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Introduction to a Newton-type method for solving nonlinear equations
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Applied Mathematics and Computation 195 (2008) 663–668 www.elsevier.com/locate/amc
Introduction to a Newton-type method for solving nonlinear equations R. Thukral Pade´ Research Centre, 39 Deanswood Hill, Leeds, West Yorkshire LS17 5JS, United Kingdom
Abstract In this paper we shall introduce a new Newton-type method, namely rational Newton, for solving a nonlinear equation. This new method is shown to converge cubically. We shall examine the effectiveness of the rational Newton method by comparing the performance with the well established methods, namely the classical Newton method, the Halley rational method, the Kou et al. and the Weerakoon and Fernando method for approximating the root of a given nonlinear equation. The approximate solution of the rational Newton method is found to be substantially more accurate than the well established methods. Ó 2007 Published by Elsevier Inc. Keywords: Rational Newton; Classical Newton; Halley rational; Kou et al.; Weerakoon and Fernando; Nonlinear equations; Order of convergence
1. Introduction In this paper, we consider a new iterative method, namely the rational Newton, to find a simple root of a nonlinear equation: f ðxÞ ¼ 0:
ð1Þ
The techniques to find the roots of these equations have many applications in science and engineering. The most well known technique is the Newton method and the other familiar methods are called the Halley rational method, the Kou et al. method and the Weerakoon and Fernando method. The rational Newton is of rational form and requires an initial estimate. Furthermore, the new method and the other methods mentioned above is dependent on the first derivative of the given function f(x) and it is essential that the denominator of the rational Newton, and other methods mentioned considered, is not equal to zero. The prime motive for the development of the new rational Newton method was to improve the precision of the well established methods of third order convergence. Consequently, we have found that the rational Newton method is consistent, stable and much more accurate than the other similar methods considered.
E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2007 Published by Elsevier Inc. doi:10.1016/j.amc.2007.05.013
664
R. Thukral / Applied Mathematics and Computation 195 (2008) 663–668
1.1. The classical Newton method (N) We begin the simple and most used expression of the classical Newton method, xn ¼ xn1
f ðxn1 Þ f 0 ðxn1 Þ
ð2Þ
where n 2 N, x0 is the initial value and is known to converge quadratically [2,6]. 1.2. The rational Newton method (RN) In this sub-section, we shall define the new iterative method, namely, the rational Newton method. This method is of rational form and utilises the classical Newton method given by (2). Therefore the expression of the new method for determining the simple root of (1) is given as: 2
xn ¼ xn1
f ðxn1 Þf 0 ðy n1 Þ þ f ðxn1 Þf 0 ðxn1 Þ
2
;
ð3Þ
f ðxn1 Þ½f 0 ðy n1 Þ þ f 0 ðxn1 Þ ; f 0 ðxn1 Þf 0 ðy n1 Þ½f 0 ðxn1 Þ þ f 0 ðy n1 Þ
ð4Þ
2
f 0 ðxn1 Þ f 0 ðy n1 Þ þ f 0 ðxn1 Þf 0 ðy n1 Þ
2
an alternative form of (3) is given as: 2
xn ¼ xn1
2
where y n1 ¼ xn1
f ðxn1 Þ ; f 0 ðxn1 Þ
ð5Þ
n 2 N, x0 is the initial value and provided f 0 (x)50. 1.3. Convergence analysis In numerical mathematics it is very useful and essential to know the behaviour of an approximate method. Therefore, in this sub-section we shall prove the order of convergence of the rational Newton. Theorem 1. Let the f(x) be a real function. Assume that f(x) has a first derivative in an interval I. If f(x) has a simple root a 2 I and x0 is sufficiently close to a then the rational Newton defined by (3) converges cubically. Proof. Let a be a simple root of f(x), i.e. f(a) = 0 and f 0 (a) 5 0, and the error is expressed as: e ¼ x a:
ð6Þ
Consider the iteration function F defined by 2
F ðxÞ ¼ x
2
f ðxÞf 0 ðyÞ þ f ðxÞf 0 ðxÞ
f 0 ðxÞ2 f 0 ðyÞ þ f 0 ðxÞf 0 ðyÞ2
;
ð7Þ
where y ¼x
f ðxÞ : f 0 ðxÞ
ð8Þ
Using (1), we find (7) becomes: F ðaÞ ¼ 0:
ð9Þ
Furthermore, we require the derivatives of (7), therefore the first and second derivatives are given as: F 0 ðaÞ ¼ 0
ð10Þ
R. Thukral / Applied Mathematics and Computation 195 (2008) 663–668
665
and F 0 ðaÞ ¼ 0:
ð11Þ
The third derivative of (7) is given as " # 1 f 0 ðaÞf 000 ðaÞ 3ff 00 ðaÞg2 000 : F ðaÞ ¼ 2 2 ff 0 ðaÞg
ð12Þ
From the Taylor expansion of F(x) around x = a is given as xnþ1 ¼ F ðaÞ þ F 0 ðaÞðxn aÞ þ
F 00 ðaÞ F 000 ðaÞ 2 3 4 ðxn aÞ þ ðxn aÞ þ O½ðxn aÞ : 2! 3!
ð13Þ
Using (6) we get in the appropriate form xnþ1 ¼ F ðaÞ þ F 0 ðaÞen þ
F 00 ðaÞ 2 F 000 ðaÞ 3 e þ e þ O½e4n : 2! n 3! n
Substituting (9)–(12) in (14), we obtain " # 2 0 000 00 1 f ðaÞf ðaÞ 3ff ðaÞg xnþ1 ¼ a þ ð12Þ e3n þ O½e4n : 2 ff 0 ðaÞg Therefore, we have " # 2 0 000 00 enþ1 1 f ðaÞf ðaÞ 3ff ðaÞg ¼ ð12Þ þ O½e4n ; 2 e3n ff 0 ðaÞg
ð14Þ
ð15Þ
ð16Þ
which establishes the third-order convergence of the rational Newton defined by (3). h The structure of this paper is as follows. In Section 2, we briefly review three well-known methods, the Halley rational method, the Kou et al method and the Weerakoon and Fernando method. Moreover, in Section 3, we examine the effectiveness of the new rational Newton method for determining the root of four different type of nonlinear equations. The rational Newton method is shown to be the most effective of the methods considered. 2. The established methods The three particular established methods considered are the Halley rational method, the Kou et al. method and the Weerakoon and Fernando method. Since these methods are well established, we shall state the essential expressions used in order to calculate the approximate solution of the given nonlinear equations and thus compare the effectiveness of the new method. 2.1. The Halley rational method (H) The Halley rational method is well established [4], hence we shall state the essential expressions used in order to evaluate the approximate root of a given nonlinear equation, xn ¼ xn1
f ðxn1 Þf 0 ðxn1 Þ 2
2f 0 ðxn1 Þ þ f ðxn1 Þf 00 ðxn1 Þ
;
ð17Þ
where n 2 N. 2.2. The Kou, Li and Wang method (KLW) The Kou et al. method is also well established [1,5], hence we shall state the essential expressions used in order to evaluate the root of a given nonlinear equation:
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R. Thukral / Applied Mathematics and Computation 195 (2008) 663–668
xn ¼ xn1
f ðwn1 Þ þ f ðxn1 Þ ; f 0 ðxn1 Þ
ð18Þ
ðxn1 Þ where wn1 ¼ xn1 þ ff0ðx and n 2 N. n1 Þ
2.3. The Weerakoon and Fernando method (WF) The Weerakoon and Fernando method is well established [1,3,7], hence we shall state the essential expressions used in order to evaluate the root of a given nonlinear equation: xn ¼ xn1
f 0 ðx
2f ðxn1 Þ ; 0 n1 Þ þ f ðy n1 Þ
ð19Þ
where yn1 is given by (5) and n 2 N. 3. Application of the new Newton-type method To demonstrate the performance of each of the five methods, we take four particular nonlinear equations. We shall determine the consistency and stability of results by examining the convergence of the new iterative method. The findings are generalised by illustrating the effectiveness of the rational Newton method for determining the simple root of a nonlinear equation. Consequently, we shall give estimates of the approximate solution produced by the five methods and list the errors obtained by each of the method. The numerical computations listed in the tables were performed on an algebraic system called Maple. In addition, the errors displayed are of absolute value. 3.1. Numerical example 1 In our first example we shall demonstrate the convergence of the new method for the following nonlinear equation: f ðxÞ ¼ ðx þ 2Þ expð1 xÞ þ 2x þ 5;
ð20Þ
and the exact value of the simple root of (20) is a = 2.043518. . . In Table 1 are the errors obtained by each of the methods described, based on the initial value x0 = 2. We observe that the rational Newton method is substantially more accurate than those from the Halley rational method, the Kou et al. method and the Weerakoon and Fernando method. We see that the rational Newton is converging cubically. 3.2. Numerical example 2 In our second example we shall demonstrate the convergence of the new method for a different type of nonlinear equation: f ðxÞ ¼ x5 þ 5x þ 1;
ð21Þ
and the exact value of the simple root of (21) is a = 0.1999361. . . Table 1 Errors occurring in the estimates of the root of (20) by the five methods described No.
WF
H
KLW
N
RN
1 2 3 4 5 6 7
0.899(4) 0.747(12) 0.428(36) 0.808(109) 0.541(327) 0.163(981) 0.446(2945)
0.218(1) 0.109(1) 0.544(2) 0.272(2) 0.136(2) 0.681(3) 0.340(3)
0.626(4) 0.176(12) 0.392(38) 0.433(115) 0.585(346) 0.144(1038) 0.214(3116)
0.176(2) 0.278(5) 0.692(11) 0.429(22) 0.165(44) 0.245(89) 0.537(179)
0.554(4) 0.988(13) 0.562(39) 0.103(117) 0.643(354) 0.155(1062) 0.216(3188)
R. Thukral / Applied Mathematics and Computation 195 (2008) 663–668
667
Table 2 Errors occurring in the estimates of the root of (21) by the five methods described No.
WF
H
KLW
N
RN
1 2 3 4 5 6 7
1.096 0.403 0.106(1) 0.504(7) 0.515(23) 0.550(71) 0.668(215)
1.358 0.887 0.355 0.169 0.839(1) 0.419(1) 0.210(1)
1.039 0.166 0.234(2) 0.199(8) 0.125(26) 0.307(81) 0.461(245)
1.318 0.853 0.355 0.211(1) 0.873(5) 0.122(11) 0.236(25)
0.721 0.183(1) 0.266(6) 0.752(21) 0.168(64) 0.189(195) 0.268(588)
In Table 2 are the errors obtained by each of the methods described, based on the initial value x0 = 2. We observe that the rational Newton method is substantially more accurate than those from the Halley rational method, the Kou et al. method and the Weerakoon and Fernando method. As expected, we see that the rational Newton method is converging cubically. 3.3. Numerical example 3 In this subsection we take another nonlinear equation. We shall demonstrate the convergence of the rational Newton method for the following nonlinear equation: f ðxÞ ¼ x3 3x þ 17;
ð22Þ
and the exact value of the simple root of (22) is a = 2.957664. . . In Table 3 are the errors obtained by each of the methods described, based on the initial value x0 = 2. We observe that the rational Newton method is substantially more accurate than those from the Halley rational method, the Kou et al. method and the Weerakoon and Fernando method. As before, we see that the rational Newton method is converging cubically. 3.4. Numerical example 4 In our last but not least of the examples we take another different type nonlinear equation. We shall demonstrate the convergence of the new iterative method for the following nonlinear equation: f ðxÞ ¼ x sinðxÞ þ cosðxÞ 1:5;
ð23Þ
and the exact value of the simple root of (23) is a = 1.246077. . . In Table 4 are the errors obtained by each of the methods described, based on the initial value x0 = 1. We observe that the rational Newton method is substantially more accurate than those from the Halley rational method, the Kou et al. method and the Weerakoon and Fernando method. Here we see that all the methods produce a poor estimate of the simple root, however the rational Newton method seems to be better than the other methods considered.
Table 3 Errors occurring in the estimates of the root of (22) by the five methods described No.
WF
H
KLW
N
RN
1 2 3 4 5 6 7
0.310 0.652(2) 0.465(7) 0.169(22) 0.802(69) 0.863(208) 0.108(624)
0.563 0.295 0.149 0.747(1) 0.374(1) 0.187(1) 0.934(2)
2.921 29 36 45 58 75 97
0.124 0.398(1) 0.132(1) 0.445(2) 0.150(2) 0.507(3) 0.172(3)
0.463 0.565(2) 0.226(7) 0.144(23) 0.369(72) 0.623(218) 0.301(655)
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R. Thukral / Applied Mathematics and Computation 195 (2008) 663–668
Table 4 Errors occurring in the estimates of the root of (23) by the five methods described No.
WF
H
KLW
N
RN
1 2 3 4 5 6 7
0.19332 0.15357 0.12299 0.99123(1) 0.80279(1) 0.65270(1) 0.53230(1)
0.220 0.197 0.176 0.159 0.143 0.129 0.116
0.233 0.221 0.210 0.199 0.189 0.180 0.172
0.662 1.011 1.245 1.405 1.521 1.610 1.679
0.19332 0.15356 0.12299 0.99119(1) 0.80276(1) 0.65267(1) 0.53228(1)
4. Remarks and conclusion In this paper, we have demonstrated the performance of the new iterative method, namely the rational Newton method. The prime motive of the development of the new rational Newton method was to increase the precision of the well established method, namely the classical Newton method, the Halley rational method, the Kou et al. method and the Weerakoon and Fernando method. We have examined the effectiveness of the rational Newton method by showing the accuracy of the simple root of a nonlinear equation. The main purpose of demonstrating the new method for four types of nonlinear equations was purely to illustrate the accuracy of the approximate solution, the stability of the convergence, the consistency of the results and to determine the efficiency of the new method. In all the numerical examples performed in this study, the nonlinear equation used in these examples were of different type, consequently we have found that the rational Newton method is very effective when compared to the classical Newton method, the Halley rational method, the Kou et al. method and the Weerakoon and Fernando method. Hence the new the rational Newton method may be considered a very good alternative method. In addition, it should be noted that like all other iterative methods, the new method has its own domain of validity and in certain circumstances should not be used. References [1] [2] [3] [4]
C. Chun, A method for obtaining iterative formulas of third order, Appl. Math. Lett., in press. S.D. Conte, C. de Boor, Elementary Numerical Analysis, McGraw-Hill, New York, 1980. M. Frontini, E. Sormani, Some variants of Newton’s method with third order convergence, Appl. Math. Comput. 140 (2003) 419–426. E. Halley, Methodus nova, Accurata and facillis inveniendi radices aequationum quarumcumque generaliter, sine praevia reductione, Philos. Trans. Roy. Soc. London 18 (1694) 136–148. [5] J. Kou, Y. Li, X. Wang, A modification of Newton method with third order convergence, Appl. Math. Comput. 181 (2006) 1106–1111. [6] A.M. Ostrowski, Solution of Equations in Euclidean and Banach space, third ed., Academic press, New York, 1973. [7] S. Weerakoon, T.G.I. Fernando, A variant of Newton’s method with accelerated third-order convergence, Appl. Math. Lett. 13 (2000) 87–93.
Applied Mathematics and Computation 195 (2008) 663–668 www.elsevier.com/locate/amc
Introduction to a Newton-type method for solving nonlinear equations R. Thukral Pade´ Research Centre, 39 Deanswood Hill, Leeds, West Yorkshire LS17 5JS, United Kingdom
Abstract In this paper we shall introduce a new Newton-type method, namely rational Newton, for solving a nonlinear equation. This new method is shown to converge cubically. We shall examine the effectiveness of the rational Newton method by comparing the performance with the well established methods, namely the classical Newton method, the Halley rational method, the Kou et al. and the Weerakoon and Fernando method for approximating the root of a given nonlinear equation. The approximate solution of the rational Newton method is found to be substantially more accurate than the well established methods. Ó 2007 Published by Elsevier Inc. Keywords: Rational Newton; Classical Newton; Halley rational; Kou et al.; Weerakoon and Fernando; Nonlinear equations; Order of convergence
1. Introduction In this paper, we consider a new iterative method, namely the rational Newton, to find a simple root of a nonlinear equation: f ðxÞ ¼ 0:
ð1Þ
The techniques to find the roots of these equations have many applications in science and engineering. The most well known technique is the Newton method and the other familiar methods are called the Halley rational method, the Kou et al. method and the Weerakoon and Fernando method. The rational Newton is of rational form and requires an initial estimate. Furthermore, the new method and the other methods mentioned above is dependent on the first derivative of the given function f(x) and it is essential that the denominator of the rational Newton, and other methods mentioned considered, is not equal to zero. The prime motive for the development of the new rational Newton method was to improve the precision of the well established methods of third order convergence. Consequently, we have found that the rational Newton method is consistent, stable and much more accurate than the other similar methods considered.
E-mail address: [email protected] 0096-3003/$ - see front matter Ó 2007 Published by Elsevier Inc. doi:10.1016/j.amc.2007.05.013
664
R. Thukral / Applied Mathematics and Computation 195 (2008) 663–668
1.1. The classical Newton method (N) We begin the simple and most used expression of the classical Newton method, xn ¼ xn1
f ðxn1 Þ f 0 ðxn1 Þ
ð2Þ
where n 2 N, x0 is the initial value and is known to converge quadratically [2,6]. 1.2. The rational Newton method (RN) In this sub-section, we shall define the new iterative method, namely, the rational Newton method. This method is of rational form and utilises the classical Newton method given by (2). Therefore the expression of the new method for determining the simple root of (1) is given as: 2
xn ¼ xn1
f ðxn1 Þf 0 ðy n1 Þ þ f ðxn1 Þf 0 ðxn1 Þ
2
;
ð3Þ
f ðxn1 Þ½f 0 ðy n1 Þ þ f 0 ðxn1 Þ ; f 0 ðxn1 Þf 0 ðy n1 Þ½f 0 ðxn1 Þ þ f 0 ðy n1 Þ
ð4Þ
2
f 0 ðxn1 Þ f 0 ðy n1 Þ þ f 0 ðxn1 Þf 0 ðy n1 Þ
2
an alternative form of (3) is given as: 2
xn ¼ xn1
2
where y n1 ¼ xn1
f ðxn1 Þ ; f 0 ðxn1 Þ
ð5Þ
n 2 N, x0 is the initial value and provided f 0 (x)50. 1.3. Convergence analysis In numerical mathematics it is very useful and essential to know the behaviour of an approximate method. Therefore, in this sub-section we shall prove the order of convergence of the rational Newton. Theorem 1. Let the f(x) be a real function. Assume that f(x) has a first derivative in an interval I. If f(x) has a simple root a 2 I and x0 is sufficiently close to a then the rational Newton defined by (3) converges cubically. Proof. Let a be a simple root of f(x), i.e. f(a) = 0 and f 0 (a) 5 0, and the error is expressed as: e ¼ x a:
ð6Þ
Consider the iteration function F defined by 2
F ðxÞ ¼ x
2
f ðxÞf 0 ðyÞ þ f ðxÞf 0 ðxÞ
f 0 ðxÞ2 f 0 ðyÞ þ f 0 ðxÞf 0 ðyÞ2
;
ð7Þ
where y ¼x
f ðxÞ : f 0 ðxÞ
ð8Þ
Using (1), we find (7) becomes: F ðaÞ ¼ 0:
ð9Þ
Furthermore, we require the derivatives of (7), therefore the first and second derivatives are given as: F 0 ðaÞ ¼ 0
ð10Þ
R. Thukral / Applied Mathematics and Computation 195 (2008) 663–668
665
and F 0 ðaÞ ¼ 0:
ð11Þ
The third derivative of (7) is given as " # 1 f 0 ðaÞf 000 ðaÞ 3ff 00 ðaÞg2 000 : F ðaÞ ¼ 2 2 ff 0 ðaÞg
ð12Þ
From the Taylor expansion of F(x) around x = a is given as xnþ1 ¼ F ðaÞ þ F 0 ðaÞðxn aÞ þ
F 00 ðaÞ F 000 ðaÞ 2 3 4 ðxn aÞ þ ðxn aÞ þ O½ðxn aÞ : 2! 3!
ð13Þ
Using (6) we get in the appropriate form xnþ1 ¼ F ðaÞ þ F 0 ðaÞen þ
F 00 ðaÞ 2 F 000 ðaÞ 3 e þ e þ O½e4n : 2! n 3! n
Substituting (9)–(12) in (14), we obtain " # 2 0 000 00 1 f ðaÞf ðaÞ 3ff ðaÞg xnþ1 ¼ a þ ð12Þ e3n þ O½e4n : 2 ff 0 ðaÞg Therefore, we have " # 2 0 000 00 enþ1 1 f ðaÞf ðaÞ 3ff ðaÞg ¼ ð12Þ þ O½e4n ; 2 e3n ff 0 ðaÞg
ð14Þ
ð15Þ
ð16Þ
which establishes the third-order convergence of the rational Newton defined by (3). h The structure of this paper is as follows. In Section 2, we briefly review three well-known methods, the Halley rational method, the Kou et al method and the Weerakoon and Fernando method. Moreover, in Section 3, we examine the effectiveness of the new rational Newton method for determining the root of four different type of nonlinear equations. The rational Newton method is shown to be the most effective of the methods considered. 2. The established methods The three particular established methods considered are the Halley rational method, the Kou et al. method and the Weerakoon and Fernando method. Since these methods are well established, we shall state the essential expressions used in order to calculate the approximate solution of the given nonlinear equations and thus compare the effectiveness of the new method. 2.1. The Halley rational method (H) The Halley rational method is well established [4], hence we shall state the essential expressions used in order to evaluate the approximate root of a given nonlinear equation, xn ¼ xn1
f ðxn1 Þf 0 ðxn1 Þ 2
2f 0 ðxn1 Þ þ f ðxn1 Þf 00 ðxn1 Þ
;
ð17Þ
where n 2 N. 2.2. The Kou, Li and Wang method (KLW) The Kou et al. method is also well established [1,5], hence we shall state the essential expressions used in order to evaluate the root of a given nonlinear equation:
666
R. Thukral / Applied Mathematics and Computation 195 (2008) 663–668
xn ¼ xn1
f ðwn1 Þ þ f ðxn1 Þ ; f 0 ðxn1 Þ
ð18Þ
ðxn1 Þ where wn1 ¼ xn1 þ ff0ðx and n 2 N. n1 Þ
2.3. The Weerakoon and Fernando method (WF) The Weerakoon and Fernando method is well established [1,3,7], hence we shall state the essential expressions used in order to evaluate the root of a given nonlinear equation: xn ¼ xn1
f 0 ðx
2f ðxn1 Þ ; 0 n1 Þ þ f ðy n1 Þ
ð19Þ
where yn1 is given by (5) and n 2 N. 3. Application of the new Newton-type method To demonstrate the performance of each of the five methods, we take four particular nonlinear equations. We shall determine the consistency and stability of results by examining the convergence of the new iterative method. The findings are generalised by illustrating the effectiveness of the rational Newton method for determining the simple root of a nonlinear equation. Consequently, we shall give estimates of the approximate solution produced by the five methods and list the errors obtained by each of the method. The numerical computations listed in the tables were performed on an algebraic system called Maple. In addition, the errors displayed are of absolute value. 3.1. Numerical example 1 In our first example we shall demonstrate the convergence of the new method for the following nonlinear equation: f ðxÞ ¼ ðx þ 2Þ expð1 xÞ þ 2x þ 5;
ð20Þ
and the exact value of the simple root of (20) is a = 2.043518. . . In Table 1 are the errors obtained by each of the methods described, based on the initial value x0 = 2. We observe that the rational Newton method is substantially more accurate than those from the Halley rational method, the Kou et al. method and the Weerakoon and Fernando method. We see that the rational Newton is converging cubically. 3.2. Numerical example 2 In our second example we shall demonstrate the convergence of the new method for a different type of nonlinear equation: f ðxÞ ¼ x5 þ 5x þ 1;
ð21Þ
and the exact value of the simple root of (21) is a = 0.1999361. . . Table 1 Errors occurring in the estimates of the root of (20) by the five methods described No.
WF
H
KLW
N
RN
1 2 3 4 5 6 7
0.899(4) 0.747(12) 0.428(36) 0.808(109) 0.541(327) 0.163(981) 0.446(2945)
0.218(1) 0.109(1) 0.544(2) 0.272(2) 0.136(2) 0.681(3) 0.340(3)
0.626(4) 0.176(12) 0.392(38) 0.433(115) 0.585(346) 0.144(1038) 0.214(3116)
0.176(2) 0.278(5) 0.692(11) 0.429(22) 0.165(44) 0.245(89) 0.537(179)
0.554(4) 0.988(13) 0.562(39) 0.103(117) 0.643(354) 0.155(1062) 0.216(3188)
R. Thukral / Applied Mathematics and Computation 195 (2008) 663–668
667
Table 2 Errors occurring in the estimates of the root of (21) by the five methods described No.
WF
H
KLW
N
RN
1 2 3 4 5 6 7
1.096 0.403 0.106(1) 0.504(7) 0.515(23) 0.550(71) 0.668(215)
1.358 0.887 0.355 0.169 0.839(1) 0.419(1) 0.210(1)
1.039 0.166 0.234(2) 0.199(8) 0.125(26) 0.307(81) 0.461(245)
1.318 0.853 0.355 0.211(1) 0.873(5) 0.122(11) 0.236(25)
0.721 0.183(1) 0.266(6) 0.752(21) 0.168(64) 0.189(195) 0.268(588)
In Table 2 are the errors obtained by each of the methods described, based on the initial value x0 = 2. We observe that the rational Newton method is substantially more accurate than those from the Halley rational method, the Kou et al. method and the Weerakoon and Fernando method. As expected, we see that the rational Newton method is converging cubically. 3.3. Numerical example 3 In this subsection we take another nonlinear equation. We shall demonstrate the convergence of the rational Newton method for the following nonlinear equation: f ðxÞ ¼ x3 3x þ 17;
ð22Þ
and the exact value of the simple root of (22) is a = 2.957664. . . In Table 3 are the errors obtained by each of the methods described, based on the initial value x0 = 2. We observe that the rational Newton method is substantially more accurate than those from the Halley rational method, the Kou et al. method and the Weerakoon and Fernando method. As before, we see that the rational Newton method is converging cubically. 3.4. Numerical example 4 In our last but not least of the examples we take another different type nonlinear equation. We shall demonstrate the convergence of the new iterative method for the following nonlinear equation: f ðxÞ ¼ x sinðxÞ þ cosðxÞ 1:5;
ð23Þ
and the exact value of the simple root of (23) is a = 1.246077. . . In Table 4 are the errors obtained by each of the methods described, based on the initial value x0 = 1. We observe that the rational Newton method is substantially more accurate than those from the Halley rational method, the Kou et al. method and the Weerakoon and Fernando method. Here we see that all the methods produce a poor estimate of the simple root, however the rational Newton method seems to be better than the other methods considered.
Table 3 Errors occurring in the estimates of the root of (22) by the five methods described No.
WF
H
KLW
N
RN
1 2 3 4 5 6 7
0.310 0.652(2) 0.465(7) 0.169(22) 0.802(69) 0.863(208) 0.108(624)
0.563 0.295 0.149 0.747(1) 0.374(1) 0.187(1) 0.934(2)
2.921 29 36 45 58 75 97
0.124 0.398(1) 0.132(1) 0.445(2) 0.150(2) 0.507(3) 0.172(3)
0.463 0.565(2) 0.226(7) 0.144(23) 0.369(72) 0.623(218) 0.301(655)
668
R. Thukral / Applied Mathematics and Computation 195 (2008) 663–668
Table 4 Errors occurring in the estimates of the root of (23) by the five methods described No.
WF
H
KLW
N
RN
1 2 3 4 5 6 7
0.19332 0.15357 0.12299 0.99123(1) 0.80279(1) 0.65270(1) 0.53230(1)
0.220 0.197 0.176 0.159 0.143 0.129 0.116
0.233 0.221 0.210 0.199 0.189 0.180 0.172
0.662 1.011 1.245 1.405 1.521 1.610 1.679
0.19332 0.15356 0.12299 0.99119(1) 0.80276(1) 0.65267(1) 0.53228(1)
4. Remarks and conclusion In this paper, we have demonstrated the performance of the new iterative method, namely the rational Newton method. The prime motive of the development of the new rational Newton method was to increase the precision of the well established method, namely the classical Newton method, the Halley rational method, the Kou et al. method and the Weerakoon and Fernando method. We have examined the effectiveness of the rational Newton method by showing the accuracy of the simple root of a nonlinear equation. The main purpose of demonstrating the new method for four types of nonlinear equations was purely to illustrate the accuracy of the approximate solution, the stability of the convergence, the consistency of the results and to determine the efficiency of the new method. In all the numerical examples performed in this study, the nonlinear equation used in these examples were of different type, consequently we have found that the rational Newton method is very effective when compared to the classical Newton method, the Halley rational method, the Kou et al. method and the Weerakoon and Fernando method. Hence the new the rational Newton method may be considered a very good alternative method. In addition, it should be noted that like all other iterative methods, the new method has its own domain of validity and in certain circumstances should not be used. References [1] [2] [3] [4]
C. Chun, A method for obtaining iterative formulas of third order, Appl. Math. Lett., in press. S.D. Conte, C. de Boor, Elementary Numerical Analysis, McGraw-Hill, New York, 1980. M. Frontini, E. Sormani, Some variants of Newton’s method with third order convergence, Appl. Math. Comput. 140 (2003) 419–426. E. Halley, Methodus nova, Accurata and facillis inveniendi radices aequationum quarumcumque generaliter, sine praevia reductione, Philos. Trans. Roy. Soc. London 18 (1694) 136–148. [5] J. Kou, Y. Li, X. Wang, A modification of Newton method with third order convergence, Appl. Math. Comput. 181 (2006) 1106–1111. [6] A.M. Ostrowski, Solution of Equations in Euclidean and Banach space, third ed., Academic press, New York, 1973. [7] S. Weerakoon, T.G.I. Fernando, A variant of Newton’s method with accelerated third-order convergence, Appl. Math. Lett. 13 (2000) 87–93.