A Class of Three-step

R

oot-solvers with Order of Convergence Five

for Nonlinear Equations

Liang Fang

1,*

, Rui Chen

1

1

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

Keywords: Iterative method, nonlinear equations, order of convergence, efficiency index, Newton's method.

Abstract: The root-finding problem of a univariate nonlinear equation is a fundamental and long-studied problem, and

it has wide applications in mathematics and engineering computation. In this paper, a class of modified

Newton-type methods for solving nonlinear equations is brought forward. Analytical discussions are

reported and the theoretical efficiency of the method is studied. The proposed algorithm requires two

evaluations of the functions and two evaluations of derivatives at each iteration. Therefore the efficiency

indices of it is 1.4953. Hence, the index of the proposed algorithm is better than that of classical Newton’s

method 1.4142. The proposed algorithm in this paper is free from second derivatives. Some numerical

results are finally provided to support the theoretical discussions of the proposed method.

1 INTRODUCTION

One of the most important questions in mathematics

is how to find a solution of the nonlinear equation

() 0,fx=

where

:

f

DR R⊆→

for an open interval

D

has

sufficient number of continuous derivatives in a

neighborhood of the root

α

. Solving nonlinear

equations is a classical problem which has

interesting applications in various branches of

science, and engineering computation. The nonlinear

equations play an important role in many fields of

science, and many numerical methods are developed.

In this paper, we apply iterative method to find a

simple root

α

of the above problem.

It is well known that Newton's method (NM for

simplicity) is one of the most important and famous

methods for computing approximations

α

by the

following iterative scheme [5]

()

.

'( )

n

nn

n

f

x

yx

f

x

=−

(1)

The Newton’s method converges quadratically in

some neighborhood of

α

for some appropriate start

value

0

x

. The main advantage of this method is

that the computation of the second derivative not

required.

In the past decades, much attention has been paid

to develop iterative methods for solving nonlinear

equations, and a large number of researchers try to

improve Newton’s method in order to get a method

with a higher order of convergence and more

accuracy in open literatures, see for example [1-18]

and the references therein for more details.

For example, the algoithm defined by

()

'( )

n

nn

n

f

x

yx

f

x

=−

(2)

22

1

2

3'( ) '( ) ( )

'( )

2 '( ) '( ) 2 '( )

nn n

nn

n

nn n

f

xfy fy

xy

f

x

fx fy fy

+

+

=−

+

(3)

is fifth-order convergent, and it satisfies the

following error equation

22 5 6

1223

1

(6 ) ( ).

2

nnn

eccceOe

+

=−+

The following fifth-order convergent iterative

scheme

22

1

2

3'( ) '( )

[

2 '( ) '( ) 2 '( )

3'( ) '( ) ( )

(1 ) ]

'( ) 5 '( ) '( )

nn

nn

nn n

nnn

nnn

fx fy

xy

fxfy fy

f

xfyfy

f

xfyfx

λ

λ

+

+

=−⋅

+

+

+− ⋅

−+

，

()

'( )

n

nn

n

f

x

yx

f

x

=− ，

(4)

1

3'( ) '( ) ( )

'( ) 5 '( ) '( )

nnn

nn

nnn

f

xfyfy

xy

f

xfyfx

+

+

=−

−+

，

(5)

satisfies error equation

22 5 6

1223

1

(3 2 ) ( ).

2

nnn

eccceOe

+

=−+

Motivated and inspired by the ongoing activities

in the direction, in this paper, to improve the local

order of convergence properties, we present a class

of modified Newton-type iterative method for

solving nonlinear equations.

Based on above two efficient five-order

convergent methods, in this paper, we construct a

new iteration formula by introducing a real

parameter

(0 1)

λλ

≤≤

. The order of convergence

of the proposed method is five, and it does not

depend on the parameter

λ

.

The proposed algorithm in this paper is free from

second derivatives. At each iteration, it requires two

evaluations of the functions and two evaluations of

derivatives. Some numerical results are given to

illustrate the advantage and effectiveness of the

methods.

The rest of the paper is organized as follows: in

Section 2 we describe a class of modified Newton-

type iterative methods and analyze its convergence.

Different numerical test confirm the theoretical

results and allow us to compare our new method

with some other known methods in Section 3.

Finally, the conclusions are given in Section 4.

2 A CLASS OF MODIFIED

NEWTON-TYPE ITERATIVE

METHODS AND

CONVERGENCE ANALYSIS

Now, we consider the following iterative scheme.

Algorithm 1. For given

0

x

, we consider the

following iteration method for solving nonlinear

equation

(6)

(7)

where

(0 1)

λλ

≤≤ is a real parameter.

For Algorithm 1, we have the following

convergence result.

THEOREM 1. Assume that the function

:

f

DR R⊆→

has a single root

D

α

∈

, where

D

is an open interval. If

()

f

x

has first, second

and third derivatives in the interval

D

, then

Algorithm 1 defined by (6)-(7) is fifth-order

convergent in a neighborhood of

α

and it satisfies

the following error equation

(8)

where

,

nn

ex

α

=− (9)

(10)

Proof. Let

α

be the simple root of

()

f

x

, and

nn

ex

α

=−,

Consider the iteration function

()Fx

defined

by

(11)

()

'( )

n

nn

n

f

x

yx

f

x

=−

2256

12 23

1

3( 1) 2 ( )

2

nnn

ec cceOe

λ

+

⎡⎤

=+−+

⎣⎦

，

()

()

,1,2,

!'()

k

k

f

ck

kf

α

α

==L

()

()

,1,2,.

!'()

k

k

f

ck

kf

α

α

==L

22

2

3'() '(())

() () [

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

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

(1 ) ]

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

fx fyx

Fx yx

f

x

fy

x

fy

x

fx fyx fyx

fx fyx fx

λ

λ

+

=−⋅

+

+

+− ⋅

−+

where

(12)

By some computations using Maple we can

obtain

(13)

Furthermore, from

the Taylor expansion of

()

n

Fx at

α

, we have

(14)

Substituting (13) into (14) yields

Therefore, we have the error expression of the

algorithm

which means the order of convergence of the

Algorithm 1 is five. The proof is completed.

3 NUMERICAL RESULTS

This section is devoted to checking the effectiveness

and efficiency of our proposed method Algorithm 1

with NM, and PPM method defined by

(15)

which is third-order convergent with efficiency

index 1.4422.

Table 1 shows the number of iterations (ITs)

required to satisfy the stopping criterion. In the

numerical experiment, we take parameter

0.5.

λ

= For orher parameter

λ

we can obain

similar results. All computations were done by using

MATLAB 7.0 and using 64 digit floating point

arithmetics. In table 1, we use the following

functions.

22

1

( ) sin ( ) 1, 1.404492.fx x x

α

=−+≈

32

2

( ) 4 10, 1.365230.fx x x

α

=+ − ≈

3

3

() ( 1) 1, 2.fx x

α

=− − =

2

4

( ) 3 1, 1.404492.

x

fx x e x

α

=−−+ ≈

2

5

( ) cos , 0.639154.

x

fx xxe x

α

=−+ ≈

6

( ) arctan( ) 1.5, 0.767653.

x

fx e x

α

=− − ≈

The computational results in Table 1 show that

Algorithm 1 requires less ITs than NM and PPM.

Therefore, the proposed method is of practical

interest and can compete with NM and PPM.

Table 1: Comparison of Algorithm 1, PPM and NM.

1

5

()

6

1

()

()

() ( ) (( )).

!

nn

k

k

nn

k

xFx

F

FxOx

k

α

ααα

+

=

=

=+ −+ −

∑

()

() .

'( )

f

x

yx x

f

x

=−

()

() ,

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

i

F

Fi

αα

α

=

==

11

2256

223

1

3( 1) 2 ( ).

2

nn

nn

xe

ccceOe

α

αλ

++

=+

⎡⎤

=+ + − +

⎣⎦

2256

12 23

1

3( 1) 2 ( ),

2

nnn

ec cceOe

λ

+

⎡⎤

=+−+

⎣⎦

1

() ()

'( )

nn

nn

n

f

xfy

xx

fx

+

+

=−

4 CONCLUSIONS

This paper presented and analyzed a class of

modified three-step Newton-type iterative methods

for solving nonlinear equations. The method is free

from second derivatives, and it requires two

evaluations of the functions and two evaluations of

derivatives at each step. Several numerical tests

demonstrate that the method proposed in the paper is

more efficient and perform better than Newton's

method, and PPM.

ACKNOWLEDGEMENTS

The work is supported by Project of Natural Science

Foundation of Shandong province (ZR2016AM06),

Excellent Young Scientist Foundation of Shandong

Province (BS2011SF024).

REFERENCES

1. S. Huang, and Z. WAN, A new nonmonotone spectral

residual method for nonsmooth nonlinear equations,

Journal of Computational and Applied Mathematics

313 (2017), pp. 82-101.

2. Z. Liu, Q. Zheng, and C-E Huang, Third- and fifth-

order Newton-Gauss methods for solving nonlinear

equations with n variables, Applied Mathematics and

Computation, 290(2016), pp. 250-257.

3. X-D. Chen, Y. Zhang, J. Shi, and Y. Wang, An

efficient method based on progressive interpolation for

solving nonlinear equations, Applied Mathematics

Letters, 61(2016), pp. 67-72.

4. S. Amat, C. Bermúdez, M.A. Hernández-Verón, and

E. Martínez, On an efficient image k-step iterative

method for nonlinear equations, Journal of

Computational and Applied Mathematics, 302(2016),

pp. 258-271.

5. X-W FANG, Q. NI, and M-L ZENG, A modified

quasi-Newton method for nonlinear equations, Journal

of Computational and Applied Mathematics 328

(2018) , pp.44-58.

6. F. Ahmad, F. Soleymani, F. Khaksar Haghani, and S.

Serra-Capizzano, Higher order derivative-free iterative

methods with and without memory for systems of

nonlinear equations, Applied Mathematics and

Computation 314 (2017), pp. 199-211.

7. A. Khare, and A. Saxena, Novel PT-invariant

solutions for a large number of real nonlinear

equations, Physics Letters A 380 (2016), pp. 856-862.

8. Y.-C, Kima, and K.-A Lee, The Evans-Krylov

theorem for nonlocal parabolic fully nonlinear

equations, Nonlinear Analysis 160 (2017), pp.79-107.

9. M. Alquran, H. M. Jaradat Muhammed, and I. Syam,

A modified approach for a reliable study of new

nonlinear equation: two-mode Korteweg-de Vries-

Burgers equation, Nonlinear Dynamics, 91(3), 2018,

pp 1619-1626.

10. Z. Wan, and W. Liu, A modifed spectral conjugate

gradient projection method for solving nonlinear

monotone symmetric equations, Pac. J. Optim.

12(2016), pp. 603-622.

11. M. Grau-Sánchez, M. Noguera, and J.L. Diaz-Barrero,

Note on the efficiency of some iterative methods for

solving nonlinear equations, SeMA J. 71 (2015),

pp.15-22.

12. M.S. Petkovic, and J.R. Sharma, On some efficient

derivative-free iterative methods with memory for

solving systems of nonlinear equations, Numer.

Algorithm 71 (2015), pp. 1017-1398.

13. J.R. Sharma, and H. Arora, Efficient derivative-free

numerical methods for solving systems of nonlinear

equations, Comput. Appl. Math. 35 (2016), pp. 269-

284.

14. X. Wang, T. Zhang, W. Qian, and M. Teng, Seventh-

order derivative-free iterative method for solving

nonlinear systems, Numer. Algorithm 70 (2015), pp.

545-558.

15. W.P. Li, F. Zhao, and T.Z. Wang, Small prime

solutions of an nonlinear equation, Acta Math. Sinica

(Chin. Ser.) 58 (2015), pp. 739-764 (in Chinese).

16. H. Chang-Lara, and D. Kriventsov, Further time

regularity for nonlocal, fully nonlinear parabolic

equations, Comm. Pure Appl. Math. 70 (2017), pp.

950-977.

17. S. Deng, Z. Wan, A three-term conjugate gradient

algorithm for large-scale unconstrained optimization

problems, Appl. Numer. Math. 92 (2015), pp. 70-81.

18. Y.H. Dai, M. Al-Baali, X.Q. Yang, A positive

Barzilai-Borwein-Like stepsize and an extension for

symmetric linear systems, Numer. Funct. Anal. Optim.

134 (2015), pp. 59-75.