Content uploaded by Mudassir Shams
Author content
All content in this area was uploaded by Mudassir Shams on Jan 04, 2022
Content may be subject to copyright.
Computer Oriented Numerical Scheme for Solving Engineering Problems
Mudassir Shams
1
, Naila Rafiq
2
, Nasreen Kausar
3
, Nazir Ahmad Mir
2
and Ahmad Alalyani
4
,
*
1
Department of Mathematica and Statistics, Riphah International University, I-14, Islamabad 44000, Pakistan
2
Department of Mathematics, NUML, Islamabad 44000, Pakistan
3
Department of Mathematics, Yildiz Technical University, Faculty of Arts and Science, Esenler, 34210, Istanbul, Turkey
4
Department of Mathematics, Faculty of Science and Arts in ALmandaq, Al-Baha University, Al-Baha, Saudi Arabia
*Corresponding Author: Ahmad Alalyani. Email: azaher@bu.edu.sa
Received: 02 August 2021; Accepted: 06 September 2021
Abstract: In this study, we construct a family of single root finding method of
optimal order four and then generalize this family for estimating of all roots of
non-linear equation simultaneously. Convergence analysis proves that the local
order of convergence is four in case of single root finding iterative method and
six for simultaneous determination of all roots of non-linear equation. Some
non-linear equations are taken from physics, chemistry and engineering to present
the performance and efficiency of the newly constructed method. Some real world
applications are taken from fluid mechanics, i.e., fluid permeability in biogels and
biomedical engineering which includes blood Rheology-Model which as an inter-
mediate result give some nonlinear equations. These non-linear equations are then
solved using newly developed simultaneous iterative schemes. Newly developed
simultaneous iterative schemes reach to exact values on initial guessed values
within given tolerance, using very less number of function evaluations in each
step. Local convergence order of single root finding method is computed using
CAS-Maple. Local computational order of convergence, CPU-time, absolute
residuals errors are calculated to elaborate the efficiency, robustness and
authentication of the iterative simultaneous method in its domain.
Keywords: Biomedical engineering; convergence order; iterative method;
CPU-time; simultaneous method
1 Introduction
Finding roots of non-linear equation
fðxÞ¼0;(1)
Is the one of the primal problems of science and engineering. Non-linear equation arise almost in all fields of
science. To approximate the root of Eq. (1), researchers and engineers look towards numerical iterative
techniques, which are further classified to approximate single [1–7] and all roots of Eq. (1). In this
research paper, we work on both types of iterative methods. The most popular method among single root
finding method is classical Newton method having locally quadratic convergence:
This work is licensed under a Creative Commons Attribution 4.0 International License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original
work is properly cited.
Computer Systems Science & Engineering
DOI:10.32604/csse.2022.022269
Article
ech
T
PressScience
vðiÞ¼xðiÞfðxðiÞÞ
f0ðxðiÞÞ
:i¼0;1;2;... (2)
Engineers and mathematician are interested in simultaneous methods due to their global convergence
region and implemented for parallel computing as well. More detail on simultaneous iterative methods
can be seen in [8–17] and reference cited there in.
The main aim of this paper is to propose a modified family of Noor et al. method and generalize it into
numerical simultaneous technique for parallel estimation of all roots of Eq. (1).
This paper is organized in five sections. In Section 2, we construct optimal fourth-order family of single
root finding method and generalize it to simultaneous method of order six. In Section 3, computational aspect
of the newly constructed simultaneous method is discussed and the method is compared with existing method
of the same convergence order existing in the literature. In Section 4, we illustrate some engineering
applications as numerical test examples to show the performance and efficiency of the simultaneous
method. Conclusion is described in Section 5.
2 Construction of Simultaneous Method
Noor et al. [18] present a two-step 4
th
order method (abbreviated as AS):
uðiÞ¼yðiÞfðyðiÞÞ
f0ðyðiÞÞbfðyðiÞÞ
;(3)
where yðiÞ¼xðiÞfðxðiÞÞ
f0ðxðiÞÞbfðxðiÞÞ
and b2<.
According to Kung and Traub [19] conjecture, the iterative method (AS) is not optimal as it requires
2 evaluations of functions and 2 of its first derivatives. To make iterative method (AS) optimal, we use
the following approximation [20]:
f0ðyðiÞÞffi2fðyðiÞÞfðxðiÞÞ
yðiÞxðiÞ
f0ðxðiÞÞ;(4)
in Eq. (3).
uðiÞ¼yðiÞfðyðiÞÞ
2fðyðiÞÞfðxðiÞÞ
yðiÞxðiÞ
f0ðxðiÞÞbfðyðiÞÞ
0
@1
A
where
yðiÞ¼xðiÞfðxðiÞÞ
f0ðxðiÞÞbfðxðiÞÞ
:
8
>
>
>
>
<
>
>
>
>
:
;(5)
The method Eq. (5) is now optimal and the convergence order of Eq. (5) is 4 if ζis simple root of
Eq. (1). Let ε=x−ζ, then by using Maple-18, we find error equation as:
uðiÞf
ðxðiÞfÞ4!ða3þ3a2C23aC2
2þC3
2þaC3C2C3Þ;CkðxÞ¼fðkÞðxÞ
k!f0ðxÞ;(6)
k¼2, 3, ::: or
uðiÞf¼Oð24Þ:(7)
690 CSSE, 2022, vol.42, no.2
Suppose, Eq. (1) has nsimple roots. Then f(x) and f
′
(x) can be written as:
fðxÞ¼ðxx1Þðxx2Þ...ðxxnÞ¼Y
n
j¼1ðxxjÞand (8)
f0ðxÞ¼ðxx2Þðxx3Þ...ðxxnÞþðxx1Þðxx3Þ...ðxxnÞþ...
þðxx1Þðxx2Þ...ðxxn1Þ
f0ðxÞ¼X
n
k¼1Y
n
j6¼k
j¼1
ðxxjÞ:
(9)
This implies,
f0ðxÞ
fðxÞ¼X
n
j¼1
1
ðxxjÞ
;or (10)
fðxÞ
f0ðxÞ¼X
n
j¼1
1
ðxxjÞ
1
¼1
1
xxkþP
n
j6¼k
j¼1
1
ðxxjÞ
;(11)
or
xxk¼1
f0ðxÞ
fðxÞX
n
j6¼k
j¼1
1
ðxxjÞ
:(12)
This gives, Albert Ehrlich method (see [21]).
vðiÞ
k¼xðiÞ
k1
1
NðxðiÞ
kÞP
n
j6¼k
j¼1
1
ðxðiÞ
kxðiÞ
jÞ
;where NðxðiÞ
kÞ¼fðxðiÞ
kÞ
f0ðxðiÞ
kÞ:(13)
Now from Eq. (11),anestimationoffðxðiÞ
kÞ
f0ðxðiÞ
kÞis formed by replacing xðiÞ
jwith uðiÞ
jfrom Eq. (5) as follows:
fðxðiÞ
kÞ
f0ðxðiÞ
kÞ¼1
1
NðxðiÞ
kÞP
n
j6¼k
j¼1
1
ðxðiÞ
kuðiÞ
jÞ
;(14)
Using Eq. (14) in Eq. (2), we have new family of simultaneous method (abbreviated as MS):
vðiÞ
k¼xðiÞ
k1
1
NðxðiÞ
kÞP
n
j6¼k
j¼1
1
ðxðiÞ
kuðiÞ
jÞ
;ðk;j¼1;...;nÞ:(15)
CSSE, 2022, vol.42, no.2 691
In case of multiple roots:
vðiÞ
k¼xðiÞ
krk
1
NðxðiÞ
kÞP
n
j6¼k
j¼1
rj
ðxðiÞ
kuðiÞ
jÞ
;(16)
where uðiÞ
j¼yðiÞ
jfðyðiÞ
jÞ
2
fðyðiÞ
jÞfðxðiÞ
jÞ
yðiÞ
jxðiÞ
j
f0ðxðiÞ
jÞbfðyðiÞ
jÞ
0
B
B
@
1
C
C
A
and yðiÞ
j¼xðiÞ
jfðxðiÞ
jÞ
f0ðxðiÞ
jÞbfðxðiÞ
jÞ
:
Convergence Analysis
Here, we discuss the convergence of simultaneous schemes (MS) as:
Theorem: Let f1;...;fnbe simple roots with multiplicity σ
1
,…,σ
n
of Eq. (1).If x
ð0Þ
1;...;xð0Þ
nbe the
initial calculations of the roots respectively and sufficiently close to actual roots, then MS has convergence
order six.
Proof: Let ɛ
k
=x
k
−ζ
k
and e0
k¼vkfkbe the errors in x
k
and v
k
estimations respectively. For
simplification, we omit iteration index i. Considering method MS, we have:
vk¼xkrk
rk
NðxkÞP
n
j6¼k
j¼1
rj
ðxkujÞ
;(17)
where NðxkÞ¼ fðxkÞ
f0ðxkÞ
:Then, obviously for distinct roots:
1
NðxkÞ¼f0ðxkÞ
fðxkÞ
¼X
n
j¼1
1
ðxkfjÞ
¼1
ðxkfkÞþX
n
j6¼k
j¼1
1
ðxkfjÞ
:(18)
Thus, for multiple roots we have from MS:
vkfk¼xkfkri
rk
ðxkfkÞþP
n
j6¼i
j¼1
rjðxkujxkþfjÞ
ðxkfjÞðxkujÞ
;(19)
e0
k¼ekri
rk
ekþP
n
j6¼i
j¼1
rjðujfjÞ
ðxkfjÞðxkujÞ
¼ekriei
rkþekP
n
j6¼i
j¼1
rjðujfjÞ
ðxkfjÞðxkujÞ
;(20)
¼ekrk:ek
rkþekP
n
j6¼k
j¼1
ðEke4
jÞ
;(21)
where ujfj¼Oðe4
jÞfrom Eq. (7) and Ei¼rj
ðxkfjÞðxkujÞ
:
Thus,
e0
k¼
e2
kP
n
j6¼k
j¼1
ðEke4
jÞ
rkþekP
n
j6¼k
j¼1
ðEke4
jÞ
:(22)
692 CSSE, 2022, vol.42, no.2
3 Computational Aspect
In this section, computational efficiency of the Petkovic et al. [22] method (abbreviated as MP)
xðiþ1Þ
k¼xðiÞ
k1
1
NkðxðiÞ
kÞP
n
j6¼k
j¼1
1
ðxðiÞ
kZðiÞ
jÞ
;(23)
where ZðiÞ
j¼xðiÞ
juðxðiÞ
jÞbjþcjtðxðiÞ
jÞ
1djtðxðiÞ
jÞ
;tðxðiÞ
jÞ¼ fðxðiÞ
jhjuðxðiÞ
jÞÞ
f0ðxðiÞ
jÞ
;hj¼2rj
rjþ2;bj¼
ðrjÞ2
2;dj¼rjþ2
rj
rj
;
cj¼rjðrjþ2Þ
2djand the new method is presented as MS. Efficiency of iterative method is given by
ELðmÞ¼logr
D;(24)
where ris the convergence order and Dis the computational cost:
D¼DðmÞ¼waqAqmþwmMmþwdDm:(25)
Thus, Eq. (24) becomes:
ELðmÞ¼ log r
waqAqmþwmMmþwdDm
:(26)
Using data given by in Tab. 1 and Eq. (26), we calculate ρ((MS), (X)) as follows:
qððMSÞ;ðMPÞÞ ¼ ELðMSÞ
ELðMPÞ1
100
qððMPÞ;MSÞÞ ¼ ELðMSÞ
ELðMPÞ1
100
8
<
:
(27)
Figs. 5–6, graphically illustrates these percentage ratios. It is evident from Figs. 5–6that MS method has
dominating efficiency as compared to MP method.
Figure 4: Shows location of exact real roots of f
5
(x) on x-axis
694 CSSE, 2022, vol.42, no.2
4 Numerical Results
Here, we compare numerical results of our newly constructed method MS with Petkovićet al. method
MP of convergence order 6. All numerical computations are performed using CAS Maple 18 with 64 digits
floating point arithmetic with stopping criteria as follows.
ðiÞek¼jfðxðiÞÞj,e
where e
k
represents the absolute error. Take ε=10
−30
as tolerance for simultaneous methods. In all Tables
stopping criteria (i) is used, CPU represents computational time in seconds and qðiÞ
krepresents local
computational order of convergence [23].
Table 1: The number of basic operations
Method AS
m
M
m
D
m
MS 9m2þOðmÞ1m2þOðmÞ2m2þOðmÞ
MP 8m
2
+O(m)6m2þOðmÞ2m2þOðmÞ
Figure 5: Shows computational efficiency of methods MS w.r.t method MP
Figure 6: Shows computational efficiency of methods MP w.r.t method MS
CSSE, 2022, vol.42, no.2 695
Applications in Engineering
In this section, we discuss some applications in engineering.
Example 1: [24] Blood Rheology Model
Blood, which is a non-Newtonian fluid is modeled as a Caisson Fluid. Caisson fluid model predicts that
simple fluid like water, blood will flow through a tube in such a way that the central core of the fluids will
move as a plug with little deformation and velocity gradient occurs near the wall.
To elaborate the plug flow of Caisson fluid flow, we used the following non-linear equation:
G¼116
7ffiffiffi
x
pþ4
3x1
21 x4
;(28)
where reduction in flow rate is measured by G. Using G= 0.40 in Eq. (28), we have:
f1ðxÞ¼ 1
441 x88
63 x50:05714285714x4þ16
9x23:624489796xþ0:36 (29)
The exact solutions of Eq. (29) are graphed using maple command smartplot3d [f
1
(x)], shown in Fig. 7.
The exact roots of Eq. (29) up to ten decimal place are:
f1¼0:1046986515;f2¼3:822389235;f3¼1:553919850 þ:9404149899i;
f4¼1:238769105 þ3:408523568i;f5¼2:278694688 þ1:987476450i
f6¼2:278694688 1:987476450i;f7¼1:238769105 3:408523568i;
f8¼1:553919850 :9404149899i:
and
x1
ð0Þ¼0:1;x2
ð0Þ¼3:8;x3
ð0Þ¼1:5þ0:9i;x4
ð0Þ¼1:2þ3:4i:
x5
ð0Þ¼2:2þ1:9i;x6
ð0Þ¼2:21:9i;x7
ð0Þ¼1:23:4i;x8
ð0Þ¼1:50:9i:
are chosen as initial guessed values. Tab. 2, clearly shows the dominance behavior of MS over MP iterative
method in terms of CPU time and absolute error. Roots of f
1
(x) are calculated at third iteration.
Figure 7: Shows the analytical solution of f
1
(x) graphically
696 CSSE, 2022, vol.42, no.2
Example 2: [25] Fluid Permeability in Biogels
Specific Hydraulic Permeability relates the pressure gradient to fluid velocity in porous medium (agarose
gel or extracellular Fiber matrix) results the following non-linear polynomial equation:
k¼Rex3
20ð1xÞ2;(30)
or Rex320kð1xÞ2¼0 (31)
where kis specific hydraulic permeability, R
e
radius of the fiber and 0 ≤x≤1 is the porosity.
Using k= 0.4655 and R
e
= 100*10
−9
, we have:
f2ðxÞ¼100 109x3þ9:3100 x218:6200 xþ9:3100 (32)
The exact solution are 3D-plot for different values of R
e
and k are graphed using maple command
smartplot3d [f
2
(x) and f
3
(x)] shown in Fig. 8 for f
2
(x) and Fig. 9 for f
3
(x) respectively. Fig. 10, shows
combined graph of f
2
(x) and f
3
(x) for −5≤k≤5, −5≤R
e
≤5, −400 ≤x≤400.
The exact roots of Eq. (32) are
ζ
1
= 0.9999999997, ζ
2
= 1.000000000, ζ
3
= 9.31*10
18
. The locations of exact root of Eq. (31) on x-axis
as shown in Fig. 1.
We choose the following initial estimates for simultaneous determination of all roots of Eq. (32):
Table 2: Simultaneous finding of all distinct roots of non-linear function f
1
(x)
Method CPU eð3Þ
1eð3Þ
2eð3Þ
3eð3Þ
4eð3Þ
5eð3Þ
6eð3Þ
7eð3Þ
8qð2Þ
k
MP 0.407 9.8e-20 1.4e-15 7.2e-14 6.8e-16 2.0e-13 4.0e-14 2.7e-15 2.4e-14 5.98
MS 0.235 1.8e-39 1.6e-31 1.6e-27 0.0 1.5e-26 0.0 0.0 1.3e-31 6.35
Figure 8: Shows graphically the analytical solution of f
2
(x) using maple command “smartplot3d [f
2
(x)] for
k¼0:4655;Re¼100
CSSE, 2022, vol.42, no.2 697
x1
ð0Þ¼0:9;x2
ð0Þ¼1:1;x3
ð0Þ¼9:31017
:
Using k= 0.3655 and R
e
= 10*10
−9
, we have:
f3ðxÞ¼100 109x3þ9:3100 x218:6200 xþ7:3100 (33)
The exact roots of Eq. (33) are
ζ
1
= 0.9999999997, ζ
2
= 1.000000000, ζ
3
= 7.31*10
18
. The locations of exact root of Eq. (33) on x-axis
are shown in Fig. 2.
We choose the following initial estimates for simultaneous determination of all roots of Eq. (33):
x1
ð0Þ¼0:9;x2
ð0Þ¼1:1;x3
ð0Þ¼7:31017
:
Tab. 3, clearly shows the dominance behavior of MS over MP iterative method in terms of CPU time and
absolute error. Roots of f
2
(x) are calculated at third iteration.
Figure 9: Shows the analytical solution of f
3
(x) in maple using command “smartplot3d [f
3
(x)]”for
5k5;Re¼10
Figure 10: Shows graphically the analytical solution of f
1
(x) using maple command “smartplot3d [f
2
(x)or
f
3
(x)].”for −5≤k≤5, −400 ≤x≤400
698 CSSE, 2022, vol.42, no.2
Tab. 4, clearly shows the dominance behavior of MS over MP iterative method in terms of CPU time and
absolute error. Roots of f
3
(x) are calculated at the third iteration. Figs. 7–10, shows analytical approximate
solution of f
1
(x)−f
3
(x) using maple command smartplot3d. Figs. 1–10, clearly show that analytical
approximate and exact solutions match.
Example 3: [26] Beam Designing Model (An Engineering Problem)
An engineer considers a problem of embedment xof a sheet-pile wall resulting a non-linear function
given as:
f4ðxÞ¼x3þ2:87x210:28
4:62 x:(34)
The exact roots of Eq. (34) are represented in Fig. 3 on x-axis and ζ
1
= 2.0021, ζ
2
=−3.3304, ζ
3
=
−1.5417. The initial guessed values are taken as:
x1
ð0Þ¼1:17;x2
ð0Þ¼7:4641;x3
ð0Þ¼0:5359:
Tab. 5, clearly shows the dominance behavior of MS over MP iterative method in terms of CPU time and
absolute error. Roots of f
4
(x) are calculated at the third iteration.
Example 4: [27]
Consider
f5ðxÞ¼sin3x1
2
sin3x2
2
sin3x2:5
2
;(35)
with multiple exact roots of Eq. (35) as represented in Fig. 4 are ζ
1
=1,ζ
2
=2,ζ
3
= 2.5. The initial guessed
values of the exact roots have been taken as:
Table 3: Simultaneous determination of all distinct roots of f
2
(x)
Method CPU eð3Þ
1eð3Þ
2eð3Þ
3qð2Þ
k
MP 0.018 0.002 0.002 3.7e-30 5.64
MS 0.015 1.2 e-7 2.5e-7 4.5e-45 6.01
Table 4: Simultaneous determination of all distinct roots of f
3
(x)
Method CPU eð3Þ
1eð3Þ
2eð3Þ
3qð2Þ
k
MP 0.019 0.002 0.002 3.7e-35 5.01
MS 0.012 1.2 e-4 2.5e-4 4.5e-45 5.91
Table 5: Simultaneous determination of all distinct roots of f
4
(x)
Method CPU eð3Þ
1eð3Þ
2eð3Þ
3qð2Þ
k
MP 0.016 5.3e-21 5.2e-20 2.2e-20 5.41
MS 0.015 0.0 0.0 0.0 6.38
CSSE, 2022, vol.42, no.2 699
x1
ð0Þ¼0:2;x2
ð0Þ¼1:7;x3
ð0Þ¼3:
For distinct roots, we have:
f51ðxÞ¼sin x1
2
sin x2
2
sin x2:5
2
:(36)
Tab. 6, clearly shows the dominance behavior of MS over MP iterative method in terms of CPU time and
absolute error. Roots of f
5−1*
(x) and f
5
(x) are calculated at the third iteration.
5 Conclusion
In this research article, we developed an optimal family of single root finding method of convergence
order 4 and then extended this family to an efficient numerical algorithm of convergence order 6 for
approximating all roots of Eq. (1). The computational efficiency of our method MS is very large as
compared to MP as given in Tab. 1, which is also obvious from Figs. 5–6. From all Figs. 1–10,Tabs. 1–
6, residual error and CPU time clearly show the dominance efficiency of iterative scheme MS as
compared to MP on same number of iterations.
Funding Statement: The authors received no specific funding for this study.
Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the
present study.
References
[1] M. Cosnard and P. Fraigniaud, “Finding the roots of a polynomial on an MIMD multicomputer,”Parallel
Computing, vol. 15, no. 1–3, pp. 75–85, 1990.
[2] A. Naseem, M. A. Rehman, T. Abdeljawad and Y. M. Chu, “Novel iteration schemes for computing zeros of non-
linear equation with engineering applications and their dynamics,”IEEE Access, vol. 9, pp. 92246–92262, 2021.
[3] J. R. Sharma and H. Arora, “A new family of optimal eight order methods with dynamics for non-linear
equations,”Applied Mathematics and Computation, vol. 273, pp. 924–933, 2016.
[4] A. Naseem, M. A. Rehman and T. Abdeljawad, “Numerical methods with engineering applications and their
visual via polynomiography,”IEEE Access, vol. 9, pp. 99287–99298, 2021.
[5] S. A. Sariman and I. Hashim, “New optimal newton-householder methods for solving nonlinear equations and
their dynamics,”Computer, Material & Continua, vol. 65, no. 1, pp. 69–85, 2020.
[6] A. Naseem, M. A. Rehman and T. Abdeljawad, “Numerical algorithms for finding zeros of nonlinear equations
and their dynamical aspects,”Journal of Mathematics, vol. 2011, pp. 1–11, 2020.
[7] A. Naseem, M. A. Rehman and T. Abdeljawad, “Some new iterative algorithms for solving one-dimensional non-
linear equations and their graphical representation,”IEEE Access, vol. 9, pp. 8615–8624, 2021.
Table 6: Simultaneous determination of all roots of f
5−1*
(x) and f
5
(x)
Method CPU eð3Þ
1eð3Þ
2eð3Þ
3qð2Þ
k
MP 0.047 0.019 4.6e-3 9.9e-10 4.91
MS 0.031 3.5e-6 1.8e-4 0.0 5.32
MP 0.094 5.0e-5 1.7e-2 3.3e-5 4.01
MS 0.062 1.9e-6 2.4e-23 0.0 5.78
700 CSSE, 2022, vol.42, no.2
[8] S. Kanno, N. Kjurkchiev and T. Yamamoto, “On some methods for the simultaneous determination of polynomial
zeros,”Japan Journal of Applied Mathematics, vol. 13, pp. 267–288, 1995.
[9] O. Albert, “Iteration methods for finding all zeros of a polynomial simultaneously,”Mathematics of Computation,
vol. 27, no. 122, pp. 339–334, 1973.
[10] P. D. Proinov and S. I. Cholakov, “Semi local convergence of chebyshev-like root-finding method for
simultaneous approximation of polynomial zeros,”Applied Mathematics and Computation, vol. 236, no. 1,
pp. 669–682, 2014.
[11] B. Sendov, A. Andereev and N. Kjurkchiev, “Numerical solutions of polynomial equations,”In: P. G. Ciarlet,
J. L. Lions (Eds.), Handbook of Numerical Analysis, Elsevier Science, New York, vol. III, pp. 629–777, 1994.
[12] T. F. Li, D. S. Li, Z. D. Xu and Y. I. Fang, “New iterative methods for non-linear equations,”Applied
Mathematics and Computations, vol. 197, no. 2, pp. 755–759, 2008.
[13] N. A. Mir, R. Muneer and I. Jabeen, “Some families of two-step simultaneous methods for determining zeros of
non-linear equations,”ISRN Applied Mathematics, vol. 2011, pp. 1–11, 2011.
[14] P. D. Proinov and M. T. Vasileva, “On the convergence of higher-order ehrlich-type iterative methods for
approximating all zeros of polynomial simultaneously,”Journal of Inequalities and Applications, vol. 336,
pp. 1–26, 2015.
[15] M. Shams, N. Rafiq, N. A. Mir, B. Ahmad, S. Abbasi et al.,“On computer implementation for comparison of
inverse numerical schemes,”Computer System Science and Engineering, vol. 36, no. 3, pp. 493–507, 2021.
[16] W. M. Nourein, “An improvement on two iteration methods for simultaneously determination of the zeros of a
polynomial,”International Journal of Computational Mathematics, vol. 6, pp. 241–252, 1977.
[17] G. H. Nedzhibov, “Iterative methods for simultaneous computing arbitrary number of multiple zeros of nonlinear
equations,”International Journal of Computer Mathematics, vol. 90, no. 5, pp. 994–1007, 2013.
[18] A. R. Alharbi, M. I. Faisal, F. A. Shah, M. Waseem, R. Ullah et al., “Higher order numerical approaches for
nonlinear equations by decomposition technique,”IEEE Access, vol. 7, pp. 44329–44337, 2019.
[19] H. T. Kung and J. F. Traub, “Optimal order of one-point and multipoint iteration,”Journal of the ACM, vol. 21,
no. 4, pp. 643–651, 1974.
[20] N. A. Mir, N. Yasmin and N. Rafiq, “Quadrature based two-step iterative methods for nonlinear equations,”
General Mathematics, vol. 16, no. 1, pp. 33–45, 2008.
[21] S. M. Ilic and L. Rancic, “On the fourth order zero-finding methods for polynomials,”Filomat, vol. 17, pp. 35–
46, 2003.
[22] M. S. Petkovic, L. D. Petkovic and J. Dμzunic, “On an efficient method for simultaneous approximation of
polynomial multiple roots,”Applicable Analysis and Discrete Mathematics, vol. 8, pp. 73–94, 2014.
[23] N. Rafiq, S. Akram, M. Shams and N. A. Mir, “Computer geometries for finding all real zeros of polynomial
equations simultaneously,”Computer Material and Continua, vol. 69, no. 2, pp. 2636–2651, 2021.
[24] R. L. Fournier, “Basic transport phenomena in biomedical engineering,”New York: Taylor & Francis, pp. 1–611,
2007.
[25] W. M. Saltzman, “Drug delivery: Engineering principal for drug therapy,”New York: Oxford University Press
2001.
[26] M. Shams, N. A. Mir, N. Rafiq, A. O. Almatroud and S. Akram, “On dynamics of iterative techniques for
nonlinear equation with application in engineering,”Mathematical Problems in Engineering, vol. 2020, pp.
1–17, 2020.
[27] M. R. Farmer, “Computing the zeros of polynomials using the divide and conquer approach,”Ph.D. dissertation,
Department of Computer Science and Information Systems, Birkbeck, University of London, 2014.
CSSE, 2022, vol.42, no.2 701
