Applied Mathematics and Computation 171 (2005) 272–280

www.elsevier.com/locate/amc

On some third-order iterative methodsfor solving nonlinear equations

Mamta, V. Kanwar *, V.K. Kukreja, Sukhjit Singh

Department of Mathematics, Sant Longowal Institute of Engineering and Technology

Longowal, Sangrur, Punjab-148 106, India

Abstract

Using the iteration formulas of second order [Mamta, V. Kanwar, V.K. Kukreja,

Sukhjit Singh, On a class of quadratically convergent iteration formulae, Appl. Math.

Comput. 2004, in press] for solving single variable nonlinear equations, two classes of

third-order multipoint methods without using second derivative are derived. The main

advantage of these classes is that they do not fail if the derivative of the function is either

zero or very small in the vicinity of the required root. Further, a new family of Secant-

like method with guaranteed super linear convergence is obtained by discrete modifica-

tions and their comparison with respect to the existing classical methods is given.

� 2005 Elsevier Inc. All rights reserved.

Keywords: Nonlinear equations; Secant method; Newton�s method; Multipoint methods; Third-

order convergence

0096-3003/$ - see front matter � 2005 Elsevier Inc. All rights reserved.

doi:10.1016/j.amc.2005.01.057

* Corresponding author.

E-mail address: vmithil@yahoo.co.in (V. Kanwar).

Mamta et al. / Appl. Math. Comput. 171 (2005) 272–280 273

1. Introduction

There is a class of third-order methods requiring the evaluation of second

derivative such as Euler�s, Halley�s, super Halley�s, Chebyshev�s etc. These

third-order methods require more computational cost than other simpler meth-

ods and this makes them disadvantageous to be used in general. The mainpractical difficulty associated with these methods is the evaluation of second

derivative. To overcome this difficulty, many researchers have developed

third-order methods which do not require the use of second or higher deriva-

tives of the function. These multipoint iterative techniques for single variable

nonlinear equations have been studied by Ortega and Rheinboldt [4], Potra

and Ptak [6], Frontini and Sormani [1,2], Weerakoon and Fernando [8], Ozban

[5] etc. We mention below only three iterative root-finding techniques to solve

nonlinear equations, given by

xnþ1 ¼ xn �f ðxnÞ

f 0ðx�nþ1Þ; n P 0; ð1:1Þ

where x�nþ1 ¼ xn � f ðxnÞ2f 0ðxnÞ.

xnþ1 ¼ xn �f ðx�nþ1Þf 0ðxnÞ

; n P 0; ð1:2Þ

where x�nþ1 ¼ xn � f ðxnÞf 0ðxnÞ and

xnþ1 ¼ xn �2f ðxnÞ

ff 0ðxnÞ þ f 0ðx�nþ1Þg; n P 0; ð1:3Þ

where x�nþ1 ¼ xn � f ðxnÞf 0ðxnÞ and x0 is an initial guess for the required root.

These techniques have been studied in [4,6] and [8]. All these techniques fail

miserably if at any stage of computation, the derivative of a function is either

zero or very small in the vicinity of a required root. The purpose of the present

work is to develop a new class of iterative techniques having cubic convergence

and which can be used as an alternative to existing techniques or in cases where

existing techniques are not successful. Recently, Mamta et al. [3] have devel-

oped one-point iterative techniques given by

xnþ1 ¼ xn �f ðxnÞ

f 0ðxnÞ � pf ðxnÞ; n P 0; ð1:4Þ

and

xnþ1 ¼ xn �2f ðxnÞ

f 0ðxnÞ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif 02ðxnÞ þ 4p2f 2ðxnÞ

p ; n P 0: ð1:5Þ

These techniques converge quadratically to the real and simple zero ofnonlinear equation. Using these iteration formulae of second order, new classes

274 Mamta et al. / Appl. Math. Comput. 171 (2005) 272–280

of third-order multipoint methods without using second derivative are

developed.

2. Third-order formulae

We shall present here two different classes of iteration techniques.

(a) In order to obtain solutions of nonlinear equations

f ðxÞ ¼ 0; ð2:1Þ

we consider first a family of iterative methods defined by the formula

xnþ1 ¼ xn �f ðxnÞ

f 0fxn þ auðxnÞg; n P 0; ð2:2Þ

where uðxnÞ ¼f ðxnÞ

f 0ðxnÞ � pf ðxnÞ, p 2 R and a is a free parameter.

For every nonzero value of the parameter a, 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 methods (2.2) by assuming a simple

and real root of Eq. (2.1) at x = r and defining the error en in nth iteration

by

xn ¼ r þ en: ð2:3ÞExpanding f(xn) and f 0(xn) about the root x = r by Taylor�s expansion, we have

f ðxnÞ ¼X1k¼1

Ckekn; ð2:4Þ

and

f 0ðxnÞ ¼X1k¼1

kCkek�1n ; ð2:5Þ

where Ck ¼ f ðkÞðrÞk! and C0 = f(r) = 0.

From these results, after some simplifications we get

uðxnÞ ¼f ðxnÞ

f 0ðxnÞ þ jpjf ðxnÞ¼ en �

C2

C1

þ jpj� �

e2n þ oðe3nÞ; ð2:6Þ

and hence

f 0fxn þ auðxnÞg ¼ C1 þ 2C2ð1þ aÞen

þ 3C3ð1þ aÞ2 � 2C22

C1

a� 2ajpjC2

� �e2n þ oðe3nÞ: ð2:7Þ

Mamta et al. / Appl. Math. Comput. 171 (2005) 272–280 275

Substituting (2.7) in (2.2) and expanding, finally we derive

enþ1 ¼C2

C1

ð1þ 2aÞ� �

e2n

þ 3C3

C1

� 4C22

C21

( )ð1þ aÞ2 � 2ajpjC2

C1

� C3

C1

þ 2C22

C21

!e3n: ð2:8Þ

A family of third-order formulae may be obtained by assigning a specific value

to free parameter a. For this we require

1þ 2a ¼ 0 or a ¼ �1=2: ð2:9Þ

Therefore, for a = �1/2, iterative method (2.2) becomes

xnþ1 ¼ xn �f ðxnÞ

f 0 xn � f ðxnÞ2ff 0ðxnÞ�pf ðxnÞg

n o ; n P 0: ð2:10Þ

The parameter p is chosen such that the corresponding functions f 0(xn) and

pf(xn) have the same sign. For a = 0, the family (2.2) reduces to Newton�s for-mula. If we let p = 0 in (2.10), we obtain iteration formula (1.1).

(b) If we consider a family of iterative method defined by the formula

xnþ1 ¼ xn �f ðxnÞ

f 0fxn þ auðxnÞg; n P 0; ð2:11Þ

where

uðxnÞ ¼2f ðxnÞ

f 0ðxnÞ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif 02ðxnÞ þ 4p2f 2ðxnÞ

p ;

then again for a = �1/2, we obtain a family of third-order formulae. Therefore,for a = �1/2, iterative method (2.11) reduces to

xnþ1 ¼ xn �f ðxnÞ

f 0 xn � f ðxnÞf 0ðxnÞ�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif 02 ðxnÞþ4p2f 2ðxnÞ

p� � : ð2:12Þ

The sign in the denominator is chosen so that xn+1 has the smallest absolute

value. For p = 0, (2.12) reduces to (1.1).

3. Numerical results

In this section, the results of some numerical tests to compare the efficiency

and accuracy of the methods are presented. The method (1.1), Weerakoon

276 Mamta et al. / Appl. Math. Comput. 171 (2005) 272–280

and Fernando (1.3) and the method that we developed (2.12) for p = 1/2 are

employed. The following test equations have been used with termination crite-

rion jf(xn)j < 1.0 · 10�15.

(a) (x � 2)23 � 1 = 0,

(b) x3 + x2 � 2 = 0,(c) x3 + 4x2 � 10 = 0,

(d) sin2x � x2 + 1 = 0,

(e) x2 � ex � 3x + 2 = 0,

(f) cosx � x = 0,

(g) ex + cos (px) � 1 = 0,

(h) (x � 1)3 � 1 = 0,

(i) x10 � 1 = 0,

(j) xex2 � sin2xþ 3 cos xþ 5 ¼ 0;

(k) tan�1x = 0,

(l) ex2þ7x�30 � 1 ¼ 0;

(m) lnx = 0,

(n) 4x4 � 4x2 = 0.

4. Discrete modification of formula (1.5)

Secant method is the variation of Newton�s method due to Newton him-

self. The method (1.5) is dependent on derivative and its applications are re-

stricted rigorously. To avoid the evaluation of derivatives one may use the

corresponding discrete modification. Replacing f 0(xn) by the gradient of the

chord joining the point {xn, f(xn)} with a neighbouring point {xn� 1, f(xn� 1)},

we have

f 0ðxnÞ ¼f ðxnÞ � f ðxn�1Þ

xn � xn�1

: ð4:1Þ

After substitution of (4.1) in the denominator of (1.5), we obtain the following

discrete modification of formula (1.5) as

xnþ1

¼ xn �2f ðxnÞðxn � xn�1Þ

ff ðxnÞ � f ðxn�1Þg�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðf ðxnÞ � f ðxn�1ÞÞ2 þ 4p2f 2ðxnÞðxn � xn�1Þ2

q ;

ð4:2Þ

where the sign should be so chosen so as to make the denominator largest in

magnitude. For p ! 0, (4.2) reduces to Secant formula. Relation (4.2) definesthe family of Secant-like method for solving nonlinear equations.

Table 1

Test equations, their initial points, number of iterations and roots

f(x) = 0 x0 Newton�s Method (1.1) Method (1.3) Halley�s method Method (2.12) Root by Newton Root by others

(a) 3.5 13 8 9 7 8

4.5 24 15 17 13 15 3.0000000000000000 3.000000000000000

5.0 28 17 19 15 18

(b) �0.5 11 Divergent Divergent 7 5

0.0 Fails Fails Fails Fails 4 1.000000000000000 1.000000000000000

2.0 5 3 3 3 3

3.0 6 4 4 4 4

(c) 0.0 Fails Fails Fails Fails 4

0.1 9 6 7 5 4

1.0 4 3 3 3 3 1.36229964256287 1.36229964256287

2.0 4 3 3 3 3

3.0 5 3 4 3 4

(d) 0.0 Fails Fails Fails Fails 4

0.5 7 6 5 4 4 1.404491602979126 1.404491602979126

1.0 5 3 3 3 3

3.0 5 3 3 4 4

(e) �4.0 5 3 4 3 4

�2.0 4 3 3 3 3

0.0 3 2 2 2 2 0.257530272006989 0.257530272006989

2.0 4 4 3 3 3

3.0 5 4 4 3 4

(f) �1.0 7 5 3 5 3

0.0 4 3 3 3 3 0.739085137844086 0.739085137844086

2.0 3 3 3 3 3

3.0 6 3 9 4 3

(continued on next page)

Mamta

etal./Appl.Math.Comput.171(2005)272–280

277

Table 1 (continued )

f(x) = 0 x0 Newton�s Method (1.1) Method (1.3) Halley�s method Method (2.12) Root by Newton Root by others

(g) 0.0 6 4 3 5 4 0.369256407022476 0.369256407022476

�0.1 71 Divergent Divergent 6 4 �7.318241119384766 0.369256407022476

�0.3 5 3 4 11 3 �0.699316561222076 �0.699316561222076

(h) �1.0 9 5 7 10 5

0.0 8 5 14 1 5

1.0 Fails Fails Fails Fails 5 2.000000000000000 2.000000000000000

2.5 3 3 3 3 3

3.5 4 4 4 4 4

(i) 0.8 8 5 5 4 5

1.5 8 6 10 4 5 1.000000000000000 1.000000000000000

(j) �3.0 13 8 9 7 8

�2.0 7 4 5 4 4

�1.0 4 3 3 2 3

�0.5 9 5 11 3 4 �1.207647800445557 �1.207647800445557

0.0 9 Divergent Divergent 4 4

0.5 10 Divergent Divergent 4 6

(k) 2.0 Divergent 4 Divergent 6 4 Divergent 0.000000000000000

(l) 3.3 8 5 5 4 5 3.000000000000000 3.000000000000000

3.5 11 7 7 6 7

(m) 0.1 6 4 4 Divergent 4 1.000000000000000 1.000000000000000

3.0 Divergent 4 Divergent 3 4

(n)ffiffiffi2

p=2 59 37 38 Divergent 4 �1.000000000000000 1.000000000000000

�ffiffiffi2

p=2 59 37 38 Divergent 4 1.000000000000000 �1.000000000000000

278

Mamta

etal./Appl.Math.Comput.171(2005)272–280

Mamta et al. / Appl. Math. Comput. 171 (2005) 272–280 279

5. Convergence analysis of formula (4.2)

Using (2.3) in (4.2) and expanding by Taylor�s expansion, we get the follow-ing error equation

enþ1 ¼ en�1enf 00ðrÞ2f 0ðrÞ : ð5:1Þ

The limiting difference Eq. (5.1) can be linearised by taking logarithm of both

sides and its initial equation

m2 � m� 1 ¼ 0; ð5:2Þhas the unique real and positive simple root 1.618. This means that the order of

convergence as well as efficiency index of family (4.2) is 1.618 for every value of

p.

6. Conclusions

In all these techniques a reasonably close starting guess is necessary for the

methods to converge and from the numerical results of Table 1, it can be seen

that the family (2.12) has cubic convergence. In applying the existing tech-

niques to solve the equation 4x4 � 4x2 = 0, problems arise if the points give

horizontal tangents. The points �ffiffi2

p

2give horizontal tangents [7] and the exist-

ing techniques do not provide the solution. However, our methods provide the

solution. Further, these techniques do not fail if the gradient of the function iseither zero or very small in the vicinity of the required root. Therefore, these

techniques can be used as an alternative to existing techniques or in cases where

existing techniques fail. Also these techniques are, in general, cheaper than any

other method of the same order requiring the evaluation of second and higher

derivatives.

References

[1] M. Frontini, E. Sormani, Some variants of Newton�s method with third-order convergence,

Appl. Math. Comput. 140 (2003) 419–426.

[2] M. Frontini, E. Sormani, Modified Newton�s method with third-order convergence and

multiple roots, J. Comput. Appl. Math. 156 (2003) 345–354.

[3] Mamta, V. Kanwar, V.K. Kukreja, Sukhjit Singh, On a class of quadratically convergent

iteration formulae, Appl. Math. Comput. in press, doi:10.1016/j.amc.2004.07.008.

[4] J.M. Ortega, W.C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables,

Academic Press, New York, 1970.

[5] A.Y. Ozban, Some new variants of Newton�s method, Appl. Math. Lett 17 (2004) 677–682.

[6] F.A. Potra, V. Ptak, Nondiscrete Induction and Iterative Processes, Research Notes in

Mathematics, vol. 103, Pitman, Boston, MA, 1984.

280 Mamta et al. / Appl. Math. Comput. 171 (2005) 272–280

[7] G.B. Thomas Jr., R.L. Finney, M.D. Wier, F.R. Giordano, Thomas� Calculus, Addison-

Wesley Publishing Company, Inc., 2001, pp. 301.

[8] S. Weerakoon, T.G.I. Fernando, A variant of Newton�s method with accelerated third-order

convergence, Appl. Math. Lett. 13 (2000) 87–93.