Variant of Two Real Parameters Chun-Kim’s Method Free Second
Derivative with Fourth-order Convergence
Rahmawati, Septia Ade Utami and Wartono
Department of Mathematics, UIN Sultan Syarif Kasim Riau, Pekanbaru, Indonesia
Keywords: Efficiency Index, Curvature, Omran’s Method, Newton’s Method, Newton-Steffensen’s Method.
Abstract: Newton’s method is one of the iterative methods that used to solve a nonlinear equation. In this paper, a new
iterative method with two parameters was developed with variant modification of Newton’s method using
curvature and second-order Taylor series expansion, then its second derivative was approximated using
equality of Newton-Steffensen’s and Halley’s Methods. The result of this study shows that this new iterative
method has fourth-order convergence and involves three evaluation of functions with the efficiency index
about 1.5874. In numerical simulation, we use several functions to test the performance of this new iterative
method and the others compared iterative methods, such as: Newton’s Method (MN), Newton-Steffensen’s
Method (MNS), Chun-Kim’s Method (MCK) and Omran’s Method (MO). The result of numerical simulation
shows that the performance of this method is better than the others.
1 INTRODUCTION
Nonlinear equation is a mathematical representation
that arises in the engineering and scientific problems.
The number of assumptions and parameters used to
construct equations will affect the complexity of
nonlinear equations (Chapra, 1998). Therefore,
mathematicians often difficult to determine the
settlement of nonlinear equations. Generally, the
problem arises when a complicated and complex
nonlinear equation cannot be solved using analytical
method. We can use repetitive computing techniques
to find an alternative solution called as iterative
method.
Classical iterative method that widely used by the
researcher to solve nonlinear equations is Newton's
method with general form,
.0)(,
)(
)(
'
'
1
n
n
n
nn
xf
xf
xf
xx
(1)
Newton's method derived from cutting the first order
Taylor’s series with quadratic order convergence and
the efficiency index about
4142.12
2/1
(Traub,
1964).
Lately, the researcher trying to develop iterative
methods with cubic convergence order using several
approaches, such as: adding new steps (Weerakoon
and Fernando, 2000) and (Omran, 2013), second
order Taylor series cutting (Traub, 1964), quadratic
function (Amat et al., 2003); (Amat et al, 2008);
(Melman, 1997); (Sharma, 2005); (Sharma, 2007),
curvature curve (Chun and Kim, 2010), and the
interpretation of two-point geometry (Ardelean,
2013).
Chun-Kim iterative method (Chun and Kim,
2010) was constructed by using curvature, this
method express is,
))(
'
1()(
'
2(
"
)(
))(
'
1)((2)(
"
)(
2*
1
22
1
nnnnn
nnnn
nn
xfxfxfxx
xfxfxfxf
xx
(2)
with
*
1n
x
defined in equation (1). Equation (2) is an
iterative method with a cubic convergence order with
three evaluation functions, and the efficiency index is
about
.4422.13
3/1
In this paper, a new method with two real
parameters is generated from the development of the
Chun-Kim Method (Chun and Kim, 2010) given in
(2) using a second order Taylor sequence expansion.
The new generated iterative method involves two real
parameters θ and λ, this condition allows us to
generate several other new iterative methods of either
two, three or four by replacing the values of the real
parameters.
Since the new iterative method that we generated
still involves second derivative of its function, the use
of the second derivative
)("
n
xf
in the new iterative
Rahmawati, ., Utami, S. and Wartono, .
Var iant of Two Real Parameters Chun-Kim’s Method Free Second Derivative with Fourth-order Convergence.
DOI: 10.5220/0008521203070313
In Proceedings of the International Conference on Mathematics and Islam (ICMIs 2018), pages 307-313
ISBN: 978-989-758-407-7
Copyright
c
2020 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
307
method can be avoided by reducing that second
derivative using the similarity of two iterative
methods as performed by (Kansal et al, 2016) and
(Wartono et al, 2016). Besides, in some cases,
reducing the second derivative by using the approach
undertaken by (Kansal et al, 2016) and (Wartono et
al, 2016) can increase the order of convergence of
iterative methods so that the order of convergence of
iterative methods becomes optimal (Kung and Traub,
1974).
At the end of this paper, numerical simulations are
provided to test the performance of new iterative
methods which include the efficiency index, the
number of iterations, accuracy level and the accuracy
that measured using absolute values of function and
relative error. In numerical simulations, the
performance measures of new iteration methods (M4)
are compared with several other iterative methods,
such as: Newton Method (Traub, 1964), Chun-Kim
Method (Chun and Kim, 2010), Newton- Steffensen
(Sharma, 2005), and the Omran Method (Omran,
2013).
2 METHODS
In this section, we use several theorems and
definitions to construct the iterative method, to
determine the convergence order, the number of
iterations and computationally we computed
convergence order. The theorems and definitions that
we used are:
Theorem 2.1. (Conte, 1980) Given a function 𝑓
which can be derived up to
1n
derivative for each
x on the open interval D containing a, the Taylor
expansion of
)(xf
around a is written ,
with
axax
n
f
xR
n
n
n
,)(
)!1(
)(
)(
)1(
)1(
(4)
Convergence order is an indicator to measure the
convergence velocity of an iterative method on
approaching the roots of nonlinear equations. To
calculated convergence order we use the Taylor’s
series expansion approach or by computation
calculations. The following is given the convergence
order definition.
Definition 2.2. Order of Convergence (Conte, 1980)
and (Mathews, 1992). Let
)(xf
be a function with
the root of the equation
and
Nnn
x
}{
is a sequence
of real numbers for
0n
converging to
for
p
and
there is a constant
c
, it is written,
,...3,2,1,lim
1
pc
x
x
p
n
n
n
(5)
Suppose
nn
xe
and
11 nn
xe
are the
errors in the iterative n and n+1 respectively on an
iterative method that produces the sequence
}{
n
x
,
then the error equation in the iteration (n+1) provided
by
),(
1
1
p
n
p
nn
eOcee
(6)
where c is the asymptotic coefficient of the
convergence order.
Definition 2.3. Computational Order of Convergence
(Weerakoon and Fernando, 2000). Let
be the root
of a nonlinear equation
)(
n
xf
,with
1n
x
,
n
x
and
1n
x
are three successive iterations close enough to
, then the Computational Order of Convergence
(COC) can be calculated using the formula
.
)/()(ln
)/()(ln
1
1
nn
nn
xx
xx
(7)
Definition 2.4. The Efficiency Index (Traub, 1964).
Let p be the order of convergence of an iteration
method, then the efficiency index (I) is given by
d
pI
/1
(8)
with d is the number of function evaluations f
(including derivatives) for each iteration.
3 RESULT AND DISCUSSION
3.1 New Iterative Method with Two
Parameters
To describe the modification of the Newton method
variant, the curve curvature equation in
))(,( xfx
n
through the X axis at
)0,(
1n
x
we reconsidered in the
form
ICMIs 2018 - International Conference on Mathematics and Islam
308
)9(.0
)("
)('1
)(2
)()(
)("
))('1()('
2)(
2
2
1
2
2
1
n
n
n
nnn
n
nn
nn
xf
xf
xf
xfxx
xf
xfxf
xx
Equation (9) can be changed in the form
)10(.
))('1()('2)(")((
)('1(2)()(")(
2*
1
2
1
nnnnn
nnnn
nn
xfxfxfxx
xfxfxfxf
xx
The right side of
*
1n
x
is substituted by one-step
iteration method. Accordingly, we reconsider again
the Newton classical method (1) with a single
parameter
is written as follows,
)('
)(
1
n
n
nn
xf
xf
xx
(11)
then substituting (11) to (10), we have
)(")())('1()('2
)('1(2)()(")(')(
22
2
1
nnnn
nnnnn
nn
xfxfxfxf
xfxfxfxfxf
xx
…(12)
Equation (12) is a cubic convergence equation
with the parameter
and involves three functions
evaluations with efficiency index is about 3
1/3
≈
1.4422.
To increase the convergence order, the iterative
method (12) we substituted into as Taylor series
expansion as did by (Behl and Kanwar, 2013).
Next, we reconsider the Taylor series expansion
of two-order
)(
1n
xf
around
n
x
written in the form,
)("
!2
)(
)(')()()(
2
1
11 n
nn
nnnnn
xf
xx
xfxxxfxf
(13)
If
1n
x
close to
, then
0)(
1
n
xf
. By simplifying
(13) we have
,
)('2)()("
)(2
1
1
nnnn
n
nn
xfxxxf
xf
xx
(14)
furthermore, by replacing
1n
x
on the right side (14)
using (12) is obtained
))('1()('4)(")()('12)(")()('
))('1()('2)(")()(2
22222
22
1
nnnnnnnn
nnnnn
nn
xfxfxfxfxfxfxfxf
xfxfxfxfxf
xx
(15)
Equation (14) still contains second derivative
)("
n
xf
in some cases it is becomes an issue in the
computational process. Therefore, the second
derivative
)("
n
xf
from (15) is reduced using
Newton-Steffensen Method (Sharma, 2005) with the
parameters
and Halley Method (Amat, et al.,
2003), (Melman, 1997) written as follows,
consecutively
,
)()()('
)(
2
1
nnn
n
nn
yfxfxf
xf
xx
(16)
and
.
)(")()('2
)(')(2
2
1
nnn
nn
nn
xfxfxf
xfxf
xx
(17)
Further, the second derivative
)("
n
xf
can be
determined by using similarities (16) and (17), then
we get
.
)(
)()('
2)("
2
2
n
nn
n
xf
yfxf
xf
(18)
Equation (18) is a second derivative which
derived from (16) and (17) containing one parameter
, and by substituting (18) to (15), a new iterative
method without second derivative is obtained
)19(,
))()(())('()('
)()(
2
2
1
nnnn
nn
nn
xfABxfAxfBxf
xfABxf
xx
with
)(,)('1
2
nn
yfBxfA
Equation (19) is the result of modification of the
curvature curve using a second-order Taylor series
expansion involving three evaluations of functions
)(),(
nn
yfxf
, and
).('
n
xf
3.2 Convergence Order Analysis
In this section, we will determine the convergence
order of the iterative method (19) as given in the
following theorem.
Theorem 3.1. Let
D
be the simplest root of the
differentiable function
RRDf :
in an open
interval D. If the initial value
0
x
close to
,
the
iterative method (19) has a four-convergence order
for
1
and
1
which satisfies the error equation
)20(.)('11
)6)('47()('27
)('1
1
4
4
2
32
23
2
2
2
1
n
n
ecf
ccfcf
f
e
Proof. Let
be the root of
)(xf
, then
0)( af
.
Assume that
nn
xe
and
.
)('
)(
!
1
)(
f
f
j
c
j
j
Next, to expand function
)(
n
xf
around
using
Taylor series expansion, we have
Variant of Two Real Parameters Chun-Kim’s Method Free Second Derivative with Fourth-order Convergence
309
)()(')(
54
4
3
3
2
2 nnnnnn
eOecececefxf
(21)
and
)(4321)(')('
43
4
2
32 nnnnn
eOecececfxf
(22)
from (21) and (22) we have
).()22(
)('
)(
432
23
2
2 nnnn
n
n
eOeccece
xf
xf
(23)
and
)24()).(
)10916()812(
)46(41()('1
1)(':
5
4
5
2
342
3
432
22
232
2
2
n
nn
nn
n
eO
ecccceccc
eccecf
xfA
Next, using (23) and
nn
ex
to determined
n
y
,
),()22(
432
23
2
2 nnnn
eOeccecy
(25)
The Taylor series expansion to
)(
n
yf
around
,
produce
)),()375(
)22(()(')(
54
432
3
2
32
23
2
2
nn
nnn
eOecccc
eccecfyf
(26)
Based on (21), (22), (24), and (26) produce
))())('73(
))('1(()('
))()(()('
542
323
2
nn
n
nnn
eOef
eff
xfAyfxf


(27)
and
)28()(
))5)1(2()(')112((
))5)1(()(')255
4)(')8(
)('1()('
))()()()(
)()('()('
5
4
3
2
2
2
22
3
2
2
223
2
222
n
n
n
n
nnn
nnn
eO
ecf
cf
ecf
eff
xfAyfxfA
yfxfxf
Furthermore, using (27),
11
nn
ex
and
nn
ex
, we have
)29(.)(...24
)('1
)('
1)1(2)21(
532
2
2
2
2
3
2
21
nn
nn
eOec
f
f
cece
Equation (29) give us information that convergence
order of the iterative method (19) increases for
1
and
1
. Therefore, by resubstituting
1
and
1
to (29), the convergence order of (19) be
)30().()('11
)6)('47()('27
)('1
1
54
4
2
32
23
2
2
2
1
nn
n
eOecf
ccfcf
f
e
Equation (30) shows that the iterative method (19)
has a four-order convergence for
1
,
1
and
involves three functions evaluation at each iteration,
with the result of efficiency index 4
1/3
≈ 1.5874.
Convergence order of the iterative method with the
efficiency index of 4
1/3
is said to be optimal (Kung
and Traub, 1974).
The new iterative method with two real
parameters
and
given in Equation (19) raises
several other iterative methods by substituting the
values of the real parameter.
If
1
and
0
, then we obtained the Newton
Method (Traub, 1964)
)('
)(
1
n
n
nn
xf
xf
xx
(31)
If
1
and
1
, we obtained four order
convergence iterative method,
.
)()(')()2)('()())('1()('
)())(1()()(
22222
22
1
nnnnnnn
nnnn
nn
yfxfyfxfxfxfxf
yfxfxfxf
xx
(32)
If
0
and
1
, we obtained a three-order convergence iterative method,
.
)()('))()()(())('1()('
))('1()(
222
23
1
nnnnnnn
nn
nn
yfxfyfxfxfxfxf
xfxf
xx
(33)
3.3 Numerical Simulation
In this section, numerical simulations are performed
to test the performance of new methods (19) for
1
and
1
(M-4) and we compare them with some
other iterative methods, such as: Newton Method
(Traub, 1964), Method Chun-Kim (Chun and Kim,
2010), Newton-Steffensen Method (Sharma, 2005),
and Omran Method (Omran, 2013) denoted by MN,
MCK, MNS, and MO respectively.
The performance of an iterative method was
measured from several indicators, they are the
number of iterations that we used, the convergence
ICMIs 2018 - International Conference on Mathematics and Islam
310
order is either computed using the Taylor’s series
expansion or computational order of convergence
(COC), and the accuracy of iteration. One of the most
important indicators of an iterative method is the
convergence order. In the process of determining the
approximation roots, the convergence order of the
iterative method gives effect to the number of
iterations used and the accuracy of the iterative
method. The higher the convergence order of an
iterative method, the number of iterations used is less
and the accuracy is better.
Therefore, the comparable indicators of the
iterative methods are the convergence order, the
number of iterations (IT), the absolute value of the
function at the
th
n
iteration (
)(
n
xf
), and the
relative error
1
nn
xx
.
Numerical simulations are performed by applying
compared iterative methods to some functions using
software Maple 13 and 850 decimal digits (digits
floating point arithmetic). Since the computation
computed convergence order (COC) in each iteration
method is calculated using at least three
approximation roots value, i.e. at the
1n
,
n
, and
1n
iterations, the accuracy is considered sufficient
and can satisfy the condition is
95
10
.
Next, let
0
x
be the initial guess value taken as
close as possible to
(displayed until 20 decimal
digits), then the iteration process can be performed.
The iteration computing process is stopped if it meets
the criteria,
,
1
nn
xx
(34)
with
.10
95
The functions used in this numerical
simulation are:
.00000000000000000000.1,)(
,15341226034044916482.1,1sin)(
,00000000000000000000.1
,1)1(cos)(
,15160641657390851332.0,)(cos)(
,20699298333065847282.4,4)(
,58962964831118325591.0,1.0)(
6
22
5
32
4
3
2
2
1
2
xxxf
xxxf
with
xxexf
xxxf
xexf
xexf
xx
x
x
To determine the performance of an iterative
method, we can see the number of iterations involved
in a certain accuracy. In this simulation the number of
iterations used (IT) and COC of the comparison
methods is shown in Table 1.
Based on Table 1, we can conclude that the M-4
method is better than the other four methods, since it
has fewer iterations. In addition, based on the value
of COC can be concluded that M-4 method has four
convergence order.
Next, we shown the value of
)(
n
xf
and
1
nn
xx
from compared method to the 12 total of
function evaluation (TNFE) which given in Table 2
and Table 3, consecutively.
Table 1: Number of iterations (IT) and COC.
)(xf
0
x
MN
MCK
MNS
MO
M-4
)(
1
xf
-0.20
0.30
8 (1.9999)
8 (1.9999)
5 (2.9999)
5 (3.0000)
5 (2.9999)
5 (3.0000)
5 (3.0000)
5 (3.0000)
4 (3.9999)
4 (3.9999)
)(
2
xf
4.10
4.50
8 (1.9999)
7 (1.9999)
5 (3.0000)
5 (2.9999)
5 (3.0000)
5 (2.9999)
5 (3.0000)
5 (3.0000)
4 (3.9999)
4 (3.9999)
)(
3
xf
-0.10
1.50
8 (1.9999)
7 (1.9999)
5 (3.0000)
5 (2.9999)
5 (3.0000)
5 (2.9999)
5 (3.0000)
5 (3.0000)
4 (3.9999)
4 (3.9999)
)(
4
xf
-1.50
0.10
7 (2.0000)
7 (2.0000)
5 (2.9999)
7 (2.9999)
5 (3.0000)
5 (2.9999)
5 (3.0000)
5 (3.0000)
4 (3.9999)
4 (3.9999)
)(
5
xf
1.20
2.00
8 (1.9999)
8 (1.9999)
5 (3.0000)
5 (2.9999)
5 (3.0000)
4 (2.9999)
5 (3.0000)
5 (3.0000)
4 (3.9999)
4 (3.9999)
)(
6
xf
0.50
1.50
8 (1.9999)
7 (1.9999)
6 (3.0000)
5 (2.9999)
5 (3.0000)
5 (2.9999)
5 (3.0000)
5 (3.0000)
4 (3.9999)
4 (3.9999)
Variant of Two Real Parameters Chun-Kim’s Method Free Second Derivative with Fourth-order Convergence
311
Table 2: The function value
)(
n
xf
for TNFE = 12.
)(xf
0
x
MN
MCK
MNS
MO
M-4
)(
1
xf
-0.20
0.30
3.0851 (e-36)
1.0736 (e-42)
1.1432 (e-40)
1.7153 (e-43)
9.0539 (e-54)
1.2725 (e-45)
4.4068 (e-46)
6.2110 (e-53)
1.4695 (e-142)
9.3099 (e-176)
)(
2
xf
4.10
4.50
2.6996 (e-45)
3.1919 (e-52)
2.7159 (e-47)
3.8771 (e-60)
1.4792 (e-57)
2.4262 (e-66)
6.4848 (e-57)
1.2177 (e-66)
6.0818 (e-159)
7.2098 (e-200)
)(
3
xf
-0.10
1.50
1.9402 (e-37)
3.7607 (e-64)
1.4134 (e-17)
9.4355 (e-50)
1.4119 (e-45)
3.5077 (e-80)
3.1919 (e-47)
3.9419 (e-77)
4.4549 (e-105)
2.3513 (e-198)
)(
4
xf
-1.50
0.10
5.7389 (e-66)
3.0579 (e-64)
7.4069 (e-51)
7.0708 (e-10)
5.1899 (e-92)
2.9369 (e-69)
1.7321 (e-95)
3.7202 (e-60)
6.3165 (e-173)
2.4217 (e-137)
)(
5
xf
1.20
2.00
2.0864 (e-47)
2.2623 (e-32)
2.1333 (e-46)
3.9133 (e-32)
7.4954 (e-60)
8.2994 (e-41)
1.4432 (e-59)
1.3569 (e-40)
9.6912 (e-165)
4.9506 (e-112)
)(
6
xf
0.50
1.50
1.5492 (e-43)
1.0649 (e-66)
1.2886 (e-12)
5.2351 (e-63)
2.2097 (e-55)
2.4094 (e-84)
3.0629 (e-57)
4.7864 (e-83)
4.1246 (e-128)
4.6824 (e-225)
Table 3: Relative error
1
nn
xx
for TNFE = 12.
)(xf
0
x
MN
MCK
MNS
MO
M-4
)(
1
xf
-0.20
0.30
1.9116 (e-18)
1.1277 (e-21)
4.3760 (e-14)
5.0098 (e-15)
2.1608 (e-18)
1.1235 (e-15)
7.8896 (e-16)
4.1058 (e-18)
3.6099 (e-36)
1.8111 (e-44)
)(
2
xf
4.10
4.50
9.0319 (e-24)
3.1057 (e-27)
8.5959 (e-17)
4.4925 (e-21)
3.7720 (e-20)
4.4484 (e-23)
6.1734 (e-20)
3.5451 (e-23)
1.1433 (e-40)
6.7084 (e-51)
)(
3
xf
-0.10
1.50
7.2458 (e-19)
3.1901 (e-32)
3.7159 (e-06)
6.9968 (e-17)
2.5865 (e-15)
7.5472 (e-27)
7.1335 (e-16)
7.8465 (e-26)
1.6788 (e-26)
8.0467 (e-50)
)(
4
xf
-1.50
0.10
2.3956 (e-33)
1.7487 (e-32)
1.5064 (e-17)
6.8799 (e-4)
6.7780 (e-31)
2.6022 (e-23)
4.7015 (e-32)
2.8156 (e-20)
1.1494 (e-43)
9.0447 (e-35)
)(
5
xf
1.20
2.00
3.2751 (e-24)
1.0784 (e-16)
4.2239 (e-16)
2.4000 (e-11)
1.7005 (e-20)
3.7903 (e-14)
2.1155 (e-20)
4.4651 (e-14)
8.2906 (e-42)
1.2464 (e-28)
)(
6
xf
0.50
1.50
1.1133 (e-21)
2.9189 (e-33)
2.1759 (e-4)
3.4727 (e-21)
1.9194 (e-18)
4.2562 (e-28)
4.6106 (e-19)
1.1527 (e-27)
3.5839 (e-32)
2.803 (e-56)
4 CONCLUSIONS
In this conclusion a new iteration method is given by
the four-order convergence for 𝜆 = 1, 𝜃 = 1 and the
index efficiency 4
1/3
≈ 1.5874. The numerical
simulations also provide information that the COC of
the new iteration method is four given by Table 1. In
addition, Tables 2 and 3 show the accuracy and the
precision of the new method better than the Newton
Method (Traub, 1964), Chun-Kim Method Chun and
Kim, 2010), Newton-Steffensen Method (Sharma,
2005), and Omran Method (Omran, 2013).
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