Hindawi Publishing Corporation ξ e Scientiο¬c World Journal Volume 2014, Article ID 890138, 5 pages http://dx.doi.org/10.1155/2014/890138
Research Article A New Sixth Order Method for Nonlinear Equations in R Sukhjit Singh and D. K. Gupta Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur 721302, India Correspondence should be addressed to Sukhjit Singh;
[email protected] Received 31 August 2013; Accepted 28 November 2013; Published 23 January 2014 Academic Editors: A. Favini, A. FoΛsner, and A. Ibeas Copyright Β© 2014 S. Singh and D. K. Gupta. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A new iterative method is described for finding the real roots of nonlinear equations in R. Starting with a suitably chosen π₯0 , the method generates a sequence of iterates converging to the root. The convergence analysis is provided to establish its sixth order of convergence. The number of iterations and the total number of function evaluations used to get a simple root are taken as performance measure of our method. The efficacy of the method is tested on a number of numerical examples and the results obtained are summarized in tables. It is observed that our method is superior to Newtonβs method and other sixth order methods considered.
1. Introduction One of the most important and challenging problems in science and engineering is to find real roots of nonlinear equations in π
. There exist a large number of applications that give rise to thousands of such equations depending on one or more parameters. For example, the kinetic theory of gases, elasticity, and other applied areas are reduced to solving these equations. With the development of computer S/W and H/W, this problem has gained added advantages. This paper is concerned with the iterative methods and their convergence analysis for finding a simple real root πΌ; that is, π(πΌ) = 0 and πσΈ (πΌ) =ΜΈ 0 of π (π₯) = 0,
(1)
where π : π· β π
β π
is the continuously differentiable real function on an open interval π·. This problem is extensively studied by many researchers [1β5] and many methods along with their convergence analysis are derived. For good reviews of these methods, one can refer to excellent books by Ortega and Rheinboldt [6], Ostrowski [2], Traub [7], and many others. The well-known Newtonβs method used to find πΌ starts with an initial approximation π₯0 near to the root πΌ and
generates a sequence of iterates {π₯π } converging quadratically to the root. It is given for π = 0, 1, 2, . . . , by π₯π+1 = π₯π β
π (π₯π ) . πσΈ (π₯π )
(2)
If the efficiency index [8] of an iterative method is defined as π1/π , where π is the order of the method and π is the number of functions evaluations per iteration, then the efficiency index of Newtonβs method is 1.414. A number of ways are considered by many researchers to improve the local order of convergence of Newtonβs method at the expense of additional evaluations of functions, derivatives, and changes in the points of iterations. All these modifications are in the direction of increasing the local order of convergence with the view of increasing their efficiency indices. For example, Frontini and Sormani [9] developed new modifications of Newtonβs method to produce iterative methods with order of convergence of three and efficiency index of 1.442. With the same efficiency index, Traub [7] developed a third order method requiring evaluations of one function and two first derivatives per iteration. In [3], a family of third order methods is given which requires evaluations of one function, one first derivative, and one second derivative per iteration.
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Starting with a suitably chosen initial approximation to the root πΌ, it is given for π = 0, 1, 2, . . . by
π₯π+1
π (π₯ ) 1 πΏ π (π₯π ) = π₯π β (1 + ) σΈ π , 2 1 β π½πΏ π (π₯π ) π (π₯π )
(3)
developed a variant of Ostrowski fourth order multipoint method defined for π = 0, 1, 2, . . ., by π§π = π₯π β π₯π+1 = π§π β
2
where πΏ π (π₯π ) = (πσΈ σΈ (π₯π )π(π₯π ))/(πσΈ (π₯π )) and π½ is a parameter. One can easily see that for particular values of π½ = 0, 1/2, and 1, Chebyshevβs, Halleyβs, and Super-Halleyβs methods are special cases of this family. These methods require evaluation of π, πσΈ , and πσΈ σΈ per iteration and hence their efficiency indices are also 1.442. But the disadvantage of these methods is that they involve the evaluation second order derivative which is either computationally difficult to compute or remains unbounded. King [1] developed a family of fourth order methods for solving (1) requiring evaluation of one function and its derivative at the starting point of each step and of the function alone at the so-called Newton point. Ostrowski [2] developed both third and fourth order methods each requiring evaluations of two functions and one derivative per iteration leading to its efficiency index equal to 1.587. Recent trend is to develop sixth and higher orders multipoint methods to solve (1). It is clear that the higher the order of the method is, the higher the rate of convergence and its operational cost will be. This requires finding an equilibrium between the high rate of convergence and the operational cost. In spite of this fact, these methods are important because many applications such as stiff systems of equations require quick convergence of their solution methods. Based on Ostrowski fourth order multipoint method, Sharma and Guha [5] described a one parameter family of sixth order methods for solving (1). Each member of the family requires three evaluations of the given function and one evaluation of its derivative per iteration. Starting with an initial approximation π₯0 near to the root πΌ, a one parameter family of sixth order methods proposed in [10] is given for π = 0, 1, 2, . . ., by
π€π = π₯π β
π§π = π€π β
π₯π+1 = π§π β
π (π₯π ) , πσΈ (π₯π )
π (π€π ) π (π₯π ) + π½π (π€π ) , πσΈ (π₯π ) π (π₯π ) + (π½ β 2) π (π€π )
(4)
π (π§π ) π (π₯π ) β π (π€π ) + πΎπ (π§π ) . πσΈ (π₯π ) π (π₯π ) β 3π (π€π ) + πΎπ (π§π )
Clearly, it requires three functions and one derivative evaluation per iteration. Starting with an initial approximation π₯0 near to the root πΌ, Miquel and DΒ΄Δ±az-Barrero [11]
π₯Μπ+1 = π₯π+1 β
π (π₯π ) , πσΈ (π₯π )
π (π§π ) π (π₯π ) , πσΈ (π₯π ) π (π₯π ) β 2π (π§π )
(5)
π (π₯π+1 ) π (π₯π ) . σΈ π (π₯π ) π (π₯π ) β 2π (π§π )
It is clear that this variant requires an additional evaluation of function π at the point iterated by Ostrowskiβs fourth order method; consequently, the local order of convergence is improved from four to six. Kou and Li [4] established a variant of Jarratt method for solving (1). Starting with a suitably chosen π₯0 , it is given for π = 0, 1, 2, . . ., by π½π (π₯π ) =
3πσΈ (π¦π ) + πσΈ (π₯π ) , 6πσΈ (π¦π ) β 2πσΈ (π₯π )
π§π = π₯π β π½π (π₯π )
π (π₯π ) , πσΈ (π₯π )
π₯π+1 = π§π β ( (π (π§π ))
(6)
3 Γ ( π½π (π₯π ) πσΈ (π¦π ) 2 β1 3 + (1 β π½π (π₯π )) πσΈ (π₯π )) ) , 2
where π¦π = π₯π β (2/3)(π(π₯π )/πσΈ (π₯π )). Per iteration the new method adds the evaluation of the function at another point in the procedure iterated by Jarratt method. As a consequence, the local order of convergence is improved from four for Jarratt method to six for the new method. In this paper, a new iterative method is described for finding the real roots of nonlinear equations in π
. Starting with a suitably chosen π₯0 , the method generates a sequence of iterates converging to the root. The convergence analysis is provided to establish its sixth order of convergence. The number of iterations and the total number of function evaluations used to get a simple root are taken as performance measure of our method. The efficacy of the method is tested on a number of numerical examples and the results obtained are summarized in tables. It is observed that our method is superior to Newtonβs method and other sixth order methods described in [4, 10, 11]. The paper is organized as follows. Section 1 is the introduction. In Section 2, the proposed method and its convergence analysis are described. The numerical examples are worked out in Section 3. Finally, conclusions are included in Section 4.
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3 Proof. Let ππ = π₯π β πΌ be the error in the iterate π₯π . Using Taylorβs series expansion, we get
2. The Proposed Method and Its Convergence Analysis
π (π₯π ) = πσΈ (πΌ) [ππ + π2 ππ2 + π3 ππ3
In this section, we shall describe our sixth order method and its convergence analysis for finding a simple root πΌ of (1). The following definition is used for convergence of our method. Let πΌ β π
, π₯π β π
, π = 0, 1, 2 . . .. Then the sequence {π₯π } is said to converge to πΌ if σ΅¨ σ΅¨ lim σ΅¨σ΅¨π₯ β πΌσ΅¨σ΅¨σ΅¨ = 0. πββ σ΅¨ π
σ΅¨ σ΅¨σ΅¨ σ΅¨ σ΅¨π σ΅¨σ΅¨π₯π+1 β πΌσ΅¨σ΅¨σ΅¨ β€ πσ΅¨σ΅¨σ΅¨π₯π β πΌσ΅¨σ΅¨σ΅¨ ,
π+1
ππ+1 = πππ + π (ππ )
π§π = π₯π β π₯π+1
π (π₯π ) , πσΈ (π₯π )
π (π§ ) π (π₯π ) = π§π β σΈ π . π (π₯π ) π (π₯π ) β 2π (π§π )
Dividing (13) by (14), we get π (π₯π ) = [ππ β π2 ππ2 + (2π22 β 2π3 ) ππ3 πσΈ (π₯π ) + (β4π23
π§π = π₯π β
π₯π+1 = π§π β
Now expanding π(π§π ) about the πΌ, we get
3
4
+ π3 (π§π β πΌ) + π4 (π§π β πΌ) +π (ππ5 )] , = πσΈ (πΌ) [π2 ππ2 + (2π3 β 2π22 ) ππ3
+(5π23 β 7π2 π3 + 3π4) ππ4 + π (ππ5 )] , π (π₯π ) β 2π (π§π ) = ππ β π2 ππ2 + (4π22 β 3π3 ) ππ3 + (β10π23 + 14π2 π3 β 5π4 ) ππ4 + π (ππ5 ) . (17)
(11) Therefore,
Now, we shall establish the convergence analysis of our method. Theorem 1. Let π : π
β π
be continuous derivatives up to third order in π
. If π(π₯) has a simple root πΌ in π
and π₯0 is near to πΌ, then the error in the method given by (11) satisfies
σΈ
(16)
+ (4π23 β 7π2 π3 + 3π4 ) ππ4 + π (ππ5 ) .
(10)
π (π₯π+1 ) (π₯π+1 β π§π ) . π (π₯π+1 ) β π (π§π )
π
π (π₯π ) , πσΈ (π₯π )
2
π (π₯π ) , πσΈ (π₯π )
ππ+1 = π22 (π23 β π2 π3 ) ππ6 + π (ππ7 ) ,
+
π (π§π ) = πσΈ (πΌ) [(π§π β πΌ) + π2 (π§π β πΌ)
π (π§π ) π (π₯π ) , σΈ π (π₯π ) π (π₯π ) β 2π (π§π )
π₯Μπ+1 = π₯π+1 β
(15) π (ππ5 )] ,
= πΌ + π2 ππ2 + (2π3 β 2π22 ) ππ3
Extending this method, our sixth order method is given for π = 0, 1, 2, . . ., by π§π = π₯π β
+ 7π2 π3 β
3π4 ) ππ4
for
(9)
is called the error equation. The value of π is called the order of this method. Starting with a suitably chosen initial approximation π₯0 , the fourth order multipoint method described in [2] for solving (1) is given for π = 0, 1, 2, . . ., by
(14)
+4π4 ππ3 + 5π5 ππ4 + π (ππ5 )] .
(8)
then {π₯π } is said to converge to root πΌ with order at least π. If π = 2 or 3, the convergence is said to be quadratic or cubic, respectively. When ππ = π₯π βπΌ is the error at the πth iterate, the relation π
πσΈ (π₯π ) = πσΈ (πΌ) [1 + 2π2 ππ + 3π3 ππ2
(7)
If, in addition, there exist a constant π β₯ 0, an integer π0 β₯ 0, and π β₯ 0 such that for all π > π0
(13)
+π4 ππ4 + π5 ππ5 + π (ππ6 )] ,
(12)
where ππ = π₯π β πΌ and ππ = (π (πΌ)/π!π (πΌ)), π = 2, 3, . . . .
π (π₯π ) = 1 + 2π2 ππ + (β2π22 + 4π3 ) ππ2 π (π₯π ) β 2π (π§π )
(18)
β 2 (2π2 π3 β 3π4 ) ππ3 + π (ππ4 ) . Also, π (π§π ) = π2 ππ2 + (β4π22 + 2π3 ) ππ3 πσΈ (π₯π ) +
(13π23
β 14π2 π3 +
3π4 ) ππ4
(19) +
π (ππ5 ) .
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The Scientific World Journal Table 1: Comparison of number of iterations (NN).
Functions
π₯0
π1 π2
GBM
KLM
NEM
PM
0.5
12
8
NC
12
8
2 3 0.5
5 6 43
3 2 15
4 3 18
3 3 NC
2 2 14
0.8 4
10 8
3 4
3 NC
NC 9
3 3
4.5 3.25 3.5
7 8 12
3 3 5
NC 4 5
3 4 6
3 3 4
π3 π4 π5
NN NM
3. Numerical Examples
Multiplication of (18) and (19) gives
In this section, our proposed method (11) is tested on the following functions for different values of the initial approximations π₯0 :
π (π§π ) π (π₯π ) = π2 ππ2 + (β2π22 + 2π3 ) ππ3 πσΈ (π₯π ) π (π₯π ) β 2π (π§π ) + 3 (π23 β 2π2 π3 + π4 ) ππ4 + π (ππ5 ) . (20) π₯π+1
π (π§ ) π (π₯π ) = π§π β σΈ π , π (π₯π ) π (π₯π ) β 2π (π§π ) = πΌ + (π23 β π2 π3 ) ππ4 ;
that is, π₯π+1 β πΌ =
β
π2 π3 ) ππ4
+
π (ππ5 ) .
(22)
Now, the Taylors expansion of π(π₯π+1 ) about the πΌ is given by 2
π (π₯π+1 ) = πσΈ (πΌ) [(π₯π+1 β πΌ) + (π₯π+1 β πΌ) + β
β
β
] , = πσΈ (πΌ) [(π23 β π2 π3 ) ππ4 + π (ππ5 )] .
(23)
Therefore, π (π₯π+1 ) (π₯π+1 β π§π ) π (π₯π+1 ) β π (π§π ) = (π23 β π2 π3 ) ππ4 β 2 (2π24 β 4π22 π3 + π32 + π2 π4 ) ππ5
(24)
+ π (ππ6 ) .
π3 (π₯) = π₯10 β 1 πΌ = 1,
(26) πΌ = 6.308777129972688,
2
π5 (π₯) = ππ₯ +7π₯β30 β 1 πΌ = 3. The comparison of the number of iterations (NN) taken by Newtonβs method (NM), Netaβs method [10] (NEM), Kou and Liβs method [4] (KLM), Miquel and DΒ΄Δ±az-Barreroβs method [11] (GBM), and our proposed method (PM) to find πΌ correct up to 15 decimal places is given in Table 1. The word NC implies that the method is not convergent. The total number of function evaluations (NOFE) taken by these methods is compared with our proposed method in Table 2. A - indicates that no function evaluations are counted since the method does not converge. From Tables 1 and 2, one can easily observe that the proposed method takes the less number of iterations and number of function evaluations compared with the other methods. For some functions, our method either is superior or behaves similarly to with other methods compared. In some cases, the proposed method converges whereas other methods diverge.
4. Conclusions
By using (21)β(24) in third step of proposed method, we get the following error equation: πΜπ+1 = π23 (π22 β π3 ) ππ6 + π (ππ7 ) .
π2 (π₯) = π₯2 β ππ₯ β 3π₯ + 2 πΌ = 0.257530285439860,
π4 (π₯) = π₯5 + π₯ β 10000 (21)
(π23
π1 (π₯) = π₯3 β π₯2 β 1 πΌ = 1.465571231876768,
(25)
Thus, the sixth order of convergence of the method is established.
A new iterative method is described for finding the real roots of nonlinear equations in π
. Starting with a suitably chosen π₯0 , the method generates a sequence of iterates converging to the root. The convergence analysis is provided to establish its sixth order of convergence. The number of iterations and the total number of function evaluations used to get a simple root are taken as performance measure of our method.
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5 Table 2: Comparison of number of function evaluations (NOFE).
Functions
π₯0
π1 π2 π3 π4 π5
NOFE NM
GBM
KLM
NEM
PM
0.5
24
32
β
48
32
2 3 0.5
10 12 86
12 8 60
16 12 72
12 12 β
8 8 56
0.8 4
20 16
12 16
12 β
β 36
12 12
4.5 3.25 3.5
14 16 24
12 12 20
β 16 20
12 16 24
12 12 16
The efficacy of the method is tested on a number of numerical examples and the results obtained are summarized in tables. It is observed that our method is superior to Newtonβs method and other sixth order methods described in [4, 10, 11].
Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments The authors would like to thank the anonymous referees for their useful comments and suggestions that led to improvement of the presentation and content of this paper. The authors thankfully acknowledge the financial assistance provided by Council of Scientific and Industrial Research (CSIR), New Delhi, India.
References [1] R. F. King, βA family of fourth order methods for nonlinear equations,β SIAM Journal on Numerical Analysis, vol. 10, pp. 876β879, 1973. [2] A. M. Ostrowski, Solutions of Equations and System of Equations, Academic Press, New York, NY, USA, 1960. [3] J. M. GutiΒ΄errez and M. A. HernΒ΄andez, βA family of chebyshevhalley type methods in banach spaces,β Bulletin of the Australian Mathematical Society, vol. 55, no. 1, pp. 113β130, 1997. [4] J. Kou and Y. Li, βAn improvement of the Jarratt method,β Applied Mathematics and Computation, vol. 189, no. 2, pp. 1816β 1821, 2007. [5] J. R. Sharma and R. K. Guha, βA family of modified Ostrowski methods with accelerated sixth order convergence,β Applied Mathematics and Computation, vol. 190, no. 1, pp. 111β115, 2007. [6] J. M. Ortega and W. C. Rheinboldt, Iterative Solutions of Nonlinear Equations in Several Variables, Academic Press, New York, NY, USA, 1970. [7] J. F. Traub, Iterative Method for the Solution of Equations, Prentice-Hall, Englewood Cliffs, NJ, USA, 1964. [8] W. Gautschi, Numerical Analysis: An Introduction, BirkhΒ¨auser, 1997.
[9] M. Frontini and E. Sormani, βSome variant of Newtonβs method with third-order convergence,β Applied Mathematics and Computation, vol. 140, no. 2-3, pp. 419β426, 2003. [10] B. Neta, βA sixth-order family of methods for nonlinear equations,β International Journal of Computer Mathematics, vol. 7, no. 2, pp. 157β161, 1979. [11] G. Miquel and J. DΒ΄Δ±az-Barrero, βAn improvement to Ostrowski root-finding method,β Applied Mathematics and Computation, vol. 173, pp. 450β456, 2006.