Two new families of sixth-order methods for solving non-linear equations

Applied Mathematics and Computation 213 (2009) 73–78

Contents lists available at ScienceDirect

Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Two new families of sixth-order methods for solving non-linear equations Xia Wang a, Liping Liu b,* a b

Department of Mathematics and Information Science, Zheng Zhou University of Light Industry, Zheng Zhou 450002, China Department of Mathematics, North Carolina Agricultural and Technical State University, Greensboro 27411, USA

a r t i c l e

i n f o

Keywords: Newton’s method Sixth-order convergence Non-linear equation Root-finding Iterative method

a b s t r a c t In this paper, we developed two new families of sixth-order methods for solving simple roots of non-linear equations. Per iteration these methods require two evaluations of the function and two evaluations of the first-order derivatives, which implies that the efficiency indexes of our methods are 1.565. These methods have more advantages than Newton’s method and other methods with the same convergence order, as shown in the illustration examples. Finally, using the developing methodology described in this paper, two new families of improvements of Jarratt method with sixth-order convergence are derived in a straightforward manner. Notice that Kou’s method in [Jisheng Kou, Yitian Li, An improvement of the Jarratt method, Appl. Math. Comput. 189 (2007) 1816–1821] and Wang’s method in [Xiuhua Wang, Jisheng Kou, Yitian Li, A variant of Jarratt method with sixth-order convergence, Appl. Math. Comput. 204 (2008) 14–19] are the special cases of the new improvements. Published by Elsevier Inc.

1. Introduction Solving non-linear equations is one of the most important problems in numerical analysis. In this paper, we consider iterative methods to find a simple root of a non-linear equation f ðxÞ ¼ 0, where f : D  R ! R for an open interval D is a scalar function. The classical Newton’s method for a single non-linear equation is written as

xnþ1 ¼ xn 

f ðxn Þ : f 0 ðxn Þ

ð1Þ

This is an important and basic method [1], which converges quadratically. Some modifications of Newton’s method with cubic convergence have been developed in [2–6]. The Jarratt method [7] is with fourth-order convergence. Recently, an improvement of Chebyshev–Halley methods with fifth-order convergence has been developed in [8]. Two improvements of the Jarratt method with sixth-order convergence have been developed by Kou [9] and by Wang [10] separately. Improvements of the Ostrowski method [1] with sixth-order convergence has been developed by Chun and Ham [11]. The aim of this paper is to develop two new families of sixth-order methods for finding real roots of non-linear equations in R by using the method of undetermined coefficients. A nice and thorough discussion of the undetermined coefficients method can be found in Chun and Neta’s study [12]. The convergence analysis is provided to establish their sixth-order of convergence. In terms of computational cost, they require the evaluations of only two functions and two first-order

* Corresponding author. E-mail addresses: [email protected] (X. Wang), [email protected] (L. Liu). 0096-3003/$ - see front matter Published by Elsevier Inc. doi:10.1016/j.amc.2009.03.007

74

X. Wang, L. Liu / Applied Mathematics and Computation 213 (2009) 73–78

derivatives per iteration. This gives 1.565 as an efficiency index of our methods. Our methods are comparable with Newton’s method and other higher order methods. The efficacy of the methods is tested on a number of numerical examples. It is observed that under the same configuration our methods need less number of iterations to converge to the root than those needed by Newton’s method and others. Compared with the other sixth-order methods, the developed methods behave either similarly or better for the examples considered. Lastly, two new families of improvements of Jarratt method are developed in Section 4. It can be seen that the methods in [9,10] are special cases of the new improvements. 2. The methods and analysis of convergence Consider the following iteration scheme:

2 f ðxn Þ ; 3 f 0 ðxn Þ 9f 0 ðxn Þ  5f 0 ðyn Þ f ðxn Þ ; zn ¼ xn  10f 0 ðxn Þ  6f 0 ðyn Þ f 0 ðyn Þ f ðzn Þ xnþ1 ¼ zn  0 : f ðzn Þ y n ¼ xn 

ð2Þ

This is a three-step method. The first and the second equations of (2) compose a fourth-order method developed by the authors in [13]. This iteration scheme consists of a fourth-order iterate method to get zn from xn , followed by a Newton iterate to calculate xnþ1 from the new point zn . It is not necessary to compute the first-order derivative at the point zn since a good approximation can be obtained. In order to approximate f 0ðzn Þ, the points ðxn ; f 0ðxn ÞÞ and ðyn ; f 0ðyn ÞÞ can be used to construct the linear interpolation

f 0 ðxÞ ’

x  xn 0 x  yn 0 f ðyn Þ þ f ðxn Þ: y n  xn xn  yn

ð3Þ

Therefore,

f 0 ðzn Þ ’

  3 f 0 ðxn Þ 9f 0 ðxn Þ  5f 0 ðyn Þ 0 3 f 0 ðxn Þ 9f 0 ðxn Þ  5f 0 ðyn Þ 0 f ðxn Þ: ðy Þ þ 1  f n 2 f 0 ðyn Þ 10f 0 ðxn Þ  6f 0 ðyn Þ 2 f 0 ðyn Þ 10f 0 ðxn Þ  6f 0 ðyn Þ

A new family of methods is then developed

2 f ðxn Þ ; 3 f 0 ðxn Þ 0 9f ðxn Þ  5f 0 ðyn Þ f ðxn Þ ; zn ¼ xn  10f 0 ðxn Þ  6f 0 ðyn Þ f 0 ðyn Þ f ðzn Þ ; xnþ1 ¼ zn  3 0 ðy Þ þ ð1  3 W ðx ÞÞf 0 ðx Þ W ðx Þf n n n f f n 2 2 y n ¼ xn 

ð4Þ

where 0

W f ðxn Þ ¼

0

af ðxn Þ þ bf ðyn Þ f 0 ðxn Þ ; 0 0 cf ðxn Þ þ df ðyn Þ f 0 ðyn Þ

yn and zn are defined by (2), and a; b; c; d 2 R are constants. With the method of undetermined coefficients, another new family of methods can be obtained:

2 f ðxn Þ ; 3 f 0 ðxn Þ 9f 0 ðxn Þ  5f 0 ðyn Þ f ðxn Þ ; zn ¼ xn  10f 0 ðxn Þ  6f 0 ðyn Þ f 0 ðyn Þ

y n ¼ xn 

0

xnþ1 ¼ zn 

ð5Þ

0

f ðzn Þðaf ðxn Þ þ bf ðyn ÞÞ 02

0

02

cf ðxn Þ þ df ðxn Þf 0 ðyn Þ þ ef ðyn Þ

;

where a ¼ 12 ð5c þ 3d þ eÞ; b ¼ 12 ð3c  d þ eÞ and c; d; e 2 R; c þ d þ e – 0 are constants. Clearly these methods require evaluations of two functions f and two derivatives f 0 per iteration. The new schemes (4) and (5) improve the order of convergence, which is shown in the following theorem. Theorem 1. Assume that function f 2 C 4 ðDÞ has a simple zero a 2 D. If the initial point x0 is sufficiently close to a, then under the conditions a ¼ b þ c þ d; c þ d – 0, the methods defined by (4) converge to a in the sixth-order, and the methods defined by (5) converge to a in the sixth-order. Proof. Considering the iteration function of (4)

75

X. Wang, L. Liu / Applied Mathematics and Computation 213 (2009) 73–78

f ðzÞ   ; 0 ðyÞ þ 1  3 W ðxÞ f 0 ðxÞ W ðxÞf f f 2 2

FðxÞ ¼ z  3

ð6Þ

where 0

0

af ðxÞ þ bf ðyÞ f 0 ðxÞ ; 0 0 cf ðxÞ þ df ðyÞ f 0 ðyÞ 2 f ðxÞ y¼x ; 3 f 0 ðxÞ 0 9f ðxÞ  5f 0 ðyÞ f ðxÞ : z¼x 10f 0 ðxÞ  6f 0 ðyÞ f 0 ðyÞ W f ðxÞ ¼

In fact, we have

xnþ1 ¼ Fðxn Þ: Expanding Fðxn Þ about a yields,

Fðxn Þ ¼ FðaÞ þ F 0 ðaÞen þ

F 00 ðaÞ 2 F 000 ðaÞ 3 F ð4Þ ðaÞ 4 F ð5Þ ðaÞ 5 F ð6Þ ðaÞ 6 e þ e þ en þ en þ en þ Oðe7n Þ; 2! n 3! n 4! 5! 6!

ð7Þ

where en ¼ xn  a. Considering f ðaÞ ¼ 0, after the algebraic manipulation, we have

FðaÞ ¼ a; F ðiÞ ðaÞ ¼ 0; ð5Þ

F ðaÞ ¼ 

i ¼ 1; 2; 3; 4;

  5ða þ b  c  dÞ 99ðf 00 Þ3  18f 0 f 00 f 000 þ ðf 0 Þ2 f ð4Þ f 00 :

9ðc þ dÞðf 0 Þ4

where f ðiÞ ¼ f ðiÞ ðaÞ; i ¼ 1; 2; 3; 4. Under the conditions a ¼ b þ c þ d and c þ d – 0, there are

F ð5Þ ðaÞ ¼ 0; F ð6Þ ðaÞ ¼ 

5ðð4b þ c þ 5dÞðf 00 Þ2 þ ðc þ dÞf 0 f 000 Þð99ðf 00 Þ3  18f 0 f 00 f 000 þ ðf 0 Þ2 f ð4Þ Þ 9ðc þ dÞðf 0 Þ5

:

The error equation of (4) is thus

enþ1 ¼ 

   ð4b þ c þ 5dÞðf 00 Þ2 þ ðc þ dÞf 0 f 000 99ðf 00 Þ3  18f 0 f 00 f 000 þ ðf 0 Þ2 f ð4Þ 1296ðc þ dÞðf 0 Þ5

e6n þ Oðe7n Þ:

Similarly, the error equation of (5) is as follows

enþ1 ¼

   ð27c þ 15d þ 11eÞðf 00 Þ2  3ðc þ d þ eÞf 0 f 000 99ðf 00 Þ3  18f 0 f 00 f 000 þ ðf 0 Þ2 f ð4Þ 3888ðc þ d þ eÞðf 0 Þ5

e6n þ Oðe7n Þ:

Hence, the methods defined by (4) and (5) are of order six. This finishes the proof of the theorem.

h

In terms of computational cost, the developed methods require evaluations of only two functions and two first-order 1 derivatives per iteration. Consider the definition of efficiency index [14] as pw , where p is the order of the method and w is the number of function evaluations per iteration required by the method. Assume that all the derivatives’ evaluations have 1 the same cost as the function’s evaluation. The new methods have the efficiency indexes 64 ¼ 1:565, which is better than 1 1 1 33 ¼ 1:442 in [2–6], 54 ¼ 1:495 in [8] and Newton’s method 22 ¼ 1:414 in [1]. 3. Numerical results and conclusions In this section, the results of the numerical tests are presented to compare the efficiency of the developed methods with that of the other sixth-order methods. The tested methods are the classical Newton’s method (NM) in [1], the WM method in [10] (a ¼ 1; b ¼ 1), the KM method in [9], the CY method in [11] (see (34) therein), and the new methods (assuming b ¼ 1; c ¼ 1; d ¼ 0 in (4) and c ¼ 1; d ¼ 3; e ¼ 0:1 in (5)). The formulas for the NM, WM and KM methods are given by (1), (13) and (11), respectively. The formula for the CY method is given as follows

f ðxn Þ ; f 0 ðxn Þ f ðxn Þ f ðyn Þ ; zn ¼ yn  f ðxn Þ  2f ðyn Þ f 0 ðxn Þ y n ¼ xn 

xnþ1 ¼ zn  1 þ

ð8Þ !

4f ðyn Þ f ðzn Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; f ðxn Þ þ signðf ðxn ÞÞ f ðxn Þ  4f ðyn Þ f 0 ðxn Þ

76

X. Wang, L. Liu / Applied Mathematics and Computation 213 (2009) 73–78

Table 1 Comparison of various six-order methods and Newton’s method. f ðxÞ

x0

N

NOFE

NM

WM

KM

CY

(4)

(5)

NM

WM

KM

CY

(4)

(5)

f1

0.5 0.5 1

151 7 5

8 4 2

NC 3 2

NC 4 2

8 3 2

8 3 2

302 14 10

32 16 8

– 12 8

– 16 8

32 12 8

32 12 8

f2

0.8 0 0.5

9 14 16

4 6 7

5 8 NC

4 6 7

4 6 7

4 6 7

18 28 32

16 24 28

20 32 –

16 24 28

16 24 28

16 24 28

f3

0.85 0

1008 5

3 3

5 2

NC 3

3 2

3 2

2016 10

12 12

20 8

– 12

12 8

12 8

f4

1 2

5 4

3 2

NC 2

3 2

3 2

3 2

10 8

12 8

– 8

12 8

12 8

12 8

f5

3 1.2

7 5

3 2

3 2

3 2

3 2

3 2

14 10

12 8

12 8

12 8

12 8

12 8

f6

3.5 4

12 19

5 8

5 9

5 8

5 8

5 8

24 38

20 32

20 36

20 32

20 32

20 32

Numerical computations reported here have been carried out in the Mathematica 4.0 environment. The stoping criterion is j f ðxnþ1 Þ j þ j xnþ1  a j< 1014 . The computing results are displayed in Table 1. The test functions are listed as follows

f1 ðxÞ ¼ x3 þ 4x2  10; f2 ðxÞ ¼

4 Y

a ¼ 1:365230013414097;

ðx  ð1 þ 0:1mÞÞ;

a ¼ 1;

m¼0

a ¼ 0:73908513321516067; a ¼ 1:7461395304080124; f5 ðxÞ ¼ sinðx  1Þ þ ðx  1Þ2 ; a ¼ 1; 2 f6 ðxÞ ¼ ex þ7x30  1; a ¼ 3: f3 ðxÞ ¼ cosðxÞ  x;

ð9Þ

f4 ðxÞ ¼ ex þ cos x;

In Table 1, f ðxÞ: the test function, x0 : the original iteration value, N: the number of iterations, NOFE: the number of function evaluations. NC in Table 1 means that the method does not converge to the root. Here the classical Newton’s method in [1] is the second-order, Wang’ method in [10], Kou’s method in [9], Changbum Chun and YoonMee Ham’s method in [11] and the new methods are the sixth-order. The results show that the new methods have advantages over the Newton’s method and the other sixth-order methods. For example, it can be seen that Kou’s method (KM) and Changbum Chun and YoonMee Ham’s method (CY) have sensitivities to the original iteration value: Kou’s method does not converge (NC) to the zero f1 for x0 ¼ 0:5, f2 for x0 ¼ 0:5, f4 for x0 ¼ 1, Changbum Chun and YoonMee Ham’s method does not converge (NC) to the zero f1 for x0 ¼ 0:5, f3 for x0 ¼ 0:85. The new methods have iteration stabilities to the original iteration value and behave either similarly or better than the methods compared. 4. Two new families of improvements of Jarratt method with sixth-order Applying the developing methodology in Section 2, we obtain two new families of improvements of Jarratt’s method as follows. The first family of improvements is

2 f ðxn Þ 3 f 0 ðxn Þ 3f 0 ðy Þ þ f 0 ðxn Þ f ðxn Þ zn ¼ xn  0 n ; 6f ðyn Þ  2f 0 ðxn Þ f 0 ðxn Þ f ðzn Þ ; xnþ1 ¼ zn  3 0 ðy Þ þ ð1  3 W ðx ÞÞf 0 ðx Þ W ðx Þf n n n f f n 2 2

y n ¼ xn 

where 0

W f ðxn Þ ¼

ðb þ c þ dÞf 0 ðxn Þ þ bf ðyn Þ 0 0 cf ðxn Þ þ df ðyn Þ

and b; c; d 2 R are constants.

ð10Þ

X. Wang, L. Liu / Applied Mathematics and Computation 213 (2009) 73–78

77

The error equation of scheme (10) is

enþ1 ¼ 

   ð4b  3c þ dÞðf 00 Þ2 þ ðc þ dÞf 0 f 000 27ðf 00 Þ3  18f 0 f 00 f 000 þ ðf 0 Þ2 f ð4Þ 1296ðc þ dÞðf 0 Þ5

e6n þ Oðe7n Þ:

It is clear that scheme (10) converges tricubically under the conditions c þ d – 0. When b ¼ 3; c ¼ 2; d ¼ 6, scheme (10) becomes Kou’s method in [9]:

2 f ðxn Þ ; 3 f 0 ðxn Þ f ðxn Þ ; zn ¼ xn  J f ðxn Þ 0 f ðxn Þ

y n ¼ xn 

ð11Þ

f ðzn Þ ; 0 ðy Þ þ ð1  3 J ðx ÞÞf 0 ðx Þ J ðx Þf n n n n 2 f 2 f

xnþ1 ¼ zn  3 where

J f ðxn Þ ¼

3f 0 ðyn Þ þ f 0 ðxn Þ : 6f 0 ðyn Þ  2f 0 ðxn Þ

Therefore, Kou’s method in [9] is a special case of the new family of improvements (10). The second family of improvements of Jarratt’s method is

2 f ðxn Þ ; 3 f 0 ðxn Þ 3f 0 ðy Þ þ f 0 ðxn Þ f ðxn Þ z n ¼ xn  0 n ; 6f ðyn Þ  2f 0 ðxn Þ f 0 ðxn Þ

y n ¼ xn 

0

xnþ1 ¼ zn 

ð12Þ 0

f ðzn Þ cf ðyn Þ þ df ðxn Þ ; 0 0 0 0 af ðyn Þ þ bf ðxn Þ ef ðyn Þ þ hf ðxn Þ

where c ¼ 12 ðae  be  ah  3bhÞ; d ¼ 12 ðae þ 3be þ 3ah þ 5bhÞ; a; b; e; h 2 R are constants. The error equation of scheme (12) is

enþ1 ¼

   gðf 00 Þ2  3ða þ bÞðe þ hÞf 0 f 000 27ðf 00 Þ3  18f 0 f 00 f 000 þ ðf 0 Þ2 f ð4Þ 3888ða þ bÞðe þ hÞðf 0 Þ5

e6n þ Oðe7n Þ;

where g ¼ 11ae þ 15be þ 15ah þ 27bh. Scheme (12) converges tricubically under the condition ða þ bÞðe þ hÞ – 0. When a ¼ 0; b ¼ 1; e ¼ b; h ¼ a, scheme (12) becomes Wang’s method in [10] (see (23) therein):

2 f ðxn Þ ; 3 f 0 ðxn Þ 3f 0 ðy Þ þ f 0 ðxn Þ f ðxn Þ ; z n ¼ xn  0 n 6f ðyn Þ  2f 0 ðxn Þ f 0 ðxn Þ ð5a þ 3bÞf 0 ðxn Þ  ð3a þ bÞf 0 ðyn Þ f ðzn Þ : xnþ1 ¼ zn  2af 0 ðxn Þ þ 2bf 0 ðyn Þ f 0 ðxn Þ y n ¼ xn 

ð13Þ

Therefore, Wang’s method in [10] is a special case of the new family of improvements (12). Acknowledgements This work is funded by the National Science Foundation of China (10701066) and the National Science Foundation of Education Department of Henan Province (2008A110022). References [1] [2] [3] [4] [5] [6] [7] [8]

A.M. Ostrowski, Solution of Equations in Euclidean and Banach Space, third ed., Academic Press, New York, 1973. A.Y. Ozban, Some new variants of Newton’s method, Appl. Math. Lett. 17 (2004) 677–682. Wang Xia, Zhao Ling-ling, Li Fei-min, Two new schemes of Newton’s iteration method, Math. Pract. Theory (in Chinese) 37 (2007) 72–76. Changbum Chun, Construction of third-order modifications of Newton’s method, Appl. Math. Comput. 189 (2007) 662–668. Jisheng Kou, Yitian Li, Xiuhua Wang, Third-order modifications of Newton’s method, J. Comput. Appl. Math. 205 (2007) 1–5. R. Thukral, Introduction to a Newton-type method for solving nonlinear equations, Appl. Math. Comput. 195 (2008) 663–668. I.K. Argyros, D. Chen, Q. Qian, The Jarratt method in Banach space setting, J. Comput. Appl. Math. 51 (1994) 103–106. Jisheng Kou, Yitian Li, The improvements of Chebyshev–Halley methods with fifth-order convergence, J. Comput. Appl. Math. 188 (2007) 143–147.

78 [9] [10] [11] [12]

X. Wang, L. Liu / Applied Mathematics and Computation 213 (2009) 73–78

Jisheng Kou, Yitian Li, An improvement of the Jarratt method, Appl. Math. Comput. 189 (2007) 1816–1821. Xiuhua Wang, Jisheng Kou, Yitian Li, A variant of Jarratt method with sixth-order convergence, Appl. Math. Comput. 204 (2008) 14–19. Changbum Chun, YoonMee Ham, Some sixth-order variants of Ostrowski root-finding methods, Appl. Math. Comput. 193 (2007) 389–394. Changbum Chun, Beny Neta, Some modification of Newton’s method by the method of undetermined coefficients, Comput. Math. Appl. 56 (10) (2008) 2528–2538. [13] Xia Wang, Liping Liu, Two new families of fourth-order methods for non-linear equations, manuscript. [14] W. Gautschi, Numerical Analysis: An Introduction, Birkhauser, 1997.