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On Chebyshev–Halley methods with sixth-order convergence for solving non-linear equations
Applied Mathematics and Computation 190 (2007) 126–131 www.elsevier.com/locate/amc
On Chebyshev–Halley methods with sixth-order convergence for solving non-linear equations Jisheng Kou Department of Mathematics, Shanghai University, Shanghai 200444, China
Abstract In this paper, we present a family of new variants of Chebyshev–Halley methods. The new methods have sixth-order convergence although they only add one evaluation of the function at the point iterated by Chebyshev–Halley methods. The numerical results presented show that the new methods work better not only in order but in efficiency. 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 where Lf ðxn Þ ¼
f 00 ðxn Þf ðxn Þ f 0 ðxn Þ
2
:
E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.01.011
J. Kou / Applied Mathematics and Computation 190 (2007) 126–131
127
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 fifthorder or sixth-order methods have been developed in [8–12]. By using Taylor approximation for the inverse function, a family of fifth-order methods has developed in [10,11] M h ðxn ; xnþ1 Þ f ðxnþ1 Þ ~xnþ1 ¼ xnþ1 1 þ ; b 2 R; ð3Þ 1 bM h ðxn ; xnþ1 Þ f 0 ðxn Þ where xn+1 is defined by (1) and M h ðxn ; xnþ1 Þ ¼
f 00 ðxn Þðf ðxn Þ hf ðxnþ1 ÞÞ f 0 ðxn Þ
2
:
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 improve the methods defined by (3) and obtain a family of new variants of Chebyshev–Halley methods with sixth-order convergence. Then analysis of convergence is supplied. In numerical tests presented, the performance of the new methods with many known methods is compared. 2. The methods Let y ¼ f ðxÞ. Similar to the derivation used in [11], we consider Newton’s theorem for the inverse function x(y) on the interval ½y nþ1 ; y Z y xðyÞ ¼ xðy nþ1 Þ þ x0 ðtÞ dt: ð4Þ y nþ1
We consider an interpolatory quadrature rule m X xc f ðfc Þ; Rm ðf Þ ¼ ðy y nþ1 Þ
ð5Þ
c¼1
where fc ¼ y nþ1 þ sc ðy y nþ1 Þ, knots sc 2 ½0; 1 and weigths xc satisfying m X xc ¼ 1; c¼1
such that Rm is at least of order 1. Applying it to (4), and using y ¼ ~y nþ1 ¼ yð~xnþ1 Þ ¼ 0, xðyÞ ¼ ~xnþ1 , and xðy nþ1 Þ ¼ xnþ1 equivalent to y nþ1 ¼ f ðxnþ1 Þ, we obtain the approximate formula m X ~xnþ1 ¼ xnþ1 y nþ1 xc x0 ðð1 sc Þy nþ1 Þ: ð6Þ c¼1
If we use Taylor expansion of x0 ðð1 sc Þy nþ1 Þ 1 2 x0 ðð1 sc Þy nþ1 Þ ’ x0 ðy n Þ þ x00 ðy n Þðð1 sc Þy nþ1 y n Þ þ x000 ðy n Þðð1 sc Þy nþ1 y n Þ 2 1 ’ x0 ðy n Þ þ x00 ðy n Þðy nþ1 y n Þ þ x000 ðy n Þðy nþ1 y n Þ2 sc x00 ðy n Þy nþ1 ; 2 in combination with Taylor approximation of xðy nþ1 Þ 1 1 2 3 xðy nþ1 Þ ’ xðy n Þ þ x0 ðy n Þðy nþ1 y n Þ þ x00 ðy n Þðy nþ1 y n Þ þ x000 ðy n Þðy nþ1 y n Þ ; 2 6 we can obtain x0 ðð1 sc Þy nþ1 Þ ’ 3
xðy nþ1 Þ xðy n Þ 1 2x0 ðy n Þ x00 ðy n Þðð2sc þ 1Þy nþ1 y n Þ: y nþ1 y n 2
ð7Þ
128
J. Kou / Applied Mathematics and Computation 190 (2007) 126–131
Since 1
x0 ðyÞ ¼ f 0 ðxðyÞÞ ; 0
1 0
x00 ðyÞ ¼ ðx0 ðyÞÞ ¼ ðf 0 ðxðyÞÞ Þ ¼
f 00 ðxðyÞÞ f 0 ðxðyÞÞ
3
;
then from (7), we can approximate x0 ðð1 sc Þy nþ1 Þ by 3ðxnþ1 xn Þ 2 f 00 ðxn Þðf ðxn Þ ð2sc þ 1Þf ðxnþ1 ÞÞ 0 : 3 f ðxnþ1 Þ f ðxn Þ f ðxn Þ 2f 0 ðxn Þ Pm So from (6) and (8), let h ¼ c¼1 xc ð2sc þ 1Þ, and then we obtain a family of new methods " # 3ðxnþ1 xn Þ 2 f 00 ðxn Þðf ðxn Þ hf ðxnþ1 ÞÞ ~xnþ1 ¼ xnþ1 f ðxnþ1 Þ ; 3 f ðxnþ1 Þ f ðxn Þ f 0 ðxn Þ 2f 0 ðxn Þ x0 ðð1 sc Þy nþ1 Þ ’
ð8Þ
ð9Þ
where h 2 R and xn+1 is defined by (2). 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 will be proved to increase from three to six in Section 3. 3. Analysis of convergence Theorem 1. Assume that the function f : D R ! R for an open interval D has a simple root x 2 D. If f(x) is sufficiently smooth in the neighborhood of the root x*, then the order of convergence of the methods defined by (9) is six. Proof. Let en ¼ xn x and ~enþ1 ¼ ~xnþ1 x . Using Taylor expansion and taking into account f ðx Þ ¼ 0, we have f ðxn Þ ¼ f 0 ðx Þ½en þ c2 e2n þ c3 e3n þ c4 e4n þ Oðe5n Þ;
ð10Þ
where ck ¼ ð1=k!Þf ðkÞ ðx Þ=f 0 ðx Þ; k ¼ 2; 3; . . .. Furthermore, we get f 0 ðxn Þ ¼ f 0 ðx Þ½1 þ 2c2 en þ 3c3 e2n þ 4c4 e3n þ Oðe4n Þ; 00
0
f ðxn Þ ¼ f ðx Þ½2c2 þ 6c3 en þ
12c4 e2n
þ
Oðe3n Þ:
ð11Þ ð12Þ
Dividing (10) by (11) gives us f ðxn Þ ¼ en c2 e2n þ 2ðc22 c3 Þe3n þ Oðe4n Þ; f 0 ðxn Þ
ð13Þ
and division of (12) by (11) is f 00 ðxn Þ ¼ 2c2 ð4c22 6c3 Þen þ ð8c32 18c2 c3 þ 12c4 Þe2n þ Oðe3n Þ: f 0 ðxn Þ
ð14Þ
From (13) and (14), it follows that Lf ðxn Þ ¼ 2c2 en 6ðc22 c3 Þe2n þ ð16c32 28c2 c3 þ 12c4 Þe3n þ Oðe4n Þ: Thus from (13) and (15), we obtain enþ1 ¼ 2ð1 aÞc22 c3 e3n þ Oðe4n Þ:
ð15Þ
ð16Þ
We expand f ðxnþ1 Þ about x* f ðxnþ1 Þ ¼ f 0 ðx Þ½enþ1 þ c2 e2nþ1 þ Oðe3nþ1 Þ:
ð17Þ
J. Kou / Applied Mathematics and Computation 190 (2007) 126–131
129
Then from (10), (11), (16) and (17), we get f 0 ðxn Þðxnþ1 xn Þ ¼ 1 þ c2 en ðc22 2c3 Þe2n þ ½ð2a 1Þc32 3c2 c3 þ 3c4 e3n þ Oðe4n Þ: f ðxnþ1 Þ f ðxn Þ
ð18Þ
Similarly, we obtain f 00 ðxn Þðf ðxn Þ hf ðxnþ1 ÞÞ 2f 0 ðxn Þ
2
¼ c2 en 3ðc22 c3 Þe2n þ ½ð8 2h þ 2haÞc32 þ ðh 14Þc2 c3 þ 6c4 e3n þ Oðe4n Þ: ð19Þ
So, from (11), (18) and (19), we can arrive at 3ðxnþ1 xn Þ 2 f 00 ðxn Þðf ðxn Þ hf ðxnþ1 ÞÞ 0 3 f ðxnþ1 Þ f ðxn Þ f ðxn Þ 2f 0 ðxn Þ ¼
1 þ ½ð6a þ 2h 2ha 11Þc32 ðh 8Þc2 c3 c4 e3n þ Oðe4n Þ : f 0 ðx Þ
Since from (9), we have ~enþ1
ð20Þ
"
# 3ðxnþ1 xn Þ 2 f 00 ðxn Þðf ðxn Þ hf ðxnþ1 ÞÞ ¼ enþ1 f ðxnþ1 Þ ; 3 f ðxnþ1 Þ f ðxn Þ f 0 ðxn Þ 2f 0 ðxn Þ
from (16), (17) and (20), we obtain ee nþ1 ¼ enþ1 ½enþ1 þ c2 e2nþ1 þ ½ð6a þ 2h 2ha 11Þc32 ðh 8Þc2 c3 c4 e3n enþ1 þ Oðe7n Þ ¼ c2 e2nþ1 ½ð6a þ 2h 2ha 11Þc32 ðh 8Þc2 c3 c4 e3n enþ1 þ Oðe7n Þ ¼ ½ð2ha 4a 2h þ 9Þc32 þ ðh 7Þc2 c3 þ c4 ½2ð1 aÞc22 c3 e6n þ Oðe7n Þ: This means that the methods defined by (9) are of sixth-order.
ð21Þ
h
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 [13] 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 evaluations the pffiffiffi have the same cost as function one, we have that the present methods have p ffiffiffi efficiency index equal to 4 6w1:565, which is better than the ones of the mentioned third-order methods 3 3w1:442. 4. Numerical results Now, we employ the present methods with a ¼ 1 and h ¼ 7 to solve some non-linear equations and compare them with NM, CM, HM, SHM and the method defined by (3) with a ¼ 1 and b ¼ 3=2. Displayed in Table 1 is the number of function evaluations (NFE) required such that jf ðxn Þj <1.E14. The results in Table 1 show that the present methods improve the computational efficiency of their classical methods CM, HM, SHM, and the method defined by (3). As far as the results we consider, in general, the present methods require the less NFEs as compared to various methods. Moreover, the present methods can compete with NM. We use the following functions, which have appeared in [8,14], respectively. f1 ðxÞ ¼x3 þ 4x2 10; f2 ðxÞ ¼x2 ex 3x þ 2;
x ¼ 1:3652300134140969: x ¼ 0:25753028543986084:
f3 ðxÞ ¼x3 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 ¼x2 þ sinðx=5Þ 1=4;
x ¼ 0:4099920179891371:
130
J. Kou / Applied Mathematics and Computation 190 (2007) 126–131
Table 1 Comparison of various iterative methods f(x)
x0
NM
CM
HM
SHM
Eq. (3)
Eq. (9)
f1
0.3 1 2
108 10 10
27 12 12
174 9 9
285 9 9
156 8 8
40 8 8
f2
0 0.5
8 8
9 9
9 9
9 9
8 8
8 8
f3
1.5 3.2
12 12
12 12
12 12
9 9
12 12
12 8
f4
0.1 1.9
10 8
12 12
9 12
9 12
8 12
8 8
f5
1.2 3.5
10 12
9 12
9 12
9 12
8 12
8 12
f6
0.2 1.8
10 12
12 12
9 12
9 9
8 12
8 12
f7
0.5 2
10 12
12 12
9 12
9 12
8 12
8 12
f8
0.5 2.4
10 10
9 12
9 12
9 9
12 8
8 8
f7 ðxÞ ¼ex 4x2 ;
x ¼ 0:7148059123627778:
f8 ðxÞ ¼ex þ cosðxÞ;
x ¼ 1:7461395304080124:
5. Conclusions We have obtained a family of new methods with sixth-order convergence. This family of methods improves the Chebyshev–Halley methods, which include some classical third-order methods, with only an additional evaluation of the function. A detailed convergence analysis of the new methods is supplied in Theorem 1. These methods can compete with the well-known classical third-order methods, such as Chebyshev’s method, Halley’s method and Super–Halley method, which is corroborated by numerical results. Acknowledgement This work was 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 Cliffs, 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). [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. [8] M. Grau, J.L. Dı´az-Barrero, An improvement of the Euler–Chebyshev iterative method, J. Math. Anal. Appl. 315 (2006) 1–7. [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.
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[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] W. Gautschi, Numerical Analysis: an Introduction, Birkha¨user, 1997. [14] S. Weerakoon, T.G.I. Fernando, A variant of Newton’s method with accelerated third-order convergence, Appl. Math. Lett. 13 (2000) 87–93.
On Chebyshev–Halley methods with sixth-order convergence for solving non-linear equations Jisheng Kou Department of Mathematics, Shanghai University, Shanghai 200444, China
Abstract In this paper, we present a family of new variants of Chebyshev–Halley methods. The new methods have sixth-order convergence although they only add one evaluation of the function at the point iterated by Chebyshev–Halley methods. The numerical results presented show that the new methods work better not only in order but in efficiency. 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 where Lf ðxn Þ ¼
f 00 ðxn Þf ðxn Þ f 0 ðxn Þ
2
:
E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.01.011
J. Kou / Applied Mathematics and Computation 190 (2007) 126–131
127
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 fifthorder or sixth-order methods have been developed in [8–12]. By using Taylor approximation for the inverse function, a family of fifth-order methods has developed in [10,11] M h ðxn ; xnþ1 Þ f ðxnþ1 Þ ~xnþ1 ¼ xnþ1 1 þ ; b 2 R; ð3Þ 1 bM h ðxn ; xnþ1 Þ f 0 ðxn Þ where xn+1 is defined by (1) and M h ðxn ; xnþ1 Þ ¼
f 00 ðxn Þðf ðxn Þ hf ðxnþ1 ÞÞ f 0 ðxn Þ
2
:
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 improve the methods defined by (3) and obtain a family of new variants of Chebyshev–Halley methods with sixth-order convergence. Then analysis of convergence is supplied. In numerical tests presented, the performance of the new methods with many known methods is compared. 2. The methods Let y ¼ f ðxÞ. Similar to the derivation used in [11], we consider Newton’s theorem for the inverse function x(y) on the interval ½y nþ1 ; y Z y xðyÞ ¼ xðy nþ1 Þ þ x0 ðtÞ dt: ð4Þ y nþ1
We consider an interpolatory quadrature rule m X xc f ðfc Þ; Rm ðf Þ ¼ ðy y nþ1 Þ
ð5Þ
c¼1
where fc ¼ y nþ1 þ sc ðy y nþ1 Þ, knots sc 2 ½0; 1 and weigths xc satisfying m X xc ¼ 1; c¼1
such that Rm is at least of order 1. Applying it to (4), and using y ¼ ~y nþ1 ¼ yð~xnþ1 Þ ¼ 0, xðyÞ ¼ ~xnþ1 , and xðy nþ1 Þ ¼ xnþ1 equivalent to y nþ1 ¼ f ðxnþ1 Þ, we obtain the approximate formula m X ~xnþ1 ¼ xnþ1 y nþ1 xc x0 ðð1 sc Þy nþ1 Þ: ð6Þ c¼1
If we use Taylor expansion of x0 ðð1 sc Þy nþ1 Þ 1 2 x0 ðð1 sc Þy nþ1 Þ ’ x0 ðy n Þ þ x00 ðy n Þðð1 sc Þy nþ1 y n Þ þ x000 ðy n Þðð1 sc Þy nþ1 y n Þ 2 1 ’ x0 ðy n Þ þ x00 ðy n Þðy nþ1 y n Þ þ x000 ðy n Þðy nþ1 y n Þ2 sc x00 ðy n Þy nþ1 ; 2 in combination with Taylor approximation of xðy nþ1 Þ 1 1 2 3 xðy nþ1 Þ ’ xðy n Þ þ x0 ðy n Þðy nþ1 y n Þ þ x00 ðy n Þðy nþ1 y n Þ þ x000 ðy n Þðy nþ1 y n Þ ; 2 6 we can obtain x0 ðð1 sc Þy nþ1 Þ ’ 3
xðy nþ1 Þ xðy n Þ 1 2x0 ðy n Þ x00 ðy n Þðð2sc þ 1Þy nþ1 y n Þ: y nþ1 y n 2
ð7Þ
128
J. Kou / Applied Mathematics and Computation 190 (2007) 126–131
Since 1
x0 ðyÞ ¼ f 0 ðxðyÞÞ ; 0
1 0
x00 ðyÞ ¼ ðx0 ðyÞÞ ¼ ðf 0 ðxðyÞÞ Þ ¼
f 00 ðxðyÞÞ f 0 ðxðyÞÞ
3
;
then from (7), we can approximate x0 ðð1 sc Þy nþ1 Þ by 3ðxnþ1 xn Þ 2 f 00 ðxn Þðf ðxn Þ ð2sc þ 1Þf ðxnþ1 ÞÞ 0 : 3 f ðxnþ1 Þ f ðxn Þ f ðxn Þ 2f 0 ðxn Þ Pm So from (6) and (8), let h ¼ c¼1 xc ð2sc þ 1Þ, and then we obtain a family of new methods " # 3ðxnþ1 xn Þ 2 f 00 ðxn Þðf ðxn Þ hf ðxnþ1 ÞÞ ~xnþ1 ¼ xnþ1 f ðxnþ1 Þ ; 3 f ðxnþ1 Þ f ðxn Þ f 0 ðxn Þ 2f 0 ðxn Þ x0 ðð1 sc Þy nþ1 Þ ’
ð8Þ
ð9Þ
where h 2 R and xn+1 is defined by (2). 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 will be proved to increase from three to six in Section 3. 3. Analysis of convergence Theorem 1. Assume that the function f : D R ! R for an open interval D has a simple root x 2 D. If f(x) is sufficiently smooth in the neighborhood of the root x*, then the order of convergence of the methods defined by (9) is six. Proof. Let en ¼ xn x and ~enþ1 ¼ ~xnþ1 x . Using Taylor expansion and taking into account f ðx Þ ¼ 0, we have f ðxn Þ ¼ f 0 ðx Þ½en þ c2 e2n þ c3 e3n þ c4 e4n þ Oðe5n Þ;
ð10Þ
where ck ¼ ð1=k!Þf ðkÞ ðx Þ=f 0 ðx Þ; k ¼ 2; 3; . . .. Furthermore, we get f 0 ðxn Þ ¼ f 0 ðx Þ½1 þ 2c2 en þ 3c3 e2n þ 4c4 e3n þ Oðe4n Þ; 00
0
f ðxn Þ ¼ f ðx Þ½2c2 þ 6c3 en þ
12c4 e2n
þ
Oðe3n Þ:
ð11Þ ð12Þ
Dividing (10) by (11) gives us f ðxn Þ ¼ en c2 e2n þ 2ðc22 c3 Þe3n þ Oðe4n Þ; f 0 ðxn Þ
ð13Þ
and division of (12) by (11) is f 00 ðxn Þ ¼ 2c2 ð4c22 6c3 Þen þ ð8c32 18c2 c3 þ 12c4 Þe2n þ Oðe3n Þ: f 0 ðxn Þ
ð14Þ
From (13) and (14), it follows that Lf ðxn Þ ¼ 2c2 en 6ðc22 c3 Þe2n þ ð16c32 28c2 c3 þ 12c4 Þe3n þ Oðe4n Þ: Thus from (13) and (15), we obtain enþ1 ¼ 2ð1 aÞc22 c3 e3n þ Oðe4n Þ:
ð15Þ
ð16Þ
We expand f ðxnþ1 Þ about x* f ðxnþ1 Þ ¼ f 0 ðx Þ½enþ1 þ c2 e2nþ1 þ Oðe3nþ1 Þ:
ð17Þ
J. Kou / Applied Mathematics and Computation 190 (2007) 126–131
129
Then from (10), (11), (16) and (17), we get f 0 ðxn Þðxnþ1 xn Þ ¼ 1 þ c2 en ðc22 2c3 Þe2n þ ½ð2a 1Þc32 3c2 c3 þ 3c4 e3n þ Oðe4n Þ: f ðxnþ1 Þ f ðxn Þ
ð18Þ
Similarly, we obtain f 00 ðxn Þðf ðxn Þ hf ðxnþ1 ÞÞ 2f 0 ðxn Þ
2
¼ c2 en 3ðc22 c3 Þe2n þ ½ð8 2h þ 2haÞc32 þ ðh 14Þc2 c3 þ 6c4 e3n þ Oðe4n Þ: ð19Þ
So, from (11), (18) and (19), we can arrive at 3ðxnþ1 xn Þ 2 f 00 ðxn Þðf ðxn Þ hf ðxnþ1 ÞÞ 0 3 f ðxnþ1 Þ f ðxn Þ f ðxn Þ 2f 0 ðxn Þ ¼
1 þ ½ð6a þ 2h 2ha 11Þc32 ðh 8Þc2 c3 c4 e3n þ Oðe4n Þ : f 0 ðx Þ
Since from (9), we have ~enþ1
ð20Þ
"
# 3ðxnþ1 xn Þ 2 f 00 ðxn Þðf ðxn Þ hf ðxnþ1 ÞÞ ¼ enþ1 f ðxnþ1 Þ ; 3 f ðxnþ1 Þ f ðxn Þ f 0 ðxn Þ 2f 0 ðxn Þ
from (16), (17) and (20), we obtain ee nþ1 ¼ enþ1 ½enþ1 þ c2 e2nþ1 þ ½ð6a þ 2h 2ha 11Þc32 ðh 8Þc2 c3 c4 e3n enþ1 þ Oðe7n Þ ¼ c2 e2nþ1 ½ð6a þ 2h 2ha 11Þc32 ðh 8Þc2 c3 c4 e3n enþ1 þ Oðe7n Þ ¼ ½ð2ha 4a 2h þ 9Þc32 þ ðh 7Þc2 c3 þ c4 ½2ð1 aÞc22 c3 e6n þ Oðe7n Þ: This means that the methods defined by (9) are of sixth-order.
ð21Þ
h
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 [13] 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 evaluations the pffiffiffi have the same cost as function one, we have that the present methods have p ffiffiffi efficiency index equal to 4 6w1:565, which is better than the ones of the mentioned third-order methods 3 3w1:442. 4. Numerical results Now, we employ the present methods with a ¼ 1 and h ¼ 7 to solve some non-linear equations and compare them with NM, CM, HM, SHM and the method defined by (3) with a ¼ 1 and b ¼ 3=2. Displayed in Table 1 is the number of function evaluations (NFE) required such that jf ðxn Þj <1.E14. The results in Table 1 show that the present methods improve the computational efficiency of their classical methods CM, HM, SHM, and the method defined by (3). As far as the results we consider, in general, the present methods require the less NFEs as compared to various methods. Moreover, the present methods can compete with NM. We use the following functions, which have appeared in [8,14], respectively. f1 ðxÞ ¼x3 þ 4x2 10; f2 ðxÞ ¼x2 ex 3x þ 2;
x ¼ 1:3652300134140969: x ¼ 0:25753028543986084:
f3 ðxÞ ¼x3 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 ¼x2 þ sinðx=5Þ 1=4;
x ¼ 0:4099920179891371:
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J. Kou / Applied Mathematics and Computation 190 (2007) 126–131
Table 1 Comparison of various iterative methods f(x)
x0
NM
CM
HM
SHM
Eq. (3)
Eq. (9)
f1
0.3 1 2
108 10 10
27 12 12
174 9 9
285 9 9
156 8 8
40 8 8
f2
0 0.5
8 8
9 9
9 9
9 9
8 8
8 8
f3
1.5 3.2
12 12
12 12
12 12
9 9
12 12
12 8
f4
0.1 1.9
10 8
12 12
9 12
9 12
8 12
8 8
f5
1.2 3.5
10 12
9 12
9 12
9 12
8 12
8 12
f6
0.2 1.8
10 12
12 12
9 12
9 9
8 12
8 12
f7
0.5 2
10 12
12 12
9 12
9 12
8 12
8 12
f8
0.5 2.4
10 10
9 12
9 12
9 9
12 8
8 8
f7 ðxÞ ¼ex 4x2 ;
x ¼ 0:7148059123627778:
f8 ðxÞ ¼ex þ cosðxÞ;
x ¼ 1:7461395304080124:
5. Conclusions We have obtained a family of new methods with sixth-order convergence. This family of methods improves the Chebyshev–Halley methods, which include some classical third-order methods, with only an additional evaluation of the function. A detailed convergence analysis of the new methods is supplied in Theorem 1. These methods can compete with the well-known classical third-order methods, such as Chebyshev’s method, Halley’s method and Super–Halley method, which is corroborated by numerical results. Acknowledgement This work was 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 Cliffs, 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). [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. [8] M. Grau, J.L. Dı´az-Barrero, An improvement of the Euler–Chebyshev iterative method, J. Math. Anal. Appl. 315 (2006) 1–7. [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.
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