Mani Sandeep Kumar Mylapalli*
| Navya Kakarlapudi | Pravin Singh | Akkapeddi Chandra Sekhar | Pragathi Chaganti
© 2024 The authors. This article is published by IIETA and is licensed under the CC BY 4.0 license (http://creativecommons.org/licenses/by/4.0/).
OPEN ACCESS
This paper provides a novel seventh-order iterative method for identifying the zeros of nonlinear equations that use the weight function and finite difference techniques. It is free of the second derivative and consists of three functional evaluations and one of its first derivatives at each iteration. Theoretical study shows that the proposed approach has a seventh order of convergence and a corresponding error equation. In some examples, the performance and effectiveness of the novel iterative method were tested and compared with a few existing equivalent order methods. We analyze a wide range of real-world situations from several fields to test the applicability and effectiveness of the proposed methods, and numerical comparisons are made with a few recently developed techniques. Finally, by investigating its complex dynamical behavior using test functions, we demonstrate that the suggested method is superior to currently used alternative methods.
nonlinear equations, iterative method, functional evaluations, efficiency index, order of convergence, basins of attraction
Iterative approaches are widely used for locating the roots of nonlinear equations. Almost all fields of the sciences, engineering, applied mathematics, and computers involve these equations. Analytical methods cannot generally solve the zeros of nonlinear equations (either algebraic or transcendental). Hence, the solutions of the equations must be approached using iterative methods. Since most application problems in every field can build a suitable mathematical model, they can be reduced to a solvable nonlinear equation. Due to their importance, various iterative techniques have been created employing tools such as Taylor’s series, interpolation formula, finite difference technique, Adomian polynomials, weight function technique, homotropy perturbation method, general quadrature formula, variational iteration method, decomposition method. Newton Raphson's Method [NR] [1] is one of the earliest techniques for locating the zeros of the nonlinear equation h(x)=0, and it is defined as:
$x_{n+1}=x_n-\frac{h\left(x_n\right)}{h^{\prime}\left(x_n\right)}, h^{\prime}\left(x_n\right) \neq 0$ (1)
with two functional evaluations, it exhibits second-order convergence, and the Efficiency Index is 1.414.
Numerous researchers have attempted to raise the order of convergence of Newton's technique to produce better outcomes. In this paper, we suggest a seventh-order iterative scheme based on the weight function and finite difference approximations for finding the solution of the nonlinear equation. Four function evaluations are performed during each iteration, and the efficiency index is 1.6265. In this study, we applied application problems in several branches to demonstrate the efficacy of our zero-finding method. We will compare our numerical calculations to well-known seventh-order methods. In addition, our suggested methods provide faster convergence and less residual error. Our approaches produce superior numerical results than those produced by the existing methods, which is how we proved the effectiveness and robustness of the suggested method. We also studied iterative processes' dynamic behavior in the complex plane.
The remainder of the study will be structured as follows: In section 1, we offered some fundamental guidelines for finding the roots of nonlinear equations. In section 2, we used a finite difference and weight function approach to create a novel seventh-order iterative algorithm. In Section 3, we theoretically demonstrate that the suggested scheme's order of convergence is seven. In section 4, we provide a number of numerical examples in various branches to highlight the benefits of the suggested technique and contrast it with other recent methods of similar and different order methods. Basins of attraction were looked at in section 5 to show the complex plane's dynamic behavior. The conclusion is provided in Section 6.
1.1 Definitions
1.1.1 Order of convergence [2]
Let $x_1, x_2, x_3 \ldots$ be a series of real integers that converges to the root $x_0$ of a real function $h(x)$, which has a simple root of $x_0$. Then, $\operatorname{Lim}_{n \rightarrow \infty} \frac{x_{n+1}-x_0}{\left(x_n-x_0\right)^p} \approx C$ is used to define the sequence's $p \in R^{+}$order of convergence. The asymptotic error constant is C≠0 in this case.
1.1.2 Error equation
The relationship $\varepsilon_{n+1}=C \varepsilon_n^p+O\left(\varepsilon_n^{p+1}\right)$ is referred to as an error equation if $\varepsilon_n=x_n$ - $x_0$ represents the error in the $\mathrm{n}^{\text {th }}$ iteration.
1.1.3 Efficiency index [3]
Let d represent how often the function was utilized in the specified algorithm, and the order of convergence is indicated by p, then the efficiency index is defined as E.I=p1/d.
1.1.4 Computational order of convergence (COC) [4]
Let $x_{n-1}, x_n, x_{n+1}$ represent three sequential iterations that are close to the root, and let $x$ be the root of the nonlinear equation $h(x)=0$. The COC can then be estimated to be below:
$\rho=\frac{\log \left(\left|x_{n+1}-x\right| /\left|x_n-x\right|\right)}{\log \left(\left|x_n-x\right| /\left|x_{n-1}-x\right|\right)}$
It is used to check the convergence order of a given iterative scheme.
1.1.5 Weight function [5]
The weight function is a real variable sufficiently differentiable function defined on a given interval. These are introduced in an iteration scheme via any arithmetical operation. The main reason for the intrusion of weight function to some specific entity in an iteration process is to enhance the behavior of the iterative method and assist in improving its order of convergence and computational efficiency.
Consider $x^*$ be the exact root of $h(x)=0$ as well as $x_n$ being the $\mathrm{n}^{\text {th }}$ approximation root and $\varepsilon_n$ being the error. Then:
$x^*=x_n+\varepsilon_n$ (2)
Now:
$h\left(x^*\right)=0$ (3)
By Taylor’s series expansion:
$h\left(x^*\right)=h\left(x_n\right)+\varepsilon_n h^{\prime}\left(x_n\right)+\frac{\varepsilon_n^2}{2 !} h^{\prime \prime}\left(x_n\right)+\ldots$ (4)
Neglecting higher powers of $\varepsilon_n$, and from Eq. (3) and Eq. (4), we have $\varepsilon_n^2 h^{\prime \prime}\left(x_n\right)+2 \varepsilon_n h^{\prime}\left(x_n\right)+2 h\left(x_n\right)=0, \varepsilon_n=$ $\left(-2 h^{\prime}\left(x_n\right) \pm \sqrt{4 h^{\prime}\left(x_n\right)-8 h\left(x_n\right) h^{\prime \prime}\left(x_n\right)}\right) \div 2 h^{\prime \prime}\left(x_n\right)$.
On simplification:
$\varepsilon_n=\frac{-2 h\left(x_n\right)}{h^{\prime}\left(x_n\right)} \cdot \frac{1}{1+\sqrt{1-2 \rho_n}}$ (5)
Putting $x^*$ by $x_{n+1}$ in $h(x)=0$ and from Eq. (5), we get:
$x_{n+1}=y_n-H(\tau)\left[\frac{2 h\left(y_n\right)}{h^{\prime}\left(y_n\right)} \cdot \frac{1}{1+\sqrt{1-2 \rho_n}}\right]$ (6)
where, $y_n=x_n-\frac{h\left(x_n\right)}{h^{\prime}\left(x_n\right)}, \rho_n=\frac{h^{\prime}\left(x_n\right)-h^{\prime}\left(y_n\right)}{h^{\prime}\left(x_n\right)}$,
$h^{\prime}\left(y_n\right)=2 h\left[y_n, x_n\right]-h^{\prime}\left(x_n\right), H(\tau)=1-\tau$ and $\tau=\frac{h\left(y_n\right)}{h\left(x_n\right)}$ is the weight function.
We may create the algorithm by employing Eq. (1) and Eq. (6) as the first and second steps, extending the above scheme by adding Newton's variation technique as the third step. Adding more sub-steps can help the method above perform better, and using multi-step iterative methods, the computational efficiency index can be raised. The motivation of this study is to determine whether it is possible to increase the computational efficiency index, basins of attraction, and rate of convergence of numerical algorithms without introducing new functions. To increase the higher order of convergence or better efficiency, we need to avoid calculating the high-order derivatives of the function. So, we compute $h^{\prime}\left(z_n\right)$ using the following finite difference approximation: $h^{\prime}\left(z_n\right)=h\left[z_n, y_n\right]+\left(z_n-y_n\right) h\left[z_n, y_n, x_n\right]$
2.1 Algorithm
The computation for the iterative scheme is xn+1:
$\begin{aligned} & \text { 1. } y_n=x_n-\frac{h\left(x_n\right)}{h^{\prime}\left(x_n\right)} \\ & \text { 2. } z_n=y_n-H(\tau)\left[\frac{2 h\left(y_n\right)}{h^{\prime}\left(y_n\right)} \cdot \frac{1}{1+\sqrt{1-2 \rho_n}}\right] \\ & \text { where, } h^{\prime}\left(y_n\right)=2 h\left[y_n, x_n\right]-h^{\prime}\left(x_n\right) \\ & \rho_n=\frac{h^{\prime}\left(x_n\right)-h^{\prime}\left(y_n\right)}{h^{\prime}\left(x_n\right)}, H(\tau)=1-\tau \text { and } \tau=\frac{h\left(y_n\right)}{h\left(x_n\right)} \\ & \text { 3. } x_{n+1}=z_n-\frac{h\left(z_n\right)}{h^{\prime}\left(z_n\right)} \\ & \text { where, } h^{\prime}\left(z_n\right)=h\left[z_n, y_n\right]+\left(z_n-y_n\right) h\left[z_n, y_n, x_n\right]\end{aligned}$ (7)
The above scheme Eq. (7) is denoted as SR. It has one of its derivatives and three functional evaluations.
In this section, we derive the convergence analysis of the developed method using Taylor’s series method.
3.1 Theorem [6]
If $x_0$ is close to $x^*$, let $x_0 \in D$ be a single zero of a sufficiently differentiable function $h(x)$ for an open interval $D$. The procedure Eq. (7) then converges to the seventh order.
Proof: Let $x^*$ represent the first zero of $h(x)=0$ and $x^*=x_n+\varepsilon_n$. Thus, $h\left(x^*\right)=0$
Using Taylor's method, expanding $h\left(x_n\right)$ and $h^{\prime}\left(x_n\right)$ about $x^*$, we have:
$h\left(x_n\right)=h^{\prime}\left(x^*\right)\left(\varepsilon_n+c_2 \varepsilon_n^2+c_3 \varepsilon_n^3+c_4 \varepsilon_n^4 +\cdots\right)$ (8)
$h^{\prime}\left(x_n\right)=h^{\prime}\left(x^*\right)\left(1+2 c_2 \varepsilon_n+3 c_3 \varepsilon_n^2+4 c_4 \varepsilon_n^3+\ldots\right)$ (9)
Dividing Eq. (8) by Eq. (9), we get:
$\frac{h\left(x_n\right)}{h^{\prime}\left(x_n\right)}=\varepsilon_n-c_2 \varepsilon_n^2-\left(2 c_3-2 c_2^2\right) \varepsilon_n^3-\left(3 c_4-7 c_2 c_3+4 c_2^3\right) \varepsilon_n^4+\ldots$ (10)
Substituting Eq. (10) in the first step of Eq. (7), we get:
$y_n=x^*+Y$ where $Y=c_2 \varepsilon_n^2+\left(2 c_3-2 c_2^2\right) \varepsilon_n^3+\left(3 c_4-7 c_2 c_3+4 c_2^3\right) \varepsilon_n{ }^4+\ldots
$ (11)
Expanding $h\left(y_n\right)$ about $x^*$ by using Taylor's method, $h\left(y_n\right)=h^{\prime}\left(x^*\right)\left(c_2 \varepsilon_n^2+\left(2 c_3-2 c_2^3\right) \varepsilon_n^3+\left(3 c_4-7 c_2 c_3+\right.\right. \left.\left.5 c_2^3\right) \varepsilon_n^4+\ldots\right) ; h^{\prime}\left(y_n\right)=h^{\prime}\left(x^*\right)\left(1+\left(2 c_2^2-c_3\right) \varepsilon_n^2+\left(6 c_2 c_3-\right.\right. \left.\left.4 c_2^3-2 c_4\right) \varepsilon_n^3+\ldots\right), \text { Thus: }$
$\frac{h\left(y_n\right)}{h^{\prime}\left(y_n\right)}=c_2 \varepsilon_n^2+\left(2 c_3-2 c_2^3\right) \varepsilon_n^3+\left(3 c_2^3-6 c_2 c_3+3 c_4\right) \varepsilon_n^4+\ldots$ (12)
$\begin{aligned} & \rho_n=2 c_2 \varepsilon_n+\left(4 c_3-6 c_2^2\right) \varepsilon_n^2+ \left(6 c_4+16 c_2^3-20 c_2 c_3\right) \varepsilon_n^3+\ldots\end{aligned}$ (13)
$H(\tau)=1-c_2 \varepsilon_n-\left(2 c_3-3 c_2^2\right) \varepsilon_n^2-\left(3 c_4-10 c_2 c_3+8 c_2^3\right) \varepsilon_n^3+\ldots$ (14)
Substitute Eqs. (12)-(14), in second step of Eq. (7), we get:
$z_n=x^*+Z$ (15)
where,
$Z=\left\{-c_2 c_3 \varepsilon_n^4+\left(c_2 c_4-c_3^2+c_2^4\right) \varepsilon_n^5+\left(\begin{array}{l}c_2 c_5+6 c_2^2 c_4+4 c_2 c_3^2 \\ +5 c_2^3 c_3-c_2^5-c_3 c_4-13 c_2 c_3 c_4\end{array}\right) \varepsilon_n^6+\ldots\right\}$
Again, expanding $h\left(z_n\right)$ about $x^*$ by using Taylor's expansion as follows:
$h\left(z_n\right)=h^{\prime}\left(x^*\right)\left(z+c_2 z^2+c_3 z^3+\ldots\right)$ (16)
$\begin{aligned} h\left[z_n, y_n\right] & =\frac{h\left(z_n\right)-h\left(y_n\right)}{z_n-y_n} \\ & =1+c_2(Y+Z)+c_3\left(Y^2+Y Z+Z^2\right)+\ldots\end{aligned}$ (17)
$\begin{aligned} h\left[y_n, x_n\right] & =\frac{h\left(y_n\right)-h\left(x_n\right)}{y_n-x_n} = 1+c_2\left(Y+\varepsilon_n\right)+c_3\left(Y^2+Y \varepsilon_n+\varepsilon_n^2\right)+\ldots\end{aligned}$ (18)
Therefore, from Eq. (17) and Eq. (18), we have:
$\begin{aligned} & h\left[z_n, y_n, x_n\right]=\frac{h\left[z_n, y_n\right]-h\left[y_n, x_n\right]}{z_n-x_n} =c_2+c_3\left(Z+Y+\varepsilon_n\right)+c_4\left(Z^2+Y^2+\varepsilon_n^2+Z \varepsilon_n+Y \varepsilon_n+Z Y\right)+\ldots\end{aligned}$ (19)
From Eq. (17) and Eq. (19), we get:
$\begin{aligned} h^{\prime}\left(z_n\right) & =h\left[z_n, y_n\right]+\left(z_n-y_n\right) h\left[z_n, y_n, x_n\right] =h^{\prime}\left(x^*\right)\left\{\begin{array}{l}1+2 c_2 Z+c_3 Z Y+c_3 Z \varepsilon_n-c_3 Y \varepsilon_n+ \\ c_4 Z \varepsilon_n^2-c_4 Y^3-c_4 Y \varepsilon_n^2-c_4 Y^2 \varepsilon_n\end{array}\right\}\end{aligned}$ (20)
Substituting Eq. (15), Eq. (16) and Eq. (20) in the third step of Eq. (7), we get $\varepsilon_{n+1}=\left(c_2^2 c_3^2\right) \varepsilon_n^7+O\left(\varepsilon_n^8\right)$.
As a result, we concluded that the order of convergence of SR is seven, and its efficiency index is 71/4=1.6265.
We will evaluate the effectiveness and convergence performance of our suggested algorithm in this part. We look at some real-time engineering uses for this. We will now compare our suggested approach (SR) to various seventh-order existing methods, and with other existing methods including SM, HM, NM, FM, PM, CM, IM, and EM regarding iterations, errors, and functional evaluations. All the numerical calculations are performed by mpmath library and an Intel(R) Core (TM) i5-10210U CPU clocked at 2.11 GHz with a 64-bit operating system, we employ PYTHON for all numerical operations. The stopping criterion $\left|f\left(x_n\right)\right|<\varepsilon$, with the required precision set to 690 decimal places and the tolerance set to $\varepsilon=10^{-199}$.
Consider the following existing seventh-order iterative methods for comparison:
In 2014, Al-Subaihi and Al-Qarni [7] suggested a seventh-order iterative method (SM) with four function evaluations given by:
$\begin{aligned} & y_n=x_n-\frac{h\left(x_n\right)}{h^{\prime}\left(x_n\right)} \\ & z_n=y_n+\frac{h\left(y_n\right)}{h^{\prime}\left(x_n\right)}-2 \frac{h\left(x_n\right) h\left(y_n\right)}{h^{\prime}\left(x_n\right)\left(h\left(x_n\right)-h\left(y_n\right)\right)} \\ & x_{n+1}=z_n-\frac{h\left(z_n\right)}{h\left[z_n, y_n\right]+h\left[z_n, x_n, x_n\right]\left(z_n-y_n\right)}\end{aligned}$ (21)
In 2021, Bawazir [8] proposed a seventh-order method (HM) with five functions given by:
$\begin{aligned} & y_n=x_n-\frac{h\left(x_n\right)}{h^{\prime}\left(x_n\right)} \\ & z_n=y_n-\frac{h\left(y_n\right)(1+\mu / 2)}{h^{\prime}\left(y_n\right)} \\ & x_{n+1}=z_n+\frac{h\left(z_n\right) h\left(y_n\right)(1+\mu / 2)}{h^{\prime}\left(y_n\right)\left(h\left(z_n\right)-h\left(y_n\right)\right)}\end{aligned}$ (22)
where, $\mu=\frac{h\left(y_n\right)\left(h^{\prime}\left(x_n\right)-h^{\prime}\left(y_n\right)\right)}{h\left(x_n\right) h^{\prime}\left(y_n\right)}$.
In 2016, Napassanan and Montri [9] developed a new scheme (NM) of order seven is given by:
$\begin{aligned} & y_n=x_n-\frac{h\left(x_n\right)}{h^{\prime}\left(x_n\right)} \\ & z_n=y_n-\frac{h\left(y_n\right)}{h^{\prime}\left(y_n\right)}-\frac{h\left(y_n\right)^2 h^{\prime \prime}\left(y_n\right)}{2\left(h^{\prime}\left(y_n\right)\right)^3} \\ & x_{n+1}=z_n-\frac{\left(x_n-z_n\right) h\left(z_n\right)}{h\left(x_n\right)-2 h\left(z_n\right)} \\ & \text { where } h^{\prime \prime}\left(y_n\right)=\frac{2}{y_n-x_n}\left(2 h^{\prime}\left(y_n\right)+h^{\prime}\left(x_n\right)-3 \frac{h\left(y_n\right)-h\left(x_n\right)}{y_n-x_n}\right)\end{aligned}$ (23)
In 2019, Francisco et al. [10] presented a novel family of iterative methods (FM) of order seven to find the root of nonlinear equations given by:
$\begin{aligned} & y_n=x_n-\frac{h\left(x_n\right)}{h^{\prime}\left(x_n\right)} \\ & z_n=x_n-G(\eta) \frac{h\left(x_n\right)}{h^{\prime}\left(x_n\right)} \\ & \text { where } G(\eta)=1+\frac{h\left(y_n\right)}{h\left(x_n\right)}+2\left(\frac{h\left(y_n\right)}{h\left(x_n\right)}\right)^2 \\ & w_n=z_n+\frac{h\left(z_n\right)}{h^{\prime}\left(x_n\right)} \\ & x_{n+1}=z_n-\left(1-4\left(\frac{h\left(z_n\right)}{h\left(w_n\right)}\right)+8\left(\frac{h\left(z_n\right)}{h\left(w_n\right)}\right)^2\right) \frac{h\left(z_n\right)}{h^{\prime}\left(x_n\right)}\end{aligned}$ (24)
In 2019, a sixth-order iterative scheme (PM) proposed by Prem Chand et al. [11] is given by:
$\begin{aligned} & y_n=x_n-\frac{h\left(x_n\right)}{h^{\prime}\left(x_n\right)} \\ & z_n=x_n-\left(1+2\left(\frac{h\left(y_n\right)}{h\left(x_n\right)}\right)^2\right)\left(\frac{h\left(x_n\right)+h\left(y_n\right)}{h^{\prime}\left(x_n\right)}\right) \\ & x_{n+1}=z_n-\left(1+2\left(\frac{h\left(y_n\right)}{h\left(x_n\right)}\right)\right) \frac{h\left(z_n\right)}{h^{\prime}\left(x_n\right)}\end{aligned}$ (25)
In 2020, Prem Chand et al. [12] presented an iterative method (CM) of order six is given by:
$\begin{gathered}y_n=x_n-\frac{2}{3} \frac{h\left(x_n\right)}{h^{\prime}\left(x_n\right)} \\ z_n=x_n-\left(-\frac{1}{4}+\frac{3}{4} \frac{h^{\prime}\left(x_n\right)}{h^{\prime}\left(\mathrm{y}_n\right)}+\frac{1}{2} \frac{h^{\prime}\left(\mathrm{y}_n\right)}{h^{\prime}\left(\mathrm{x}_n\right)}\right) \cdot \frac{2 h\left(x_n\right)}{h^{\prime}\left(x_n\right)+h^{\prime}\left(y_n\right)} \\ x_{n+1}=z_n-\frac{h\left(z_n\right)}{h^{\prime}\left(y_n\right)} \cdot\left(\frac{1}{2}+\frac{1}{2} \frac{h^{\prime}\left(x_n\right)}{h^{\prime}\left(\mathrm{y}_n\right)}\right)\end{gathered}$ (26)
In 2017, A sixth-order Iterative method (IM) is free from derivative for solving nonlinear equation is developed by Rahma et al. [13] and given by:
$\begin{gathered}y_n=x_n-\frac{h\left(x_n\right)}{h^{\prime}\left(x_n\right)}, x_{n+1}=y_n-\frac{h\left(y_n\right)}{h^{\prime}\left(y_n\right)} -\frac{2 h\left(y_n\right)^2 h^{\prime}\left(y_n\right) h^{\prime \prime}\left(y_n\right)}{4\left(h^{\prime}\left(y_n\right)\right)^4-4 h\left(y_n\right)\left(h^{\prime}\left(y_n\right)\right)^2 h^{\prime \prime}\left(y_n\right)+\left(h\left(y_n\right)\right)^2\left(h^{\prime \prime}\left(y_n\right)\right)^2}\end{gathered}$ (27)
where, $h^{\prime\prime}\left(y_n\right) \frac{2}{x_n-y_n}\left[3 \frac{h\left(x_n\right)-h\left(y_n\right)}{x_n-y_n}-2 h^{\prime}\left(y_n\right)-h^{\prime}\left(x_n\right)\right]$.
In 2022. An efficient sixth-order Iterative method (EM) suggested by Sharma and Sunil [14] is given by:
$\begin{aligned} & y_n=x_n-\frac{h\left(x_n\right)}{h^{\prime}\left(x_n\right)} \\ & x_{n+1}=y_n-\frac{h\left(y_n\right)}{h^{\prime}\left(y_n\right)}-\left(\frac{h\left(y_n\right)}{h^{\prime}\left(y_n\right)}\left(\frac{-2 M_1+M_2+M_3 * M_4}{2 M_1-2 M_2-M_3 * M_4}\right)\right)^2 \frac{h^{\prime \prime}\left(y_n\right)}{2 h^{\prime}\left(y_n\right)}\end{aligned}$ (28)
where,
$\begin{aligned} & M_1=h\left(x_n\right)^3 h^{\prime}\left(y_n\right)^2 h^{\prime \prime}\left(y_n\right), M_2=h\left(x_n\right)^3 h\left(y_n\right)\left(h^{\prime \prime}\left(y_n\right)\right)^2 \\ & M_3=h^{\prime}\left(x_n\right) h\left(y_n\right) h^{\prime}\left(y_n\right), M_4=h\left(x_n\right)^2 h^{\prime \prime}\left(y_n\right)-2 h\left(y_n\right) h^{\prime}\left(x_n\right)^2 \\ & h^{\prime \prime}\left(y_n\right)=\frac{2}{x_n-y_n}\left[3 \frac{h\left(x_n\right)-h\left(y_n\right)}{x_n-y_n}-2 h^{\prime}\left(y_n\right)-h^{\prime}\left(x_n\right)\right]\end{aligned}$
The analogy of an efficiency index of different iterative methods is shown in Table 1. Table 2 shows the roots of the test functions. For each test function and application, we offer the number of iterations, functional evaluations and first five absolute residual errors $\left|e_1\right|,\left|e_2\right|,\left|e_3\right|,\left|e_4\right|$, and $\left|e_5\right|$ in order to facilitate comparisons. Table 3 reports the comparison results for a few other techniques applied to application problems from various fields based on the number of function evaluations and inaccuracy.
Table 1. The analogy of the efficiency index
|
Method |
P |
N |
E. I |
|
SM |
7 |
4 |
1.626 |
|
HM |
7 |
5 |
1.475 |
|
NM |
7 |
5 |
1.475 |
|
FM |
7 |
5 |
1.475 |
|
PM |
6 |
5 |
1.430 |
|
CM |
6 |
4 |
1.565 |
|
IM |
6 |
4 |
1.565 |
|
EM |
6 |
4 |
1.565 |
|
SR |
7 |
4 |
1.626 |
Note: P represents the order of the method, N represents the number of functional evaluations per iteration and E. I represent the efficiency index.
The following test functions (algebraic or transcendental) and their simple zeros are provided for our investigation [6]:
Table 2. Roots of the test functions
|
h(z) |
Root |
|
$\begin{gathered}h_1(x)=\sin (2 \cos x)-1-x^2 +e^{\sin \left(x^3\right)}\end{gathered}$ |
-0.784895987661 |
|
$h_2(x)=(x+2) e^x-1$ |
-0.442854010023 |
|
$h_3(x)=x^2+\sin \left(\frac{x}{5}\right)-\frac{1}{4}$ |
0.409992017989 |
|
$h_4(x)=x^3-10$ |
2.154434690031 |
|
$h_5(x)=\sin ^2 x-x^2+1$ |
-1.404491648215 |
The efficacy and computational behavior of the provided methods are examined using the following application problems:
4.1 Azeotropic point of a binary solution [15]
To determine the azeotropic point of the nonlinear equation:
$h_6(x)=\frac{P Q\left[Q(1-x)^2-P x^2\right]}{[x(P-Q)+Q]^2}+0.14845$
where, the Van Laar equation's coefficients, which describe the phase equilibria of liquid solutions, are P and Q. We took, P=0.38969 and Q=0.55954 were used. The root of the above equation is 0.69147373574714144, is displayed in the table below.
4.2 Fractional conversion [11]
When using hydrogen feed that is transformed into ammonia at 500℃ and 250 atm, the fractional conversion of nitrogen can be calculated using the following equation:
$h_7(x)=x^4-7.79075 x^3+14.7445 x^2+2.511 x-1.674$
The root lies in the range of 0 and 1 according to the definition of fractional conversion. Therefore, the real root is 0.2777595428417206.
4.3 Ideal and non-ideal gas laws [15]
The equation:
$p x=n R T$
Stands for the ideal gas law, also known as the universal gas equation. Where p, x, n, R, T are pressure, volume, number of moles, a gas's universal constant, and temperature respectively. By resolving:
$h_8(x)=\left(p+a / x^2\right)(x-b)-R T$
can be used to calculate the molal volume.
For carbon dioxide, we use the values R=0.082054L atm/(mol K), where b=0.04267, T=300K and. Therefore, the nonlinear equation's root is 24.5125881284415006.
4.4 Parachutist’s problem [15]
The total force for parachutists is calculated as:
$F=m g-x v$
where, $m$ is the mass, $g$ refers the acceleration due to gravity, $x$ is the drag coefficient, and $v$ is the parachutist's velocity, and from the above equation, we obtain the nonlinear equation:
$h_9(x)=\frac{g m}{x}\left(1-e^{\frac{-x}{m} t}\right)-v$
We suppose that the parameters will have values of g=9.8m/s2, v=41m/s, m=68kg, and t=8s. Therefore, 12.533522848184467 is the root of the above nonlinear equation.
4.5 Study of multifactor effect problem [16]
The equation:
$\begin{aligned} x(\mathrm{t}) & =x_0+\left(v_0+e E_0(\mathrm{~m} w)^{-1} \sin \left(w t_0+\eta\right)\right)\left(t-t_0\right) +e E_0\left(\mathrm{~m} w^2\right)^{-1}\left(\cos \left(w t_0+\eta\right)+\sin \left(w t_0+\eta\right)\right)\end{aligned}$
describes the moment of an electron in the space between two parallel plates. Where $x_0$ is the position of the electron, $v_0$ is the velocity, $\underline{\underline{e}}$ refers the charge, RF electric field between plates at time $t_0$ is represented by $E_0 \sin \left(w t_0+\eta\right)$, and the resting mass of an electron is $m$. Regarding the specific values, it can be reduced in polynomial form as:
$h_{10}(x)=x-0.5 \cos x-\frac{\pi}{4}$
This function has a simple root at x=-0.3094661392082.
4.6 The vertical stress [15]
The vertical stress $\sigma_x$, which is generated at a place in the elastic continuum beneath the edge of a strip footing sustaining a constant pressure $q$ and is specified by:
$h_{11}(x)=\frac{x+\cos x \sin x}{\pi}-\frac{1}{4}$
is one of the fundamental stresses experienced by finite subsurface constructions.
The nonlinear equation $h_{11}(x)$=0 has a root of 0.4160444988100767043.
4.7 Volume from Van der Waals equation [6]
An equation:
$h(V)=p V^3-n(R T+B p) V^2+n^2 A V-n^3 A B$
represents the non-ideal gas in the Van der Waals equation.
Put V=x and for instance, the nonlinear polynomial function h12(x)=40x3-95.26535116x2+35.28x-5.6998368.
It can be used to figure out how much 1.4 moles of benzene vapor will weigh at 500 C and 40 atm of pressure. This is due to the fact that benzene, A=18 and B=0.1154 are the Van der Waals constants. It has three roots, one of which, x≈1.9707842194070294, is real.
4.8 Blood rheology model [17]
The scientific field of blood rheology investigates the physical and flow characteristics of blood. Blood is considered a Caisson fluid since it is a non-Newtonian fluid. We assume the following function as a nonlinear equation to test the plug flow of Caisson fluids:
$h_{13}(x)=\frac{1}{441} x^8-\frac{8}{63} x^5-0.0571428571 x^4+\frac{16}{9} x^2-3.624489796 x+0.3$
The findings are displayed in the Table 3 as 0.0864335580522467, which is the root of h13(x)=0.
4.9 A stirred tank reactor [18]
Think about the stirred tank's reactor. The reactor receives materials at rates of $\beta$ and $q-\beta$, respectively. The equipment enhances mixed reaction as follows:
$\begin{aligned} & H_1+H_2 \rightarrow H_3, H_3+H_2 \rightarrow H_4 \\ & H_4+H_2 \rightarrow H_5, H_4+H_2 \rightarrow H_6\end{aligned}$
In their initial investigation of this intricate control system, Douglas [19] discovered the nonlinear polynomial equation shown below:
$\frac{2.98 \times(x+2.25)}{(x+1.45) \times(x+2.85)^2 \times(x+4.35)}=\frac{1}{T_C}$
where, $T_c$ is the proportional controller's gain. By taking, we have $T_c=0$.
$h_{14}(x)=x^4+11.50 x^3+47.49 x^2+83.06325 x-51.23266875=0$
The root of the above nonlinear equation is -1.45.
Table 3. The analogy of different methods
|
Method |
x0 |
n |
|e1| |
|e2| |
|e3| |
|e4| |
|e5| |
|h(xn+1)| |
|
|
h1(x) |
-0.9 |
|
|
|
|
|
|
|
|
|
SM HM NM FM PM CM IM EM SR |
|
4 4 4 4 5 5 5 5 4 |
0.115103 0.115104 0.115104 0.115104 0.1151 0.115102 0.115108 0.115042 0.115104 |
1.2E-07 4.5E-08 2.5E-08 1.5E-07 3.8E-06 1.8E-06 3.7E-06 6.2E-05 2.3E-08 |
3.00E-49 1.76E-45 1.62E-54 2.60E-48 1.80E-32 5.60E-35 9.41E-32 1.00E-17 4.30E-55 |
2.36E-340 6.23E-270 7.78E-378 1.31E-333 2.00E-190 6.01E-136 2.50E-185 6.81E-69 3.29E-382 |
--- --- --- --- 5.47E-691 6.48E-439 4.78E-690 1.38E-273 --- |
6.67E-340 1.75E-269 2.19E-377 3.71E-333 1.64E-690 1.83E-438 1.09E-690 3.89E-273 9.27E-382 |
|
|
|
-0.5 |
|
|
|
|
|
|
|
|
|
SM HM NM FM PM CM IM EM SR |
|
5 5 5 5 5 5 6 4 |
Divergent 0.277695 0.284602 0.23325 0.25856 0.29106 0.350526 0.29887 0.28495 |
0.007201 0.000294 0.051650 0.026330 0.006160 0.065629 0.013980 4.9E-050 |
3.13E-14 4.96E-26 2.70E-09 2.50E-09 9.60E-14 9.10E-07 2.40E-08 7.80E-32 |
1.99E-82 1.90E-178 1.90E-60 1.30E-51 6.00E-69 2.07E-35 2.10E-31 2.00E-219 |
1.32E-491 1.77E-690 1.45E-418 3.17E-305 7.22E-238 2.86E-207 1.00E-123 --- |
3.73E-491 1.09E-690 4.10E-418 8.94E-305 2.03E-237 8.10E-207 3.56E-492 6.01E-219 |
|
|
h2(x) |
-0.2 |
|
|
|
|
|
|
|
|
|
SM HM NM FM PM CM IM EM SR |
|
4 4 4 4 5 5 5 5 4 |
0.15715 0.157148 0.157145 0.15714 0.1572 0.15715 0.157181 0.15723 0.15715 |
1.50E-06 2.60E-06 5.91E-07 1.20E-06 5.10E-05 7.50E-06 3.51E-05 8.80E-05 1.10E-07 |
4.80E-42 4.99E-35 4.73E-45 4.50E-42 3.90E-26 7.10E-32 3.12E-27 5.50E-18 9.70E-51 |
2.03E-290 2.50E-207 9.99E-312 3.49E-290 7.00E-153 1.00E-126 1.50E-159 9.00E-71 3.34E-352 |
--- --- --- --- 3.14E-689 3.85E-411 1.09E-690 6.11E-282 --- |
3.34E-290 4.10E-207 1.64E-311 5.74E-290 5.47E-691 6.32E-411 5.47E-691 1.00E-281 5.48E-352 |
|
|
|
-0.6 |
|
|
|
|
|
|
|
|
|
SM HM NM FM PM CM IM EM SR |
|
4 4 4 4 5 5 5 5 4 |
0.15715 0.157148 0.157145 0.15714 0.1572 0.15715 0.157181 0.15723 0.15715 |
1.50E-06 2.60E-06 5.91E-07 1.20E-06 5.10E-05 7.50E-06 3.51E-05 8.80E-05 1.10E-07 |
4.80E-42 4.99E-35 4.73E-45 4.50E-42 3.90E-26 7.10E-32 3.12E-27 5.50E-18 9.65E-51 |
2.03E-290 2.51E-207 9.99E-312 3.49E-290 7.00E-153 1.00E-126 1.50E-159 9.01E-71 3.34E-352 |
--- --- --- --- 3.14E-689 3.85E-411 1.09E-690 6.11E-282 --- |
3.34E-290 4.14E-207 1.64E-311 5.74E-290 5.47E-691 6.32E-411 5.47E-691 1.00E-281 5.48E-352 |
|
|
h3(x) |
0.3 |
|
|
|
|
|
|
|
|
|
SM HM NM FM PM CM IM EM SR |
|
4 5 4 4 5 5 5 5 4 |
0.109992 0.109995 0.109991 0.109991 0.110080 0.110008 0.110196 0.110110 0.109992 |
6.96E-07 3.12E-06 7.48E-07 9.43E-07 8.82E-05 1.63E-05 0.000204 0.000118 9.22E-10 |
5.72E-46 8.32E-34 2.32E-43 5.86E-42 7.62E-24 7.93E-29 1.97E-21 8.93E-17 9.26E-70 |
1.46E-319 3.03E-199 6.49E-299 2.11E-288 3.23E-138 2.90E-117 1.60E-123 2.93E-65 9.55E-490 |
--- 3.42E-691 --- --- 2.73E-691 1.51E-382 1.37E-690 3.38E-259 --- |
1.48E-319 5.47E-691 6.62E-299 2.15E-288 5.41E-691 1.54E-382 5.47E-691 3.45E-259 9.74E-490 |
|
|
|
0.5 |
|
|
|
|
|
|
|
|
|
SM HM NM FM PM CM IM EM SR |
|
4 4 4 4 5 5 5 5 4 |
0.090008 0.090008 0.090008 0.090008 0.090004 0.090007 0.090002 0.089988 0.090008 |
1.27E-08 2.86E-07 4.63E-08 2.22E-07 3.74E-06 1.20E-06 5.50E-06 1.95E-05 1.37E-11 |
3.87E-58 4.92E-40 8.08E-52 2.38E-46 4.47E-32 1.23E-35 7.53E-31 6.71E-20 1.51E-82 |
9.44E-405 1.28E-236 3.96E-358 3.83E-319 1.30E-187 1.11E-137 4.89E-180 9.31E-78 2.96E-579 |
--- --- --- --- 8.89E-691 7.69E-444 1.36E-690 3.45E-309 --- |
9.63E-405 1.31E-236 4.04E-358 3.91E-319 4.78E-691 7.84E-444 5.47E-691 3.52E-309 3.02E-579 |
|
|
h4(x) |
3 |
|
|
|
|
|
|
|
|
|
SM HM NM FM PM CM IM EM SR |
|
4
5 5 5 5 7 6 4 |
0.154435 Divergent 0.844733 0.842430 0.835893 0.841309 0.301894 0.861193 0.845483 |
0.000106
0.000833 0.003135 0.009672 0.004257 0.365484 0.015628 8.21E-05 |
2.76E-30
4.62E-24 2.77E-19 2.58E-13 4.17E-16 0.178198 4.22E-10 2.79E-32 |
2.10E-209
7.50E-166 1.20E-131 9.61E-77 4.50E-79 1.07E-05 2.46E-40 1.49E-224 |
---
0 1.69E-689 2.56E-457 1.21E-268 1.17E-28 2.80E-161 --- |
2.99E-208
1.31E-689 1.31E-688 3.57E-456 1.69E-267 1.31E-689 6.77E-644 2.10E-223 |
|
|
|
1 |
|
|
|
|
|
|
|
|
|
SM HM NM FM PM CM IM EM SR |
|
6
4 4 5 5 6 5 4 |
0.154435 Divergent 0.154435 0.154434 0.154442 0.154436 0.658368 0.154434 0.154435 |
7.85E-08
4.86E-08 1.98E-07 7.39E-06 1.33E-06 0.488293 2.79E-07 3.24E-09 |
3.26E-52
1.07E-53 1.10E-48 5.31E-32 3.95E-37 0.007774 4.69E-29 4.19E-63 |
5.24E-35
2.64E-373 1.82E-337 7.30E-189 8.20E-143 1.36E-11 3.70E-116 3.50E-441 |
5.24E-35
--- --- 2.29E-689 7.31E-460 4.91E-64 1.48E-464 --- |
2.69E-241
3.68E-372 2.54E-336 1.31E-689 1.47E-377 1.47E-377 2.07E-463 3.52E-439 |
|
|
h5(x) |
-1 |
|
|
|
|
|
|
|
|
|
SM HM NM FM PM CM IM EM SR |
|
4 5 4
5 5 6 4 |
0.204531 0.204454 0.204478 Divergent Divergent 0.426101 0.468371 0.451963 0.204491 |
3.89E-05 3.77E-05 1.36E-05
0.021610 0.063879 0.047468 3.00E-07 |
2.54E-32 8.44E-28 3.20E-35
1.08E-10 2.93E-07 3.57E-06 1.03E-48 |
1.27E-222 1.10E-163 1.27E-242
8.32E-57 3.71E-39 1.28E-22 5.65E-339 |
--- 4.37E-690 ---
2.20E-201 1.51E-230 2.10E-88 --- |
3.20E-222 8.48E-690 3.17E-242
5.40E-201 3.75E-230 3.82E-351 1.40E-338 |
|
|
|
0.7 |
|
|
|
|
|
|
|
|
|
SM HM NM FM PM CM IM EM SR |
|
6
5 13 18 6 8
5 |
8.124874 Divergent 0.651895 191511.5 1699089 1.384208 16.95328 Divergent 2.128140 |
7.410812
0.052597 1548910 1306983 0.673621 13.33576
0.023648 |
0.00957
5.25E-10 29617.30 3016178 0.006096 2.791647
1.45E-14 |
1.19E-15
4.04E-66 5663.092 6960553 5.83E-14 0.121366
6.40E-100 |
6.40E-106
6.48E-459 1082.98 1606314 7.09E-70 1.00E-05
2.73E-691 |
1.64E-689
5.45E-458 1.64E-689 2.49E-206 3.34E-240 8.48E-690
6.45E-690 |
|
|
h6(x) |
0.9 |
|
|
|
|
|
|
|
|
|
SM HM NM FM PM CM IM EM SR |
|
4 4 4 4 5 5 5 5 4 |
0.208528 0.208522 0.208525 0.208521 0.208486 0.208518 0.208472 0.208517 0.208526 |
1.84E-06 4.41E-06 1.03E-06 4.99E-06 4.01E-05 8.18E-06 5.39E-05 9.54E-06 6.64E-07 |
9.90E-42 4.70E-34 8.62E-44 2.31E-38 2.86E-27 3.20E-32 1.91E-26 2.10E-22 1.64E-45 |
1.28E-288 6.90E-202 2.39E-303 1.06E-264 3.70E-160 6.70E-128 3.80E-155 4.91E-89 9.33E-316 |
--- --- --- --- 3.01E-690 6.09E-415 1.50E-690 1.48E-355 --- |
1.41E-288 7.57E-202 2.63E-303 1.17E-264 5.47E-691 6.71E-415 7.18E-691 1.62E-355 1.03E-315 |
|
|
|
0.2 |
|
|
|
|
|
|
|
|
|
SM HM NM FM PM CM IM EM SR |
|
5 5 5 5 5 5 6 4 |
Divergent 0.492151 0.490938 0.495749 0.498268 0.492947 0.522265 0.496602 0.491330 |
0.000678 0.000535 0.004275 0.006794 0.001473 0.030791 0.005128 0.000138 |
6.15E-21 8.55E-25 7.88E-18 6.68E-14 1.09E-18 6.38E-10 1.72E-11 2.79E-29 |
3.40E-123 2.30E-170 5.70E-121 6.08E-80 2.69E-87 5.27E-56 2.20E-45 3.80E-202 |
4.10E-691 6.84E-691 1.91E-690 3.46E-476 3.99E-293 1.66E-332 5.90E-181 --- |
7.18E-691 7.18E-691 1.57E-690 3.81E-476 4.39E-293 1.83E-332 7.18E-691 4.23E-202 |
|
|
h7(x) |
0.1 |
|
|
|
|
|
|
|
|
|
SM HM NM FM PM CM IM EM SR |
|
5 6 5 5 5 5 6
4 |
0.178041 0.290677 0.177624 0.178224 0.183736 0.179821 0.179596 Divergent 0.177759 |
0.000282 0.112961 0.000135 0.000464 0.005976 0.002062 0.001837
9.62E-08 |
6.97E-25 4.40E-05 2.03E-27 4.47E-23 9.21E-13 5.62E-16 4.96E-12
3.98E-50 |
4.00E-169 6.24E-27 3.40E-187 3.39E-156 1.35E-71 1.73E-77 2.68E-46
9.23E-348 |
1.98E-690 5.10E-158 4.10E-691 8.21E-691 1.35E-424 3.02E-263 2.30E-183
--- |
6.29E-690 2.74E-690 6.29E-690 6.29E-690 1.21E-423 2.71E-262 2.74E-690
7.34E-346 |
|
|
|
0.5 |
|
|
|
|
|
|
|
|
|
SM HM NM FM PM CM IM EM SR |
|
4 5 4 4 5 5 25 5 4 |
0.222227 0.226725 0.222235 0.222225 0.222075 0.222141 3.678586 0.222204 0.222237 |
1.31E-05 0.004485 5.37E-06 1.57E-05 0.000165 0.000099 0.192571 0.000196 3.79E-06 |
3.36E-34 6.36E-15 3.12E-37 2.22E-33 4.40E-22 7.30E-24 0.518845 6.56E-16 5.79E-39 |
2.43E-234 5.68E-86 7.06E-256 2.60E-228 1.60E-127 2.30E-102 0.4208610 8.22E-62 1.13E-268 |
--- 2.88E-512 --- --- 2.02E-689 6.69E-338 0.209340 2.02E-245 --- |
2.18E-233 2.59E-511 6.34E-255 2.32E-227 2.73E-690 6.01E-337 1.91E-690 1.81E-244 1.02E-267 |
|
|
h8(x) |
4 |
|
|
|
|
|
|
|
|
|
SM HM NM FM PM CM IM EM SR |
|
4 5 4 4 5 5 5 5 4 |
20.47773 20.56040 20.50939 20.50714 20.83466 20.64364 20.50714 19.14509 20.51028 |
0.034860 0.047813 0.003197 0.005451 0.322077 0.131050 0.005447 1.367500 0.002304 |
7.57E-28 1.02E-26 2.02E-35 1.04E-35 2.62E-17 2.36E-18 2.58E-27 1.31E-07 2.02E-36 |
1.70E-207 9.80E-175 8.19E-261 9.55E-265 8.20E-114 4.86E-93 2.90E-173 8.80E-36 8.33E-268 |
--- 4.38e-690 --- --- 5.25E-689 4.21E-317 4.07E-688 1.80E-148 --- |
1.71E-207 3.94E-689 8.14E-261 9.49E-265 6.13E-689 4.18E-317 3.94E-689 3.00E-599 8.28E-268 |
|
|
|
30 |
|
|
|
|
|
|
|
|
|
SM HM NM FM PM CM IM EM SR |
|
4 4 4 4 4 4 4 5 4 |
5.487412 5.487412 5.487412 5.487412 5.487412 5.487412 5.487412 5.487424 5.487412 |
3.93E-13 1.46E-13 2.15E-13 2.44E-15 2.03E-10 5.17E-09 4.80E-10 1.21E-05 2.14E-13 |
1.70E-104 8.37E-96 1.20E-106 3.80E-122 1.79E-72 9.75E-59 1.20E-69 6.32E-28 1.20E-106 |
3.11E-688 2.97E-589 0 1.22E-688 8.32E-445 3.41E-214 3.01E-427 4.70E-117 1.75E-689 |
--- --- --- --- --- --- --- 1.51E-473 --- |
6.13E-689 2.95E-589 3.94E-689 6.13E-689 8.27E-445 3.40E-214 2.98E-427 1.51E-473 6.13E-689 |
|
|
h9(x) |
7 |
|
|
|
|
|
|
|
|
|
SM HM NM FM PM CM IM EM SR |
|
4 5 4 4 5 5 5 6 4 |
5.533826 6.081900 5.533054 5.531875
5.533150 |
0.000474 0.548548 0.000298 0.001477 0.007453 0.001890 0.015058 0.298131 0.000202 |
2.91E-32 3.01E-09 5.54E-34 1.99E-28 8.94E-20 3.94E-24 2.04E-18 4.87E-06 1.35E-35 |
9.51E-230 3.36E-59 4.25E-242 1.60E-202 4.40E-106 1.20E-113 3.47E-25 2.70E-121 7.91E-254 |
--- 6.59E-359 --- --- 6.25E-686 5.93E-352 6.44E-685 8.90E-102 --- |
1.75E-229 1.21E-358 7.82E-242 2.99E-202 2.63E-689 1.09E-351 1.18E-684 7.15E-408 1.46E-253 |
|
|
|
15 |
|
|
|
|
|
|
|
|
|
SM HM NM FM PM CM IM EM SR |
|
4
5 4 5 5 4 5 4 |
2.466652 Divergent 10.01989 2.466643 2.466799 2.466669 2.466679 2.485734 2.466647 |
3.99E-06
0.013463 5.14E-06 0.000151 2.10E-05 0.000321 0.019086 5.08E-07 |
8.74E-47
2.14E-22 1.24E-45 6.14E-30 7.40E-36 1.89E-34 8.19E-11 8.48E-54 |
2.11E-331
5.40E-161 5.70E-323 2.80E-182 2.80E-182 8.10E-210 2.78E-44 3.07E-381 |
---
0 --- 1.71E-688 1.72E-457 --- 3.70E-178 --- |
3.88E-331
4.12E-352 1.05E-322 2.63E-689 3.16E-457 1.50E-209 2.63E-689 5.65E-381 |
|
|
h10(x) |
-0.7 |
|
|
|
|
|
|
|
|
|
SM HM NM FM PM CM IM EM SR |
|
4 4 4 4 5 5
5 4 |
0.390538 0.390541 0.390533 0.390531 0.390669 0.390552 Divergent 0.390149 0.390534 |
3.65E-06 7.45E-06 4.96E-07 3.34E-06 0.000135 1.79E-05
0.000385 2.36E-07 |
8.48E-42 2.99E-34 2.05E-48 2.12E-41 1.55E-25 1.66E-31
3.81E-16 2.90E-51 |
3.06E-291 1.30E-204 4.17E-338 8.59E-288 3.60E-151 2.20E-126
3.62E-64 1.22E-358 |
--- --- --- --- 1.01E-689 5.46E-411
3.00E-256 --- |
2.60E-291 1.07E-204 3.54E-338 7.29E-288 2.74E-689 4.63E-411
2.54E-256 1.03E-358 |
|
|
|
0.5 |
|
|
|
|
|
|
|
|
|
SM HM NM FM PM CM IM EM SR |
|
4 5 4 5 5 5 5 6 4 |
0.809435 0.809378 0.809449 0.809244 0.808076 0.808913 0.807733 0.815672 0.809466 |
3.07E-05 8.84E-05 1.72E-05 0.000222 0.001390 0.000553 0.001733 0.006206 4.49E-07 |
2.52E-35 8.35E-28 1.23E-37 1.22E-28 1.86E-19 1.44E-22 1.10E-18 2.57E-11 2.64E-49 |
6.22E-246 5.90E-166 1.16E-262 1.80E-198 1.10E-114 1.40E-99 7.10E-110 7.49E-45 6.36E-345 |
--- 4.78E-691 --- 4.65E-690 3.55E-686 1.48E-330 5.15E-657 5.40E-179 --- |
5.28E-246 5.47E-691 9.85E-263 1.37E-690 3.01E-686 1.20E-330 4.37E-657 5.47E-691 5.39E-345 |
|
|
h11(x) |
-0.6 |
|
|
|
|
|
|
|
|
|
SM HM NM FM PM CM IM EM SR |
|
4 4 4 4 5 5 6 7 4 |
0.183954 0.183957 0.183955 0.183954 1.071820 1.046144 1.152809 0.288316 0.183955 |
1.45E-06 1.87E-06 3.24E-07 1.23E-06 0.055776 0.030100 0.136785 0.726026 1.17E-07 |
1.18E-42 7.43E-37 4.68E-48 4.04E-43 2.52E-08 1.73E-10 2.02E-05 0.001703 4.23E-51 |
2.80E-295 2.82E-219 6.06E-334 1.67E-298 1.39E-46 1.64E-56 6.01E-29 1.53E-12 3.39E-355 |
--- --- --- --- 3.99E-276 5.30E-201 4.20E-170 1.02E-48 --- |
1.49E-295 1.51E-219 3.23E-234 8.91E-299 2.13E-276 2.80E-201 3.42E-691 5.47E-691 1.81E-355 |
|
|
|
0.9 |
|
|
|
|
|
|
|
|
|
SM HM NM FM PM CM IM EM SR |
|
5 5 5 8 9 5 7 6 5 |
0.497056 0.485554 0.482772 1.380297 0.227427 0.509435 2.824133 0.527256 0.484290 |
0.013100 0.001598 0.001183 0.949824 991.2025 0.025479 10.48625 0.0433 0.000334 |
5.51E-15 2.78E-19 4.06E-23 3582.672 982.6279 4.71E-11 8.285516 5.42E-07 6.63E-27 |
1.40E-101 7.80E-114 2.30E-159 3583.486 8.940183 8.97E-59 0.139444 1.59E-26 7.91E-186 |
1.43E-690 3.67E-681 0 0.101104 0.355279 8.70E-208 1.98E-06 1.20E-104 1.37E-691 |
1.37E-691 1.95E-681 1.37E-691 1.20E-387 5.57E-450 4.60E-208 1.00E-206 1.83E-417 3.42E-691 |
|
|
h12(x) |
2.5 |
|
|
|
|
|
|
|
|
|
SM HM NM FM PM CM IM EM SR |
|
8 8 8 7 10 13
14 5 |
0.528856 0.362811 0.526889 0.171201 0.513484 0.520835 Divergent 0.521721 0.529028 |
0.000360 0.142116 0.002327 0.000417 0.015732 0.008381
0.007495 0.000187 |
2.17E-24 0.02434 6.25E-18 4.11E-23 3.70E-10 2.10E-12
2.39E-09 1.06E-27 |
2.09E-62 5.14E-05 4.24E-54 3.10E-71 2.65E-38 2.06E-31
2.25E-27 1.23E-92 |
2.01E-100 1.42E-26 4.08E-92 2.30E-119 8.13E-67 2.02E-50
2.20E-46 1.63E-222 |
2.24E-212 1.18E-404 4.57E-204 5.07E-205 2.80E-207 2.20E-200
2.30E-215 2.05E-220 |
|
|
|
1.8 |
|
|
|
|
|
|
|
|
|
SM HM NM FM PM CM IM EM SR |
|
8 7 8 7 10 13 16 13 5 |
0.170925 0.216648 0.170745 0.171201 0.173303 0.171210 0.009717 0.172196 0.170782 |
0.000141 0.048712 3.97E-05 0.000417 0.002518 0.000426 0.010149 0.001412 0.000001 |
3.06E-27 0.002847 2.46E-23 4.11E-23 7.01E-15 1.05E-19 0.010655 3.09E-12 6.71E-40 |
2.90E-65 1.95E-12 2.37E-61 3.10E-71 2.15E-43 1.03E-38 0.011259 3.07E-31 4.86E-117 |
2.83E-103 1.30E-52 2.28E-99 2.30E-119 6.61E-72 1.01E-57 0.011994 0.011994 2.55E-271 |
3.16E-215 1.45E-291 2.56E-211 1.66E-213 2.30E-212 1.10E-207 1.90E-104 3.20E-200 3.20E-269 |
|
|
h13(x) |
-0.6 |
|
|
|
|
|
|
|
|
|
SM HM NM FM PM CM IM EM SR |
|
4 5 5 5 5 5 6 6 4 |
0.686277 0.466648 0.686026 0.685246 0.681939 0.684256 0.422264 0.666446 0.686433 |
0.000157 6.93E-10 0.000407 0.001187 0.004494 0.002177 0.264083 0.019987 1.11E-07 |
9.83E-30 5.05E-63 7.61E-26 7.70E-22 6.25E-15 2.05E-17 8.61E-05 4.56E-08 4.52E-54 |
3.80E-206 7.55E-382 6.10E-178 3.80E-149 4.62E-86 1.88E-83 4.63E-25 1.28E-30 1.77E-379 |
--- 1.56E-333 5.13E-692 1.03E-691 7.52E-513 1.16E-281 1.10E-146 7.90E-121 --- |
1.26E-205 5.17E-333 3.05E-691 4.79E-691 2.49E-512 3.87E-281 2.05E-691 3.74E-481 2.86E-378 |
|
|
|
-0.2 |
|
|
|
|
|
|
|
|
|
SM HM NM FM PM CM IM EM SR |
|
4 4 4 4 5 5 5 6 4 |
1.03E-691 0.113566 0.113566 0.113566 0.286325 0.286394 0.286309 0.285258 0.113566 |
9.25E-09 7.86E-08 1.45E-08 6.88E-08 0.000109 3.96E-05 0.000125 0.001175 3.68E-11 |
2.49E-59 1.03E-44 5.69E-57 1.69E-51 1.29E-24 7.52E-28 4.35E-24 5.63E-13 2.02E-78 |
2.51E-413 5.30E-266 7.91E-396 9.29E-357 3.00E-144 7.50E-115 7.70E-141 2.98E-50 6.36E-550 |
--- --- --- --- 1.71E-692 7.46E-376 4.45E-691 2.34E-199 --- |
8.34E-413 1.76E-265 2.62E-395 8.08E-356 2.05E-691 2.47E-375 1.36E-691 2.05E-691 1.03E-548 |
|
|
h14(x) |
-1.3 |
|
|
|
|
|
|
|
|
|
SM HM NM FM PM CM IM EM SR |
|
4 5 4 5 5 5 5 6 4 |
0.150007 0.152445 0.149974 0.149887 0.149496 0.149812 0.14476 0.149065 0.149998 |
7.05E-06 0.002445 2.58E-05 0.000113 0.000504 0.000188 0.00524 0.000935 2.00E-06 |
4.63E-35 3.79E-15 4.10E-31 6.95E-26 4.34E-18 2.60E-21 8.29E-12 3.05E-12 4.01E-40 |
2.41E-239 5.22E-86 1.04E-211 2.30E-174 1.80E-102 3.42E-94 1.47E-64 3.47E-46 5.25E-276 |
--- 3.53E-511 --- 1.37E-689 8.94E-609 7.73E-313 4.67E-381 5.80E-182 --- |
1.37E-238 2.01E-510 5.90E-211 2.62E-689 5.08E-608 4.39E-313 2.65E-380 2.62E-689 2.99E-275 |
|
|
|
-1.5 |
|
|
|
|
|
|
|
|
|
SM HM NM FM PM CM IM EM SR |
|
4 4 4 4 5 5 5 5 4 |
0.05 0.049999 0.149974 0.05 0.050008 0.050001 0.050005 0.050033 0.05 |
1.08E-07 6.35E-07 0.000026 2.01E-07 8.02E-06 1.40E-06 4.67E-06 3.32E-05 3.71E-09 |
9.03E-48 1.15E-36 4.10E-31 3.83E-45 7.11E-29 4.43E-34 4.71E-30 4.85E-18 3.08E-59 |
2.61E-328 4.04E-215 1.04E-211 3.52E-309 3.50E-167 1.70E-132 5.00E-174 2.22E-69 5.36E-412 |
--- --- --- --- 1.25E-689 9.18E-428 1.69E-689 9.69E-275 --- |
1.48E-327 2.29E-214 7.49E-339 2.00E-308 4.69E-409 6.12E-689 5.22E-427 2.62E-689 5.51E-274 |
Note: x0 represents the starting approximation, n number of iterations, |ei| represents error and |h(xn+1)| represents nth functional evaluation.
In this study, we will use basins of attraction to visually compare a few test polynomials in the complex domain. The starting points in the complex plane that travel to the root are known as the basins of attraction. For computations, let us consider a square region $[-2,2] \times[-2,2] \in C^2$ with 250×250 mesh points. We examine the iterative methods in all the mesh points z0 in the square part. The stopping criterion is used, and 100 iterations are the most that can be made without reaching an attractive root. To illustrate the basins of attraction, we present three test polynomials to discuss the proposed method’s efficiency in solving the roots of a complex function. The roots of f1(z)=1-z2, f2(z)=1-z3 and f3(z)=1-z4 are mapped with a dark violet color. The regions where the colors are purple are assigned to a more significant number of iterations to converge than with blue and yellow, representing that the method cannot find the root in that region. We compare our newly developed method (SR) with well-known seventh-order methods such as SM, HM, NM, FM, PM, CM, IM, and EM. All the computations(graphs) are made by PYTHON programming using the mpmath library. Figures 1-3 demonstrate that the suggested approach (SR) exhibits the greatest results and is rapidly convergent with the fewest iterations when compared to other methods.
Figure 1. The polynomiographs obtained by the suggested method SR, SM, HM, NM, FM, PM, CM, IM, and EM for f1(z)
Figure 2. The polynomiographs obtained by the suggested method SR, SM, HM, NM, FM, PM, CM, IM, and EM for f2(z)
5.1 Example 1
The first forecasts at or near the limits require the most iterations, as shown in Figures 1-3 (yellow lines in the figures). The smallest number of iterations is required when the first guess, z0, is relatively close to the exact result.
The polynomiographs created using the polynomial approaches are displayed in Figure 1. We can see that the SR approach works well. Near the root, the methods SM, HM, IM, and EM exhibit some chaotic behaviour. In this scenario, the initial guess selection affects the performance of the procedures NM, FM, PM, and CM.
5.2 Example 2
The polynomiographs produced using the polynomial approaches are shown in Figure 2. We can see that the SR approach works well. Near the root, the methods SM, HM, NM, FM, PM, IM, and EM exhibit some chaotic behaviour.
5.3 Example 3
Figure 3 displays the polynomiographs created using the polynomial techniques. We can observe that the SR strategy is effective. The methods SM, HM, NM, FM, PM, CM, IM, and EM display some chaotic behaviour close to the root.
Figure 3. The polynomiographs obtained by the suggested methods SR, SM, HM, NM, FM, PM, CM, IM, and EM for f3(z)
In this study, we have devised an effective finite difference approximation-based seventh-order iterative approach for solving nonlinear equations. The approaches call for four functions' evaluations to be computed, with a seven-fold order of convergence. This suggested method's order of convergence has undergone development. We test the proposed scheme and some other known schemes on a few examples, demonstrating the superiority of the proposed technique SR. Additionally, the suggested strategy and a few others already in use have been used to solve real-world issues. The outcomes show promise for the new approach SR and are intriguing. The results of the numerical tests indicate that the novel approach would be a valuable substitute for solving nonlinear equations. Finally, we also contrasted the complex plane basins of attraction of several seventh and sixth order approaches.
Future study will include:
• We are currently looking at the idea of using Newton's method to create optimal methods of any order.
• Additionally, we are looking into ways to solve systems of nonlinear equations without the need of derivatives.
The authors appreciate the assistance and materials provided by GITAM (Deemed to be University).
[1] Traub, J.F. (1977). Iterative Methods for the Solution of Equations. Chelsea Publishing Company, New York.
[2] Wait, R. (1979). The Numerical Solution of Algebraic Equations. John Wiley and Sons, New York.
[3] Gautschi, W. (1997). Numerical Analysis: An Introduction. Birkhauser, Boston, M.A.
[4] Weerakoon, S., Fernando, T.G.I. (2000). A variant of Newton's method with accelerated third-order convergence. Applied Mathematical Letters, 13(8): 87-93. https://doi.org/10.1016/S0893-9659(00)00100-2
[5] Petković, M.S., Neta, B., Petković, L.D., Džunić, J. (2014). Multipoint methods for solving nonlinear equations: A survey. Applied Mathematics and Computation, 226: 635-660. https://doi.org/10.1016/j.amc.2013.10.072
[6] Kumar Mylapalli, M.S., Palli, R.K., Chaganti, P., Sri, R. (2022). An optimal fourth order iterative method for solving non-linear equations. IAENG International Journal of Applied Mathematics, 52(3): 732-741.
[7] Al-Subaihi, I.A., Al-Qarni, A.J. (2014). Higher-order iterative methods for solving nonlinear equations. Life Science Journal, 11(12): 85-91.
[8] Bawazir, H.M. (2021). Seventh and twelfth-order iterative methods for roots of nonlinear equations. Hadhramout University Journal of Natural & Applied Science, 18(1): 9-15.
[9] Napassanan, S., Montri, T. (2016). A note on three-step iterative method with seventh-order of convergence for solving nonlinear equations. The Thai Journal of Mathematics, 14(3): 565-573.
[10] Francisco, I.C., Cordero, A., Garrido, N., Torregrosa, J.R. (2019). Wide stability in a new family of optimal fourth-order iterative methods. Computational and Mathematical Methods, 2019(1): 1-14. https://doi.org/10.1002/cmm4.1023
[11] Prem Chand, B., Francisco, I.C., Garrido, N., Pankaj, J. (2019). Design and complex dynamics of Potra–Pták-type optimal methods for solving nonlinear equations and its applications. Mathematics, 7(942): 1-21. https://doi.org/10.3390/math7100942
[12] Prem Chand, B., Francisco, I.C., Pankaj, J. (2020). On the design and analysis of high-order Weerakoon-Fernando methods based on weight functions. Computational and Mathematical Methods, 2020(2): 1-17. https://doi.org/10.1002/cmm4.1114
[13] Rahma, Q., Imran, M., Syam, S. (2017). Another sixth-order iterative method free from derivative for solving multiple roots of a nonlinear equation. Applied Mathematical Sciences, 11(43): 2121-2129. https://doi.org/10.12988/ams.2017.76208
[14] Sharma, S., Sunil, P. (2022). Efficient sixth order iterative method free from higher derivatives for nonlinear equations. Journal of Mathematical and Computational Science, 12(46): 1-13. https://doi.org/10.28919/jmcs/695013
[15] Navya, K., Mani Sandeep Kumar, M., Pravin, S. (2023). Dynamical analysis of an optimal iterative scheme and its real-life applications. Engineering Letters, 14(3): 1160-1170.
[16] Navya, K., Mani Sandeep Kumar, M., Ramadevi Sri, Soujanya, M. (2023). Applications of an efficient iterative scheme for finding zeros of nonlinear equations and its basins of attraction. Communications in Mathematics and Applications, 14(1): 67-79. https://doi.org/10.26713/cma.v14i1.2113
[17] Amir, N., Rehman, M.A., Jihad, Y. (2021). Some real-life applications of a newly designed algorithm for nonlinear equations and its dynamics via computer tools. Complexity, 2021: 1-9. https://doi.org/10.1155/2021/9234932
[18] Mudassir, S., Rafiq, N., Nasreen, K. (2021). Inverse family of numerical methods for approximating all simple and roots with the multiplicity of nonlinear polynomial equations with engineering applications. Mathematical Problems in Engineering, 2021: 1-9. https://doi.org/10.1155/2021/3124615
[19] Douglas, J.M. (1972). Process Dynamics and Control. Prentice-Hall, Englewood Cliffs, NJ, USA.
