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A fifth-order iterative method for solving nonlinear equations
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Applied Mathematics and Computation 194 (2007) 287–290 www.elsevier.com/locate/amc
A fifth-order iterative method for solving nonlinear equations YoonMee Ham
a,1
, Changbum Chun
b,*
a
b
Department of Mathematics, Kyonggi University, Suwon 443-760, Republic of Korea School of Liberal Arts, Korea University of Technology and Education, Cheonan City, Chungnam 330-708, Republic of Korea
Abstract In this paper, we present a new fifth-order method for solving nonlinear equations. Per iteration the new method requires two function and two first derivative evaluations. It is shown that the new method is fifth-order convergent. Several numerical examples are given to illustrate the performance of the presented method. 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. 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 fifth-order iterative methods have been proposed and analyzed for solving nonlinear equations that improve some classical methods such as the Newton method, Chebyshev–Halley methods, Ostrowski’s method. It has been demonstrated that the methods are efficient, and can compete with Newton’s method. For more details about them and the other fifth-order methods, see [2–6] and the reference therein. Motivated by the recent activities in this direction, in this paper we present and analyze a new two-step iterative method for solving nonlinear equations. The method is shown to be fifth-order convergent, and several numerical examples are given to show the performance of the new method presented in this contribution.
*
1
Corresponding author. E-mail addresses: [email protected] (Y. Ham), [email protected] (C. Chun). 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.2007.04.005
288
Y. Ham, C. Chun / Applied Mathematics and Computation 194 (2007) 287–290
2. Development of method and convergence analysis To develop the new method, let us consider the iteration scheme in the form f ðxn Þ ; f 0 ðxn Þ Af 0 ðy Þ þ Bf 0 ðxn Þ f ðy n Þ ; ¼ yn 0 n Cf ðy n Þ þ Df 0 ðxn Þ f 0 ðxn Þ
y n ¼ xn
ð2Þ
xnþ1
ð3Þ
where A; B; C; D are the disposable parameters. For the iterative method defined by (2) and (3), we have the following convergence result. Theorem 2.1. Let a 2 I be a simple zero of a sufficiently differentiable function f : I ! R for an open interval I. If x0 is sufficiently close to a, and A; B; C; D satisfy the condition A þ B ¼ C þ D;
C ¼ B þ 2A;
B ¼ 3A;
C þ D 6¼ 0;
ð4Þ
then the method defined by (2) and (3) is of fifth-order. Proof. Let a be a simple zero of f. Consider the iteration function F defined by F ðxÞ ¼ yðxÞ
Af 0 ðyðxÞÞ þ Bf 0 ðxÞ f ðyðxÞÞ ; Cf 0 ðyðxÞÞ þ Df 0 ðxÞ f 0 ðxÞ
ð5Þ
where yðxÞ ¼ x f ðxÞ=f 0 ðxÞ. 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 ðxn Þ around x ¼ a (see [7] for details). We find after simplifying that CþDAB 2 2 c2 en þ ½ðB þ 2A CÞc22 þ ðC þ D A BÞc3 e3n CþD CþD 1 þ ½K 1 c32 þ K 2 c2 c3 þ K 3 c4 e4n þ Oðe5n Þ; 2 ðC þ DÞ
xnþ1 ¼ F ðxn Þ ¼ a þ
ð6Þ
where en ¼ xn a and ck ¼ f ðkÞ ðaÞ=k!f 0 ðaÞ, K 1 ¼ 12CD þ 4D2 þ 4C 2 13AC 27AD 3BC 13BD; K 2 ¼ 7½CðB þ 2A C 2DÞ þ Dð3A þ 2B DÞ;
ð7Þ ð8Þ
K 3 ¼ 3ðC þ DÞðC þ D A BÞ:
ð9Þ
An easy manipulation shows that K 1 ¼ K 2 ¼ K 3 ¼ 0 when A þ B ¼ C þ D; C ¼ B þ 2A; B ¼ 3A, which thus completes the proof. h Solving system of the equations of condition (4), we find that A ¼ t, B ¼ 3t, C ¼ 5t, D ¼ t, (t 2 RÞ, thereby we obtain the new fifth-order iterative method f ðxn Þ ; f 0 ðxn Þ f 0 ðy Þ þ 3f 0 ðxn Þ f ðy n Þ : ¼ yn 0 n 5f ðy n Þ f 0 ðxn Þ f 0 ðxn Þ
y n ¼ xn
ð10Þ
xnþ1
ð11Þ
Per iteration this method requires two function and two first derivative evaluations. If we consider the defini1 tion of efficiency index [8] 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 iteration formula defined byp(10) ffiffiffi and (11) has 1 the efficiency index equal to 54 1:495, which is better than the one of Newton’s method 2 1:414.
Y. Ham, C. Chun / Applied Mathematics and Computation 194 (2007) 287–290
289
3. Numerical examples All computations were done using MAPLE using 64 digit floating point arithmetics (Digits :¼ 64). 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) jxnþ1 xn j < , (ii) jf ðxnþ1 Þj < ; and so, when the stopping criterion is satisfied, xnþ1 is taken as the exact root a computed. We used ¼ 1015 . We present some numerical test results for various fifth-order convergent iterative schemes in Table 1. Compared were the Grau and Dı´az-Barrero method (GM) [2] defined by f 00 ðxn Þðf ðxn Þ þ f ðzn ÞÞ f ðxn Þ þ f ðzn Þ xnþ1 ¼ xn 1 þ ; ð12Þ 2f 02 ðxn Þ f 0 ðxn Þ 00 where zn ¼ xn 1 þ 12 f fðx02nðxÞfnðxÞ n Þ ff0ðxðxnnÞÞ, Noor and Noor’s method (NNM) [4] defined by xnþ1 ¼ xn
ð13Þ
hðxÞ ¼ f ðxÞ f ðxn Þ ðx xn Þf 0 ðxn Þ 12 ðx xn Þ2 f 00 ðxn Þ
where tðxn Þ ¼
2½f ðxn Þ þ hðzn Þf 0 ðxn Þ ; 2f 02 ðxn Þ ½f ðxn Þ þ hðzn Þf 00 ðxn Þ and
f 00 ðxn Þf ðxn Þ , f 02 ðxn Þ
tðxn Þ f ðxn Þ zn ¼ xn 1 þ 2tðx , f 0 ðxn Þ nÞ
where
the method of Kou and Li (KM) [5] defined by Mðxn Þ f ðzn Þ xnþ1 ¼ zn 1 þ ; ð14Þ 1 þ Mðxn Þ f 0 ðxn Þ 00 tðxn Þ f ðxn Þ ðzn ÞÞ where zn ¼ xn 1 þ 12 1tðx and Mðxn Þ ¼ f ðxn Þðff 02ðxðxn nÞf , and the method (10) and (11) (FM) introduced f 0 ðxn Þ Þ nÞ 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; x
2
f4 ðxÞ ¼ sinðxÞe þ lnðx þ 1Þ;
x ¼ 1:2076478271309189270094167584;
x ¼ 0;
Table 1 Comparison of various fifth-order convergent iterative methods f(x)
IT GM
NNM
KM
FM
f1 ; x0 ¼ 0:3 f1 ; x0 ¼ 1
27 4
div 4
24 4
11 3
f2 ; x0 ¼ 0 f2 ; x0 ¼ 1
3 4
4 4
3 4
3 3
f3 ; x0 ¼ 1 f3 ; x0 ¼ 2
4 5
4 div
4 div
4 5
f4 ; x0 ¼ 2 f4 ; x0 ¼ 5
4 6
4 6
4 4
4 4
f5 ; x0 ¼ 3 f5 ; x0 ¼ 4
4 5
5 6
4 4
4 4
f6 ; x0 ¼ 2 f6 ; x0 ¼ 3:5
5 6
9 15
div div
5 5
31 4
5 5
5 4
4 4
f7 ; x0 ¼ 1 f7 ; x0 ¼ 2
290
Y. Ham, C. Chun / Applied Mathematics and Computation 194 (2007) 287–290 3
f5 ðxÞ ¼ ðx 1Þ 2; x ¼ 2:2599210498948731647672106073; f6 ðxÞ ¼ ðx þ 2Þex 1; x ¼ 0:44285440100238858314132800000; f7 ðxÞ ¼ sin2 ðxÞ x2 þ 1;
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 is the number of iterations to approximate the zero (IT). In the Table, the ‘div’ means that the sequence of approximate zeros produced from the corresponding method does not converge within the maximum iteration number set to n = 200 under the stopping criterion ¼ 1015 , so the sequence may slow in convergence. As far as the results presented in Table 1 are concerned, for most of the functions we tested, the proposed method is more stable and works better in its performance as compared with the other methods of the same order. We refer to [5] for some numerical results showing that the Kou and Li method (KM) can compete with Newton’s method. 4. Conclusion In this work we presented a new fifth-order iterative method for solving nonlinear equations which requires two function and two first derivative evaluations per step. It was compared in its performance to some fifthorder methods, and the proposed method has been observed to have at least better performance and more stability. References [1] A.M. Ostrowski, Solution of Equations in Euclidean and Banach Space, Academic Press, New York, 1973. [2] M. Grau, J.L. Dı´az-Barrero, An improvement of the Euler–Chebyshev iterative method, J. Math. Anal. Appl. 315 (2006) 1–7. [3] J.M. Gutie´rrez, M.A. Herna´ndez, An acceleration of Newton’s method: super-Halley method, Appl. Math. Comput. 117 (2001) 223– 239. [4] M.A. Noor, K.I. Noor, Fifth-order iterative methods for solving nonlinear equations, Appl. Math. Comput., doi:10.1016/ j.amc.2006.10.007. [5] J. Kou, Y. Li, The improvements of Chebyshev–Halley methods with fifth-order convergence, Appl. Math. Comput., 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] C. Chun, Iterative methods improving Newton’s method by the decomposition method, Comput. Math. Appl. 50 (2005) 1559–1568. [8] W. Gautschi, Numerical Analysis: An Introduction, Birkha¨user, 1997.
Applied Mathematics and Computation 194 (2007) 287–290 www.elsevier.com/locate/amc
A fifth-order iterative method for solving nonlinear equations YoonMee Ham
a,1
, Changbum Chun
b,*
a
b
Department of Mathematics, Kyonggi University, Suwon 443-760, Republic of Korea School of Liberal Arts, Korea University of Technology and Education, Cheonan City, Chungnam 330-708, Republic of Korea
Abstract In this paper, we present a new fifth-order method for solving nonlinear equations. Per iteration the new method requires two function and two first derivative evaluations. It is shown that the new method is fifth-order convergent. Several numerical examples are given to illustrate the performance of the presented method. 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. 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 fifth-order iterative methods have been proposed and analyzed for solving nonlinear equations that improve some classical methods such as the Newton method, Chebyshev–Halley methods, Ostrowski’s method. It has been demonstrated that the methods are efficient, and can compete with Newton’s method. For more details about them and the other fifth-order methods, see [2–6] and the reference therein. Motivated by the recent activities in this direction, in this paper we present and analyze a new two-step iterative method for solving nonlinear equations. The method is shown to be fifth-order convergent, and several numerical examples are given to show the performance of the new method presented in this contribution.
*
1
Corresponding author. E-mail addresses: [email protected] (Y. Ham), [email protected] (C. Chun). 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.2007.04.005
288
Y. Ham, C. Chun / Applied Mathematics and Computation 194 (2007) 287–290
2. Development of method and convergence analysis To develop the new method, let us consider the iteration scheme in the form f ðxn Þ ; f 0 ðxn Þ Af 0 ðy Þ þ Bf 0 ðxn Þ f ðy n Þ ; ¼ yn 0 n Cf ðy n Þ þ Df 0 ðxn Þ f 0 ðxn Þ
y n ¼ xn
ð2Þ
xnþ1
ð3Þ
where A; B; C; D are the disposable parameters. For the iterative method defined by (2) and (3), we have the following convergence result. Theorem 2.1. Let a 2 I be a simple zero of a sufficiently differentiable function f : I ! R for an open interval I. If x0 is sufficiently close to a, and A; B; C; D satisfy the condition A þ B ¼ C þ D;
C ¼ B þ 2A;
B ¼ 3A;
C þ D 6¼ 0;
ð4Þ
then the method defined by (2) and (3) is of fifth-order. Proof. Let a be a simple zero of f. Consider the iteration function F defined by F ðxÞ ¼ yðxÞ
Af 0 ðyðxÞÞ þ Bf 0 ðxÞ f ðyðxÞÞ ; Cf 0 ðyðxÞÞ þ Df 0 ðxÞ f 0 ðxÞ
ð5Þ
where yðxÞ ¼ x f ðxÞ=f 0 ðxÞ. 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 ðxn Þ around x ¼ a (see [7] for details). We find after simplifying that CþDAB 2 2 c2 en þ ½ðB þ 2A CÞc22 þ ðC þ D A BÞc3 e3n CþD CþD 1 þ ½K 1 c32 þ K 2 c2 c3 þ K 3 c4 e4n þ Oðe5n Þ; 2 ðC þ DÞ
xnþ1 ¼ F ðxn Þ ¼ a þ
ð6Þ
where en ¼ xn a and ck ¼ f ðkÞ ðaÞ=k!f 0 ðaÞ, K 1 ¼ 12CD þ 4D2 þ 4C 2 13AC 27AD 3BC 13BD; K 2 ¼ 7½CðB þ 2A C 2DÞ þ Dð3A þ 2B DÞ;
ð7Þ ð8Þ
K 3 ¼ 3ðC þ DÞðC þ D A BÞ:
ð9Þ
An easy manipulation shows that K 1 ¼ K 2 ¼ K 3 ¼ 0 when A þ B ¼ C þ D; C ¼ B þ 2A; B ¼ 3A, which thus completes the proof. h Solving system of the equations of condition (4), we find that A ¼ t, B ¼ 3t, C ¼ 5t, D ¼ t, (t 2 RÞ, thereby we obtain the new fifth-order iterative method f ðxn Þ ; f 0 ðxn Þ f 0 ðy Þ þ 3f 0 ðxn Þ f ðy n Þ : ¼ yn 0 n 5f ðy n Þ f 0 ðxn Þ f 0 ðxn Þ
y n ¼ xn
ð10Þ
xnþ1
ð11Þ
Per iteration this method requires two function and two first derivative evaluations. If we consider the defini1 tion of efficiency index [8] 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 iteration formula defined byp(10) ffiffiffi and (11) has 1 the efficiency index equal to 54 1:495, which is better than the one of Newton’s method 2 1:414.
Y. Ham, C. Chun / Applied Mathematics and Computation 194 (2007) 287–290
289
3. Numerical examples All computations were done using MAPLE using 64 digit floating point arithmetics (Digits :¼ 64). 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) jxnþ1 xn j < , (ii) jf ðxnþ1 Þj < ; and so, when the stopping criterion is satisfied, xnþ1 is taken as the exact root a computed. We used ¼ 1015 . We present some numerical test results for various fifth-order convergent iterative schemes in Table 1. Compared were the Grau and Dı´az-Barrero method (GM) [2] defined by f 00 ðxn Þðf ðxn Þ þ f ðzn ÞÞ f ðxn Þ þ f ðzn Þ xnþ1 ¼ xn 1 þ ; ð12Þ 2f 02 ðxn Þ f 0 ðxn Þ 00 where zn ¼ xn 1 þ 12 f fðx02nðxÞfnðxÞ n Þ ff0ðxðxnnÞÞ, Noor and Noor’s method (NNM) [4] defined by xnþ1 ¼ xn
ð13Þ
hðxÞ ¼ f ðxÞ f ðxn Þ ðx xn Þf 0 ðxn Þ 12 ðx xn Þ2 f 00 ðxn Þ
where tðxn Þ ¼
2½f ðxn Þ þ hðzn Þf 0 ðxn Þ ; 2f 02 ðxn Þ ½f ðxn Þ þ hðzn Þf 00 ðxn Þ and
f 00 ðxn Þf ðxn Þ , f 02 ðxn Þ
tðxn Þ f ðxn Þ zn ¼ xn 1 þ 2tðx , f 0 ðxn Þ nÞ
where
the method of Kou and Li (KM) [5] defined by Mðxn Þ f ðzn Þ xnþ1 ¼ zn 1 þ ; ð14Þ 1 þ Mðxn Þ f 0 ðxn Þ 00 tðxn Þ f ðxn Þ ðzn ÞÞ where zn ¼ xn 1 þ 12 1tðx and Mðxn Þ ¼ f ðxn Þðff 02ðxðxn nÞf , and the method (10) and (11) (FM) introduced f 0 ðxn Þ Þ nÞ 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; x
2
f4 ðxÞ ¼ sinðxÞe þ lnðx þ 1Þ;
x ¼ 1:2076478271309189270094167584;
x ¼ 0;
Table 1 Comparison of various fifth-order convergent iterative methods f(x)
IT GM
NNM
KM
FM
f1 ; x0 ¼ 0:3 f1 ; x0 ¼ 1
27 4
div 4
24 4
11 3
f2 ; x0 ¼ 0 f2 ; x0 ¼ 1
3 4
4 4
3 4
3 3
f3 ; x0 ¼ 1 f3 ; x0 ¼ 2
4 5
4 div
4 div
4 5
f4 ; x0 ¼ 2 f4 ; x0 ¼ 5
4 6
4 6
4 4
4 4
f5 ; x0 ¼ 3 f5 ; x0 ¼ 4
4 5
5 6
4 4
4 4
f6 ; x0 ¼ 2 f6 ; x0 ¼ 3:5
5 6
9 15
div div
5 5
31 4
5 5
5 4
4 4
f7 ; x0 ¼ 1 f7 ; x0 ¼ 2
290
Y. Ham, C. Chun / Applied Mathematics and Computation 194 (2007) 287–290 3
f5 ðxÞ ¼ ðx 1Þ 2; x ¼ 2:2599210498948731647672106073; f6 ðxÞ ¼ ðx þ 2Þex 1; x ¼ 0:44285440100238858314132800000; f7 ðxÞ ¼ sin2 ðxÞ x2 þ 1;
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 is the number of iterations to approximate the zero (IT). In the Table, the ‘div’ means that the sequence of approximate zeros produced from the corresponding method does not converge within the maximum iteration number set to n = 200 under the stopping criterion ¼ 1015 , so the sequence may slow in convergence. As far as the results presented in Table 1 are concerned, for most of the functions we tested, the proposed method is more stable and works better in its performance as compared with the other methods of the same order. We refer to [5] for some numerical results showing that the Kou and Li method (KM) can compete with Newton’s method. 4. Conclusion In this work we presented a new fifth-order iterative method for solving nonlinear equations which requires two function and two first derivative evaluations per step. It was compared in its performance to some fifthorder methods, and the proposed method has been observed to have at least better performance and more stability. References [1] A.M. Ostrowski, Solution of Equations in Euclidean and Banach Space, Academic Press, New York, 1973. [2] M. Grau, J.L. Dı´az-Barrero, An improvement of the Euler–Chebyshev iterative method, J. Math. Anal. Appl. 315 (2006) 1–7. [3] J.M. Gutie´rrez, M.A. Herna´ndez, An acceleration of Newton’s method: super-Halley method, Appl. Math. Comput. 117 (2001) 223– 239. [4] M.A. Noor, K.I. Noor, Fifth-order iterative methods for solving nonlinear equations, Appl. Math. Comput., doi:10.1016/ j.amc.2006.10.007. [5] J. Kou, Y. Li, The improvements of Chebyshev–Halley methods with fifth-order convergence, Appl. Math. Comput., 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] C. Chun, Iterative methods improving Newton’s method by the decomposition method, Comput. Math. Appl. 50 (2005) 1559–1568. [8] W. Gautschi, Numerical Analysis: An Introduction, Birkha¨user, 1997.