Two Modified Three-Step Iterative Methods for Solving Nonlinear

Equations

Liang Fang

*, Rui Chen

and Jing Meng

College of Mathematics and Statistics, Taishan University, Tai'an, China

Keywords: Nonlinear equation, iterative method, Newton's method, efficiency index.

Abstract: With the rapid development of information and engineering technology and wide application of science and

technology, nonlinear problems become an important direction of research in the field of numerical

calculation and analysis. In this paper, we mainly study modified iterative methods for solving nonlinear

equations. We present and analyze a sixth-order convergent modified three-step Newton-type method for

solving nonlinear equations. Then we give a seventh-order convergence algorithm. The convergence

analysis of the presented algorithms are given. Both of the given methods are free from second derivatives.

The efficiency indices of the presented methods are 1.431 and 1.476, respectively, which are better than that

of the classical Newton’s method 1.414. Some numerical experiments illustrate the efficiency and

performance of the proposed two methods.

1 INTRODUCTION

Nonlinear problems are an important direction of

research in the field of numerical calculation and

analysis. The solution of nonlinear equations is one

of the most investigated topics in applied

mathematics, numerical analysis, and the problem of

solving nonlinear equations by numerical methods

has gained more importance than before, since many

practical problems in the applied information

technology, as well as in intelligent materials and

mechanical engineering, can build a suitable

mathematical model, and then be transformed into

nonlinear equations to solve.

In this paper, in order to improve the efficiency,

we consider iterative methods to find a simple root

of a nonlinear equation

() 0fx= , (1)

where

RRΩ⊆ →

for an open interval

a scalar function and it is sufficiently differentiable

in a neighborhood of

It is well known that the classical Newton's

method (NM) is an important and basic method for

solving nonlinear equation [1] by the iterative

scheme

()

'( )

=− (2)

which is quadratically convergent in the

neighborhood of

In recent years, much attention has been given to

develop iterative methods for solving nonlinear

equations and a vast literature has been produced [2-

12].

Motivated and inspired by the on-going activities

in this direction, in this paper, we present a sixth-

order convergent three-step iterative method and a

seventh-order convergent method. Both of the three-

step methods are free from second derivatives.

Several numerical results are given to illustrate the

efficiency and advantage of the algorithms.

2 TWO MODIFIED THREE-STEP

METHODS AND THEIR

CONVERGENCE ANALYSIS

Let us consider the following two three-step iterative

methods.

Algorithm 1. For given

, we consider the

following iteration scheme

()

'( )

=−

(3)

2( ) ( ) '( )

'( )

nnn

yfyfy

=− + (4)

5'( ) 3'( ) ( )

'( )

'( ) 7 '( )

nnn

xfyfz

fx fy

=−

(5)

For Algorithm 1, we have the following

convergence result.

THEOREM 1. Assume that the function

RRΩ⊆ → has a single root

∈Ω

, where

Ω is an open interval. If ()

x has first, second

and third derivatives in the interval

Ω , then

Algorithm 1 defined by (3)-(5) is sixth-order

convergent in a neighborhood of

and it satisfies

error equation

56 7

5()

nnn

eceOe

=+ (6)

where

=−

(7)

()

,1,2,

!'()

==L . (8)

Proof. Let

be the simple root of ()

x , and

()

,1,2,

!'()

==L

=−.

Consider the iteration function ()Fx defined by

5'() 3'(()) (())

() ()

'( )

'( ) 7 '( ( ))

xfyxfzx

Fx zx

fx fyx

=−

(9)

where

By some computations using Maple we can

obtain

()

( ) , ( ) 0, 1,2,3,4,5,

FF i

αα α

===

(10)

(11)

Further more, from the Taylor expansion of

()

around

, we get

(4) (5)

(6)

() ()

4! 5!

()

()(()).

xOx

αα

+−+−

(12)

Substituting (11) into (12) yields

56 7

112

5().

nn nn

eceOe

αα

=+ =+ +

Therefore, we have

56 7

5()

nnn

eceOe

which shows that Algorithm 1 defined by (3)-(5)

is sixth-order convergent.

Algorithm 2. For given

, we consider the

three-step Newton-type iteration scheme

()

'( )

=−

(13)

2( ) ( ) '( )

'( )

nnn

yfyfy

=− +

(14)

'( ) '( ) ( )

'( ) 3 '( ) '( )

nnn

xfyfz

xfyfx

=−

−+

(15)

THEOREM 2. Assume that the function

RRΩ⊆ →

has a single root

∈Ω

, where

is an open interval. If

()

has first, second

(2) 5

(6)

225 ( )

() .

2'()

(2) (3)

() () '()( )

() ()

2! 3!

nn n

xFxFFx

ααα

αα

==+ −

+−+−

2 (()) (()) '(())

() () ,

'( )

()

() .

'( )

yx f yx f yx

zx yx

yx x

=− +

=−

and third derivatives in the interval

, then

Algorithm 2 is seventh-order convergent in a

neighborhood of

and it satisfies error equation

42 7 8

1223

10 ( ) ( )

nnn

eccceOe

=−+,

where

=−

()

,1,2,

!'()

==L .

Proof. Let

be the simple root of

()

, and

()

,1,2,,

!'()

==L

=−.

Consider the iteration function

()Fx

defined by

'( ) '( ( )) ( ( ))

() ()

'( ) 3 '( ( )) '( )

xfyx fzx

Fx zx

xfyxfx

=−

−+

(16)

Where

By some computations using Maple we can

obtain

()

( ) , ( ) 0, 1,2,3,4,5,6

FF i

αα α

===

(17)

(18)

Furthermore, from the Taylor expansion of

()

around

, we get

Substituting (18) into (19) yields

42 7 8

22 3

10 ( ) ( ).

cc ce Oe

=+ − +

Therefore, we have

42 7 8

1223

10 ( ) ( )

nnn

eccceOe

=−+

which shows the seventh order of convergence.

To obtain an assessment of the efficiency of the

proposed method, we shall make use of efficiency

index, according to which the efficiency of an

iterative method is given by

, where

is the

order of the method and

is the number of

function evaluations per iteration required by the

method. It is not hard to see that the efficiency

indices of the Algorithm 1 and Algorithm 2 are

1.431 and 1.476 respectively, which are better than

that of the classical Newton's method 1.414.

3 NUMERICAL RESULTS

Now, we employ Algorithm 1 and Algorithm 2 to

solve some nonlinear equations and compare them

with NM and the iterative method (PPM for short)

Potra and Pták presented in [7]

() ()

'( )

xfy

=−

(19)

which is cubically convergent with efficiency index

1.442, where

()

'( )

=−

Displayed in Table 1 are the number of iterations

(ITs) required such that

| ( ) | 1. 14.

fx E<−

In table 1, we use the following functions.

( ) 4 10, 1.36523001341410.fx x x

=+ − =

( ) cos , 0.73908513321516.fx xx

=−=

() ( 1) 1, 2.fx x

=− − =

730

() 1, 3.

fx e

+−

=−=

( ) sin ( ) 1, 1.40449164885154.fx x x

=−+=

( ) ( 2) 1, 0.44285440096708.

fx x e

=+ − =−

7()

()

() () ( )

(( ) )

nn n

xFxF x

αα

==+ −

+−

∑

2(()) (()) '(())

() () ,

'( )

()

() .

'( )

yx f yx f yx

zx yx

yx x

=− +

=−

(2) 4 (2) 2 (3)

(7)

525 ()[3 () 2'() ()]

() .

2'()

ffff

αααα

−+

=−

The computational results in Table 1 show that

Algorithm 1 requires less ITs than NM. Therefore,

Algorithm is of practical interest and can compete

with NM.

Table 1: Comparison of Algorithm 1, Algorithm 2, PPM

and NM

Func

tions

NM PPM

Algorithm

Algorith

m 2

1 5 3 3 3

1.46 4 3 3 3

0.5 4 3 3 3

3.12 7 4 3 3

2.5 6 3 3 3

1.45 7 5 4 4

4.2 16 8 6 6

3.4 12 5 4 4

1.8 5 4 3 3

2.1 5 4 4 3

7.6 12 9 5 5

-0.5 4 4 3 3

4 CONCLUSIONS

In order to obtain efficient iterative methods for the

nonlinear equations which come from the practical

problems in the materials science and manufacturing

technology field, in this paper, we present and

analyze two modified Newton-type iterative

methods for solving nonlinear equations. Both of the

algorithms are free from second derivatives. Several

numerical results illustrate the convergence behavior

and computational efficiency of the method

proposed in this paper. Computational results

demonstrate that they are more efficient and

performs better than the classical NM.

ACKNOWLEDGEMENTS

The work is supported by Project of Natural

Science Foundation of Shandong province

(ZR2016AM06), Excellent Young Scientist

Foundation of Shandong Province (BS2011SF024),

National Natural Science Foundation of China

(11601365).

REFERENCES

1. J. F. Traub, Iterative Methods for Solution of

Equations, Prentice-Hall, Englewood Clis, NJ(1964).

2. M.A. Noor and K. I. Noor, Fifth-order iterative

methods for solving nonlinear equations, Appl. Math.

Comput., 188(1), 406-410, 2007.

3. J. Kou, The improvements of modified Newton's

method, Appl. Math. Comput., 189(1), 602-609, 2007.

4. F.A. Potra and Potra-Pták, Nondiscrete induction and

iterative processes, Research Notes in Mathematics,

103(ISBN-10: 0273086278) (1-250), Pitman, Boston,

1984.

5. J. Kou, Y. Li and X. Wang, Some variants of

Ostrowski's method with seventh-order convergence,

J. Comput. Appl. Math., 209(2), 153-159, 2007.

6. C. Chun, Construction of Newton-like iteration

methods for solving nonlinear equations. Numer.

Math., 104 (3):297-315, 2006.

7. C. Chun: Some fourth-order iterative methods for

solvingnonlinear equations, Applied Mathematics and

Computation 195 (2008) 454-459.

8. S. Huang and Z. Wan, A new nonmonotone spectral

residual method for nonsmooth nonlinear equations,

Journal of Computational and Applied Mathematics,

313:82-101, 2017.

9. L. Fang and G. He, Some modifications of Newton's

method with higher-order convergence for solving

nonlinear equations, J. Comput. Appl. Math., 228(1),

296-303, 2009.

10. A. Galantai, The theory of Newton’s method. J.

Comput. Appl. Math., 124:25-44, 2000.

11. A. N. Muhammad and I. N. Khalida: Modified

iterative methods with cubic convergence for

solving nonlinear equations, Applied Mathematics and

Computation 184 (2007) 322-325.

12. Mamta, V. Kanwar, V.K. Kukreja: On some third-

order iterative methods for solving nonlinear equations,

Applied Mathematics and Computation 171 (2005)

272–280.