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Some sixth-order variants of Ostrowski root-finding methods
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Applied Mathematics and Computation 193 (2007) 389–394 www.elsevier.com/locate/amc
Some sixth-order variants of Ostrowski root-finding methods Changbum Chun a
a,*
, YoonMee Ham
b,1
School of Liberal Arts, Korea University of Technology and Education, Cheonan City, Chungnam 330-708, Republic of Korea b Department of Mathematics Kyonggi University, Suwon 443-760, Republic of Korea
Abstract In this paper, we present some sixth-order class of modified Ostrowski’s methods for solving nonlinear equations. Per iteration each class member requires three function and one first derivative evaluations, and is shown to be at least sixthorder convergent. Several numerical examples are given to illustrate the performance of some of the presented methods. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Newton’s method; Iterative methods; Nonlinear equations; Order of convergence; Ostrowski’s method
1. Introduction In this paper, we consider iterative methods to find a simple root a, i.e., f(a) = 0 and f 0 (a) 5 0, of a nonlinear equation f(x) = 0. Newton’s method is the well-known iterative method for finding a by using xnþ1 ¼ xn
f ðxn Þ ; f 0 ðxn Þ
ð1Þ
that converges quadratically in some neighborhood of a [1]. In recent years, some iterative methods have been proposed and analyzed for solving nonlinear equations that improve order of convergence of the classical methods such as Jarratt’s method, Euler–Chebyshev methods, Ostrowski’s method [1] given by
*
1
f ðxn Þ ; f 0 ðxn Þ f ðxn Þ f ðy n Þ ; ¼ yn f ðxn Þ 2f ðy n Þ f 0 ðxn Þ
y n ¼ xn
ð2Þ
xnþ1
ð3Þ
Corresponding author. E-mail addresses: [email protected] (C. Chun), [email protected] (Y. Ham). This work was supported by University of Northern Iowa.
0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.03.074
390
C. Chun, Y. Ham / Applied Mathematics and Computation 193 (2007) 389–394
which are improvements of Newton’s method; the order increases by at least two at the expense of additional function evaluation at another point iterated by the classical methods, see [2–5] and the references therein. Grau et al. [6] developed the sixth-order variant of Ostrowski’s method which is given by f ðxn Þ ; f 0 ðxn Þ f ðxn Þ f ðy n Þ zn ¼ y n ; f ðxn Þ 2f ðy n Þ f 0 ðxn Þ f ðxn Þ f ðzn Þ xnþ1 ¼ zn : f ðxn Þ 2f ðy n Þ f 0 ðxn Þ
y n ¼ xn
ð4Þ ð5Þ ð6Þ
Another sixth-order family of modified Ostrowski’s method was considered by Sharma et al. in [7] f ðxn Þ ; f 0 ðxn Þ f ðxn Þ f ðy n Þ zn ¼ y n ; f ðxn Þ 2f ðy n Þ f 0 ðxn Þ f ðxn Þ þ ðb 2Þf ðy n Þ f ðzn Þ xnþ1 ¼ zn ; f ðxn Þ þ bf ðy n Þ f 0 ðxn Þ y n ¼ xn
ð7Þ ð8Þ ð9Þ
where b 2 R. Motivated by the recent activities in this direction, in this paper, we consider developing a class of methods which includes the above-mentioned sixth-order variants as particular cases and can also be employed to obtain many other new sixth-order variants. Per iteration the methods derived by our result require three function evaluations and one first derivative evaluation, analysis of convergence shows that they are sixth-order convergent. Finally, some numerical examples are provided to show the performance of the methods presented in this contribution. 2. Derivation of methods and convergence analysis In the iteration scheme of the form f ðxn Þ y n ¼ xn 0 ; ð10Þ f ðxn Þ f ðxn Þ f ðy n Þ zn ¼ y n ; ð11Þ f ðxn Þ 2f ðy n Þ f 0 ðxn Þ f ðzn Þ xnþ1 ¼ zn H ðun Þ 0 ; ð12Þ f ðxn Þ 1 nÞ where un ¼ ff ðy and H(t) represents a real-valued function, it is observed that if we take H ðtÞ ¼ 12t and ðxn Þ H ðtÞ ¼ 1þðbþ2Þt , then the substep defined by (12) reduces to (6) and (9), respectively. Further, we note that these 1þbt functions H(t) satisfy the properties H ð0Þ ¼ 1;
H 0 ð0Þ ¼ 2:
ð13Þ
Now, a simple question occurs; does any function H(t) satisfying conditions (13) produce a sixth-order variant of Ostrowski’s method? The answer indeed is yes, and the results are presented in the following theorem. Theorem 2.1. Let a 2 I be a simple zero of sufficiently differentiable function f:I ! R for an open interval I and H any function with H(0) = 1, H 0 (0) = 2 and jH00 (0)j < 1. If x0 is sufficiently close to a, then the the method defined by (10)–(12) is of sixth-order, and satisfy the error equation 1 1 00 H ð0Þ 7 c32 c3 e6n þ Oðe7n Þ; enþ1 ¼ 6 H 00 ð0Þ c52 þ c2 c23 þ ð14Þ 2 2 where en ¼ xn a, ck ¼ f ðkÞ ðaÞ=k!f 0 ðaÞ.
C. Chun, Y. Ham / Applied Mathematics and Computation 193 (2007) 389–394
Proof. Let a be a simple zero of f, en ¼ xn a, ck ¼ f ðkÞ ðaÞ=k!f 0 ðaÞ. We let f ðxÞ yðxÞ ¼ x 0 ; f ðxÞ f ðxÞ f ðyðxÞÞ zðxÞ ¼ yðxÞ ; f ðxÞ 2f ðyðxÞÞ f 0 ðxÞ
391
ð15Þ ð16Þ
and consider the iteration function F defined by F ðxÞ ¼ zðxÞ H ðuðxÞÞ
f ðzðxÞÞ ; f 0 ðxÞ
ð17Þ
where u(x) = f(y(x))/f(x). In view of an elementary, tedious evaluation of derivatives of F, we employ the symbolic computation of the Maple package to compute the Taylor expansion of F(xn) around x = a (see [8] for details). We find xnþ1 ¼ F ðxn Þ ¼ a þ K 4 e4n þ K 5 e5n þ K 6 e6n þ Oðe7n Þ;
ð18Þ
where K 4 ¼ ½1 H ð0Þc2 ðc22 c3 Þ; 0
4Þc42
ð19Þ 1Þc23
K 5 ¼ ð6H ð0Þ H ð0Þ þ 2ðH ð0Þ 1 0 þ 2 4 5H ð0Þ þ H ð0Þ c22 c3 þ 2ðH ð0Þ 1Þc2 c4 ; 2 1 00 0 K 6 ¼ 10 22H ð0Þ þ 9H ð0Þ H ð0Þ c52 þ ð18 25H ð0Þ þ 4H 0 ð0ÞÞc2 c23 2 1 00 0 þ 3ðH ð0Þ 1Þc2 c5 þ 53H ð0Þ 30 15H ð0Þ þ H ð0Þ c32 c3 2 þ 2ð6 8H ð0Þ þ H 0 ð0ÞÞc22 c4 þ 7ðH ð0Þ 1Þc3 c4 ; so that if H(0) = 1, H 0 (0) = 2 and jH00 (0)j < 1, then from (18)–(21), we obtain the error equation 1 1 00 H ð0Þ 7 c32 c3 e6n þ Oðe7n Þ: enþ1 ¼ 6 H 00 ð0Þ c52 þ c2 c23 þ 2 2 This completes the proof.
ð20Þ
ð21Þ
ð22Þ
h
3. Some examples Some existing methods and also many other sixth-order iterative methods are special cases of Theorem 2.1. 1 It is clear that if we take H ðtÞ ¼ 12t and H ðtÞ ¼ 1þðbþ2Þt , then we obtain Grau et al.’s method and Sharma 1þbt et al.’s family defined by (6) and (9), respectively. In particular, the following methods are obtained as particular cases. Example 3.1. For the function H defined by H ðtÞ ¼ 1 þ 2t þ lt2 þ ct3 ;
ð23Þ
where l, c 2 R, we obtain the new two-parameter sixth-order family of methods f ðxn Þ ; f 0 ðxn Þ f ðxn Þ f ðy n Þ zn ¼ y n ; f ðxn Þ 2f ðy n Þ f 0 ðxn Þ f ðy n Þ f 2 ðy n Þ f 3 ðy n Þ f ðzn Þ xnþ1 ¼ zn 1 þ 2 þl 2 þc 3 : f ðxn Þ f ðxn Þ f ðxn Þ f 0 ðxn Þ
y n ¼ xn
ð24Þ ð25Þ ð26Þ
392
C. Chun, Y. Ham / Applied Mathematics and Computation 193 (2007) 389–394
Example 3.2. The function H defined by H ðtÞ ¼
c þ ðb þ 2cÞt ; c þ bt
ð27Þ
where c, b 2 R, can easily be shown to satisfy conditions of Theorem 2.1. Hence we obtain the new twoparameter sixth-order family of methods f ðxn Þ ; f 0 ðxn Þ f ðxn Þ f ðy n Þ zn ¼ y n ; f ðxn Þ 2f ðy n Þ f 0 ðxn Þ cf ðxn Þ þ ðb þ 2cÞf ðy n Þ f ðzn Þ : xnþ1 ¼ zn cf ðxn Þ þ bf ðy n Þ f 0 ðxn Þ y n ¼ xn
ð28Þ ð29Þ ð30Þ
We remark that this family contains the Grau et al. method defined by (6) as a special case, which is obtained when c = 1, b = 2. Example 3.3. The function H defined by H ðtÞ ¼ 1 þ
4t pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 1 4t
ð31Þ
can easily be shown to satisfy conditions of Theorem 2.1. Thus, we obtain the new sixth-order method f ðxn Þ ; f 0 ðxn Þ f ðxn Þ f ðy n Þ zn ¼ y n ; f ðxn Þ 2f ðy n Þ f 0 ðxn Þ " y n ¼ xn
xnþ1
# 4f ðy n Þ f ðzn Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 ; ¼ zn 1 þ f ðxn Þ þ signðf ðxn ÞÞ f ðxn Þ 4f ðy n Þ f ðxn Þ
where the sign function is 8 if > <1 signðxÞ ¼ 1 if > : 0 if
ð32Þ ð33Þ ð34Þ
defined by x > 0; x < 0; x ¼ 0:
Each class member defined by (10)–(12) requires three function and one first derivative evaluations per iteration. In particular, it improves the order of convergence of Ostrowski’s method from four to six with additional function evaluation at the point iterated by Ostrowski’s method. If we consider the definition of 1 efficiency index [9] as pm , where p is the order of the method and m is the number of functional evaluations per iteration required by the method, we have that the methods obtained by the formula pffiffiffi (10)–(12) have the 1 efficiency index equal to 64 1:565, which is better than the one of Newton’s method 2 1:414. 4. Numerical examples We present some numerical test results for various iterative scheme which improve Ostrowski’s method in Table 1. Compared were Newton’s method (NM), Ostrowski’s method defined by (3) (OM), Grau et al.’s method defined by (6) (GM), Sharma et al.’s method (9) with b = 3 (SM), and (26) with l = c = 1 (CM1), (30) with c = b = 1 (CM2), (34) (CM3) introduced in the present contribution. All computations were done using MAPLE using 128 digit floating point arithmetics (Digits:¼128). Displayed in Table 1 is the number of iterations (IT) required such that j f ðxn Þ j< 1050 . We used the following test functions and display the approximate zero x* found up to the 28th decimal places.
C. Chun, Y. Ham / Applied Mathematics and Computation 193 (2007) 389–394
393
Table 1 Comparison of various iterative methods and Newton’s method f(x)
IT NM
OM
GM
SM
CM1
CM2
CM3
f1, x0 = 2 f1, x0 = 1
8 8
5 5
4 4
4 5
4 4
4 4
4 4
f2, x0 = 0 f2, x0 = 1
7 7
4 4
4 4
4 4
3 4
4 4
4 4
f3, x0 = 1 f3, x0 = 2
8 11
4 6
4 5
4 5
4 5
4 5
4 5
f4, x0 = 2 f4, x0 = 5
8 10
5 6
4 5
4 5
4 5
4 5
4 5
f5, x0 = 3 f5, x0 = 4
9 10
5 5
4 5
5 5
4 5
4 5
4 5
f6, x0 = 2 f6, x0 = 4
11 13
6 7
5 6
6 7
5 6
5 6
5 6
9 9
5 5
4 4
5 5
5 4
6 4
5 5
f7, x0 = 1 f7, x0 = 2.5
f1 ðxÞ ¼ x3 þ 4x2 10; 2
x ¼ 1:3652300134140968457608068290;
x
f2 ðxÞ ¼ x e 3x þ 2; x2
x ¼ 0:25753028543986076045536730494;
2
f3 ðxÞ ¼ xe sin x þ 3 cos x þ 5; x
2
f4 ðxÞ ¼ sinðxÞe þ lnðx þ 1Þ; 3
f5 ðxÞ ¼ ðx 1Þ 2;
x ¼ 0;
x ¼ 2:2599210498948731647672106073;
f6 ðxÞ ¼ ðx þ 2Þex 1; 2
x ¼ 1:2076478271309189270094167584;
f7 ðxÞ ¼ sin ðxÞ x2 þ 1;
x ¼ 0:44285440100238858314132800000; x ¼ 1:4044916482153412260350868178:
The computational results presented in Table 1 show that for most of the functions we tested, the presented methods are efficient and have at least equal performance as compared with the other methods of the same order which have been shown to really improve the computational efficiency of Ostrowski’s method (see [6,7]). Moreover, the methods can compete with Newton’s method. Thus, presented methods in this contribution can be considered as improvements of Ostrowski’s method. 5. Conclusion In this work, we presented a new sixth-order class of iterative methods which improves the order of convergence of Ostrowski’s method from four to six with an additional function evaluation. Per iteration each class member requires three function and one first derivative evaluations. Some existing sixth-order variants of Ostrowski’s method are obtained as special cases of our result. Some of the obtained methods were also compared in their performance to various other iterative methods of the same order, and it was observed that they demonstrate at least equal performance. References [1] A.M. Ostrowski, Solution of equations in Euclidean and Banach space, Academic Press, New York, 1973. [2] C. Chun, Some improvements of Jarratt’s method with sixth-order convergence, Appl. Math. Comput. (in press). doi:10.1016/ j.amc.2007.02.023. [3] M. Grau, J.L. Dı´az-Barrero, An improvement of the Euler–Chebyshev iterative method, J. Math. Anal. Appl. 315 (2006) 1–7. [4] J. Kou, Y. Li, X. Wang, An improvement of the Jarrat method, Appl. Math. Comput. (in press). doi:10.1016/j.amc.2006.12.062.
394
C. Chun, Y. Ham / Applied Mathematics and Computation 193 (2007) 389–394
[5] J. Kou, Y. Li, The improvements of Chebyshev–Halley methods with fifth-order convergence, Appl. Math. Comput. (in press). doi:10.1016/j.amc.2006.09.097. [6] M. Grau, J.L. Dı´az-Barrero, An improvement to Ostrowski root-finding method, J. Math. Anal. Appl. 173 (2006) 450–456. [7] J.R. Sharma, R.K. Guha, A family of modified Ostrowski methods with accelerated sixth-order convergence, Appl. Math. Comput. (in press). doi:10.1016/j.amc.2007.01.009. [8] C. Chun, Iterative methods improving Newton’s method by the decomposition method, Comput. Math. Appl. 50 (2005) 1559–1568. [9] W. Gautschi, Numerical Analysis: An introduction, Birkha¨user, 1997.
Applied Mathematics and Computation 193 (2007) 389–394 www.elsevier.com/locate/amc
Some sixth-order variants of Ostrowski root-finding methods Changbum Chun a
a,*
, YoonMee Ham
b,1
School of Liberal Arts, Korea University of Technology and Education, Cheonan City, Chungnam 330-708, Republic of Korea b Department of Mathematics Kyonggi University, Suwon 443-760, Republic of Korea
Abstract In this paper, we present some sixth-order class of modified Ostrowski’s methods for solving nonlinear equations. Per iteration each class member requires three function and one first derivative evaluations, and is shown to be at least sixthorder convergent. Several numerical examples are given to illustrate the performance of some of the presented methods. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Newton’s method; Iterative methods; Nonlinear equations; Order of convergence; Ostrowski’s method
1. Introduction In this paper, we consider iterative methods to find a simple root a, i.e., f(a) = 0 and f 0 (a) 5 0, of a nonlinear equation f(x) = 0. Newton’s method is the well-known iterative method for finding a by using xnþ1 ¼ xn
f ðxn Þ ; f 0 ðxn Þ
ð1Þ
that converges quadratically in some neighborhood of a [1]. In recent years, some iterative methods have been proposed and analyzed for solving nonlinear equations that improve order of convergence of the classical methods such as Jarratt’s method, Euler–Chebyshev methods, Ostrowski’s method [1] given by
*
1
f ðxn Þ ; f 0 ðxn Þ f ðxn Þ f ðy n Þ ; ¼ yn f ðxn Þ 2f ðy n Þ f 0 ðxn Þ
y n ¼ xn
ð2Þ
xnþ1
ð3Þ
Corresponding author. E-mail addresses: [email protected] (C. Chun), [email protected] (Y. Ham). This work was supported by University of Northern Iowa.
0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.03.074
390
C. Chun, Y. Ham / Applied Mathematics and Computation 193 (2007) 389–394
which are improvements of Newton’s method; the order increases by at least two at the expense of additional function evaluation at another point iterated by the classical methods, see [2–5] and the references therein. Grau et al. [6] developed the sixth-order variant of Ostrowski’s method which is given by f ðxn Þ ; f 0 ðxn Þ f ðxn Þ f ðy n Þ zn ¼ y n ; f ðxn Þ 2f ðy n Þ f 0 ðxn Þ f ðxn Þ f ðzn Þ xnþ1 ¼ zn : f ðxn Þ 2f ðy n Þ f 0 ðxn Þ
y n ¼ xn
ð4Þ ð5Þ ð6Þ
Another sixth-order family of modified Ostrowski’s method was considered by Sharma et al. in [7] f ðxn Þ ; f 0 ðxn Þ f ðxn Þ f ðy n Þ zn ¼ y n ; f ðxn Þ 2f ðy n Þ f 0 ðxn Þ f ðxn Þ þ ðb 2Þf ðy n Þ f ðzn Þ xnþ1 ¼ zn ; f ðxn Þ þ bf ðy n Þ f 0 ðxn Þ y n ¼ xn
ð7Þ ð8Þ ð9Þ
where b 2 R. Motivated by the recent activities in this direction, in this paper, we consider developing a class of methods which includes the above-mentioned sixth-order variants as particular cases and can also be employed to obtain many other new sixth-order variants. Per iteration the methods derived by our result require three function evaluations and one first derivative evaluation, analysis of convergence shows that they are sixth-order convergent. Finally, some numerical examples are provided to show the performance of the methods presented in this contribution. 2. Derivation of methods and convergence analysis In the iteration scheme of the form f ðxn Þ y n ¼ xn 0 ; ð10Þ f ðxn Þ f ðxn Þ f ðy n Þ zn ¼ y n ; ð11Þ f ðxn Þ 2f ðy n Þ f 0 ðxn Þ f ðzn Þ xnþ1 ¼ zn H ðun Þ 0 ; ð12Þ f ðxn Þ 1 nÞ where un ¼ ff ðy and H(t) represents a real-valued function, it is observed that if we take H ðtÞ ¼ 12t and ðxn Þ H ðtÞ ¼ 1þðbþ2Þt , then the substep defined by (12) reduces to (6) and (9), respectively. Further, we note that these 1þbt functions H(t) satisfy the properties H ð0Þ ¼ 1;
H 0 ð0Þ ¼ 2:
ð13Þ
Now, a simple question occurs; does any function H(t) satisfying conditions (13) produce a sixth-order variant of Ostrowski’s method? The answer indeed is yes, and the results are presented in the following theorem. Theorem 2.1. Let a 2 I be a simple zero of sufficiently differentiable function f:I ! R for an open interval I and H any function with H(0) = 1, H 0 (0) = 2 and jH00 (0)j < 1. If x0 is sufficiently close to a, then the the method defined by (10)–(12) is of sixth-order, and satisfy the error equation 1 1 00 H ð0Þ 7 c32 c3 e6n þ Oðe7n Þ; enþ1 ¼ 6 H 00 ð0Þ c52 þ c2 c23 þ ð14Þ 2 2 where en ¼ xn a, ck ¼ f ðkÞ ðaÞ=k!f 0 ðaÞ.
C. Chun, Y. Ham / Applied Mathematics and Computation 193 (2007) 389–394
Proof. Let a be a simple zero of f, en ¼ xn a, ck ¼ f ðkÞ ðaÞ=k!f 0 ðaÞ. We let f ðxÞ yðxÞ ¼ x 0 ; f ðxÞ f ðxÞ f ðyðxÞÞ zðxÞ ¼ yðxÞ ; f ðxÞ 2f ðyðxÞÞ f 0 ðxÞ
391
ð15Þ ð16Þ
and consider the iteration function F defined by F ðxÞ ¼ zðxÞ H ðuðxÞÞ
f ðzðxÞÞ ; f 0 ðxÞ
ð17Þ
where u(x) = f(y(x))/f(x). In view of an elementary, tedious evaluation of derivatives of F, we employ the symbolic computation of the Maple package to compute the Taylor expansion of F(xn) around x = a (see [8] for details). We find xnþ1 ¼ F ðxn Þ ¼ a þ K 4 e4n þ K 5 e5n þ K 6 e6n þ Oðe7n Þ;
ð18Þ
where K 4 ¼ ½1 H ð0Þc2 ðc22 c3 Þ; 0
4Þc42
ð19Þ 1Þc23
K 5 ¼ ð6H ð0Þ H ð0Þ þ 2ðH ð0Þ 1 0 þ 2 4 5H ð0Þ þ H ð0Þ c22 c3 þ 2ðH ð0Þ 1Þc2 c4 ; 2 1 00 0 K 6 ¼ 10 22H ð0Þ þ 9H ð0Þ H ð0Þ c52 þ ð18 25H ð0Þ þ 4H 0 ð0ÞÞc2 c23 2 1 00 0 þ 3ðH ð0Þ 1Þc2 c5 þ 53H ð0Þ 30 15H ð0Þ þ H ð0Þ c32 c3 2 þ 2ð6 8H ð0Þ þ H 0 ð0ÞÞc22 c4 þ 7ðH ð0Þ 1Þc3 c4 ; so that if H(0) = 1, H 0 (0) = 2 and jH00 (0)j < 1, then from (18)–(21), we obtain the error equation 1 1 00 H ð0Þ 7 c32 c3 e6n þ Oðe7n Þ: enþ1 ¼ 6 H 00 ð0Þ c52 þ c2 c23 þ 2 2 This completes the proof.
ð20Þ
ð21Þ
ð22Þ
h
3. Some examples Some existing methods and also many other sixth-order iterative methods are special cases of Theorem 2.1. 1 It is clear that if we take H ðtÞ ¼ 12t and H ðtÞ ¼ 1þðbþ2Þt , then we obtain Grau et al.’s method and Sharma 1þbt et al.’s family defined by (6) and (9), respectively. In particular, the following methods are obtained as particular cases. Example 3.1. For the function H defined by H ðtÞ ¼ 1 þ 2t þ lt2 þ ct3 ;
ð23Þ
where l, c 2 R, we obtain the new two-parameter sixth-order family of methods f ðxn Þ ; f 0 ðxn Þ f ðxn Þ f ðy n Þ zn ¼ y n ; f ðxn Þ 2f ðy n Þ f 0 ðxn Þ f ðy n Þ f 2 ðy n Þ f 3 ðy n Þ f ðzn Þ xnþ1 ¼ zn 1 þ 2 þl 2 þc 3 : f ðxn Þ f ðxn Þ f ðxn Þ f 0 ðxn Þ
y n ¼ xn
ð24Þ ð25Þ ð26Þ
392
C. Chun, Y. Ham / Applied Mathematics and Computation 193 (2007) 389–394
Example 3.2. The function H defined by H ðtÞ ¼
c þ ðb þ 2cÞt ; c þ bt
ð27Þ
where c, b 2 R, can easily be shown to satisfy conditions of Theorem 2.1. Hence we obtain the new twoparameter sixth-order family of methods f ðxn Þ ; f 0 ðxn Þ f ðxn Þ f ðy n Þ zn ¼ y n ; f ðxn Þ 2f ðy n Þ f 0 ðxn Þ cf ðxn Þ þ ðb þ 2cÞf ðy n Þ f ðzn Þ : xnþ1 ¼ zn cf ðxn Þ þ bf ðy n Þ f 0 ðxn Þ y n ¼ xn
ð28Þ ð29Þ ð30Þ
We remark that this family contains the Grau et al. method defined by (6) as a special case, which is obtained when c = 1, b = 2. Example 3.3. The function H defined by H ðtÞ ¼ 1 þ
4t pffiffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 1 4t
ð31Þ
can easily be shown to satisfy conditions of Theorem 2.1. Thus, we obtain the new sixth-order method f ðxn Þ ; f 0 ðxn Þ f ðxn Þ f ðy n Þ zn ¼ y n ; f ðxn Þ 2f ðy n Þ f 0 ðxn Þ " y n ¼ xn
xnþ1
# 4f ðy n Þ f ðzn Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 ; ¼ zn 1 þ f ðxn Þ þ signðf ðxn ÞÞ f ðxn Þ 4f ðy n Þ f ðxn Þ
where the sign function is 8 if > <1 signðxÞ ¼ 1 if > : 0 if
ð32Þ ð33Þ ð34Þ
defined by x > 0; x < 0; x ¼ 0:
Each class member defined by (10)–(12) requires three function and one first derivative evaluations per iteration. In particular, it improves the order of convergence of Ostrowski’s method from four to six with additional function evaluation at the point iterated by Ostrowski’s method. If we consider the definition of 1 efficiency index [9] as pm , where p is the order of the method and m is the number of functional evaluations per iteration required by the method, we have that the methods obtained by the formula pffiffiffi (10)–(12) have the 1 efficiency index equal to 64 1:565, which is better than the one of Newton’s method 2 1:414. 4. Numerical examples We present some numerical test results for various iterative scheme which improve Ostrowski’s method in Table 1. Compared were Newton’s method (NM), Ostrowski’s method defined by (3) (OM), Grau et al.’s method defined by (6) (GM), Sharma et al.’s method (9) with b = 3 (SM), and (26) with l = c = 1 (CM1), (30) with c = b = 1 (CM2), (34) (CM3) introduced in the present contribution. All computations were done using MAPLE using 128 digit floating point arithmetics (Digits:¼128). Displayed in Table 1 is the number of iterations (IT) required such that j f ðxn Þ j< 1050 . We used the following test functions and display the approximate zero x* found up to the 28th decimal places.
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Table 1 Comparison of various iterative methods and Newton’s method f(x)
IT NM
OM
GM
SM
CM1
CM2
CM3
f1, x0 = 2 f1, x0 = 1
8 8
5 5
4 4
4 5
4 4
4 4
4 4
f2, x0 = 0 f2, x0 = 1
7 7
4 4
4 4
4 4
3 4
4 4
4 4
f3, x0 = 1 f3, x0 = 2
8 11
4 6
4 5
4 5
4 5
4 5
4 5
f4, x0 = 2 f4, x0 = 5
8 10
5 6
4 5
4 5
4 5
4 5
4 5
f5, x0 = 3 f5, x0 = 4
9 10
5 5
4 5
5 5
4 5
4 5
4 5
f6, x0 = 2 f6, x0 = 4
11 13
6 7
5 6
6 7
5 6
5 6
5 6
9 9
5 5
4 4
5 5
5 4
6 4
5 5
f7, x0 = 1 f7, x0 = 2.5
f1 ðxÞ ¼ x3 þ 4x2 10; 2
x ¼ 1:3652300134140968457608068290;
x
f2 ðxÞ ¼ x e 3x þ 2; x2
x ¼ 0:25753028543986076045536730494;
2
f3 ðxÞ ¼ xe sin x þ 3 cos x þ 5; x
2
f4 ðxÞ ¼ sinðxÞe þ lnðx þ 1Þ; 3
f5 ðxÞ ¼ ðx 1Þ 2;
x ¼ 0;
x ¼ 2:2599210498948731647672106073;
f6 ðxÞ ¼ ðx þ 2Þex 1; 2
x ¼ 1:2076478271309189270094167584;
f7 ðxÞ ¼ sin ðxÞ x2 þ 1;
x ¼ 0:44285440100238858314132800000; x ¼ 1:4044916482153412260350868178:
The computational results presented in Table 1 show that for most of the functions we tested, the presented methods are efficient and have at least equal performance as compared with the other methods of the same order which have been shown to really improve the computational efficiency of Ostrowski’s method (see [6,7]). Moreover, the methods can compete with Newton’s method. Thus, presented methods in this contribution can be considered as improvements of Ostrowski’s method. 5. Conclusion In this work, we presented a new sixth-order class of iterative methods which improves the order of convergence of Ostrowski’s method from four to six with an additional function evaluation. Per iteration each class member requires three function and one first derivative evaluations. Some existing sixth-order variants of Ostrowski’s method are obtained as special cases of our result. Some of the obtained methods were also compared in their performance to various other iterative methods of the same order, and it was observed that they demonstrate at least equal performance. References [1] A.M. Ostrowski, Solution of equations in Euclidean and Banach space, Academic Press, New York, 1973. [2] C. Chun, Some improvements of Jarratt’s method with sixth-order convergence, Appl. Math. Comput. (in press). doi:10.1016/ j.amc.2007.02.023. [3] M. Grau, J.L. Dı´az-Barrero, An improvement of the Euler–Chebyshev iterative method, J. Math. Anal. Appl. 315 (2006) 1–7. [4] J. Kou, Y. Li, X. Wang, An improvement of the Jarrat method, Appl. Math. Comput. (in press). doi:10.1016/j.amc.2006.12.062.
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[5] J. Kou, Y. Li, The improvements of Chebyshev–Halley methods with fifth-order convergence, Appl. Math. Comput. (in press). doi:10.1016/j.amc.2006.09.097. [6] M. Grau, J.L. Dı´az-Barrero, An improvement to Ostrowski root-finding method, J. Math. Anal. Appl. 173 (2006) 450–456. [7] J.R. Sharma, R.K. Guha, A family of modified Ostrowski methods with accelerated sixth-order convergence, Appl. Math. Comput. (in press). doi:10.1016/j.amc.2007.01.009. [8] C. Chun, Iterative methods improving Newton’s method by the decomposition method, Comput. Math. Appl. 50 (2005) 1559–1568. [9] W. Gautschi, Numerical Analysis: An introduction, Birkha¨user, 1997.