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symmetry
S
S
Article
Improving the Convergence of Interval Single-Step Method for
Simultaneous Approximation of Polynomial Zeros
Nur Raidah Salim 1,*,† , Chuei Yee Chen 1,2,† , Zahari Mahad 1and Siti Hasana Sapar 1,2
Citation: Salim, N.R.; Chen, C.Y.;
Mahad, Z.; Sapar, S.H. Improving the
Convergence of Interval Single-Step
Method for Simultaneous
Approximation of Polynomial Zeros.
Symmetry 2021,13, 1971. https://
doi.org/10.3390/sym13101971
Academic Editor: Clemente Cesarano
Received: 20 September 2021
Accepted: 14 October 2021
Published: 19 October 2021
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1Institute for Mathematical Research, Universiti Putra Malaysia, Serdang 43400, Selangor, Malaysia;
cychen@upm.edu.my (C.Y.C.); zaharimahad@upm.edu.my (Z.M.); sitihas@upm.edu.my (S.H.S.)
2Department of Mathematics and Statistics, Faculty of Science, Universiti Putra Malaysia,
Serdang 43400, Selangor, Malaysia
*Correspondence: nurraidah@upm.edu.my
† These authors contributed equally to this work.
Abstract:
This paper describes the extended method of solving real polynomial zeros problems using
the single-step method, namely, the interval trio midpoint symmetric single-step (ITMSS) method,
which updates the midpoint at each forward-backward-forward step. The proposed algorithm will
constantly update the value of the midpoint of each interval of the previous roots before entering
the preceding steps; hence, it always generate intervals that decrease toward the polynomial zeros.
Theoretically, the proposed method possesses a superior rate of convergence at 16, while the existing
methods are known to have, at most, 9. To validate its efficiency, we perform numerical experiments
on 52 polynomials, and the results are presented, using performance profiles. The numerical results
indicate that the proposed method surpasses the other three methods by fine-tuning the midpoint,
which reduces the final interval width upon convergence with fewer iterations.
Keywords:
single-step method; interval arithmetic; R-order of convergence; performance profile;
polynomial zeros
1. Introduction
The widespread application of interval arithmetic was inhibited in the past by a lack
of hardware and software. Nonetheless, more real-world applications have appeared
in recent years. In the twenty-first century, interval arithmetic was discovered to have
substantially contributed to the steganography field, notably in improving the quality of
watermarked images [
1
]. Interval arithmetic also plays a significant role in enhancing
the power of higher performance computing, such as GPUs [
2
]. Furthermore, in the era
of data science and artificial intelligence, many scholars and practitioners from various
backgrounds have incorporated the interval analysis concepts into their existing models or
methods in order to investigate uncertainty propagation in specific data or systems [
3
–
5
].
These applications have led to increased attention and more rigorous studies on interval
methods over the recent years.
The history of simultaneous inclusion of polynomial zeros can be traced back to the
Weierstrass’ function [
6
], where the iterative procedures for finding polynomial zeros is
guaranteed to be of quadratic convergence. Numerous studies by many scholars have
contributed to the development of polynomial inclusion studies over the decades. In re-
cent years, the approach for bounding polynomial zeroes simultaneously was studied
via different strategies, such as the Chebyshev-like root-finding method [
7
], Weierstrass
root-finding method [
8
,
9
], Halley’s method [
10
], Ehrlich method [
11
,
12
] and also by con-
sidering various types of initial conditions to solve this problem [
13
]. In 2014, Proinov
and Petcova [
14
] obtained the important result related to the semi-local convergence of
the Weierstrass root finding method. Later, Cholakov and Vasileva [
15
] introduced and
Symmetry 2021,13, 1971. https://doi.org/10.3390/sym13101971 https://www.mdpi.com/journal/symmetry
Symmetry 2021,13, 1971 2 of 14
studied a new fourth-order iterative method for finding all zeros of a polynomial simul-
taneously to achieve semi-local convergence. Correspondingly, Kyncheva et al. [
16
] also
transformed the convergence theorem of Newton, Halley, and Chebychev into semi-local
convergence theorems for simultaneous determination of multiple polynomial zeroes with
accurate initial conditions. In other respects, Proinov and Ivanov [
17
] elaborated in detail
the analysis of any local convergence of the Sakurai–Torii–Sugiura method specifically for
high-order iterative problems in finding that polynomial zeroes can be transformed into a
semi-local convergence.
Among the classical schemes for finding polynomial roots in the sense of interval
arithmetic are those by Gargantini, and Henrici [
18
], and Petkovic [
19
]. In 1988, Monsi
and Wolfe [
20
] stated that instead of using the standard point single-step method [
21
],
which involved specifics points in the algorithm, they proposed to tackle the problem
by computing sets on the real line that exactly resembled the fundamental of intervals
arithmetic [
22
]. The point iterative methods might be very efficient but somehow have
many disadvantages, such as that the sequence obtained only converges for perfect initial
estimates of the zeros. On the other hand, several modifications on the point total-step
method using interval has improved the order of convergence and computational analysis
in terms of the number of iterations, such as the interval single-step (IS1) procedure [
23
],
and interval single-step (IS2) procedure [
24
]. The difference between these two methods
is discussed in Section 2. Later, Salim et al. collaborated the idea of [
20
,
21
] and [
24
]
by proving that the new extension approach known as the interval single-step (ISS2)
method [
25
], has a superior order of convergence (R = 9) and lesser iterations generated.
In 2018, Jamaludin et al. [
26
] showed a significant reduction in the computational time
when compared to its prior modifications, somehow; the R-order of convergence was not
clear and has not been discussed by the authors.
Therefore, this paper proposes an interval iterative procedure for finding polyno-
mial zeros with superior convergence. This method imposes that the value of midpoints
computed for use in the first loop are renewed in the next loop. While the standard
interval single-step procedures compute the midpoint only once at each iteration, our
procedure updates the midpoints at each step, leading to reduced iterations with smaller
final intervals.
This paper is organized accordingly as follows. In Section 2, we first present the stan-
dard formulation of finding the root problems that lead to interval single-step procedures.
This is then followed by establishing our proposed iterative procedure, and we end the
section by proving its inclusion property. Next, in Section 3, the convergence analysis and
numerical results are displayed, using the performance profile to compare the efficiency of
the methods. The following section includes sufficient arguments based on the findings.
Finally, Section 5provides a concise conclusion.
2. Materials and Method
In this section, we will discuss the standard formulation of finding the root problem.
2.1. Interval Single-Step Method
We begin this section with the class of simultaneous methods, which leads to our
proposed method. Let p:R→Rbe a polynomial of degree n:
p(x)=
n
∑
i=0
aixi(1)
where
ai∈R
,
i=
0,
· · ·
,
n
are given and
an6=
0. A zero of polynomials is equivalent to a
root of the equation p(x)=0.
Symmetry 2021,13, 1971 3 of 14
If polynomial (1) has distinct zeros
x∗
i
,
i=
1, 2,
· · ·
,
n
, then it can be written as follows:
p(x)=
n
∏
j=1x−x∗
j(2)
by letting an=1. It follows from (2) that any zero can be expressed as follows:
x∗
i=xi−p(xi)
∏j6=ixi−x∗
j. (3)
If xj≈x∗
j(j=1, · · · ,n)so by (3), we have
x∗
i≈xi−p(xi)
∏j6=ixi−xj,i=1, · · · ,n.
This gives us the point total-step iterative procedure defined by the following:
x(k+1)
i=x(k)
i−
px(k)
i
∏n
j=1,j6=ix(k)
i−x(k)
j,i=1, · · · ,n;k=0, 1, 2, . . . (4)
where
x(0)
i
is some initial guess. The value
x(k+1)
i
can be updated one at a time or simul-
taneously. It is firstly mentioned by Weierstrass [
6
], Durand [
27
] and Kerner [
28
], which
is also known as the WDK method. When compared to Newton’s method, the WDK
method is much more robust, i.e., regardless of the initial assumption, it converges to the
zeros approximately.
The simultaneous computation of the polynomial zeros in the sense of (4) was pro-
posed by [21] and is known as the point single-step (PS1):
x(k+1)
i=x(k)
i−
px(k)
i
∏i−1
j=1x(k)
i−x(k+1)
j∏n
j=i+1x(k)
i−x(k)
j,i=1, · · · ,n;k=0, 1, 2, . . . (5)
Due to its simplicity in implementation, the PS1 has been studied and extended in
various ways. For example, Alefeld and Herzberger [
23
] improvised PS1 using the interval
approach and named it the interval single-step (IS1) method. Later, the modifications of
the variants were tested for five polynomials, producing a fewer number of iterations and
quicker CPU time [
29
]. In addition, Chen et al. [
30
] demonstrated that adding a scaling
function to IS1 outperforms the existing procedures, leading to a more significant reduction
in the final interval width with fewer iterations.
An alternative expression of (5) is as follows. We first differentiate (2) with respect to
xto give the following:
p0(x)=
n
∑
i=1
n
∏
j6=ix−x∗
j. (6)
From (2) and (6), we have the following:
p0(xi)
p(xi)=
n
∑
j=1
1
xi−x∗
j
=1
xi−x∗
i
+
n
∑
j6=i
1
xi−x∗
j
. (7)
Rearranging Equation (7), we obtain the following:
1
xi−x∗
i
=p0(xi)
p(xi)−
n
∑
j6=i
1
xi−x∗
j
Symmetry 2021,13, 1971 4 of 14
and therefore
xi−x∗
i=1
p0(xi)
p(xi)−∑n
j6=i1
xi−x∗
j
. (8)
Finally, we have the following equation:
x∗
i=xi−1
p0(xi)
p(xi)−∑n
j6=i1
xi−x∗
j
. (9)
By implementing (4) to (6) and (9), a revised point single-step (PS2) method [
21
] can be
expressed as follows:
x(k+1)
i=x(k)
i−g(k)
i
"1−g(k)
i ∑i−1
j=11
x(k)
i−x(k+1)
j
+∑n
j=i+11
x(k)
i−x(k)
j!#,i=1, . . . , n;k=0, 1, 2, . . . (10)
where
g(k)
i=gx(k)
i=px(k)
i
p0x(k)
i.
Incorporating the interval computation into (10), Salim [
24
] proposed the following
iterative formula:
X(k+1)
i=x(k)
i−g(k)
i
"1−g(k)
i ∑i−1
j=11
x(k)
i−X(k+1)
j
+∑n
j=i+11
x(k)
i−X(k)
j!#,i=1, . . . , n;k=0, 1, 2, . . . (11)
where
x(k)
i
is the midpoint of the interval
X(k)
i
. Known as the alternative interval single step
procedure (IS2), the R-order of convergence of the iterative procedure (11) is more than
3. The proof of this holds with the corresponding proof of the PS2 method and is almost
identical. Later, Salim et al. [
25
] improved the corresponding R-order of convergence of the
ISS2 method, which is at least 9.
2.2. Interval Trio Symmetric Single-Step (ITMSS) Method
This study proposes the interval trio midpoint symmetric single-step (ITMSS) method
for bounding real zeros of a polynomial simultaneously. The proposed method will update
each interval’s midpoint value of
X
, denoted by
mid(X)
, and revise the value of
g(k)
i
before
entering the next step. Enforcing this strategy will allow us to narrow the computed
bounds rigorously. The process is repeated until smaller intervals with guaranteed roots
are generated, and the the stopping condition imposed on the interval width is satisfied.
We denote the width of interval
X
as
w(X)
. Table 1describes the proposed algorithm in
detail.
Symmetry 2021,13, 1971 5 of 14
Table 1. Interval trio symmetric single-step (ITMSS).
Step 0: Given initial intervals X(0)
1,X(0)
2, . . . , X(0)
nand X(0)
i∩X(0)
j=∅,i6=j.
Set the stopping criterion ε
Step 1: For k≥0, set x(k)
i=mid(X(k)
i),i=1, . . . , n. Compute g(k)
i=g(x(k)
i) = px(k)
i
p0x(k)
i.
Step 2.1: Compute
X(k,1)
i=
x(k)
i−g(k)
i
"1−g(k)
i ∑i−1
j=11
x(k)
i−X(k,1)
j
+∑n
j=i+11
x(k)
i−X(k)
j!#
∩X(k)
i,i=1, . . . , n,
x(k,1)
i=mid(X(k,1)
i),g(k,1)=px(k,1)
i
p0x(k,1)
i
Step 2.2: Compute
X(k,2)
i=
x(k,1)
i−g(k,1)
i
"1−g(k,1)
i ∑i−1
j=11
x(k,1)
i−X(k,1)
j
+∑n
j=i+11
x(k,1)
i−X(k,2)
j!#
∩X(k,1)
i,i=n, . . . , 1,
x(k,2)
i=mid(X(k,2)
i),g(k,2)=px(k,2)
i
p0x(k,2)
i
Step 2.3: Compute
X(k,3)
i=
x(k,2)
i−g(k,2)
i
"1−g(k,2)
i ∑i−1
j=11
x(k,2)
i−X(k,3)
j
+∑n
j=i+11
x(k,2)
i−X(k,2)
j!#
∩X(k,2)
i,i=1, . . . , n,
x(k,3)
i=mid(X(k,3)
i),g(k,3)=px(k,3)
i
p0x(k,3)
i
Step 2.4: Set X(k+1)
i=X(k,3)
i
Step 3: If w(X(k+1)
i)<ε, for every i=1, 2, . . . , nthen stop. Else, set k=k+1 and
g(k+1)=g(k,3), and go to Step 1.
The significance of the ITMSS algorithm is that the values of
g(k)
i
,
i=
1,
. . .
,
n
in
Step 1, which are computed for use in Step 2.1, are renewed every time it completes the
inner loop of Step 2. This means that we will compute the new midpoint values from
the final interval width for the used in the next internal loop of Step 2, generating the
updating values of
g(k,1)
i
,
g(k,2)
i
, and
g(k,3)
i
for every root,
i
, where
i=
1,
. . .
,
n
. It also has the
following forward-backward-forward attractive features, where the value of summations
∑i−1
j=11
x(k)
i−X(k,1)
j
,
i=
2,
. . .
,
n
, which are computed in Step 2.1, will be used in Step 2.2.
Hence, it will constantly update the value of the midpoint of each interval of the previous
roots before entering the following steps that will always generate intervals that decrease
toward the polynomial zeros.
Moreover, the value from summations
∑i−1
j=11
x(k)
i−X(k,2)
j
,
i=
2,
. . .
,
n
, which are com-
puted in Step 2.2, will be used in Step 2.3. The iteration stops when
w(X(k+1)
i)<ε
for
some fixed stopping criterion
ε=10−16
. Otherwise, set
k=k+
1 and then subsequently
X(k+1)
i=X(k,3)
i
and
g(k+1)=g(k,3)
. The iteration stops when the stopping condition is
satisfied. The renewing of midpoint values before computing the new
g(k)
i
are repeated
three times in the internal loop to increase the convergence to the zeros simultaneously
in each iteration, hence the name, interval trio midpoint symmetric single-step (ITMSS)
method.
The following shows that the proposed method with updating the midpoint of the
enclosing intervals will always generate intervals that decrease toward the polynomial zeros.
Theorem 1.
Let
p(x)=a(n)xn+a(n−1)xn−1+· · · +a(0)
be a polynomial with
n
simple roots
x∗
i
,
(1≤i≤n)
. Inclusion intervals
X(0,j)3ξ∗
i
, 1
≤i≤n
, are furthermore known for which
Symmetry 2021,13, 1971 6 of 14
X(0,j)∩X(0,k)=φ
, 1
≤j≤k≤n
holds. It follows that the sequence
nX(k)o∞
k=0
, 1
≤i≤n
,
generated from the ITMSS Algorithm, satisfies the following:
x∗
i∈X(k,i),k≥0
and
X(0,i)⊃X(1,i)⊃X(2,i)⊃ · · · with lim
k→∞X(k,i)=x∗
i
or the sequence comes to rest at x∗
i,x∗
iafter a finite number of steps.
Proof of Theorem 1. Let X= [x,x]. By substituting
m(X(k)
i) = 1
2x(k)
i+¯
x(k)
i=x∗
i
in this ITMSS method and considering the construction of (11), it follows immediately that
the width of the inclusions for each zero is at least halved at each calculation of a new
iteration.
Theorem 1 partially holds when the polynomial has multiple roots. If we collect these
multiple roots together as
x∗
1
,
x∗
2
,
· · ·
,
x∗
n
, this approach must be altered so that the new
calculations of the included intervals are only done for the indices 1
≤i≤n
. Theorem 1 is
then only valid for simple zeros, where the included intervals are recomputed at each step.
Meanwhile, the other intervals remain unchanged [21].
3. Results
In this section, we present the theoretical convergence results of the proposed method,
which is then followed by numerical results on real polynomials.
3.1. Convergence Analysis
The following theorem is about the inclusion of the generated intervals
nX(k)
io
,
i=
1,
. . .
,
n
and their convergence to
x∗
i
,
i=
1,
. . .
,
n
. Finally, we establish the R-order of
convergence of the proposed method, and the theoretical analysis of convergence will
be discussed.
Theorem 2.
Let
I(R)
be the set of all closed intervals on the real line and
Di
be subset of
I(R)
for
i=
1,
. . .
,
n
. If the assumptions of Theorem 1 are valid, then it follows that 0
/∈Di∈I(R)
such
that p0(x)∈Di,i=1, . . . , n, and then
w(X(k+1)
i)≤1
21−diI
diSwX(k)
i
are satisfied. Finally, the R-Order of convergence of ITMSS is given by the following:
OR(ITMSS,x∗)≥16, i=1, . . . , n.
Proof of Theorem 2.
The proof that
w(X(k+1)
i)≤1
21−diI
diSwX(k)
i
and that
X(k)
i→
x∗
i
,
k→∞
,
i=
1,
. . .
,
n
holds is almost identical with the corresponding proofs in [
21
],
and is therefore omitted. It remains to be proven that R-order of convergence is at least 16.
Symmetry 2021,13, 1971 7 of 14
As in the proof of Theorem [
21
], it may be shown that there exists
α>
0 such that for
every k≥0, the following holds:
w(k,1)
i≤βw(k,0)
i5(i−1
∑
j=1
w(k,1)
j+
n
∑
j=i+1
w(k,0)
j),i=1, . . . , n(12)
and
w(k,2)
i≤βw(k,0)
i5(i−1
∑
j=i+1
w(k,1)
j+
n
∑
j=i+1
w(k,2)
j),i=n, . . . , 1 (13)
and
w(k,3)
i≤βw(k,0)
i5(i−1
∑
j=i+1
w(k,2)
j+
n
∑
j=i+1
w(k,3)
j),i=1, . . . , n(14)
where
w(k,s)
i=1(n−1)αwX(k,s)
i,s=0, 1, 2, 3, (15)
and
β=1
n−1.
Let the following hold:
u(1,1)
i=(6, i=1, . . . , n−1
11, i=n(16)
u(1,2)
i=(16, i=1
11, i=2, . . . , n−1(17)
u(1,3)
i=(16, i=1, . . . , n−1
21, i=n(18)
and for r=1, 2, 3, with
u(k+1,r)
i=(16u(k)
i,i=1, . . . , n−1
16u(k)
i+5(k+1),i=n. (19)
Then, by (16)–(19) for every k≥0, we have the following:
u(k,1)
i=(6(16(k−1)),i=1, . . . , n−1
11(16(k−1)) + ∑k−2
j=05(k−j)16j,i=n(20)
and
u(k,2)
i=
11(16(k−1)) + ∑k−2
j=05(k−j)16j,i=n
11(16(k−1)),i=n−1, . . . , 2
16(k),i=1
(21)
and
u(k,3)
i=(16(k),i=1, . . . , n−1
21(16(k−1)) + ∑k−2
j=05(k−j)16j,i=n. (22)
Suppose without loss of generality, we have the following:
w(0,0)
i≤h<1, i=1, . . . , n. (23)
Symmetry 2021,13, 1971 8 of 14
Then, by an inductive argument, it follows from (12)–(23) that for
i=
1,
. . .
,
n
and
k≥0, we have the following:
w(k,1)
i≤hu(k+1,1)
i,
and
w(k,2)
i≤hu(k+1,2)
i,
and
w(k,3)
i≤hu(k+1,3)
i.
Hence, by (18) and Step 2.4, for every k≥0, we have the following:
w(k+1)
i≤h42(k+1)=h42k+2.
Thus, the following is true:
w(k)≤h42k=h16k.
Then for every k≥0, by (12)–(23),
wX(k)
i≤β
αh16k,i=1, . . . , n(24)
Let the following hold:
w(k)=max
1≤i≤nnwX(k)
io,
and by (24), we have the following:
w(k)≤β
αh16k.
Hence,
R16w(k)=lim
k→∞sup (w(k)1
(16k))
=lim
k→∞sup (β
α1
(16k)h)
=h<1.
Therefore, it follows from [21] and [31] that, ORITMSS,x∗
i≥16, i=1, . . . , n.
3.2. Numerical Experiments
We analyze the efficiency of the proposed method by comparing the computational
results of the 52 test problems with the interval single-steps (IS2) method and its variants re-
garding the number of iterations and the largest final width of the interval
generated [26,30,32]
.
The selected test examples are arranged starting from a real polynomial
p(x)
with degree
n=
3 up to
n=
12. Furthermore, we only consider real and simple zeros in this experi-
ment. Next, these algorithms are implemented using MATLAB R2017b cooperated with the
Intlab V12.1 toolbox, specifically for interval arithmetic developed by Rump [
33
]. Mean-
while, the stopping criterion used is
w(X(k)
i)≤10−16
,
i=
1,
. . .
,
n
. Then, we observe the
results using the performance profile to test the degree of efficiency of our algorithms.
The algorithms that are considered include the following:
1. Interval single-step (IS2) method [20];
2. Interval symmetric single-step (ISS2) method [25];
3. Interval zoro-symmetric single-step (IZSS2) method [32];
4. Interval trio midpoint symmetric single-step (ITMSS) method.
Symmetry 2021,13, 1971 9 of 14
We begin with setting up the initial intervals for each
n
in each test example into
all four algorithms stated above in order to compute the number of iterations,
k
, and the
largest final interval width,
w(X(k)
i)
. There are 52 test examples that consist of 343 starting
points, multiplied by four algorithms and then multiplied by two output categories (
k
and
w
). Thus, 416 of the output findings must be assessed using a performance profile.
According to Dolan and Moré [
34
], when a large number of tested examples are employed,
such as 100, 250, 500, or 1000, the output analyzing phases become extremely tough to
evaluate, motivating researchers to apply performance profile comparison. In a nutshell,
a performance profile is a visualization-based analytical tool used to evaluate the results
of a benchmark experiment, allowing the user to compare the performance of each solver.
Therefore, it is also known as a (cumulative) distribution function of a performance indica-
tor, where
ρ(τ)
is the probability that a performance ratio is at most
τ
(the best possible
ratio). The solver ’s likelihood will prevail over the other solvers, and is represented by
ρ(
1
)
.
If we are mainly concerned about the number of triumphs, we can compare the values
of
ρ(
1
)
for each solver. Hence, the preferable solver is the one with the highest number
of winning results.
The performance profiles for the number of iterations,
k
, and the maximum final
interval width,
w
, are shown in Figures 1and 2, respectively. The lines indicate the
methods on the graphs. From Figure 1, it is noticeable that the ITMSS method performs
better than IS2, ISS2, and IZSS2, respectively. In other words, the midpoint renewing
procedures require fewer iterations, and converge to the zeros in terms of the number of
iterations. As previously stated, the ITMSS method has a better chance of winning and
is the most preferred method of all. Note that the order of convergence for IS2 is at least
3 [
24
], and the graph shows that this method is the least efficient in terms of the number
of iterations. Meanwhile, the order of convergence for ISS2 and IZSS2 is at least 9 and 13,
respectively [
25
,
32
]. Therefore, from an overall view of Figure 1, the method’s efficiency
resembles the order of convergence of the methods.
0 0.5 1 1.5 2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
p( )
Comparison of methods for number of iterations, k
IS2
ISS2
IZSS2
ITMSS
Figure 1. Comparison of methods for number of iterations, k.
Symmetry 2021,13, 1971 10 of 14
0 2 4 6 8 10 12 14 16 18
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
p( )
Comparison of methods for final largest interval width, w
IS2
ISS2
IZSS2
ITMSS
Figure 2. Comparison of methods for final largest interval width, w.
4. Discussion
In this study, we considered the polynomial zeros inclusion problem specifically by
using a single-step procedure. The idea is to propose a more dynamic approach to finding
the zeros. Considering that adjusting the midpoint of the intervals at every inner loop of
the algorithm will encapsulate all real potential results, the interval arithmetic method’s
precision is guaranteed. It fulfills the stopping condition faster without neglecting other
preliminary concepts of interval arithmetic. Based on the idea, we name the proposed
method as the interval trio midpoint symmetric single-step (ITMSS) method. Then, accord-
ing to the order of convergence analysis, the ITMSS method has a high order of convergence
of at least 16. The results show that the ITMSS method converges faster than any single-step
(IS2) procedure antecedently.
Furthermore, to evaluate the performance of the ITMSS method, we conducted a
numerical experiment using 52 test examples, comparing it with IS2, ISS2 and IZSS2. We
visualized the numerical results using the performance profile regarding the number of
iterations and the largest final interval width. From the numerical results explained above,
it is evident that the ITMSS method performs better than the IS2, ISS2, and IZSS2 methods,
respectively. Although the proposed method is highly likely to win in Figure 1, this is not
the case with the largest final interval width findings, as shown in Figure 2.
Figure 2shows the comparison of all four methods in terms of the largest final interval
width generated when reaching the stopping condition. From the graph, the ITMSS method
has a high probability of winning for most of the scenarios, but the graph shows a less
satisfactory outcome in some circumstances. In certain situations, the number of iterations
for the ITMSS method is equivalent to the iterations by the IZSS2 method. Furthermore,
there are also situations where even the ITMSS method has the fewest number of iterations,
compared to the other three methods, but the value of its largest final interval width is
similar. Next, we provide the details of the above situations in the tables.
From Table 2, we selected one of the test examples
p(x)=y4+40
3y3−
0.02
y2−
0.4
y
,
which happened to have a less satisfactory outcome for ITMSS in terms of the number of
Symmetry 2021,13, 1971 11 of 14
iteration generated. The table displays the interval width of every iteration by comparing
all of the methods under consideration. At iteration
k
, the largest width of the final
interval is highlighted in grey. As shown in Table 2, the polynomial required four iterations
to complete for the IS2 method, whereas the ISS2 method only needed three iterations.
However, both IZSS2 and ITMSS methods stopped at the second iteration. This table shows
that the ITMSS method yields smaller intervals than other methods. For example, the width
of
i=
2 for the ITMSS method reached the stopping condition at
k=
1; meanwhile, other
ith
did not. All the
ith
will continually run until the next iteration. Even though the
ITMSS iterates twice as compared to the IZSS2 method, from the table, this method can
obtain
[x∗
1
,
x∗
1]
and
[x∗
4
,
x∗
4]
, which are essentially the roots for the first and fourth intervals,
respectively. It can be seen that the largest final interval width for ITMSS is significantly
smaller. This proved that the ITMSS method converges faster than other methods; it can
also approximately reduce the final interval width for every iteration.
In Table 3, the polynomial p(x)=20000y8+16080000y7+551830000y6+1053409
3200
y5+
122028205260
y4+
875779839648
y3+
3789351757513
y2+
8998687954893
y+
8930
298867308 gives the final interval width generated for all methods as 8.88178419700125
×
10
−16
. However, the ITMSS method has fulfilled the stopping conditions at
k=
1. This
observation, depending only on the largest final interval width value, does not significantly
interpret the algorithm. However, indeed, the largest final interval width and the number
of iterations are correlated to each other. From the overall view, the proposed algorithm
can converge to the roots simultaneously and fulfill the inclusion theorem better than the
other three methods. This means that the final width of the intervals generated is minimal.
Hence, the ITMSS algorithm almost always converges to zero, irrespective of the initial
estimates. From an overall view, imposing the value
g(x)
and midpoint at every inner loop
reduces the number of iterations and lessens the final interval width upon convergence.
That is, the proposed method ensures that zero is contained within a suitably narrow
final interval.
Table 2. Interval width of every iteration for polynomial p(x)=y4+40
3y3−0.02y2−0.4y.
k i Largest Interval Width in Every Iteration
IS2 Method ISS2 Method IZSS2 Method ITMSS Method
1
19.31439152433 ×10−22.38797944655 ×10−42.387979446556 ×10−42.48750330555 ×10−7
21.08570553860 ×10−21.23524990965 ×10−31.234908302445 ×10−32.21779953451 ×10−16
33.33618893631 ×10−14.98845687881 ×10−21.795679442520 ×10−26.61625643161 ×10−10
41.95646289260 ×10−21.95646289260 ×10−21.582165843121 ×10−36.66966482043 ×10−14
2
14.32639438052 ×10−61.77635683940 ×10−15 1.776356839400 ×10−16 0.00000000000000000
21.54184787449 ×10−63.65010213894 ×10−16 2.775557561562 ×10−17 2.21779953451 ×10−17
31.04796990859 ×10−35.58678381334 ×10−11 2.775557561562 ×10−17 2.77555756156 ×10−17
48.47766782607 ×10−82.13209447319 ×10−92.775557561562 ×10−17 0.00000000000000000
3
1
1.77635683940
×
10
−15 1.77635683940 ×10−16
Already Converge Already Converge
2
1.52727551189
×
10
−17 3.65010213894 ×10−16
3
1.70696790036
×
10
−14 2.77555756156 ×10−17
4
2.77555756156
×
10
−17 2.77555756156 ×10−17
4
1
1.77635683940
×
10
−16
Already Converge
2
1.52727551189
×
10
−17
3
2.77555756156
×
10
−17
4
2.77555756156
×
10
−17
Symmetry 2021,13, 1971 12 of 14
Table 3.
Interval width of every iteration for polynomial
p(x)=
20000
y8+
16080000
y7+
551830000
y6+
10534093200
y5+
122028205260y4+875779839648y3+3789351757513y2+8998687954893y+8930298867308.
k i Largest Interval Width in Every Iteration
IS2 Method ISS2 Method IZSS2 Method ITMSS Method
1
1 8.790842687792 ×10−16.848788945959 ×10−36.848788945959 ×10−31.684182283412 ×10−6
2 7.416880512258 ×10−22.425248230748 ×10−21.741756142319 ×10−47.105427357601 ×10−15
3 9.554594425101 ×10−31.499200540824 ×10−31.702508947332 ×10−53.552713678800 ×10−15
4 3.380631011767 ×10−21.910729522056 ×10−33.766384543979 ×10−53.552713678800 ×10−15
5 3.691891521136 ×10−21.304172267009 ×10−33.477536074925 ×10−53.552713678800 ×10−15
6 1.384620531206 ×10−26.131183766120 ×10−45.649886381409 ×10−51.776356839400 ×10−15
7 1.554485059640 ×10−14.396253377089 ×10−37.726031239547 ×10−41.776356839400 ×10−15
8 7.723378231853 ×10−37.723378231853 ×10−34.005897095238 ×10−58.881784197001 ×10−16
2
1 1.045554681976 ×10−37.105427357601 ×10−15 3.552713678800 ×10−15 0.00000000000000
2 7.953034852903 ×10−77.993605777301 ×10−13 1.776356839400 ×10−15 0.00000000000000
3 1.077414779615 ×10−85.329070518200 ×10−15 1.776356839400 ×10−15 0.00000000000000
4 3.946566735635 ×10−73.552713678800 ×10−15 1.776356839400 ×10−15 0.00000000000000
5 6.842841298038 ×10−73.552713678800 ×10−15 1.776356839400 ×10−15 0.00000000000000
6 3.241325474689 ×10−71.776356839400 ×10−15 8.881784197001 ×10−16 0.00000000000000
7 3.632269596209 ×10−71.776356839400 ×10−15 8.881784197001 ×10−16 0.00000000000000
8 8.304468224196 ×10−14 8.881784197001 ×10−16 4.440892098500 ×10−16 8.881784197001 ×10−16
3
1 7.105427357601 ×10−15 0.00000000000000 0.00000000000000
Already Converge
2 3.552713678800 ×10−15 1.776356839400 ×10−15 0.00000000000000
3 1.776356839400 ×10−15 0.00000000000000 0.00000000000000
4 3.552713678800 ×10−15 0.00000000000000 1.776356839400 ×10−16
5 3.552713678800 ×10−15 0.00000000000000 0.00000000000000
6 1.776356839400 ×10−15 0.00000000000000 8.881784197001 ×10−16
7 8.881784197001 ×10−16 0.00000000000000 8.881784197001 ×10−16
8 4.440892098500 ×10−16 8.881784197001 ×10−16 4.440892098500 ×10−16
4
1 0.00000000000000 0.00000000000000
Already Converge
2 0.00000000000000 0.00000000000000
3 0.00000000000000 0.00000000000000
4 0.00000000000000 0.00000000000000
5 0.00000000000000 0.00000000000000
6 0.00000000000000 0.00000000000000
7 8.881784197001 ×10−16 0.00000000000000
8 4.440892098500 ×10−16 8.881784197001 ×10−16
5. Conclusions
In this study, we investigated the interval iterative methods for the inclusion of
polynomial zeros specifically for solving simple roots problems simultaneously. We provide
the numerical results, using a performance profile to validate the efficiency of all four
methods: IS2, ISS2, IZSS2 and ITMSS. Theoretically, the proposed ITMSS method has
an R-order of convergence of 16, which means it can bound the zeros rigorously at a
superior rate. The numerical results indicate that the ITMSS method surpassed the other
three methods by fine-tuning the midpoint and decreasing the final interval width with
fewer iterations. However, it is suggested that further investigations can be conducted for
polynomials with complex roots or with complex coefficients while solving the complexity
of the computational CPU time running by implementing the interval arithmetic technique.
Author Contributions:
Conceptualization, N.R.S. and C.Y.C.; methodology, N.R.S. and C.Y.C.; formal
analysis, N.R.S. and Z.M.; software, N.R.S. and Z.M.; writing—original draft, N.R.S.; writing—review
and editing, N.R.S., C.Y.C. and S.H.S.; visualization, N.R.S. and C.Y.C.; supervision, C.Y.C.; validation,
C.Y.C. and S.H.S. All authors have read and agreed to the published version of the manuscript.
Symmetry 2021,13, 1971 13 of 14
Funding:
This research was funded by UNIVERSITI PUTRA MALAYSIA grant number GP-IPM/2021
/9699800.
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Conflicts of Interest: No potential conflict of interest was reported by the authors.
Abbreviations
The following abbreviations are used in this manuscript:
IS2 Interval single-step method
ISS2 Interval symmetric single-step method
IZSS2 Interval zoro-symmetric single-step method
ITMSS Interval trio midpoint symmetric single-step method
References
1.
Ohura, R.; Minamoto, T. A blind digital image watermarking method based on the dyadic wavelet packet transform and fast
interval arithmetic techniques. Int. J. Wavelets Multiresolution Inf. Process. 2015,13, 1550040. [CrossRef]
2.
Rump, S.M.; Ogita, T.; Morikura, Y.; Oishi, S.I. Interval arithmetic with fixed rounding mode. Nonlinear Theory and Its
Applications. IEICE 2016,7, 362–373.
3.
Pereira, D.R.; Papa, J.P.; Saraiva, G.F.R.; Souza, G.M. Automatic classification of plant electrophysiological responses to environ-
mental stimuli using machine learning and interval arithmetic. Comput. Electron. Agric. 2018,145, 35–42. [CrossRef]
4.
Orozco-Gutierrez, M.L. An Interval-Arithmetic-Based Approach to the Parametric Identification of the Single-Diode Model of
Photovoltaic Generators. Energies 2020,13, 932. [CrossRef]
5.
Pan, W.; Feng, L.; Zhang, L.; Cai, L.; Shen, C. Time-series interval prediction under uncertainty using modified double multiplica-
tive neuron network. Expert Syst. Appl. 2021,184, 115478. [CrossRef]
6.
Weierstrass, K. Neuer Beweis des Satzes, dass jede ganze rationale Funktion einer Veranderlichen dargestellt werden kann als ein
Product aus lineare Funktionen derselben Veranderlichen. Gesammelte Werke 1967,3, 251–269.
7.
Proinov, P.D.; Cholakov, S.I. Semilocal Convergence of Chebyshev-like Root-finding Method for Simultaneous Approximation of
Polynomial Zeros. Appl. Math. Comput. 2014,236, 669–682. [CrossRef]
8.
Proinov, P.D.; Petkova, M.D. Convergence of The Two-point Weierstrass Root-finding Method. Jpn. J. Ind. Appl. Math.
2014
,31,
279–292. [CrossRef]
9.
Proinov, P.D.; Vasileva, M.T. On a Family of Weierstrass-type Root-finding Methods with Accelerated Convergence. Appl. Math.
Comput. 2014,273, 957–968. [CrossRef]
10.
Proinov, P.D.; Ivanov, S.I. On The Convergence of Halley’s Method for Simultaneous Computation of Polynomial Zeros. J. Numer.
Math. 2015,23, 379–394. [CrossRef]
11.
Proinov, P.D.; Vasileva, M.T. On The Convergence of High-order Ehrlich-type Iterative Methods for Approximating All Zeros of
A Polynomial Simultaneously. J. Inequalities Appl. 2015,2015, 336. [CrossRef]
12. Proinov, P.D. On The Local Convergence of Ehrlich Method for Numerical Computation of Polynomial Zeros. Calcolo 2016,253,
413–426. [CrossRef]
13.
Proinov, P.D. Relationships Between Different Types of Initial Conditions for Simultaneous Root Finding Methods. Appl. Math.
Lett. 2016,52, 102–111. [CrossRef]
14.
Proinov, P.D.; Petkova, M.D. A New Semilocal Convergence Theorem for the Weierstrass Method for Finding Zeros of A
Polynomial Simultaneously. J. Complex. 2014,30, 366–380. [CrossRef]
15.
Cholakov, S.I.; Vasileva, M.T. A Convergence Analysis of A Fourth-order Method for Computing All Zeros of A Polynomial
Simultaneously. J. Comput. Appl. Math. 2017,321, 270–283. [CrossRef]
16.
Kyncheva, V.K.; Yotov, V.V.; Ivanov, S.I. Convergence of Newton, Halley and Chebyshev Iterative Methods as Methods for
Simultaneous Determination of Multiple Polynomial Zeros. J. Appl. Numer. Math. 2017,112, 146–154. [CrossRef]
17.
Proinov, P.D.; Ivanov, S.I. Convergence Analysis of Sakurai–Torii–Sugiura Iterative Method for Simultaneous Approximation of
Polynomial Zeros. J. Comput. Appl. Math. 2019,357, 56–70. [CrossRef]
18.
Gargantini, I.; Henrici, P. Circular Arithmetic and The Determination of Polynomial Zeros. Numer. Math.
1971
,18, 305–320.
[CrossRef]
19.
Petkovi´c, M.S. On an iterative method for simultaneous inclusion of polynomial complex zeros. J. Comput. Appl. Math.
1982
,8,
51–56. [CrossRef]
20.
Monsi, M.; Wolfe, M.A. Interval Versions of Some Procedures for The Simultaneous Estimation of Complex Polynomial Zeros.
Appl. Math. Comput. 1988,28, 191–209. [CrossRef]
21.
Alefeld, G.; Herzberger, J. On the Convergence Speed of Some Algorithms for The Simultaneous Approximation of Polynomial
Roots. SIAM J. Numer. Anal. 1974,11, 237–243. [CrossRef]
Symmetry 2021,13, 1971 14 of 14
22. Moore, R.E. Methods and Applications of Interval Analysis, 1st ed.; SIAM: Philadelphia, PA, USA, 1979.
23. Alefeld, G.; Herzberger, J. Introduction to Interval Computations, 1st ed.; Academic Press: New York, NY, USA, 1983.
24.
Salim, N.R. Convergence of Interval Symmetric Single-step Method for Simultaneous Inclusion of Real Polynomial Zeros. Ph.D.
Thesis, Universiti Putra Malaysia, Seri Kembangan, Malaysia, 2012.
25.
Salim, N.R.; Monsi, M.; Hassan, M.A.; Leong, W.J. On The Convergence Rate of Symmetric Single-step Method ISS for Simultane-
ous Bounding Polynomial Zeros. Appl. Math. Sci. 2011,5, 3731–3741.
26.
Jamaludin, N.; Monsi, M.; Hassan, N. The Performance of The Interval Midpoint Zoro Symmetric Single-step (IMZSS2-5D)
Procedure to Converge Simultaneously to The Zeros. In AIP Conference Proceedings, Proceeding of The International Conference
on Mathematics, Engineering and Industrial Applications 2018 (ICoMEIA 2018), Kuala Lumpur, Malaysia, 24–26 July 2018; Zin, S.M.,
Abdullah, N., Khazali, K.A.M., Roslan, N., Rusdi, N.A., Saad, R.M., Yazid, N.M., Zain, N.A.M., Eds.; AIP Publishing LLC:
Melville, NY, USA, 2018; Volume 2013, p. 020033.
27.
Durand, E. Solutions numéRiques des Équations algéBriques: Systèmes de Plusieurs Équations; Masson: Paris, France, 1960; Volume 2.
28.
Kerner, I.O. Ein gesamtschrittverfahren zur berechnung der nullstellen von polynomen [A complete procedure for calculating the
zeros of polynomials]. Numer. Mathl Sci. 2015,8, 290–294. [CrossRef]
29.
Rusli, S.F.; Monsi, M.; Hassan, M.A.; Leong, W.J. On the interval zoro symmetric single-step procedure for simultaneous finding
of real polynomial zeros. Appl. Math. Sci. 2011,5, 3693–3706.
30.
Chen, C.Y.; Ghazali, A.H.; Leong, W.J. Scaled parallel iterative method for finding real roots of nonlinear equations. Optimization
2021, 1–17. [CrossRef]
31.
Ortega, J.M.; Rheinboldt, W.C. Numerical Solution of Nonlinear Problems: Studies in Numerical Analysis 2. Symp. Spons. Nav.
Res. 1970,2, 122–143.
32.
Salim, N.R.; Monsi, M.; Hassan, N. On The Performances of IMZSS2 Method for Bounding Polynomial Zeros Simultaneously. In
Proceedings of the 7th International Conference on Research and Education in Mathematics (ICREM7), Kuala Lumpur, Malaysia,
25–27 August 2015; Majid, Z.A., Salim, N.R., Laham, M.F., Gopal, K.; Phang, P.S., Mahad, Z., Eds.; IEEE: Piscataway, NJ, USA,
2015; pp. 5–9.
33. Rump, S.M. INTLAB — INTerval LABoratory. In Developments in Reliable Computing; Csendes, T., Ed.; Springer: Dordrecht, The
Netherlands, 1999; pp. 77–104.
34.
Dolan, E.D.; Moré, J.J. Benchmarking optimization software with performance profiles. Math. Program.
2002
,91, 201–213.
[CrossRef]
