev–Halley methods, the new methods require one additional evaluation of the function. The numerical results presented show that the new methods compete with Chebyshev–Halley methods. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Chebyshev–Halley methods; Newton’s method; Non-linear equations; Iterative method; Root-finding
1. Introduction Solving non-linear equations is one of the most important problems in numerical analysis. In this paper, we consider iterative methods to find a simple root of a non-linear equation f ðxÞ ¼ 0, where f : D R ! R for an open interval D is a scalar function. Newton’s method (NM) for a single non-linear equation is written as xnþ1 ¼ xn
f ðxn Þ : f 0 ðxn Þ
ð1Þ
This is an important and basic method [1], which converges quadratically. To improve the local order of convergence of Newton’s method, many modified methods have been proposed. A family of third-order methods, called Chebyshev–Halley methods [2], is defined by 1 Lf ðxn Þ f ðxn Þ xnþ1 ¼ xn 1 þ ; ð2Þ 2 1 aLf ðxn Þ f 0 ðxn Þ
*
Corresponding author. E-mail address: [email protected] (J. Kou).
0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.02.069
1840
J. Kou, X. Wang / Applied Mathematics and Computation 190 (2007) 1839–1843
where Lf ðxn Þ ¼
f 00 ðxn Þf ðxn Þ f 0 ðxn Þ2
:
This family includes the classical Chebyshev’s method (CM) (a ¼ 0), Halley’s method (HM) (a ¼ 1=2) and Super–Halley method (SHM) (a ¼ 1) (for the details of these methods, see [3–6] or a recent review [7]). Recently, in order to improve the local order of convergence of Chebyshev–Halley methods, many higher methods have been developed in [8–15]. A family of fifth-order methods is introduced in [11] Lf ðxn Þ f ðzn Þ ; ð3Þ xnþ1 ¼ zn 1 þ 1 bLf ðxn Þ f 0 ðxn Þ where
1 Lf ðxn Þ f ðxn Þ : zn ¼ x n 1 þ 2 1 aLf ðxn Þ f 0 ðxn Þ
ð4Þ
These methods are very interesting because they can improve the order of convergence and computational efficiency of the classical third-order methods with an additional evaluation of the function. In this paper, we will further improve the the methods defined by (3) and the local order of convergence is increased from five for (3) to six for the new methods. Then the the order of convergence is analyzed. Their better performance is also demonstrated by numerical results. 2. The methods and analysis of convergence Now, we combine f ðzn Þ in (3) to find a method with an improved order of convergence. We consider the following iterations Lf ðxn Þ f ðzn Þ f ðzn Þ þh ; ð5Þ xnþ1 ¼ zn 1 þ 1 bLf ðxn Þ f ðxn Þ cf ðzn Þ f 0 ðxn Þ where zn is defined by (4). The choices of the real parameters a, b, h and c are determined in the following theorem. Theorem 1. Assume that the function f : D R ! R for an open interval D has a simple root x 2 D. Let f ðxÞ be sufficiently smooth in the neighborhood of the root x*, then the order of convergence of the methods defined by (5) is six if h ¼ 3, b ¼ 32 a. Proof. Let en ¼ xn x and d n ¼ zn x , where zn is defined by (4). Using Taylor expansion and taking into account f ðx Þ ¼ 0, we get f ðxn Þ ¼ f 0 ðx Þ½en þ c2 e2n þ c3 e3n þ c4 e4n þ Oðe5n Þ; where ck ¼ ð1=k!Þf
ðkÞ
0
ðx Þ=f ðx Þ; k ¼ 2; 3; . . .. Furthermore, we have
f 0 ðxn Þ ¼ f 0 ðx Þ½1 þ 2c2 en þ 3c3 e2n þ 4c4 e3n þ Oðe4n Þ; 00
0
ð6Þ
f ðxn Þ ¼ f ðx Þ½2c2 þ 6c3 en þ
12c4 e2n
þ
Oðe3n Þ:
ð7Þ ð8Þ
Dividing (6) by (7) gives us f ðxn Þ ¼ en c2 e2n þ 2ðc22 c3 Þe3n þ Oðe4n Þ; f 0 ðxn Þ
ð9Þ
and division of (8) by (7) is f 00 ðxn Þ ¼ 2c2 ð4c22 6c3 Þen þ ð8c32 18c2 c3 þ 12c4 Þe2n þ Oðe3n Þ: f 0 ðxn Þ
ð10Þ
From (9) and (10), we obtain Lf ðxn Þ ¼ 2c2 en 6ðc22 c3 Þe2n þ ð16c32 28c2 c3 þ 12c4 Þe3n þ Oðe4n Þ;
ð11Þ
J. Kou, X. Wang / Applied Mathematics and Computation 190 (2007) 1839–1843
1841
and then we have Lf ðxn Þ ¼ 2c2 en þ ½ð4a 6Þc22 þ 6c3 e2n þ ½ð8a2 24a þ 16Þc32 þ ð24a 28Þc2 c3 þ 12c4 e3n þ Oðe4n Þ: 1 aLf ðxn Þ ð12Þ Thus from (9) and (12), we can obtain d n ¼ ½2ð1 aÞc22 c3 e3n þ Oðe4n Þ:
ð13Þ
We expand f ðzn Þ about x* f ðzn Þ ¼ f 0 ðx Þ½d n þ c2 d 2n þ Oðd 3n Þ:
ð14Þ
Using (6), (14) and combining with (13), we get f ðzn Þ ¼ ½2ð1 aÞc22 c3 e2n ½2ð1 aÞc32 c2 c3 e3n þ Oðe4n Þ: f ðxn Þ cf ðzn Þ
ð15Þ
From (12) and (15), we have Lf ðxn Þ hf ðzn Þ þ 1 bLf ðxn Þ f ðxn Þ cf ðzn Þ ¼ 2c2 en þ ½ð4b þ 2h 2ha 6Þc22 þ ð6 hÞc3 e2n þ ½ð8a2 24a þ 2ha 2h þ 16Þc32 þ ð24a þ h 28Þc2 c3 þ 12c4 e3n þ Oðe4n Þ:
ð16Þ
So, using (7) and (16), we obtain Lf ðxn Þ hf ðzn Þ 1 1þ þ ¼ f ðx Þf1 þ ½ð4b þ 2h 2ha 6Þc22 þ ð3 hÞc3 e2n 0 1 bLf ðxn Þ f ðxn Þ cf ðzn Þ f ðxn Þ þ ½ð8a2 24a þ 6ha 6h 8b þ 28Þc32 þ ð24a þ 3h 34Þc2 c3 þ 8c4 e3n þ Oðe4n Þg:
ð17Þ
Since from (5), we have Lf ðxn Þ hf ðzn Þ f ðzn Þ enþ1 ¼ d n 1 þ þ ; 1 bLf ðxn Þ f ðxn Þ cf ðzn Þ f 0 ðxn Þ then from (13), (14) and (17), we obtain enþ1 ¼ d n fd n þ c2 d 2n þ ½ð4b þ 2h 2ha 6Þc22 þ ð3 hÞc3 e2n d n þ ½ð8a2 24a þ 6ha 6h 8b þ 28Þc32 þ ð24a þ 3h 34Þc2 c3 þ 8c4 e3n d n þ Oðe7n Þg ¼ c2 d 2n ½ð4b þ 2h 2ha 6Þc22 þ ð3 hÞc3 e2n d n ½ð8a2 24a þ 6ha 6h 8b þ 28Þc32 þ ð24a þ 3h 34Þc2 c3 þ 8c4 e3n d n þ Oðe7n Þ: 2
¼ ½ð4b þ 2h 2ha 6Þc22 þ ð3 hÞc3 ½2ð1 aÞc22 c3 e5n c2 ½2ð1 aÞc22 c3 e6n ½ð8a2 24a þ 6ha 6h 8b þ 28Þc32 þ ð24a þ 3h 34Þc2 c3 þ 8c4 ½2ð1 aÞc22 c3 e6n þ Oðe7n Þ:
ð18Þ
If the order is six, then we have 3 h ¼ 0; 4b þ 2h 2ha 6 ¼ 0; which implies that h ¼ 3; b ¼ 32 a:
ð19Þ
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J. Kou, X. Wang / Applied Mathematics and Computation 190 (2007) 1839–1843
Thus, substituting (19) in (18), we obtain the following error equation: enþ1 ¼ ½ð8a2 20a þ 12Þc32 þ ð24a 26Þc2 c3 þ 8c4 ½2ða 1Þc22 þ c3 e6n þ Oðe7n Þ: This ends the proof.
ð20Þ
h
Theorem 1 shows that for h ¼ 3 and b ¼ 32 a, we can obtain a family of new sixth-order methods 1 Lf ðxn Þ f ðxn Þ zn ¼ xn 1 þ ; 2 1 aLf ðxn Þ f 0 ðxn Þ ! Lf ðxn Þ f ðzn Þ f ðzn Þ ; c 2 R: xnþ1 ¼ zn 1 þ þ3 3 f ðxn Þ cf ðzn Þ f 0 ðxn Þ 1 2 aLf ðxn Þ
ð21Þ
We can see that this family of new methods only adds one evaluation of the function at another point iterated by their classical third-order methods but their order of convergence increases from three to six. This family defined by (21) includes the variants of Chebyshev’s method (VCM) (a ¼ 0), Halley’s method (VHM) (a ¼ 1=2) and Super–Halley method (VSHM) (a ¼ 1). In what follows, we use these logograms to represent the present methods. Per iteration the present methods require two evaluations of the function, one of its first derivative and one of its second derivative. We consider the definition of efficiency index [16] as p1/w, where p is the order of the method and w is the number of function evaluations per iteration required by the method. If we assume that all the evaluationsphave the same cost as function one, we have that the present methods have the ffiffiffi p ffiffiefficiency ffi 3 indexes equal to 4 6 w p 1:565, which are better than the ones of the Chebyshev–Halley methods 3 w 1:442 ffiffiffi and Newton’s method 2 w 1:414. 3. Numerical results The performance of the present methods VCM, VHM and VSHM (c = 1) with NM, CM, HM and SHM is compared. Displayed in Table 1 is the number of function evaluations (NFE) required such that j f ðxn Þ j < 1.E 14. As far as the results we consider, for most of the cases, the sixth-order methods VCM, VHM and VSHM require the less NFEs as compared to their classical methods CM, HM and SHM. These results corroborate the theory that the present methods improve the computational efficiency of their classical methods. Moreover, the present methods can also compete with NM.
Table 1 Comparison of various iterative methods f ðxÞ
x0
NM
CM
VCM
HM
VHM
SHM
VSHM
f1
1 2 0 0.5 1.7 2.7 0 1 3.5 6 0.15 1 1 2 0.5 1.5
10 10 8 8 10 10 10 8 12 14 10 12 10 12 10 8
12 12 9 9 12 9 12 9 12 15 15 12 9 12 9 9
12 12 8 8 12 8 12 8 12 12 16 12 8 12 8 8
9 9 9 9 9 9 12 9 12 15 12 12 9 12 9 9
8 8 8 8 8 8 12 8 12 12 12 12 8 12 8 8
9 9 9 9 9 9 9 9 12 15 9 9 9 12 9 9
8 8 8 8 8 8 8 8 12 12 8 8 8 12 8 8
f2 f3 f4 f5 f6 f7 f8
J. Kou, X. Wang / Applied Mathematics and Computation 190 (2007) 1839–1843
1843
We use the following functions, which are the same as in [17,8], respectively. x ¼ 1:3652300134140969;
f1 ðxÞ ¼ x3 þ 4x2 10; f2 ðxÞ ¼ x2 ex 3x þ 2;
x ¼ 0:25753028543986084;
f3 ðxÞ ¼ x3 10; x ¼ 2:1544346900318837; f4 ðxÞ ¼ cosðxÞ x; x ¼ 0:73908513321516067; f5 ðxÞ ¼ sin2 ðxÞ x2 þ 1;
x ¼ 1:4044916482153411;
f6 ðxÞ ¼ x2 þ sinðx=5Þ 1=4; f7 ðxÞ ¼ ex 4x2 ;
x ¼ 0:4099920179891371;
x ¼ 0:7148059123627778;
f8 ðxÞ ¼ ex þ cosðxÞ;
x ¼ 1:7461395304080124:
Acknowledgement Work supported by National Natural Science Foundation of China (50379038). References [1] A.M. Ostrowski, Solutions of Equations and System of Equations, Academic Press, New York, 1960. [2] J.M. Gutie´rrez, M.A. Herna´ndez, A family of Chebyshev–Halley type methods in Banach spaces, Bull. Aust. Math. Soc. 55 (1997) 113–130. [3] J.F. Traub, Iterative Methods for Solution of Equations, Prentice-Hall, Englewood CliJs, NJ, 1964. [4] I.K. Argyros, A note on the Halley method in Banach spaces, Appl. Math. Comput. 58 (1993) 215–224. [5] D. Chen, I.K. Argyros, Q.S. Qian, A local convergence theorem for the Super–Halley method in a Banach space, App. Math. Lett. 7 (5) (1994) 49–52. [6] J.M. Gutie´rrez, M.A. Herna´ndez, An acceleration of Newton’s method: Super–Halley method, Appl. Math. Comput. 117 (2001) 223– 239. [7] S. Amat, S. Busquier, J.M. Gutie´rrez, Geometric constructions of iterative functions to solve nonlinear equations, J. Comput. Appl. Math. 157 (2003) 197–205. ´ az-Barrero, An improvement of the Euler–Chebyshev iterative method, J. Math. Anal. Appl. 315 (2006) 1–7. [8] M. Grau, J.L. Dı [9] Jisheng Kou, Yitian Li, Xiuhua Wang, A family of fifth-order iterations composed of Newton and third-order methods, Appl. Math. Comput. (2006), doi:10.1016/j.amc.2006.07.150. [10] Jisheng Kou, Yitian Li, The improvements of Chebyshev–Halley methods with fifth-order convergence, Appl. Math. Comput. (2006), doi:10.1016/j.amc.2006.09.097. [11] Jisheng Kou, Yitian Li, Some variants of Chebyshev–Halley methods with fifth-order convergence, Appl. Math. Comput. (2006), doi:10.1016/j.amc.2006.11.068. [12] Jisheng Kou, Yitian Li, Modified Chebyshev–Halley methods with sixth-order convergence, Appl. Math. Comput. (2006), doi:10.1016/j.amc.2006.10.018. [13] Jisheng Kou, Some new sixth-order methods for solving non-linear equations, Appl. Math. Comput. (2006), doi:10.1016/ j.amc.2006.11.117. [14] Jisheng Kou, On Chebyshev–Halley methods with sixth-order convergence for solving nonlinear equations, Appl. Math. Comput. (2007), doi:10.1016/j.amc.2007.01.011. [15] C. Chun, Certain improvements of Chebyshev–Halley methods with accelerated fourth-order convergence, Appl. Math. Comput. (2007), doi:10.1016/j.amc.2006.11.118. [16] W. Gautschi, Numerical Analysis: An Introduction, Birkha¨user, 1997. [17] S. Weerakoon, T.G.I. Fernando, A variant of Newton’s method with accelerated third-order convergence, Appl. Math. Lett. 13 (2000) 87–93.
Sixth-order variants of Chebyshev–Halley methods for solving non-linear equations Jisheng Kou
a,*
, Xiuhua Wang
b
a
b
Department of Mathematics, Shanghai University, Shanghai 200444, China Department of Mathematics, Xiaogan University, Xiaogan 432100, Hubei, China
Abstract In this paper, we present a family of new variants of Chebyshev–Halley methods with sixth-order convergence. Compared with Chebyshev–Halley methods, the new methods require one additional evaluation of the function. The numerical results presented show that the new methods compete with Chebyshev–Halley methods. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Chebyshev–Halley methods; Newton’s method; Non-linear equations; Iterative method; Root-finding
1. Introduction Solving non-linear equations is one of the most important problems in numerical analysis. In this paper, we consider iterative methods to find a simple root of a non-linear equation f ðxÞ ¼ 0, where f : D R ! R for an open interval D is a scalar function. Newton’s method (NM) for a single non-linear equation is written as xnþ1 ¼ xn
f ðxn Þ : f 0 ðxn Þ
ð1Þ
This is an important and basic method [1], which converges quadratically. To improve the local order of convergence of Newton’s method, many modified methods have been proposed. A family of third-order methods, called Chebyshev–Halley methods [2], is defined by 1 Lf ðxn Þ f ðxn Þ xnþ1 ¼ xn 1 þ ; ð2Þ 2 1 aLf ðxn Þ f 0 ðxn Þ
*
Corresponding author. E-mail address: [email protected] (J. Kou).
0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.02.069
1840
J. Kou, X. Wang / Applied Mathematics and Computation 190 (2007) 1839–1843
where Lf ðxn Þ ¼
f 00 ðxn Þf ðxn Þ f 0 ðxn Þ2
:
This family includes the classical Chebyshev’s method (CM) (a ¼ 0), Halley’s method (HM) (a ¼ 1=2) and Super–Halley method (SHM) (a ¼ 1) (for the details of these methods, see [3–6] or a recent review [7]). Recently, in order to improve the local order of convergence of Chebyshev–Halley methods, many higher methods have been developed in [8–15]. A family of fifth-order methods is introduced in [11] Lf ðxn Þ f ðzn Þ ; ð3Þ xnþ1 ¼ zn 1 þ 1 bLf ðxn Þ f 0 ðxn Þ where
1 Lf ðxn Þ f ðxn Þ : zn ¼ x n 1 þ 2 1 aLf ðxn Þ f 0 ðxn Þ
ð4Þ
These methods are very interesting because they can improve the order of convergence and computational efficiency of the classical third-order methods with an additional evaluation of the function. In this paper, we will further improve the the methods defined by (3) and the local order of convergence is increased from five for (3) to six for the new methods. Then the the order of convergence is analyzed. Their better performance is also demonstrated by numerical results. 2. The methods and analysis of convergence Now, we combine f ðzn Þ in (3) to find a method with an improved order of convergence. We consider the following iterations Lf ðxn Þ f ðzn Þ f ðzn Þ þh ; ð5Þ xnþ1 ¼ zn 1 þ 1 bLf ðxn Þ f ðxn Þ cf ðzn Þ f 0 ðxn Þ where zn is defined by (4). The choices of the real parameters a, b, h and c are determined in the following theorem. Theorem 1. Assume that the function f : D R ! R for an open interval D has a simple root x 2 D. Let f ðxÞ be sufficiently smooth in the neighborhood of the root x*, then the order of convergence of the methods defined by (5) is six if h ¼ 3, b ¼ 32 a. Proof. Let en ¼ xn x and d n ¼ zn x , where zn is defined by (4). Using Taylor expansion and taking into account f ðx Þ ¼ 0, we get f ðxn Þ ¼ f 0 ðx Þ½en þ c2 e2n þ c3 e3n þ c4 e4n þ Oðe5n Þ; where ck ¼ ð1=k!Þf
ðkÞ
0
ðx Þ=f ðx Þ; k ¼ 2; 3; . . .. Furthermore, we have
f 0 ðxn Þ ¼ f 0 ðx Þ½1 þ 2c2 en þ 3c3 e2n þ 4c4 e3n þ Oðe4n Þ; 00
0
ð6Þ
f ðxn Þ ¼ f ðx Þ½2c2 þ 6c3 en þ
12c4 e2n
þ
Oðe3n Þ:
ð7Þ ð8Þ
Dividing (6) by (7) gives us f ðxn Þ ¼ en c2 e2n þ 2ðc22 c3 Þe3n þ Oðe4n Þ; f 0 ðxn Þ
ð9Þ
and division of (8) by (7) is f 00 ðxn Þ ¼ 2c2 ð4c22 6c3 Þen þ ð8c32 18c2 c3 þ 12c4 Þe2n þ Oðe3n Þ: f 0 ðxn Þ
ð10Þ
From (9) and (10), we obtain Lf ðxn Þ ¼ 2c2 en 6ðc22 c3 Þe2n þ ð16c32 28c2 c3 þ 12c4 Þe3n þ Oðe4n Þ;
ð11Þ
J. Kou, X. Wang / Applied Mathematics and Computation 190 (2007) 1839–1843
1841
and then we have Lf ðxn Þ ¼ 2c2 en þ ½ð4a 6Þc22 þ 6c3 e2n þ ½ð8a2 24a þ 16Þc32 þ ð24a 28Þc2 c3 þ 12c4 e3n þ Oðe4n Þ: 1 aLf ðxn Þ ð12Þ Thus from (9) and (12), we can obtain d n ¼ ½2ð1 aÞc22 c3 e3n þ Oðe4n Þ:
ð13Þ
We expand f ðzn Þ about x* f ðzn Þ ¼ f 0 ðx Þ½d n þ c2 d 2n þ Oðd 3n Þ:
ð14Þ
Using (6), (14) and combining with (13), we get f ðzn Þ ¼ ½2ð1 aÞc22 c3 e2n ½2ð1 aÞc32 c2 c3 e3n þ Oðe4n Þ: f ðxn Þ cf ðzn Þ
ð15Þ
From (12) and (15), we have Lf ðxn Þ hf ðzn Þ þ 1 bLf ðxn Þ f ðxn Þ cf ðzn Þ ¼ 2c2 en þ ½ð4b þ 2h 2ha 6Þc22 þ ð6 hÞc3 e2n þ ½ð8a2 24a þ 2ha 2h þ 16Þc32 þ ð24a þ h 28Þc2 c3 þ 12c4 e3n þ Oðe4n Þ:
ð16Þ
So, using (7) and (16), we obtain Lf ðxn Þ hf ðzn Þ 1 1þ þ ¼ f ðx Þf1 þ ½ð4b þ 2h 2ha 6Þc22 þ ð3 hÞc3 e2n 0 1 bLf ðxn Þ f ðxn Þ cf ðzn Þ f ðxn Þ þ ½ð8a2 24a þ 6ha 6h 8b þ 28Þc32 þ ð24a þ 3h 34Þc2 c3 þ 8c4 e3n þ Oðe4n Þg:
ð17Þ
Since from (5), we have Lf ðxn Þ hf ðzn Þ f ðzn Þ enþ1 ¼ d n 1 þ þ ; 1 bLf ðxn Þ f ðxn Þ cf ðzn Þ f 0 ðxn Þ then from (13), (14) and (17), we obtain enþ1 ¼ d n fd n þ c2 d 2n þ ½ð4b þ 2h 2ha 6Þc22 þ ð3 hÞc3 e2n d n þ ½ð8a2 24a þ 6ha 6h 8b þ 28Þc32 þ ð24a þ 3h 34Þc2 c3 þ 8c4 e3n d n þ Oðe7n Þg ¼ c2 d 2n ½ð4b þ 2h 2ha 6Þc22 þ ð3 hÞc3 e2n d n ½ð8a2 24a þ 6ha 6h 8b þ 28Þc32 þ ð24a þ 3h 34Þc2 c3 þ 8c4 e3n d n þ Oðe7n Þ: 2
¼ ½ð4b þ 2h 2ha 6Þc22 þ ð3 hÞc3 ½2ð1 aÞc22 c3 e5n c2 ½2ð1 aÞc22 c3 e6n ½ð8a2 24a þ 6ha 6h 8b þ 28Þc32 þ ð24a þ 3h 34Þc2 c3 þ 8c4 ½2ð1 aÞc22 c3 e6n þ Oðe7n Þ:
ð18Þ
If the order is six, then we have 3 h ¼ 0; 4b þ 2h 2ha 6 ¼ 0; which implies that h ¼ 3; b ¼ 32 a:
ð19Þ
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J. Kou, X. Wang / Applied Mathematics and Computation 190 (2007) 1839–1843
Thus, substituting (19) in (18), we obtain the following error equation: enþ1 ¼ ½ð8a2 20a þ 12Þc32 þ ð24a 26Þc2 c3 þ 8c4 ½2ða 1Þc22 þ c3 e6n þ Oðe7n Þ: This ends the proof.
ð20Þ
h
Theorem 1 shows that for h ¼ 3 and b ¼ 32 a, we can obtain a family of new sixth-order methods 1 Lf ðxn Þ f ðxn Þ zn ¼ xn 1 þ ; 2 1 aLf ðxn Þ f 0 ðxn Þ ! Lf ðxn Þ f ðzn Þ f ðzn Þ ; c 2 R: xnþ1 ¼ zn 1 þ þ3 3 f ðxn Þ cf ðzn Þ f 0 ðxn Þ 1 2 aLf ðxn Þ
ð21Þ
We can see that this family of new methods only adds one evaluation of the function at another point iterated by their classical third-order methods but their order of convergence increases from three to six. This family defined by (21) includes the variants of Chebyshev’s method (VCM) (a ¼ 0), Halley’s method (VHM) (a ¼ 1=2) and Super–Halley method (VSHM) (a ¼ 1). In what follows, we use these logograms to represent the present methods. Per iteration the present methods require two evaluations of the function, one of its first derivative and one of its second derivative. We consider the definition of efficiency index [16] as p1/w, where p is the order of the method and w is the number of function evaluations per iteration required by the method. If we assume that all the evaluationsphave the same cost as function one, we have that the present methods have the ffiffiffi p ffiffiefficiency ffi 3 indexes equal to 4 6 w p 1:565, which are better than the ones of the Chebyshev–Halley methods 3 w 1:442 ffiffiffi and Newton’s method 2 w 1:414. 3. Numerical results The performance of the present methods VCM, VHM and VSHM (c = 1) with NM, CM, HM and SHM is compared. Displayed in Table 1 is the number of function evaluations (NFE) required such that j f ðxn Þ j < 1.E 14. As far as the results we consider, for most of the cases, the sixth-order methods VCM, VHM and VSHM require the less NFEs as compared to their classical methods CM, HM and SHM. These results corroborate the theory that the present methods improve the computational efficiency of their classical methods. Moreover, the present methods can also compete with NM.
Table 1 Comparison of various iterative methods f ðxÞ
x0
NM
CM
VCM
HM
VHM
SHM
VSHM
f1
1 2 0 0.5 1.7 2.7 0 1 3.5 6 0.15 1 1 2 0.5 1.5
10 10 8 8 10 10 10 8 12 14 10 12 10 12 10 8
12 12 9 9 12 9 12 9 12 15 15 12 9 12 9 9
12 12 8 8 12 8 12 8 12 12 16 12 8 12 8 8
9 9 9 9 9 9 12 9 12 15 12 12 9 12 9 9
8 8 8 8 8 8 12 8 12 12 12 12 8 12 8 8
9 9 9 9 9 9 9 9 12 15 9 9 9 12 9 9
8 8 8 8 8 8 8 8 12 12 8 8 8 12 8 8
f2 f3 f4 f5 f6 f7 f8
J. Kou, X. Wang / Applied Mathematics and Computation 190 (2007) 1839–1843
1843
We use the following functions, which are the same as in [17,8], respectively. x ¼ 1:3652300134140969;
f1 ðxÞ ¼ x3 þ 4x2 10; f2 ðxÞ ¼ x2 ex 3x þ 2;
x ¼ 0:25753028543986084;
f3 ðxÞ ¼ x3 10; x ¼ 2:1544346900318837; f4 ðxÞ ¼ cosðxÞ x; x ¼ 0:73908513321516067; f5 ðxÞ ¼ sin2 ðxÞ x2 þ 1;
x ¼ 1:4044916482153411;
f6 ðxÞ ¼ x2 þ sinðx=5Þ 1=4; f7 ðxÞ ¼ ex 4x2 ;
x ¼ 0:4099920179891371;
x ¼ 0:7148059123627778;
f8 ðxÞ ¼ ex þ cosðxÞ;
x ¼ 1:7461395304080124:
Acknowledgement Work supported by National Natural Science Foundation of China (50379038). References [1] A.M. Ostrowski, Solutions of Equations and System of Equations, Academic Press, New York, 1960. [2] J.M. Gutie´rrez, M.A. Herna´ndez, A family of Chebyshev–Halley type methods in Banach spaces, Bull. Aust. Math. Soc. 55 (1997) 113–130. [3] J.F. Traub, Iterative Methods for Solution of Equations, Prentice-Hall, Englewood CliJs, NJ, 1964. [4] I.K. Argyros, A note on the Halley method in Banach spaces, Appl. Math. Comput. 58 (1993) 215–224. [5] D. Chen, I.K. Argyros, Q.S. Qian, A local convergence theorem for the Super–Halley method in a Banach space, App. Math. Lett. 7 (5) (1994) 49–52. [6] J.M. Gutie´rrez, M.A. Herna´ndez, An acceleration of Newton’s method: Super–Halley method, Appl. Math. Comput. 117 (2001) 223– 239. [7] S. Amat, S. Busquier, J.M. Gutie´rrez, Geometric constructions of iterative functions to solve nonlinear equations, J. Comput. Appl. Math. 157 (2003) 197–205. ´ az-Barrero, An improvement of the Euler–Chebyshev iterative method, J. Math. Anal. Appl. 315 (2006) 1–7. [8] M. Grau, J.L. Dı [9] Jisheng Kou, Yitian Li, Xiuhua Wang, A family of fifth-order iterations composed of Newton and third-order methods, Appl. Math. Comput. (2006), doi:10.1016/j.amc.2006.07.150. [10] Jisheng Kou, Yitian Li, The improvements of Chebyshev–Halley methods with fifth-order convergence, Appl. Math. Comput. (2006), doi:10.1016/j.amc.2006.09.097. [11] Jisheng Kou, Yitian Li, Some variants of Chebyshev–Halley methods with fifth-order convergence, Appl. Math. Comput. (2006), doi:10.1016/j.amc.2006.11.068. [12] Jisheng Kou, Yitian Li, Modified Chebyshev–Halley methods with sixth-order convergence, Appl. Math. Comput. (2006), doi:10.1016/j.amc.2006.10.018. [13] Jisheng Kou, Some new sixth-order methods for solving non-linear equations, Appl. Math. Comput. (2006), doi:10.1016/ j.amc.2006.11.117. [14] Jisheng Kou, On Chebyshev–Halley methods with sixth-order convergence for solving nonlinear equations, Appl. Math. Comput. (2007), doi:10.1016/j.amc.2007.01.011. [15] C. Chun, Certain improvements of Chebyshev–Halley methods with accelerated fourth-order convergence, Appl. Math. Comput. (2007), doi:10.1016/j.amc.2006.11.118. [16] W. Gautschi, Numerical Analysis: An Introduction, Birkha¨user, 1997. [17] S. Weerakoon, T.G.I. Fernando, A variant of Newton’s method with accelerated third-order convergence, Appl. Math. Lett. 13 (2000) 87–93.