1. Introduction
When analytical approaches are not available, iterative schemes are the only viable strategy for numerically approximating the roots of nonlinear equations
in a stable manner. They start with an initial approximation and iteratively refine the solution using algebraic equations until a satisfactory approximation is obtained. This approximation of the solution is carried out in this manner until every root is identified. There are two types of iterative root-finding schemes: simultaneous techniques, which approximate all roots simultaneously, and methods which approximate one root at a time (see, for example, Traub’s method [
1], Jarratt’s method [
2], King’s method [
3], Ostrowski’s method [
4], Chun et al.’s method [
5], and many others). In recent years, simultaneous techniques have grown in popularity as a result of their global convergence and inherent parallelism (see, for example, the works by Weierstrass [
6], Kanno [
7], Proinov [
8], Mir [
9], Farmer [
10], Nourein [
11], Aberth [
12], and Cholakov [
13] and the references therein). On the other hand, because of the intrinsic difficulties of these equations, such as the non-linearity and non-locality, standard analytical, semi-analytical, and classical numerical approaches are typically ineffective.
In order to decrease the overall computational time, parallel numerical schemes utilize parallel computing [
14] to solve nonlinear equations. This is achieved by decomposing the problem into smaller tasks, which can be executed simultaneously on multiple processors or cores. Therefore, these schemes are particularly useful when dealing with large-scale or computationally intensive engineering problems [
15]. A comprehensive understanding of parallel programming techniques, algorithms, and the specific characteristics of the problem at hand is necessary for the effective implementation of parallel numerical schemes. Furthermore, the selection of a parallel scheme is often influenced by the nature of the nonlinear equations being solved, the hardware at hand, and the size of the problem. An overview of parallel numerical methods for solving nonlinear equations can be found in [
16,
17,
18].
The performance of simultaneous root-finding algorithms varies depending on the initial guess and the problem at hand, and convergence is not always guaranteed [
19,
20,
21]. As a result, efforts have been made to develop more robust and efficient procedures. In this research, we propose highly efficient fractional numerical techniques for simultaneously approximating all the roots of nonlinear equations. Fractional simultaneous methods utilize fractional-order derivatives of the function to solve (
1). Fractional calculus, which is concerned with non-integer-order derivatives and integrals, is used in many areas, including physics, engineering, and finance [
22,
23,
24]. A comprehensive analysis of the convergence and of the computational complexity of our method is derived. The performance and global convergence behavior of the algorithm is assessed for solving some practical engineering applications by considering various factors, including CPU time, maximum computational time on random initial guess values, maximum residual error, and local computational order of convergence.
The structure of the paper is outlined as follows. After the introduction, we discuss some basic definitions in
Section 2. In
Section 3, parallel computing schemes are developed and analyzed to solve (
1).
Section 4 compares the computational aspects of newly proposed simultaneous techniques to existing methods in the literature. In
Section 5, we discuss the numerical results of the newly developed scheme. The conclusion of the paper is in
Section 6.
3. Construction of Fractional Parallel Computing Scheme for Estimating All Distinct and Multiple Roots
Weierstrass-Dochive [
18] presents the following local quadratic convergence scheme:
where
is Weierstrass’ Correction.
In [
19], Nedzibove et al. present two new modifications to (
14) as:
In order to construct an iterative process for approximating all the multiple roots of polynomial, let us assume a monic polynomial of degree n with roots
having known multiplicities
such that
Consider the Newton correction
and
This implies that
where
is the exact root and
is its approximation. This gives
Substituting the roots
by its approximations
in (
19), we obtain the third-order convergent Ehrlich–Aberth method [
36] for the roots with multiplicities
where
is new approximation to the root
. Instead of simple approximation
, we can apply some better approximation to
. The main goal in this accelerating process is to improve convergence. The aim can be achieved by choosing Newton’s approximation
instead of
in (
22):
Now, we derive a new
-order method for the determination of all the roots of (
1). Let
be reasonably close approximations to the roots
, respectively, of polynomial
, which means that
is a sufficiently small quantity. Let us return to the relation (
19) by replacing
with
. We have:
Assuming that
is small enough to provide
, we use the development into geometric series and obtain:
Neglecting terms of a higher order in the last relations, we obtain:
Replacing
by
,
by
in (
29) and using in (
23), we have
We name the method introduced in (
30) as the SFM
-Method. Now, we calculate the convergence order of the SFM
-Method. Firstly, we introduce some notations as:
Now, we suppose the condition
where
The conditions hold for each, where
Convergence Analysis: Here, we prove the following lemma:
Lemma 1. Let be reasonably close approximations of roots , respectively. Let , , where are the new approximations produced by the iterative SFM. If (32) is satisfied, then the following estimate is also true: - (i)
- (ii)
Proof. Taking into account (
31), we find:
Now, considering (
33) and (
34), we have:
Now, we introduce some new notations:
Thus, using (
35), we obtain:
as
for every
i, this implies
As
Using (
32) and (
35) in the above result, we have:
As
From (
34), we have
. Therefore,
and therefore,
using Newton’s correction, we obtain,
where,
From (
55), we get:
Now, applying Equations (
31), (
32) and (
39) in the above relation in (
70), we obtain:
Therefore,
where
and therefore,
Since
is a monotonically decreasing sequence, let us estimate from the above the absolute values of
. We have:
Since
and
for all
Using
for all
i, we obtain:
also from (
84)
Using Equations (
81) and (
85) in Equation (
77), we obtain
Hence, we have the proof of Lemma 1 (i). Now, from Equation (
86), we have
which completes the proof of Lemma 1 (ii). □
Let be the good initial guesses to roots of an algebraic polynomial f and suppose where approximations are obtained in the iterative step by the simultaneous SFM-method. Using the conditions of Lemma 1, now we state the main convergence theorem of our SFM-Method.
Theorem 2. According to the following assumptionsthe iterative formula SFM is convergent, having convergent order . Proof. In Lemma 1 (i), we develop the results (
86) under the assumptions (
32). Using the same arguments under condition (
88) of theorem 1, we have from (
86):
So according to Lemma 1 (ii), we have:
We prove the theorem by mathematical induction; condition (
88) implies
for every,
and
Using
(
91) becomes
Let
, then from assumptions (
93), it follows that
. For all
and from (
93), we obtain
for each
and
Therefore, from (
93), we obtain:
which shows that the proposition
converges to zero. Consequently, the sequence
also converges to zero. That is,
for all
i as
increases. Finally, from (
94), it can be concluded that the method (SFM
-Method) has convergence order
. □
5. Numerical Outcomes
To compare our recently developed simultaneous methods SFM
SFM
of order
to SFM
, we look at a few numerical test examples in this section. With Maple 18’s 64 digits floating point arithmetic, all calculations were completed. The parallel computer algorithm was terminated based on the following conditions:
where
denotes the absolute error of consecutive iterations. In Table, 2-21, the numerical schemes for various fractional parameter values, i.e., 0.1, 0.3, 0.5, 0.8, 1.0, are represented by SFM
SFM
, SFM
respectively, and B** denotes digits floating point arithmetic. In all tables, we use the following computer terminating criteria (Algorithm 1).
Algorithm 1 For the fractional numerical scheme SFMs |
|
Engineering Applications
This section presents many problems in engineering whose solutions are approximated by our newly created parallel approaches SFMSFM and SFM.
Engineering Application 1: Emden–Fowler equation
The Emden–Fowler second-order nonlinear differential equation arises in various fields of physics and engineering, fluid dynamics, heat transfer, and astrophysics, in particular, to model the structure of self-gravitating, spherically symmetric objects, such as stars. The equation is named in honor of Ralph H. Fowler and Robert Emden, two German astrophysicists who made significant contributions to its formulation. The general form of the Emden–Fowler equation is given by [
38,
39]:
Because of its nonlinearity, solving the Emden–Fowler equation is often difficult, and closed-form solutions exists only in specific cases. Choosing
and
in (
97), we obtain the following nonlinear initial value problem:
Using the procedure described in [
40], the numerical solution of (
98) can be performed by solving the following polynomial:
The Caputo-type derivative of (
99) is given as:
The exact solution of (
99) up to four decimal places is:
In order to determine the global convergence behavior of the parallel scheme, we generate a random initial guess ranging from
to
using Matlab as explained in
Appendix A Table A1. According to the results presented in
Table 2, when an arbitrary starting value is used, SFM
SFM
SFM
converges to exact zeros after 19, 17, 13, 10, and 10 iterations for fraction parameters 0.1, 0.3, 0.5, 0.8, and 1.0, respectively. The corresponding CPU times are 2.1254, 1.0874, 1.0078, 0.0784 and 0.0078 as shown in
Table 3. The acceleration of the convergence rate of SFM
SFM
SFM
as the fractional parameter value increases from 0.1 to 1.0 can be clearly seen in
Table 4. Global convergence is demonstrated by the fact that the newly developed method converges to exact roots for randomly generated initial guess values.
Table 2 shows the number of iterations of the fractional simultaneous scheme SFM
SFM
SFM
for different choices of the random initial vector given in
Appendix A,
Table A1.
Table 2 clearly shows that the number of iterations decreased as the fractional parameter values increased from 0.1 to 1.0.
Table 5 shows the maximum error (Max-Err) computed by the fractional simultaneous scheme SFM
SFM
SFM
for different selections of the random initial vector given in
Appendix A Table A1 to approximate all roots of the polynomial equations used in application 1.
Table 5 clearly demonstrates that as the fractional parameter values increased from 0.1 to 1.0, the accuracy computed by the simultaneous scheme increased significantly (
Figure 2).
Table 4 shows the approximate local computational order of convergence. The approximate local computational order of convergence increases as the fractional parameter values increase from 0.1 to 1.0.
Table 3 shows the computational CPU time in seconds to approximate all roots of the polynomial equation used in application 1 employing the fractional simultaneous scheme.
The rate of convergence increases as the initial guess values are chosen to be sufficiently close to the exact root of (
99) as:
If we start with initial guessed values that are close to the exact root,
Table 6 demonstrates that the fractional simultaneous scheme’s accuracy and convergence order improve. As the fractional parameter value was increased from 0.1 to 1.0, the residual error calculated using numerical methods also increased.
Engineering Application 2: Under Conservative Force—Mass Spring System
Let us now examine an external force acting on a vibrating mass on a spring. A driving force that causes the spring support to oscillate vertically, for instance, could be represented by
. If the mechanical system is conservative, the following nonlinear equation arises [
41,
42]:
Using the method described in [
40], the following polynomial is used to simulate (
101) as:
The Caputo-type derivative of (
102) is given as:
The exact solution up to four decimal places is written as:
To determine the global convergence component of the parallel scheme, use Matlab to generate a random initial guess value ranging from
as specified in
Appendix A Table A2. With an arbitrary starting value, SFM
SFM
SFM
converges to exact zeros after 19, 16, 14, 10 and 10 iterations as indicated in
Table 7 for fraction parameters values 0.1, 0.3, 0.5, 0.8, and 1.0, respectively. As described in
Table 8, the corresponding CPU times are 3.1254, 1.0729, 1.0137, 0.0881 and 0.0141, respectively.
Table 9 clearly illustrates how the rate of convergence of SFM
SFM
SFM
accelerates as the value of the fractional parameter increases from 0.1 to 1.0. The newly developed method converges to exact roots for randomly generated initial guess values, demonstrating its global convergence.
Table 7 shows the number of iterations of fractional simultaneous scheme SFM
SFM
SFM
for different random initial vectors given in
Appendix A Table A2.
Table 7 clearly shows that the number of iterations decreased as the fractional parameter values increased from 0.1 to 1.0.
Table 9 shows the maximum error (Max-Err) computed by fractional simultaneous scheme SFM
SFM
SFM
for different random initial vectors given in
Appendix A Table A2 to approximate all the roots of polynomial equations used in application 2.
Table 9 clearly demonstrates that as the fractional parameter values increased from 0.1 to 1.0, the accuracy computed by simultaneous scheme increased significantly (
Figure 3).
The approximate local computational order of convergence is shown in
Table 10. From 0.1 to 1.0, as the fractional parameter values increase, the approximate local computational order of convergence increases.
According to the fractional simultaneous scheme,
Table 10 displays the local computational order of convergence needed for the approximation of all roots of the polynomial equation used in application 2. Convergence rates increase as the following initial estimations are sufficiently adjusted to the exact roots of the engineering application 2:
are chosen as initial guessed values.
Table 11 shows that the convergence order and accuracy of the fractional simultaneous scheme are increased if we take the initial guessed values close to the exact root. The residual error computed by numerical methods also increased as we increased the fractional parameter value from 0.1 to 1.0.
Engineering Application 3: Series Circuit Analogue
Consider a flexible spring that is stretched vertically from a rigid support and has a mass
m attached to its free end. Naturally, the mass will determine how much the spring elongates or stretches; different weight masses will result in different ways that the spring will stretch. Hooke’s law states that the spring itself generates a restoring force F that is opposed to the direction of elongation and proportional to the amount of elongation s. In short, a proportionality constant is defined as
, where k is the spring constant. In an undamped spring/mass system, the differential equation represents
, which is mathematically modeled as [
40,
42]:
Using the method described in [
40], the following polynomial is used to simulate (
104) as:
The Caputo-type derivative of (
105) is given as:
The exact solution of (
105) up to the decimal places is written as follows:
To determine the global convergence component of the parallel scheme, use Matlab to generate a random initial guess value ranging from
as specified in
Appendix A Table A3. With an arbitrary starting value, SFM
SFM
SFM
converges to exact zeros after 19, 16, 13, 8, and 8 iterations as indicated in
Table 12 for various fractional parameters, i.e., 0.1, 0.3, 0.5, 0.8, and 1.0. As described in
Table 13, the corresponding CPU times are 3.1364, 1.0701, 1.0078, 0.0874 and 0.0975, respectively.
Table 14 clearly illustrates how the rate of convergence of SFM
SFM
SFM
accelerates as the value of the fractional parameter increases from 0.1 to 1.0. The newly developed method converges to exact roots for randomly generated initial guess values, demonstrating its global convergence.
Table 12 shows the number of iterations of fractional simultaneous scheme SFM
SFM
SFM
for different random initial vectors given in
Appendix A Table A3.
Table 12 clearly shows that the number of iterations decreased as the fractional parameter values increased from 0.1 to 1.0.
Table 14 shows the maximum error (Max-Err) computed by fractional simultaneous scheme SFM
SFM
SFM
for different random initial vectors given in
Appendix A Table A3 to approximate all roots of the polynomial equations used in application 3.
Table 14 clearly demonstrates that as the fractional parameter values increased from 0.1 to 1.0, the accuracy computed by simultaneous scheme increased significantly (
Figure 4).
The approximate local computational order of convergence is shown in
Table 15. As the fractional parameter values increase from 0.1 to 1.0, the approximate local computational order of convergence increases.
Table 13 displays the computational CPU time in seconds required to approximate all roots of the polynomial equation used in application 3 using the fractional simultaneous scheme.
Convergence rates increase as the following initial estimations are sufficiently adjusted to the exact root of engineering application 3:
are chosen as the initial guessed values.
Table 16 shows how the convergence order and accuracy of the fractional simultaneous scheme increase when we use initial guessed values close to the exact root. The residual error computed by numerical schemes increased as we increased the fractional parameter value from 0.1 to 1.0.
Application 4: Hanging Object
A chain attached to an object on the ground is pulled vertically upward by constant forces against gravity, causing the following nonlinear initial value problem:
Using the method described in [
40], the following polynomial is used to simulate (
107) as:
The Caputo-type derivative of (
108) is given as:
The exact solution of (
109) up to 4 decimal places is written as follows:
To determine the global convergence component of the parallel scheme, use Matlab to generate a random initial guess value ranging from
as specified in
Appendix A Table A4. With an arbitrary starting value, SFM
SFM
SFM
converges to exact zeros after 19, 16, 14, 8 and 8 iterations as indicated in
Table 17 for various fractional parameters, i.e., 0.1, 0.3, 0.5, 0.8, and 1.0. The newly developed method converges to exact roots for randomly generated initial guess values, demonstrating its global convergence.
Table 17 shows the number of iterations of fractional simultaneous scheme SFM
SFM
SFM
for different random initial vectors given in
Appendix A Table A4.
Table 18 shows the maximum error (Max-Err) computed by fractional simultaneous scheme SFM
SFM
SFM
for different random initial vectors given in
Appendix A Table A4 to approximate all roots of the polynomial equations used in application 4.
Table 18 clearly demonstrates that as the fractional parameter values increased from 0.1 to 1.0, the accuracy computed by the simultaneous scheme increased significantly (
Figure 5). This indicates the behavior of our recently developed simultaneous scheme in terms of global convergence.
Table 19 clearly illustrates how the computational order of convergence of SFM
SFM
SFM
increase as the value of the fractional parameter increases from 0.1 to 1.0. As described in
Table 20, the corresponding CPU times are 2.1254, 1.0874, 1.0078, 0.0874, and 0.0078, are consumed respectively.
The approximate local computational order of convergence is shown in
Table 19. As the fractional parameter values increase from 0.1 to 1.0, the approximate local computational order of convergence increases.
Table 20 displays the computational CPU time in seconds required to approximate all roots of the polynomial equation used in application 4 using the fractional simultaneous scheme.
Convergence rates increase as the following initial estimations are sufficiently adjusted to the exact answer of engineering application 4:
are chosen as initial guessed values.
Table 21 shows how the convergence order and accuracy of the fractional simultaneous scheme increase when we use initial guessed values close to the exact root. The residual error computed by the numerical schemes increased as we increased the fractional parameter value from 0.1 to 1.0.