A family of third-order multipoint methods for solving nonlinear equations

Applied Mathematics and Computation 176 (2006) 409–413 www.elsevier.com/locate/amc

A family of third-order multipoint methods for solving nonlinear equations V. Kanwar Department of Mathematics, Sant Longowal Institute of Engineering and Technology, Longowal, Sangrur 148 106, Punjab, India

Abstract We further present a new modification to the quadratically convergent iteration formulae proposed by Mamta et al. [Mamta, V. Kanwar, V.K. Kukreja, S. Singh, On a class of quadratically convergent iteration formulae, Appl. Math. Comput. 166 (2005) 633–637] for solving single variable nonlinear equations. It is proven that the modification converges cubically. Further, a new family with cubic convergence is obtained by discrete modification and the experiments show that the method is suitable in the cases where Steffensen or Newton–Steffensen methods fail.  2005 Elsevier Inc. All rights reserved. Keywords: Multipoint methods; Root finding; NewtonÕs method; SteffensenÕs method; Newton–Steffensen method; Cubic convergence

1. Introduction Multipoint iterative methods for finding simple zeros of a nonlinear equations have been studied recently ¨ zban [6], Homeier [3,4] and Sharma [7]. These by Frontini and Sormani [1,2], Weerakoon and Fernando [10], O techniques calculate the new approximations to a zero of the given function by sampling per iteration the function and possibly its derivatives for a number of values of the independent variables. All these techniques are variants of NewtonÕs method and the main practical difficulty associated with these techniques is that they fail miserably if at any stage of computation, the derivative of the function is either zero or very small in the vicinity of the required root. We mention below only one root-finding technique given by xnþ1 ¼ xn 

f 2 ðxn Þ ; f 0 ðxn Þff ðxn Þ  f ðaÞg

n P 0;

ð1:1Þ

where a ¼ xn  ff0ðxðxnnÞÞ and x0 is an initial approximation. This technique is developed recently by Sharma [7] and is a composite of Newton and Steffensen [8] methods.

E-mail address: [email protected] 0096-3003/$ - see front matter  2005 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2005.09.029

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V. Kanwar / Applied Mathematics and Computation 176 (2006) 409–413

The purpose of the present work is to develop a new technique having cubic convergence and which can be used as an alternative to existing techniques or in cases where existing techniques are not successful. Mamta et al. [5] have developed one point iterative techniques and one of them is given by xnþ1 ¼ xn 

f ðxn Þ ; f 0 ðxn Þ  pf ðxn Þ

ð1:2Þ

n P 0.

Using this formula of quadratic convergence, a new family of third-order multipoint methods without using second derivative is constructed. 2. Development of the method and its convergence Consider the equation f ðxÞ ¼ 0;

ð2:1Þ

whose solution is required. In order to obtain solutions of nonlinear Eq. (2.1), we consider the iteration scheme of the type xnþ1 ¼ xn 

af ðxn Þ ; f 0 ðxn Þ þ f 0 fxn þ buðxn Þg

n P 0;

ð2:2Þ

ðxn Þ where uðxn Þ ¼ f 0 ðxnfÞpf ; p 2 R and a, b are the disposable parameters. ðxn Þ For every non-zero values of the disposable parameters, each application of (2.2) will require one evaluation of function and two evaluations of its derivatives. We shall now study the properties of iterative method (2.2) by assuming a simple and real root of Eq. (2.1) at x = r and defining the error in nth iteration by

xn ¼ r þ en .

ð2:3Þ

We shall now prove the following theorem: Theorem 1. Assume that f(x) has first, second and third derivatives in (a, b). If f(x) has a simple root r 2 (a, b) and x0 is sufficiently close to r, then the family of methods defined by (2.2) is cubically convergent if a = 2 and b = 1. Proof. Expanding f(x) and f 0 (x) around x = r by TaylorÕs expansion, we have   f ðxn Þ ¼ f ðr þ en Þ ¼ f 0 ðrÞ en þ c2 e2n þ c3 e3n þ oðe4n Þ ; and

  f 0 ðxn Þ ¼ f 0 ðr þ en Þ ¼ f 0 ðrÞ 1 þ 2c2 en þ 3c3 e2n þ oðe3n Þ ;

ð2:4Þ

ð2:5Þ

f k ðrÞ

where ck ¼ k!f 0 ðrÞ ; k ¼ 1; 2; 3 . . . From these results, after some simplifications, we get uðxn Þ ¼

f ðxn Þ ¼ en  fc2 þ j p jge2n þ oðe3n Þ; f 0 ðxn Þþ j p j f ðxn Þ

ð2:6Þ

and hence f 0 fxn þ buðxn Þg ¼ f 0 ðrÞ½1 þ 2ð1 þ bÞc2 en þ f3ð1 þ bÞ2 c3  2bc22  2b j p j c2 ge2n þ oðe3n Þ.

ð2:7Þ

From (2.4) and (2.7), we get f 0 ðxn Þ þ f 0 fxn þ buðxn Þg     3 2 ð1 þ ð1 þ bÞ Þc3  bc22  b j p j c2 e2n þ oðe3n Þ. ¼ 2f 0 ðrÞ 1 þ ð2 þ bÞc2 en þ 2

ð2:8Þ

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Substituting (2.8) in (2.2) and expanding, finally we derive  

  a a a 3 2 2 2 2 2 þ b þ 3b c3  ð2 þ b þ 4bÞc2  j p j bc2 e3n þ oðe4n Þ. enþ1 ¼ 1  en þ ð1 þ bÞc2 en þ 2 2 2 2 ð2:9Þ We can now use these disposable parameters a and b to ensure that the iteration (2.2) will be cubically convergent for simple and real roots of nonlinear Eq. (2.1). For this we require  1  a2 ¼ 0 ð2:10Þ 1þb¼0 which implies a = 2 and b = 1. Therefore, for a = 2 and b = 1, iterative method (2.2) becomes xnþ1 ¼ xn 

f 0 ðx

nÞ

þ

f0

2f ðx Þ  n ; ðxn Þ xn  f 0 ðxnfÞpf ðxn Þ

ð2:11Þ

n P 0.

Here, the parameter p in the denominator is chosen so that the corresponding functions f 0 (xn) and pf(xn) have the same sign. In order to make this happen, we take  ¼ þve; if pf ðxn Þf 0 ðxn Þ 6 0; ð2:12Þ ¼ ve; if pf ðxn Þf 0 ðxn Þ P 0. Furthermore, using a = 2 and b = 1, the error Eq. (2.9) reduces to n c3 o 3 enþ1 ¼ c2 ðj p j þc2 Þ þ e þ oðe4n Þ. 2 n If we let p = 0, then (2.11) reduces to method given by Weerakoon and Fernando [10]. h

ð2:13Þ

3. Discrete modification of formula (2.11) SteffensenÕs method is the discrete modification of NewtonÕs method. Each application of the method (2.11) will require one function and two derivative evaluations per iteration. To reduce the number of the computations of derivatives, one may use the corresponding discrete modifications. Replacing the denominator of (2.11) namely   f ðxn Þ f 0 ðxn Þ þ f 0 xn  0 ; with its discrete modifications f ðxn Þ  pf ðxn Þ 

 ð3:1Þ 2ff 0 ðxn Þ  pf ðxn Þg f ðxn Þ f ðxn Þ  f xn  0 ; f ðxn Þ f ðxn Þ  pf ðxn Þ we get the following discrete modified formula of (2.11) as xnþ1 ¼ xn 

n

f 2 ðxn Þ

 o ; ðxn Þ ff 0 ðxn Þ  pf ðxn Þg f ðxn Þ  f xn  f 0 ðxnfÞpf ðxn Þ

n P 0.

ð3:2Þ

If we let p = 0, then (3.2) reduces to (1.1) which is the composite of Newton and Steffensen methods. This discrete modification requires the information of two functions and one derivative per iteration. Again in (3.2), the sign in the denominator should be so chosen as to make the denominator largest in magnitude. In order to make this happen, we take  ¼ þve; if f ðxn Þf 0 ðxn Þ  f fxn  f ðxn Þg 6 0; ð3:3Þ ¼ ve; if f ðxn Þf 0 ðxn Þ  f fxn  f ðxn Þg P 0.

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V. Kanwar / Applied Mathematics and Computation 176 (2006) 409–413

Table 1 Test functions, their initial points, number of iterations and roots f(x)

x0

SteffensenÕs method

Method (1.1)

f1

pffiffiffi  2=2 pffiffiffi 2=2 pffiffiffiffiffi 21=7

12

Divergent

5

1.000000000000000

1.000000000000000

12

Divergent

5

1.000000000000000

1.000000000000000

7

Divergent

13

1.000000000000000

1.000000000000000

4 4 5

1.895494341850281 1.895494341850281 1.895494222640991

0.000000000000000 1.895494222640991 1.895494222640991

1.000000000000000 Divergent Divergent

1.000000000000000 1.000000000000000 1.000000000000000

f2

1.0 2.0 3.0

f3

0.0 0.5 0.7

1 Divergent Divergent

Fails Divergent Divergent

1 20 8

f4

2.8 3.5

Divergent Divergent

Divergent 7

9 7

f5

1.5 2.0 3.0

f6

f7

f8

f9

f10

5 2 4

Method (3.2)

4 3 4

5 3 3

4 3 3

Root by Steffensen

Divergent Divergent

Root by others

3.0000000000000000 3.0000000000000000

4 3 3

2.740646123886108

2.740646123886108

0.699316561222076 0.699316561222076 0.699316561222076 0.699316561222076 Divergent

0.369256407022476

0.18 0.1 0.0 0.11 0.17

5 5 4 4 Divergent

Divergent Divergent 3 3 3

3 3 5 2 3

0.5 0.0 0.1 3.0

12 12 37 62

9 Fails 7 3

5 4 4 4

3.0 1.0 1 3.0

4 4 3 4

5 3 2 3

2 4 2 3

1.0 1.1 1.5 3.5

1 9 Divergent 24

Fails 45 4 4

1 3 3 4

3.0 0.5 0.1 0.1

Divergent Divergent Divergent Divergent

8 7 Divergent Divergent

9 4 6 6

1.365230083465576 1.365229964256287

0.739085137844086

0.739085137844086

2.000000000000000

2.000000000000000

Divergent

1.207647800445557

4. Numerical results In this section, the results of some numerical tests to compare the efficiency and accuracy of the methods are presented. The SteffensenÕs method, method (1.1) and the method that we developed by discrete modification (3.2) for p = 1 are employed. The following test functions have been used with termination criterion jf(xn)j < 1.0 · 1015: f1 : 4x4  4x2 ;

f 5 : x log 10ðxÞ  1:2; 3

f 3 : x10  1;

f 2 : sin x  x=2;

f 9 : ðx  1Þ  1;

f 6 : ex þ cosðpxÞ  1; x2

2

f 4 : ex

2 þ7x30

 1;

f 7 : x3 þ 4x2  10;

f 10 : xe  sin x þ 3 cos x þ 5.

f 8 : cos x  x;

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5. Conclusions In all these techniques a reasonably close initial guess is necessary for the methods to converge and from the numerical results of Table 1, it can be seen that the method (3.2) has cubic convergence. The method (2.11) has cubic convergence and requires one evaluation of function and two evaluations of its derivatives per iteration. The Steffensen method requires two evaluations of the function and is quadratically convergent. However, the new proposed method (3.2) and the method proposed by Sharma [7] require two evaluations of the function and one evaluation of the derivative term per iteration unlike SteffensenÕs method. In applying the existing techniques to solve the equation 4x4  4x2 = 0, problems arise if the points give horizontal pffiffiffi pffiffiffiffiffi tangents or cycle back and forth from one to another. The points  2=2 give horizontal tangents and  21=7 cycle, each leading to the other and back [9]. On the other hand, our method provides the root 1 and 1 except at the point pffiffiffiffiffi  21=7 whatever starting value is chosen. Therefore, this technique can be used as an alternative to existing techniques or in cases where existing techniques fail. References [1] M. Frontini, E. Sormani, Some variants of NewtonÕs method with third-order convergence, Appl. Math. Comput. 140 (2003) 419– 426. [2] M. Frontini, E. Sormani, Modified NewtonÕs method with third-order convergence and multiple roots, J. Comput. Appl. Math. 156 (2003) 345–354. [3] H.H.H. Homeier, A modified Newton method for rootfinding with cubic convergence, J. Comput. Appl. Math. 157 (2003) 227–230. [4] H.H.H. Homeier, A modified Newton method with cubic convergence: the multivariate case, J. Comput. Appl. Math. 169 (1) (2004) 161–169. [5] Mamta, V. Kanwar, V.K. Kukreja, S. Singh, On a class of quadratically convergent iteration formulae, Appl. Math. Comput. 166 (2005) 633–637. ¨ zban, Some new variants of NewtonÕs method, Appl. Math. Lett. 17 (2004) 677–682. [6] A.Y. O [7] J.R. Sharma, A composite third order Newton–Steffensen method for solving nonlinear equations, Appl. Math. Comput. 169 (1) (2005) 242–246. [8] I.F. Steffensen, Remarks on iteration, Scand. Aktuarietidskr. 16 (1933) 64–72. [9] G.B. Thomas Jr., R.L. Finney, M.D. Wier, F.R. Giordano, Thomas Calculus, Addison–Wesley Publishing Company, Inc, 2001, p. 301. [10] S. Weerakoon, T.G.I. Fernando, A variant of NewtonÕs method with accelerated third-order convergence, Appl. Math. Lett. 13 (2000) 87–93.