Acta Appl Math (2010) 109: 1053–1063 DOI 10.1007/s10440-008-9359-3
Several New Third-Order Iterative Methods for Solving Nonlinear Equations Changbum Chun · Yong-Il Kim
Received: 22 August 2008 / Accepted: 30 October 2008 / Published online: 7 November 2008 © Springer Science+Business Media B.V. 2008
Abstract In this paper, we present some new third-order iterative methods for finding a simple root α of nonlinear scalar equation f (x) = 0 in R. A geometric approach based on the circle of curvature is used to construct the new methods. Analysis of convergence shows that the new methods have third-order convergence, that is, the sequence {xn }∞ 0 generated by each of the presented methods converges to α with the order of convergence three. The efficiency of the methods are tested on several numerical examples. It is observed that our methods can compete with Newton’s method and the classical third-order methods. Keywords Newton’s method · Iterative methods · Nonlinear equations · Order of convergence · Circle of curvature · Efficiency index Mathematics Subject Classification (2000) 41A25 · 65D99 1 Introduction One of the most important problems in scientific and engineering applications is to solve nonlinear equations [1–5]. Many optimization problems such as searching for a local minimizer of function [5], the problems to solve the potential equations in the transonic regime of dense gases in gasdynamics [2] and the boundary value problems encountered in kinetic theory of gases, elasticity [3] and problems in other applied areas can be reduced to nonlinear equations. In general, to compute their roots in a finite number of arithmetic operations are impossible, and this requires an iterative method. This paper is concerned with iterative methods to find a simple root α, i.e., f (α) = 0 and f (α) = 0, of a nonlinear equation This paper was supported by Faculty Research Fund, Sungkyunkwan University, 2008. C. Chun () Department of Mathematics, Sungkyunkwan University, Suwon 440-746, Republic of Korea e-mail:
[email protected] Y.-I. Kim School of Liberal Arts, Korea University of Technology and Education, Cheonan, Chungnam 330-708, Republic of Korea
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f (x) = 0 that uses no higher than the second derivative of f . There are many iterative methods such as Newton’s method and its variants [6–30], Secant method [31], Halley’s method [6, 8–10], Chebyshev method [6, 7, 9, 10] and super-Halley method [9, 10]. Among these methods, Newton’s method is the best known and probably the most used algorithm for solving nonlinear equations. It is given by xn+1 = xn −
f (xn ) , f (xn )
n = 0, 1, 2, . . . .
(1)
It is well-known [6] that the sequence of successive iterates {xn }∞ 0 generated from (1) converges quadratically to a root α, which means that there exists a positive constant λ such −α| |x that limn→∞ |xn+1−α|2 = λ. More generally, for the sequence {xn }∞ 0 generated by an iterative n method, if there exist positive constants λ and p such that lim
n→∞
|xn+1 − α| =λ |xn − α|p
(2)
then the method is said to converge to α with the (local) order of convergence p or we say that the method has the (local) order of convergence p [4]. To guarantee the convergence of Newton’s method, the initial approximation x0 must be chosen sufficiently close to a true solution. Finding a criterion for choosing x0 is quite difficult, demands an extensive investigation [11, 12] and therefore effective and globally convergent algorithms are required. For some overview of some of these developments on this issue, see, e.g., the survey article by Yamamoto [13]. When considering a practical utility of any method, the study of its efficiency is needed. The efficiency of a method may be measured by the efficiency index introduced by Ostrowski [6], which is defined by I = p 1/d
(3)
where p is the order of the method and d is the number of the function- (and derivative-) evaluations per step. The efficiency index of Newton’s method is 1.414. A systematic treatment of iterative methods, both old and new, are provided in [6] and [7]. Many researchers developed modifications of Newton’s method or Newton-like methods in a number of ways to improve the order of convergence of Newton’s method at the expense of additional evaluations of functions and/or derivatives mostly at the point iterated by the method. All these modifications are targeted at increasing the local order of convergence with a view of increasing their efficiency index. Even though many efficient higher order methods [14–16] and their convergence results in Banach spaces [6, 10, 13] are available in the literature, in this paper, we focus only on the development of third-order methods for solving nonlinear scalar equations. There are many novel approaches developed for solving various kinds of differential equations. Among them, few powerful methods are the Adomian decomposition method [17, 18], the homotopy perturbation method [19] and the homotopy analysis method [20]. These methods are successfully adapted and applied in deriving an abundant and interesting modifications of Newton’s method [21–26] and the references therein. Also, there are other class of iterative methods available, which are developed based on quadrature formulas for the computation of the integral arising from Newton’s theorem x f (t)dt. (4) f (x) = f (xn ) + xn
Several New Third-Order Iterative Methods
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Weerakoon and Fernando [27] applied the rectangular and trapezoidal rules to the integral of (4) to rederive the Newton method and arrive at the cubically convergent method xn+1 = xn −
f (x
n
2f (xn ) , n − f (xn )/f (xn ))
) + f (x
(5)
while Frontini et al. [28] obtained the cubically convergent method xn+1 = xn −
f (xn ) f (xn − f (xn )/(2f (xn )))
by considering the midpoint rule. In [29], Homeier derived the following cubically convergent iteration scheme 1 1 f (xn ) xn+1 = xn − + 2 f (xn ) f (xn − f (xn )/f (xn ))
(6)
(7)
by considering Newton’s theorem for the inverse function x = f (y) instead of y = f (x). Kou et al. in [30] considered Newton’s theorem on a new interval of integration and arrived at the following cubically convergent iterative scheme xn+1 = xn −
f (xn + f (xn )/f (xn )) − f (xn ) . f (xn )
(8)
All of the above-mentioned third-order methods require three function and first derivative evaluations per iteration to improve the order of convergence, so that their efficiency index is 1.442, which is better than the Newton method. On the other hand, a simple geometric observation can apply to the development of many new iterative methods. Newton’s method can be geometrically constructed by the point of intersection of the tangent line to the graph of f (x) at the point (xn , f (xn )) with x-axis [9, 32]. The classical Halley’s method and Chebyshev method also admit their geometric derivation from a hyperbola and parabola, respectively under appropriately imposed tangency conditions by taking the intersection of the considered curve with x-axis as next iterate [32]. Both methods are of third-order. Some more iterative methods were obtained by considering more general form of curves, see [9, 32, 33] for more details in this direction. The problem with the existing methods is that these methods may fail to converge in some cases if the initial guess is far from a true root or if the derivative of the function is small or even zero in the vicinity of the true zero. There are some iterative techniques to provide an alternative to the failure case, which are developed based on geometric observations such as osculating circle method [34], ellipse method [35] and modified curves [32, 36]. It is worth mentioning that the iterative methods based on quadrature formulas can be geometrically interpreted due to the observation that they can be obtained by replacing the derivative f (xn ) in Newton’s method by appropriate approximations. Motivated and inspired by the ongoing research with the iterative methods, in this paper we are concerned with the construction of the iterative methods based on a geometrical approach, especially by using the circle of curvature, which is an underlying fundamental concept in differential geometry to analyze the curvature analysis. As a result, we present some new interesting methods. By analysis of convergence we prove that the proposed methods have third-order convergence, and their efficiency are tested on several numerical examples. It is observed that our methods are competitive to Newton’s method and other methods of the same order.
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2 Iterative Methods and Convergence Analysis Let yn = f (xn ), yn = f (xn ) and yn = f (xn ), where xn is an n-th iterate. In this section we’ll use the circle of curvature, which has the same tangent line at the point (xn , yn ) as the curve y = f (x), to develop new methods. By an elementary calculation, the circle of curvature at (xn , yn ) can be found to be x − xn +
yn [1 + yn2 ] yn
2
2 1 + yn2 (1 + yn2 )3 + y − yn − = . yn yn2
(9)
At the intersection point (xn+1 , 0) of (9) with the x-axis, we get (xn+1 − xn )2 + 2
yn (1 + yn2 ) 1 + yn2 (xn+1 − xn ) + yn2 + 2yn = 0. yn yn
(10)
Equation (9) can further be rewritten as follows xn+1 = xn −
yn2 + 2yn
1+yn2 yn
2
n) xn+1 − xn + 2 yn (1+y y
.
(11)
n
By replacing xn+1 on the right-hand side of (11) by the Newton iterate, we obtain the new iterative method xn+1 = xn −
yn2 + 2yn
1+yn2 yn
2
∗ n) xn+1 − xn + 2 yn (1+y y
,
(12)
n
∗ = xn − where xn+1
yn , yn
or more simply yn yn yn2 + 2yn yn (1 + yn2 ) , 2yn2 (1 + yn2 ) − yn yn
(13)
f (xn )f (xn )[f (xn )f (xn ) + 2 + 2f 2 (xn )] . 2f 2 (xn )[1 + f 2 (xn )] − f (xn )f (xn )
(14)
xn+1 = xn − or equivalently xn+1 = xn −
We observe that the method (14) requires evaluation of the second derivative. To derive its second-derivative-free variant, which is important from the practical point of view, we consider the approximation f (xn ) ≈ where zn = xn −
f (xn ) , f (xn )
f (zn ) − f (xn ) , zn − xn
(15)
to obtain
xn+1 = xn −
f (xn )[2 + 3f 2 (xn ) − f (xn )f (zn )] . f (xn ) + 2f 3 (xn ) + f (zn )
(16)
Several New Third-Order Iterative Methods
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Another new method may be derived by manipulating equation (10) in a different way. Replacing the first term of (10), (xn+1 − xn )2 , with ( yyn )2 from Newton’s iterate (1), results n in the following method yn2 yn + 2yn yn2 , 2yn3
(17)
f 2 (xn )f (xn ) + 2f (xn )f 2 (xn ) . 2f 3 (xn )
(18)
xn+1 = xn − or equivalently, xn+1 = xn −
By using the approximation defined by (15) the following new second-derivative-free variant of (18) is obtained 1 f (zn ) f (xn ) . (19) xn+1 = xn − 3− 2 f (xn ) f (xn ) For the method defined by (14) and (19), we have the following analysis of convergence. A similar analysis can be done for (16) and (18), respectively. Theorem 2.1 Let α ∈ I be a simple zero of sufficiently differentiable function f : I → R for an open interval I . If x0 is sufficiently close to α, then the order of convergence of the methods defined by (14) and (19) is three. Proof Let α be a simple zero of f . Using Taylor expansion around xn = α and taking into account f (α) = 0, we have (20) f (xn ) = f (α) en + c2 en2 + c3 en3 + O(en4 ) , 2 3 f (xn ) = f (α) 1 + 2c2 en + 3c3 en + O(en ) , (21) f (xn ) = f (α) 2c2 + 6c3 en + O(en2 ) , (22) where en = xn − α and ck =
1 f (k) (α) , k! f (α)
k = 1, 2, . . . . Furthermore, we have
f 2 (xn ) = f 2 (α) en2 + 2c2 en3 + O(en4 ) , f 2 (xn ) = f 2 (α) 1 + 4c2 en + (4c22 + 6c3 )en2 + O(en3 ) , f 3 (xn ) = f 2 (α) 1 + 6c2 en + (12c22 + 9c3 )en2 + O(en3 ) , f 4 (xn ) = f 4 (α) 1 + 8c2 en + (24c22 + 12c3 )en2 + O(en3 ) . By a simple manipulation with (20)–(23) and (25), we obtain f (xn )f (xn )f 2 (xn ) = f 4 (α) 2c2 en2 + (8c22 + 6c3 )en3 + O(en4 ) , f (xn )f (xn ) = f 2 (α) en + 3c2 en2 + (2c22 + 4c3 )en3 + O(en4 ) , f (xn )f (xn ) = f 2 (α) 2c2 en + 2(c22 + 3c3 )en2 + O(en3 ) , f 3 (xn )f (xn ) = f 4 (α) en + 7c2 en2 + (18c22 + 10c3 )en3 + O(en4 ) ,
(23) (24) (25) (26)
(27) (28) (29) (30)
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so that f (xn )f (xn )f 2 (xn ) + 2f (xn )f (xn ) + 2f 3 (xn )f (xn ) = 2 f 2 (α) + f 2 (α) en + 2c2 3f 2 (α) + 8f 4 (α) en2 + 4(c22 + 2c3 )f 2 (α) + 2(22c22 + 13c3 )f 4 (α) en3 + O(en4 ).
(31)
From (24), (26) and (29) we also easily find 2f 2 (xn ) + 2f 4 (xn ) − f (xn )f (xn ) = 2[f 2 (α) + f 4 (α)] + 2c2 3f 2 (α) + 8f 4 (α) en + 6(c22 + c3 )f 2 (α) + 24(2c22 + c3 )f 4 (α) en2 + O(en3 ).
(32)
Dividing (31) by (32) gives us f (xn )f (xn )[f (xn )f (xn ) + 2 + 2f 2 (xn )] 2f 2 (xn )[1 + f 2 (xn )] − f (xn )f (xn ) = en +
(c3 − c22 )f 2 (α) + (c3 − 2c22 )f 4 (α) 3 en + O(en4 ). f 2 (α) + f 4 (α)
(33)
We thus obtain
en+1
(c3 − c22 )f 2 (α) + (c3 − 2c22 )f 4 (α) 3 4 en + O(en ) = en − en + f 2 (α) + f 4 (α) =
(c22 − c3 )f 2 (α) + (2c22 − c3 )f 4 (α) 3 en + O(en4 ). f 2 (α) + f 4 (α)
(34)
This means that the method defined by (14) is of third-order. On the other hand, by a simple calculation, we get f (xn ) = en − c2 en2 + 2(c22 − c3 )en3 + O(en4 ), f (xn )
(35)
so that zn = xn −
f (xn ) = α + c2 en2 − 2(c22 − c3 )en3 + O(en4 ), f (xn )
(36)
whence f (zn ) = f (α)[1 + 2c2 (zn − α) + O((zn − α)2 )] = f (α)[1 + 2c22 en2 + O(en3 )].
(37)
Dividing (37) by (21) gives us f (zn ) = 1 − 2c2 en + 3(2c22 − c3 )en2 + O(en3 ). f (xn )
(38)
Several New Third-Order Iterative Methods
1059
Multiplying (35) by (38) gives us f (xn ) f (zn ) = en − 3c2 en2 + 5(2c22 − c3 )en3 + O(en4 ). f (xn ) f (xn )
(39)
From (35) and (39) it thus follows that 1 f (xn ) f (zn ) 3 f (xn ) + 2 f (xn ) 2 f (xn ) f (xn ) 1 = 2c22 + c3 en3 + O(en4 ). 2
en+1 = en −
(40)
This shows that the method defined by (19) is of third-order. This completes the proof.
3 Numerical Examples We present some numerical test results for various cubically convergent iterative methods in Table 1. The following methods were compared: the Newton method (NM), the method of Weerakoon and Fernando (5) (WF), the midpoint method (6) (MP), Homeier’s method (7) (HM), the method of Kou et al. (8) (KM), and our new curvature methods (14) (CM1), (16) (CM2), (18) (CM3) and (19) (CM4). All computations were done using MAPLE using 64 digit floating point arithmetics (Digits := 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) |xn+1 − xn | < , (ii) |f (xn+1 )| < , and so, when the stopping criterion is satisfied, xn+1 is taken as the exact root α computed. We used = 10−15 . We used the following test functions and display the computed approximate zero x∗ f1 (x) = x 3 + 4x 2 − 10,
x∗ = 1.3652300134140968457608068290,
f2 (x) = sin2 x − x 2 + 1,
x∗ = 1.4044916482153412260350868178,
f3 (x) = x − e − 3x + 2, 2
x
f4 (x) = cos x − x,
x∗ = 0.73908513321516064165531208767,
f5 (x) = (x − 1)3 − 1, f6 (x) = sin x − x/2, x2
x∗ = 0.25753028543986076045536730494,
x∗ = 2, x∗ = 1.8954942670339809471440357381,
f7 (x) = xe − sin2 x + 3 cos x + 5,
x∗ = −1.2076478271309189270094167584.
As convergence criterion, it was required that the distance of two consecutive approximations δ for the zero was less than 10−15 . Also displayed are the number of iterations to approximate the zero (IT), the number of functional evaluations (NFE) counted as the sum of the number of evaluations of the function itself plus the number of evaluations of the derivative, and the value f (x∗ ). The numerical results presented in Table 1 show that the proposed methods in this contribution have at least equal performance as compared with the other methods of the same order. Thus, the new methods can compete with other third-order methods in literature.
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Table 1 Comparison of various cubically convergent iterative methods and the Newton method
C. Chun, Y.-I. Kim
IT
NFE
f (x∗ )
δ
f1 , x0 = 3.0 NM
7
14
4.60e−39
2.38e−20
WF
5
15
0
2.96e−24
MP
5
15
0
2.25e−26
HM
5
15
0
1.80e−42
KM
5
15
0
2.48e−22
CM1
5
15
2.61e−60
7.22e−21
CM2
5
15
1.87e−54
6.05e−19
CM3
5
15
2.76e−60
7.35e−21
CM4
5
15
1.99e−54
6.18e−19
NM
7
14
−3.03e−43
3.95e−22
WF
5
15
−2.0e−63
2.12e−30
MP
5
15
−4.56e−61
6.76e−21
HM
5
15
−2.0e−63
KM
5
15
1.18e−45
7.67e−16
CM1
5
15
−2.34e−58
4.47e−20
CM2
4
12
−2.0e−63
2.57e−29
CM3
5
15
−1.47e−57
8.03e−20
CM4
4
12
9.32e−60
1.43e−20
1.76e−26
f2 , x0 = 3.5
9.30e−33
f3 , x0 = −1.0 NM
6
12
1.10e−52
WF
4
12
3.72e−54
2.98e−18
MP
4
12
0
3.26e−24
HM
4
12
1.0e−63
3.79e−22
KM
4
12
−1.77e−54
1.69e−18
CM1
4
12
−1.99e−56
5.08e−19
CM2
4
12
8.92e−50
8.40e−17
CM3
4
12
−2.86e−56
5.76e−19
CM4
4
12
1.02e−48
1.80e−16
−1.90e−35
f4 , x0 = 1.2 NM
5
10
WF
4
12
0
1.97e−34
MP
4
12
0
2.72e−27
HM
4
12
0
4.0e−29
KM
4
12
CM1
4
12
0
3.50e−22
CM2
4
12
0
6.95e−29
CM3
4
12
0
5.54e−22
CM4
4
12
0
1.84e−34
−6.07e−57
7.16e−18
2.50e−19
Several New Third-Order Iterative Methods Table 1 (Continued)
1061 IT
NFE
f (x∗ )
δ
f5 , x0 = 2.5 NM
7
14
5.03e−56
1.29e−28
WF
5
15
0
4.57e−34
MP
5
15
0
4.46e−37
HM
4
12
5.98e−54
2.29e−18
KM
5
15
0
7.99e−33
CM1
5
15
0
7.18e−31
CM2
5
15
0
4.73e−28
CM3
5
15
0
1.59e−30
CM4
5
15
0
1.18e−27
NM
10
20
−1.25e−44
1.21e−22
WF
5
15
2.0e−64
2.47e−23
5
15
7.98e−50
4.66e−17 2.11e−31
f6 , x0 = 1.3
MP HM
Divergent
KM
18
54
0
CM1
9
27
2.05e−27
9.56e−20
CM2
8
24
3.0e−64
6.26e−22
0
2.26e−31
2.0e−64
7.0e−27
CM3
Divergent
CM4
27
81
f7 , x0 = −2 NM
9
18
−2.27e−40
2.73e−21
WF
7
21
−4.0e−63
3.11e−44
MP
6
18
−4.0e−63
2.12e−23
HM
6
18
−4.0e−63
2.57e−32
KM
6
18
−4.0e−63
8.87e−34
CM1
6
18
−1.4e−50
6.50e−18
CM2
6
18
−4.0e−63
4.25e−30
CM3
6
18
−1.46e−50
6.59e−18
CM4
7
21
−4.0e−63
4.40e−30
4 Conclusion In this paper, we presented four new third-order methods for solving nonlinear equations based on the circle of curvature. We observed from numerical examples that the proposed methods have at least equal performance as compared with the other methods of the same order.
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Several New Third-Order Iterative Methods
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