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Some fourth-order iterative methods for solving nonlinear equations
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Applied Mathematics and Computation 195 (2008) 454–459 www.elsevier.com/locate/amc
Some fourth-order iterative methods for solving nonlinear equations Changbum Chun School of Liberal Arts, Korea University of Technology and Education, Cheonan City, Chungnam 330-708, Republic of Korea
Abstract In this paper we present some fourth-order iterative methods for solving nonlinear equations, which contains the wellknown King’s fourth-order family as a particular case. Per iteration the methods require two function and one first derivative evaluations. Numerical comparisons are made with several other existing methods to show the performance of the presented methods. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Newton’s method; Iterative methods; Nonlinear equations; Order of convergence
1. Introduction In this paper, we consider iterative methods to find a simple root a, i.e., f ðaÞ ¼ 0 and f 0 ðaÞ 6¼ 0, of a nonlinear equation f ðxÞ ¼ 0. The case for multiple roots will not be considered in the present contribution. 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 fourth-order iterative methods have been proposed and analyzed for solving nonlinear equations which improve such classical methods as Newton’s method, Halley-like methods, etc. in a number of ways. They usually require three evaluations of the given function and its first derivative per iteration, see [2–9] and the references therein. These methods can also be viewed as obtained by taking an appropriate approximation to f 0 ðwn Þ in the following iteration scheme: xnþ1 ¼ wn
f ðwn Þ ; f 0 ðwn Þ
where wn ¼ /ðxn Þ, /ðxÞ is usually an iteration function such as the Newton iteration function. E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.04.105
ð2Þ
C. Chun / Applied Mathematics and Computation 195 (2008) 454–459
455
In this paper the search for new approximation to f 0 ðwn Þ in (2) when /ðxÞ is taken as Newton’s iteration function is continued. As a result, we present and analyze some fourth-order iterative methods. The presented methods require only two function and one first derivative evaluations per iteration. Several numerical comparisons are made with some other known iterative methods to show the performance of the methods presented in this contribution. 2. Development of methods and convergence analysis It is well known [10] that the two-substep Newton iteration scheme given by f ðxn Þ wn ¼ xn 0 ; f ðxn Þ f ðwn Þ xnþ1 ¼ wn 0 f ðwn Þ has fourth-order convergence in some neighborhood of a . To derive new methods, we approximate f 0 ðwn Þ given in (4) as follows: f 0 ðwn Þ f 0 ðxn Þhðun Þ;
ð3Þ ð4Þ
ð5Þ
nÞ and h(t) is a real valued function to be determined later so that the two-step iteration scheme where un ¼ ff ðw ðxn Þ defined by
f ðxn Þ ; f 0 ðxn Þ f ðwn Þ ; ¼ wn 0 f ðxn Þhðun Þ
wn ¼ xn
ð6Þ
xnþ1
ð7Þ
where f ðwn Þ ð8Þ f ðxn Þ may yield a fourth-order iteration scheme, in which the derivative is computed every other substep only. We then have the following convergence result. un ¼
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 hð0Þ ¼ 1, h0 ð0Þ ¼ 2 and jh00 ð0Þj < 1. If x0 is sufficiently close to a, then the two-step method defined by (6) and (7) is of fourth-order, and satisfy the error equation h00 ð0Þ 2 ð9Þ enþ1 ¼ c2 1 þ c2 c3 e4n þ Oðe5n Þ; 2 where en ¼ xn a, ck ¼ f ðkÞ ðaÞ=k!f 0 ðaÞ. Proof. Let a be a simple zero of f and h a function with hð0Þ ¼ 1, h0 ð0Þ ¼ 2 and jh00 ð0Þj < 1. We let un ¼ f ðwn Þ=f ðxn Þ. Using Taylor expansion around xn ¼ a and taking into account f ðaÞ ¼ 0, we have f ðxn Þ ¼ f 0 ðaÞ½en þ c2 e2n þ c3 e3n þ c4 e4n þ Oðe5n Þ; 0
0
f ðxn Þ ¼ f ðaÞ½1 þ 2c2 en þ where en ¼ xn a and ck ¼
3c3 e2n
ðkÞ 1 f ðaÞ ;k k! f 0ðaÞ
þ
4c4 e3n
þ
Oðe4n Þ;
ð10Þ ð11Þ
¼ 1; 2; . . . By a simple calculation, we get
f ðxn Þ ¼ en c2 e2n þ 2ðc22 c3 Þe3n þ ð7c2 c3 4c32 3c4 Þe4n þ Oðe5n Þ; f 0 ðxn Þ
ð12Þ
so that wn ¼ xn
f ðxn Þ ¼ a þ c2 e2n þ 2ðc3 c22 Þe3n þ ð7c2 c3 þ 4c32 þ 3c4 Þe4n þ Oðe5n Þ; f 0 ðxn Þ
ð13Þ
456
C. Chun / Applied Mathematics and Computation 195 (2008) 454–459
whence by expanding f ðwn Þ about a, we have f ðwn Þ ¼ f 0 ðaÞ½c2 e2n þ 2ðc3 c22 Þe3n þ ð7c2 c3 þ 5c32 þ 3c4 Þe4n þ Oðe5n Þ:
ð14Þ
Dividing (14) by (10), we have un ¼
f ðwn Þ ¼ c2 en þ ð2c3 3c22 Þe2n þ Oðe3n Þ: f ðxn Þ
ð15Þ
From the assumption on h and (15), we have h00 ð0Þ 2 un þ Oðu3n Þ hðun Þ ¼ hð0Þ þ h0 ð0Þun þ 2 h00 ð0Þ 2 2 ¼ 1 2c2 en þ 4c3 þ 6 þ c2 en þ Oðe3n Þ; 2 whence, we easily obtain from (11) h00 ð0Þ 2 f 0 ðxn Þhðun Þ ¼ f 0 ðaÞ 1 þ 2 þ c2 c3 e2n þ Oðe3n Þ : 2 Dividing (14) by (18) gives us f ðwn Þ ¼ c2 e2n þ 2ðc3 c22 Þe3n þ 0 f ðxn Þhðun Þ
h00 ð0Þ 3 3 c2 6c2 c3 þ 3c4 e4n þ Oðe5n Þ: 2
From (13) and (19), we now have xnþ1 ¼ wn f ðwn Þf 0 ðxn Þhðun Þ ¼ a þ c2
h00 ð0Þ 2 1þ c2 c3 e4n þ Oðe5n Þ: 2
ð16Þ ð17Þ
ð18Þ
ð19Þ
ð20Þ
Since enþ1 ¼ xnþ1 a, we thus obtain the error equation h00 ð0Þ 2 c2 c3 e4n þ Oðe5n Þ: enþ1 ¼ c2 1 þ 2 This means that the two-step method defined by (6) and (7) has fourth-order convergence. This completes the proof. h 3. Some examples Some well-known methods and also many other fourth-order iterative methods are special cases of Theorem 2.1. In particular, the following methods are obtained as particular cases. Example 3.1. For h given by hðtÞ ¼
1 þ ðb 2Þt ¼ 1 2t þ 2bt2 2b2 t3 þ 2b3 t4 þ ; 1 þ bt
ð21Þ
we obtain King’s fourth-order family, which contains Traub–Ostrowski’s fourth-order method as a particular case when b = 0. Example 3.2. For h given by hðtÞ ¼ 1 2t þ bt2 þ ct3 ;
ð22Þ
we obtain the new two-parameter fourth-order family xnþ1 ¼ wn
f ðwn Þ f ðx Þ h i 0 n : 2 ðw Þ 3 ðw Þ f ðw Þ f f f 0 ðxn Þ 1 2 f ðxnnÞ þ b f 2 ðxnnÞ þ c f 3 ðxnnÞ f ðxn Þ
ð23Þ
C. Chun / Applied Mathematics and Computation 195 (2008) 454–459
457
Example 3.3. For h given by hðtÞ ¼
1 ¼ 1 2t þ 4t2 ; 1 þ 2t
ð24Þ
we obtain the new fourth-order method f ðwn Þ f 2 ðwn Þ f ðxn Þ þ2 2 : xnþ1 ¼ xn 1 þ f ðxn Þ f ðxn Þ f 0 ðxn Þ
ð25Þ
Example 3.4. For h given by hðtÞ ¼ ð1 þ tÞ2 ¼ 1 2t þ 3t2 4t3 þ ;
ð26Þ
we obtain the new fourth-order method f ðwn Þ f 2 ðwn Þ f 3 ðwn Þ f ðxn Þ þ2 2 þ : xnþ1 ¼ xn 1 þ f ðxn Þ f ðxn Þ f 3 ðxn Þ f 0 ðxn Þ
ð27Þ
It should be mentioned that the methods defined by (6) and (7) are of fourth-order even though they require but two function and one first derivative evaluations per iteration. If we consider the definition of efficiency 1 index [11] as pm , where p is the order of the method and m is the number of function evaluations per iteration 1 3 required by the method, then the presented pffiffiffi methods have the efficiency index equal to 4 1:587, which is better than the one of Newton’s method 2 1:414. 4. Numerical examples All computations were done using the Maple package using 64 digit floating point arithmetics. We accept an approximate solution rather than the exact root, depending on the precision () of the computer. We use the following stopping criteria for computer programs: (i) j xnþ1 xn j< ; (ii) j f ðxnþ1 Þ j< ; and so, when the stopping criterion is satisfied, xnþ1 is taken as the exact root a computed. For numerical illustrations we used the fixed stopping criterion ¼ 1015 . We present some numerical test results for various quadratically convergent classical iterative schemes in Table 1. Compared were Newton’s method (NM), King’s method with b = 3 [4] (KM) defined by Table 1 Comparison of various quadratically convergent iterative methods and Newton’s method f(x)
IT
NFE
NM
JM
TM
KM
CM1
CM2
NM
JM
TM
KM
CM1
CM2
f1 ; x0 ¼ 0:3 f1 ; x0 ¼ 1
55 6
46 4
46 4
49 4
25 4
57 4
110 12
138 12
138 12
147 12
75 12
171 12
f2 ; x0 ¼ 0 f2 ; x0 ¼ 1
5 5
3 3
3 3
3 3
3 3
3 3
10 10
9 9
9 9
9 9
9 9
9 9
f3 ; x0 ¼ 1 f3 ; x0 ¼ 2
6 9
4 5
4 5
5 6
5 6
4 6
12 18
12 15
12 15
15 18
15 18
12 18
f4 ; x0 ¼ 2 f4 ; x0 ¼ 5
6 8
4 5
4 5
6 5
5 5
5 5
12 16
12 15
12 15
18 15
15 15
15 15
f5 ; x0 ¼ 3 f5 ; x0 ¼ 4
7 8
4 5
4 5
4 5
4 5
4 5
14 16
12 15
12 15
12 15
12 15
12 15
f6 ; x0 ¼ 2 f6 ; x0 ¼ 3:5
9 11
5 6
5 6
6 7
6 7
6 7
18 22
15 18
15 18
18 21
18 21
18 21
f7 ; x0 ¼ 1 f7 ; x0 ¼ 2
7 6
4 4
4 4
8 4
5 4
5 4
14 12
12 12
12 12
24 12
15 12
15 12
458
C. Chun / Applied Mathematics and Computation 195 (2008) 454–459
f ðwn Þ f ðxn Þ þ bf ðwn Þ ; f 0 ðxn Þ f ðxn Þ þ ðb 2Þf ðwn Þ
xnþ1 ¼ wn
ð28Þ
where wn ¼ xn f ðxn Þ=f 0 ðxn Þ, Traub–Ostrowski’s method [10] (TM) defined by xnþ1 ¼ xn
f ðwn Þ f ðxn Þ f ðxn Þ ; 2f ðwn Þ f ðxn Þ f 0 ðxn Þ
ð29Þ
where wn ¼ xn f ðxn Þ=f 0 ðxn Þ, Jarratt’s method [5] (JM) defined by 3 f 0 ðzn Þ f 0 ðxn Þ f ðxn Þ ; xnþ1 ¼ xn 1 2 3f 0 ðzn Þ f 0 ðxn Þ f 0 ðxn Þ
ð30Þ
where zn ¼ xn 2f ðxn Þ=3f 0 ðxn Þ, and the methods (25) (CM1) and (27) (CM2) introduced in the present contribution. We used the following test functions and display the approximate zero x* found up to the 28th decimal places: 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; 2
x
f4 ðxÞ ¼ sinðxÞe þ lnðx þ 1Þ; 3
f5 ðxÞ ¼ ðx 1Þ 2; 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 x þ 1;
x ¼ 0:44285440100238858314132800000; x ¼ 1:4044916482153412260350868178:
As convergence criterion, it was required that the distance of two consecutive approximations for the zero was less than 1015. Displayed in Table 1 are the number of iterations to approximate the zero (IT) and the number of function evaluations (NFE) counted as the sum of the number of evaluations of the function itself plus the number of evaluations of the derivative. The computational results presented in Table 1 show that in almost all of the cases the presented methods converge more rapidly than Newton’s method and require the less number of function evaluations, so that they can compete with Newton’s method. Furthermore, for most of the functions we tested, the new methods have at least equal performance compared to the other existing methods of the same order. 5. Conclusion In this work we presented some new fourth-order iterative methods for solving nonlinear equations. The presented methods require two function and one first derivative evaluations per iteration. Some of the obtained methods were compared in their performance to several existing fourth-order methods, and it was observed that they have at least equal performance. Our approach can be continuously applied to develop higher order iterative methods. References [1] A.M. Ostrowski, Solution of Equations in Euclidean and Banach Space, Academic Press, New York, 1973. [2] C. Chun, Construction of Newton-like iteration methods for solving nonlinear equations, Numer. Math. 104 (3) (2006) 297–315. [3] C. Chun, Certain improvements of Chebyshev–Halley methods with accelerated fourth-order convergence, Appl. Math. Comput. (2006), doi:10.1016/j.amc.2006.11.118. [4] R. King, A family of fourth-order methods for nonlinear equations, SIAM J. Numer. Anal. 10 (5) (1973) 876–879. [5] I.K. Argyros, D. Chen, Q. Qian, The Jarratt method in Banach space setting, J. Comput. Appl. Math. 51 (1994) 103–106. [6] J. Kou, Y. Li, X. Wang, A variant of super-Halley method with accelerated fourth-order convergence, Appl. Math. Comput. (2006), doi:10.1016/j.amc.2006.07.118. [7] J. Kou, Y. Li, X. Wang, Fourth-order iterative methods free from second derivative, Appl. Math. Comput. (2006), doi:10.1016/ j.amc.2006.05.189.
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[8] M.A. Noor, F. Ahmad, Fourth-order convergent iterative method for nonlinear equation, Appl. Math. Comput. (2006), doi:10.1016/ j.amc.2006.04.068. [9] J.R. Sharma, R.K. Goyal, Fourth-order derivative-free methods for solving non-linear equations, Int. J. Comput. Math. 83 (1) (2006) 101–106. [10] J.F. Traub, Iterative Methods for the Solution of Equations, Chelsea Publishing Company, New York, 1977. [11] W. Gautschi, Numerical Analysis: An Introduction, Birkha¨user, 1997.
Applied Mathematics and Computation 195 (2008) 454–459 www.elsevier.com/locate/amc
Some fourth-order iterative methods for solving nonlinear equations Changbum Chun School of Liberal Arts, Korea University of Technology and Education, Cheonan City, Chungnam 330-708, Republic of Korea
Abstract In this paper we present some fourth-order iterative methods for solving nonlinear equations, which contains the wellknown King’s fourth-order family as a particular case. Per iteration the methods require two function and one first derivative evaluations. Numerical comparisons are made with several other existing methods to show the performance of the presented methods. Ó 2007 Elsevier Inc. All rights reserved. Keywords: Newton’s method; Iterative methods; Nonlinear equations; Order of convergence
1. Introduction In this paper, we consider iterative methods to find a simple root a, i.e., f ðaÞ ¼ 0 and f 0 ðaÞ 6¼ 0, of a nonlinear equation f ðxÞ ¼ 0. The case for multiple roots will not be considered in the present contribution. 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 fourth-order iterative methods have been proposed and analyzed for solving nonlinear equations which improve such classical methods as Newton’s method, Halley-like methods, etc. in a number of ways. They usually require three evaluations of the given function and its first derivative per iteration, see [2–9] and the references therein. These methods can also be viewed as obtained by taking an appropriate approximation to f 0 ðwn Þ in the following iteration scheme: xnþ1 ¼ wn
f ðwn Þ ; f 0 ðwn Þ
where wn ¼ /ðxn Þ, /ðxÞ is usually an iteration function such as the Newton iteration function. E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.04.105
ð2Þ
C. Chun / Applied Mathematics and Computation 195 (2008) 454–459
455
In this paper the search for new approximation to f 0 ðwn Þ in (2) when /ðxÞ is taken as Newton’s iteration function is continued. As a result, we present and analyze some fourth-order iterative methods. The presented methods require only two function and one first derivative evaluations per iteration. Several numerical comparisons are made with some other known iterative methods to show the performance of the methods presented in this contribution. 2. Development of methods and convergence analysis It is well known [10] that the two-substep Newton iteration scheme given by f ðxn Þ wn ¼ xn 0 ; f ðxn Þ f ðwn Þ xnþ1 ¼ wn 0 f ðwn Þ has fourth-order convergence in some neighborhood of a . To derive new methods, we approximate f 0 ðwn Þ given in (4) as follows: f 0 ðwn Þ f 0 ðxn Þhðun Þ;
ð3Þ ð4Þ
ð5Þ
nÞ and h(t) is a real valued function to be determined later so that the two-step iteration scheme where un ¼ ff ðw ðxn Þ defined by
f ðxn Þ ; f 0 ðxn Þ f ðwn Þ ; ¼ wn 0 f ðxn Þhðun Þ
wn ¼ xn
ð6Þ
xnþ1
ð7Þ
where f ðwn Þ ð8Þ f ðxn Þ may yield a fourth-order iteration scheme, in which the derivative is computed every other substep only. We then have the following convergence result. un ¼
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 hð0Þ ¼ 1, h0 ð0Þ ¼ 2 and jh00 ð0Þj < 1. If x0 is sufficiently close to a, then the two-step method defined by (6) and (7) is of fourth-order, and satisfy the error equation h00 ð0Þ 2 ð9Þ enþ1 ¼ c2 1 þ c2 c3 e4n þ Oðe5n Þ; 2 where en ¼ xn a, ck ¼ f ðkÞ ðaÞ=k!f 0 ðaÞ. Proof. Let a be a simple zero of f and h a function with hð0Þ ¼ 1, h0 ð0Þ ¼ 2 and jh00 ð0Þj < 1. We let un ¼ f ðwn Þ=f ðxn Þ. Using Taylor expansion around xn ¼ a and taking into account f ðaÞ ¼ 0, we have f ðxn Þ ¼ f 0 ðaÞ½en þ c2 e2n þ c3 e3n þ c4 e4n þ Oðe5n Þ; 0
0
f ðxn Þ ¼ f ðaÞ½1 þ 2c2 en þ where en ¼ xn a and ck ¼
3c3 e2n
ðkÞ 1 f ðaÞ ;k k! f 0ðaÞ
þ
4c4 e3n
þ
Oðe4n Þ;
ð10Þ ð11Þ
¼ 1; 2; . . . By a simple calculation, we get
f ðxn Þ ¼ en c2 e2n þ 2ðc22 c3 Þe3n þ ð7c2 c3 4c32 3c4 Þe4n þ Oðe5n Þ; f 0 ðxn Þ
ð12Þ
so that wn ¼ xn
f ðxn Þ ¼ a þ c2 e2n þ 2ðc3 c22 Þe3n þ ð7c2 c3 þ 4c32 þ 3c4 Þe4n þ Oðe5n Þ; f 0 ðxn Þ
ð13Þ
456
C. Chun / Applied Mathematics and Computation 195 (2008) 454–459
whence by expanding f ðwn Þ about a, we have f ðwn Þ ¼ f 0 ðaÞ½c2 e2n þ 2ðc3 c22 Þe3n þ ð7c2 c3 þ 5c32 þ 3c4 Þe4n þ Oðe5n Þ:
ð14Þ
Dividing (14) by (10), we have un ¼
f ðwn Þ ¼ c2 en þ ð2c3 3c22 Þe2n þ Oðe3n Þ: f ðxn Þ
ð15Þ
From the assumption on h and (15), we have h00 ð0Þ 2 un þ Oðu3n Þ hðun Þ ¼ hð0Þ þ h0 ð0Þun þ 2 h00 ð0Þ 2 2 ¼ 1 2c2 en þ 4c3 þ 6 þ c2 en þ Oðe3n Þ; 2 whence, we easily obtain from (11) h00 ð0Þ 2 f 0 ðxn Þhðun Þ ¼ f 0 ðaÞ 1 þ 2 þ c2 c3 e2n þ Oðe3n Þ : 2 Dividing (14) by (18) gives us f ðwn Þ ¼ c2 e2n þ 2ðc3 c22 Þe3n þ 0 f ðxn Þhðun Þ
h00 ð0Þ 3 3 c2 6c2 c3 þ 3c4 e4n þ Oðe5n Þ: 2
From (13) and (19), we now have xnþ1 ¼ wn f ðwn Þf 0 ðxn Þhðun Þ ¼ a þ c2
h00 ð0Þ 2 1þ c2 c3 e4n þ Oðe5n Þ: 2
ð16Þ ð17Þ
ð18Þ
ð19Þ
ð20Þ
Since enþ1 ¼ xnþ1 a, we thus obtain the error equation h00 ð0Þ 2 c2 c3 e4n þ Oðe5n Þ: enþ1 ¼ c2 1 þ 2 This means that the two-step method defined by (6) and (7) has fourth-order convergence. This completes the proof. h 3. Some examples Some well-known methods and also many other fourth-order iterative methods are special cases of Theorem 2.1. In particular, the following methods are obtained as particular cases. Example 3.1. For h given by hðtÞ ¼
1 þ ðb 2Þt ¼ 1 2t þ 2bt2 2b2 t3 þ 2b3 t4 þ ; 1 þ bt
ð21Þ
we obtain King’s fourth-order family, which contains Traub–Ostrowski’s fourth-order method as a particular case when b = 0. Example 3.2. For h given by hðtÞ ¼ 1 2t þ bt2 þ ct3 ;
ð22Þ
we obtain the new two-parameter fourth-order family xnþ1 ¼ wn
f ðwn Þ f ðx Þ h i 0 n : 2 ðw Þ 3 ðw Þ f ðw Þ f f f 0 ðxn Þ 1 2 f ðxnnÞ þ b f 2 ðxnnÞ þ c f 3 ðxnnÞ f ðxn Þ
ð23Þ
C. Chun / Applied Mathematics and Computation 195 (2008) 454–459
457
Example 3.3. For h given by hðtÞ ¼
1 ¼ 1 2t þ 4t2 ; 1 þ 2t
ð24Þ
we obtain the new fourth-order method f ðwn Þ f 2 ðwn Þ f ðxn Þ þ2 2 : xnþ1 ¼ xn 1 þ f ðxn Þ f ðxn Þ f 0 ðxn Þ
ð25Þ
Example 3.4. For h given by hðtÞ ¼ ð1 þ tÞ2 ¼ 1 2t þ 3t2 4t3 þ ;
ð26Þ
we obtain the new fourth-order method f ðwn Þ f 2 ðwn Þ f 3 ðwn Þ f ðxn Þ þ2 2 þ : xnþ1 ¼ xn 1 þ f ðxn Þ f ðxn Þ f 3 ðxn Þ f 0 ðxn Þ
ð27Þ
It should be mentioned that the methods defined by (6) and (7) are of fourth-order even though they require but two function and one first derivative evaluations per iteration. If we consider the definition of efficiency 1 index [11] as pm , where p is the order of the method and m is the number of function evaluations per iteration 1 3 required by the method, then the presented pffiffiffi methods have the efficiency index equal to 4 1:587, which is better than the one of Newton’s method 2 1:414. 4. Numerical examples All computations were done using the Maple package using 64 digit floating point arithmetics. We accept an approximate solution rather than the exact root, depending on the precision () of the computer. We use the following stopping criteria for computer programs: (i) j xnþ1 xn j< ; (ii) j f ðxnþ1 Þ j< ; and so, when the stopping criterion is satisfied, xnþ1 is taken as the exact root a computed. For numerical illustrations we used the fixed stopping criterion ¼ 1015 . We present some numerical test results for various quadratically convergent classical iterative schemes in Table 1. Compared were Newton’s method (NM), King’s method with b = 3 [4] (KM) defined by Table 1 Comparison of various quadratically convergent iterative methods and Newton’s method f(x)
IT
NFE
NM
JM
TM
KM
CM1
CM2
NM
JM
TM
KM
CM1
CM2
f1 ; x0 ¼ 0:3 f1 ; x0 ¼ 1
55 6
46 4
46 4
49 4
25 4
57 4
110 12
138 12
138 12
147 12
75 12
171 12
f2 ; x0 ¼ 0 f2 ; x0 ¼ 1
5 5
3 3
3 3
3 3
3 3
3 3
10 10
9 9
9 9
9 9
9 9
9 9
f3 ; x0 ¼ 1 f3 ; x0 ¼ 2
6 9
4 5
4 5
5 6
5 6
4 6
12 18
12 15
12 15
15 18
15 18
12 18
f4 ; x0 ¼ 2 f4 ; x0 ¼ 5
6 8
4 5
4 5
6 5
5 5
5 5
12 16
12 15
12 15
18 15
15 15
15 15
f5 ; x0 ¼ 3 f5 ; x0 ¼ 4
7 8
4 5
4 5
4 5
4 5
4 5
14 16
12 15
12 15
12 15
12 15
12 15
f6 ; x0 ¼ 2 f6 ; x0 ¼ 3:5
9 11
5 6
5 6
6 7
6 7
6 7
18 22
15 18
15 18
18 21
18 21
18 21
f7 ; x0 ¼ 1 f7 ; x0 ¼ 2
7 6
4 4
4 4
8 4
5 4
5 4
14 12
12 12
12 12
24 12
15 12
15 12
458
C. Chun / Applied Mathematics and Computation 195 (2008) 454–459
f ðwn Þ f ðxn Þ þ bf ðwn Þ ; f 0 ðxn Þ f ðxn Þ þ ðb 2Þf ðwn Þ
xnþ1 ¼ wn
ð28Þ
where wn ¼ xn f ðxn Þ=f 0 ðxn Þ, Traub–Ostrowski’s method [10] (TM) defined by xnþ1 ¼ xn
f ðwn Þ f ðxn Þ f ðxn Þ ; 2f ðwn Þ f ðxn Þ f 0 ðxn Þ
ð29Þ
where wn ¼ xn f ðxn Þ=f 0 ðxn Þ, Jarratt’s method [5] (JM) defined by 3 f 0 ðzn Þ f 0 ðxn Þ f ðxn Þ ; xnþ1 ¼ xn 1 2 3f 0 ðzn Þ f 0 ðxn Þ f 0 ðxn Þ
ð30Þ
where zn ¼ xn 2f ðxn Þ=3f 0 ðxn Þ, and the methods (25) (CM1) and (27) (CM2) introduced in the present contribution. We used the following test functions and display the approximate zero x* found up to the 28th decimal places: 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; 2
x
f4 ðxÞ ¼ sinðxÞe þ lnðx þ 1Þ; 3
f5 ðxÞ ¼ ðx 1Þ 2; 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 x þ 1;
x ¼ 0:44285440100238858314132800000; x ¼ 1:4044916482153412260350868178:
As convergence criterion, it was required that the distance of two consecutive approximations for the zero was less than 1015. Displayed in Table 1 are the number of iterations to approximate the zero (IT) and the number of function evaluations (NFE) counted as the sum of the number of evaluations of the function itself plus the number of evaluations of the derivative. The computational results presented in Table 1 show that in almost all of the cases the presented methods converge more rapidly than Newton’s method and require the less number of function evaluations, so that they can compete with Newton’s method. Furthermore, for most of the functions we tested, the new methods have at least equal performance compared to the other existing methods of the same order. 5. Conclusion In this work we presented some new fourth-order iterative methods for solving nonlinear equations. The presented methods require two function and one first derivative evaluations per iteration. Some of the obtained methods were compared in their performance to several existing fourth-order methods, and it was observed that they have at least equal performance. Our approach can be continuously applied to develop higher order iterative methods. References [1] A.M. Ostrowski, Solution of Equations in Euclidean and Banach Space, Academic Press, New York, 1973. [2] C. Chun, Construction of Newton-like iteration methods for solving nonlinear equations, Numer. Math. 104 (3) (2006) 297–315. [3] C. Chun, Certain improvements of Chebyshev–Halley methods with accelerated fourth-order convergence, Appl. Math. Comput. (2006), doi:10.1016/j.amc.2006.11.118. [4] R. King, A family of fourth-order methods for nonlinear equations, SIAM J. Numer. Anal. 10 (5) (1973) 876–879. [5] I.K. Argyros, D. Chen, Q. Qian, The Jarratt method in Banach space setting, J. Comput. Appl. Math. 51 (1994) 103–106. [6] J. Kou, Y. Li, X. Wang, A variant of super-Halley method with accelerated fourth-order convergence, Appl. Math. Comput. (2006), doi:10.1016/j.amc.2006.07.118. [7] J. Kou, Y. Li, X. Wang, Fourth-order iterative methods free from second derivative, Appl. Math. Comput. (2006), doi:10.1016/ j.amc.2006.05.189.
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[8] M.A. Noor, F. Ahmad, Fourth-order convergent iterative method for nonlinear equation, Appl. Math. Comput. (2006), doi:10.1016/ j.amc.2006.04.068. [9] J.R. Sharma, R.K. Goyal, Fourth-order derivative-free methods for solving non-linear equations, Int. J. Comput. Math. 83 (1) (2006) 101–106. [10] J.F. Traub, Iterative Methods for the Solution of Equations, Chelsea Publishing Company, New York, 1977. [11] W. Gautschi, Numerical Analysis: An Introduction, Birkha¨user, 1997.