 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 
()
.
'( )
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
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. 