An Optimal Family of Eighth-Order Methods for Multiple-Roots and Their Complex Dynamics

Abstract

We present a new family of optimal eighth-order numerical methods for finding the multiple zeros of nonlinear functions. The methodology used for constructing the iterative scheme is based on the approach called the ‘weight factor approach’. This approach ingeniously combines weight functions to enhance convergence properties and stability. An extensive convergence analysis is conducted to prove that the proposed scheme achieves optimal eighth-order convergence, providing a significant improvement in efficiency over lower-order methods. Furthermore, the applicability of these novel methods to some real-world problems is demonstrated, showcasing their superior performance in terms of speed and accuracy. This is illustrated through a series of three examples involving basins of attraction with reflection symmetry, confirming the dominance of the new methods over existing counterparts. The examples highlight not only the robustness and precision of the proposed methods but also their practical utility in solving the complex nonlinear equations encountered in various scientific and engineering domains. Consequently, these eighth-order methods hold great promise for advancing computational techniques in fields that require the resolution of multiple roots with high precision.

1. Introduction

The problem of solving nonlinear equations is very old and recognized in history as many practical problems that arise are nonlinear in nature. Nonlinear equations can be solved using a variety of one-point and multi-point techniques [1,2,3]. The aforementioned techniques are intended for the simple roots of nonlinear equations; however, when handling the multiple roots of nonlinear equations, these techniques behave differently. When working with the multiple roots of nonlinear equations, the well-known Newton’s method with quadratic convergence for simple roots of nonlinear equations decays to first order. These issues give rise to both minor issues like increased computing cost and major issues like a complete lack of convergence. The strange characteristics of iterative techniques while handling multiple roots have been widely recognized since the 19th century, when Schröder [4] created a modification of the classical Newton’s method to maintain its second order of convergence when dealing with multiple roots. Many subjects, including complex variables, fractional diffusion, image processing, and applications in statistics and economics, can give rise to nonlinear equations having many roots. In the recent past, a variety of one-point and multi-point root solvers have been developed due to the practical nature of multiple root finders.

The fractional versions of different phenomena in nature and biology involve the use of fractional calculus, which extends traditional calculus to non-integer orders, allowing the more accurate modeling of complex systems. For example, fractional differential equations can describe anomalous diffusion processes, such as the movement of molecules in heterogeneous environments, more precisely than classical models [5,6]. In biology, they help in modeling memory and hereditary properties in tissues and organs. Iterative methods for multiple roots, on the other hand, are numerical techniques used to find solutions to equations with multiple roots. These methods, such as the Newton–Raphson method and its variations, are crucial for refining the approximations of roots in equations where standard techniques may fail or converge slowly. By iteratively applying corrections based on the function’s derivatives, these methods improve the accuracy and efficiency of finding multiple roots in complex biological and natural systems.

Finding efficient higher order multi-point iteration functions for the multiple zeros of the univariate function $f\left(r\right)$, where $f:\mathbb{C}\to \mathbb{C}$ is analytic in neighborhood of the required zero, is one of the most important and challenging tasks in the field of numerical analysis. The advantages of multi-point iterative techniques over one-point iterative techniques are very well illustrated in [1].

By taking advantage of the symmetry of the function, it is possible to construct methods that are efficient and accurate in finding roots [7,8]. Numerous higher-order methods have been developed and analyzed in the literature, see [9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25]. The methods have been developed either independently or based on the well-known Newton’s method [4]

$$r_{k+1}=r_k-m{\displaystyle \frac{f\left(r_k\right)}{f^{\prime }\left(r_k\right)}}.$$

(1)

Here, m is the multiplicity of a zero (say, $r^{*}$) of the function $f\left(r\right)$, that is, $f^{\left(j\right)}\left(r^{*}\right)=0,j=0,1,\dots ,m-1$ and $f^{\left(m\right)}\left(r^{*}\right)\ne 0$. These techniques require either the evaluation of first-order derivatives or both first- and second-order derivatives.

Our goal in this work is to develop multiple root solvers of a higher convergence order that employ less computations than other methods. Following this approach, we present an eighth-order multi-point Newton-type scheme for multiple zeros which produces several methods as its special cases. Each method of the family also satisfies the classical Kung–Traub hypothesis [26], which states that ‘the multi-point techniques without memory based on n function evaluations are said to have an optimal order $2^{n-1}$. Usually, the methods that possess this property are referred to as optimal methods.

The methodology used here for constructing the iterative scheme is based on the approach called the weight factor approach. This approach involves weighting the Newton steps with specific factors, thereby generating Newton-type methods with higher convergence orders and improved efficiency. The presented iterative algorithms consist of three steps: the first step employs a traditional Newton’s iteration (1), which serves as the foundation, and the succeeding steps are the enhanced Newton-type iterations that incorporate the weight factors to accelerate convergence. This structured progression ensures that each iteration builds upon the accuracy of the previous one, effectively reducing the number of iterations needed to reach a solution and enhancing the overall computational efficiency. Numerical experiments on various problems in applied science illustrate the utility of these methodologies. These problems, selected from diverse fields, test the robustness and versatility of the proposed methods. The experiments include solving nonlinear equations with multiple roots, where some existing methods struggle with slow convergence or fail to find all roots. By applying the new iterative algorithms, we observed a marked improvement in the speed and accuracy of the solutions.

Investigating the local and global features of the convergence of the family of suggested methods is one of the motivations of this study. An iterative approach to solving nonlinear equations typically ensures local convergence if the starting guess is made in close enough proximity to the desired zero. However, under normal conditions, effective formation on its global convergence is rarely attained. Attractor basins provide us with valuable information about global convergence, and it is worthwhile to investigate the underlying dynamics. The better performance of the methods in comparison to the existing ones is also observed using the basins of attraction which is a useful technique to check the convergence regions visually.

Here is a summary of the remaining portion of the paper. The development and study of the family of optimal eighth-order iterative solvers is performed in Section 2. In Section 3, a few numerical experiments are conducted to validate the theoretical outcomes and assess the stability of the new scheme. This section also includes a comparison with the currently in use procedures of the same order. Section 4 shows the basins of attractors in order to visualize the convergence domain of the methods applied on certain polynomials. The concluding remarks are included in Section 5.

2. The Method

Let us consider the following three-step iterative scheme to compute a multiple root with multiplicity $m\ge 1$:

$$\begin{array}{cc}\hfill s_k=& r_k-m{\displaystyle \frac{f\left(r_k\right)}{f^{\prime }\left(r_k\right)}},\hfill \\ \hfill t_k=& s_k-mh(A_1+A_{2}h){\displaystyle \frac{f\left(r_k\right)}{f^{\prime }\left(r_k\right)}},\hfill \\ \hfill r_{k+1}=& t_k-muvH(h,v){\displaystyle \frac{f\left(r_k\right)}{f^{\prime }\left(r_k\right)}},\hfill \end{array}$$

(2)

where $h={\displaystyle \frac{u}{1+u}}$, $u=\sqrt[m]{{\displaystyle \frac{f\left(s_k\right)}{f\left(r_k\right)}}}$, $v=\sqrt[m]{{\displaystyle \frac{f\left(t_k\right)}{f\left(s_k\right)}}}$ and $H:{\mathbb{C}}^2\to \mathbb{C}$ is analytic in a neighborhood of $(0,0)$. Notice that this is a three-step process, with the Newton iteration (1) serving as the first step. The next two stages are weighted by various factors, which are referred to as weight factors or weight functions. We examine the convergence outcomes of the suggested iterative technique (2) in the following theorem.

Theorem 1.

Consider an analytic function $f:\mathbb{C}\to \mathbb{C}$ in a region containing a multiple zero (let us say, $r^{*}$) with multiplicity m. The local order of convergence of scheme (2) is at least eight if the initial guess $r_0$ is sufficiently near to $r^{*}$ and $A_1=1$, $A_2=3$, $H_{00}=1$, $H_{10}=2$, $H_{01}=1$, $H_{20}=-4$, $H_{11}=4$, $H_{30}=-72$ and $\left\{\right|H_{02}|,|H_{21}\left|\right\}<\infty$, where $H_{ij}={\displaystyle \frac{{\partial }^{i+j}}{\partial h^i\partial t^j}}{H\left(h,v\right)|}_{(0,0)}$, $i,j\in \{0,1,2,3\}.$

Proof. 

Let ${ϵ}_k=r_k-r^{*}$ be the error at k-th iteration. By applying Taylor’s expansion to $f\left(r_k\right)$ about $r^{*}$ and considering $f^{\left(j\right)}\left(r^{*}\right)=0,j=0,1,\dots ,m-1$ and $f^{\left(m\right)}\left(r^{*}\right)\ne 0$, we obtain that

$$f\left(r_k\right)={\displaystyle \frac{f^m\left(r^{*}\right)}{m!}}{ϵ}_k^m\left(1+B_1{ϵ}_k+B_2{ϵ}_k^2+B_3{ϵ}_k^3+B_4{ϵ}_k^4+B_5{ϵ}_k^5+B_6{ϵ}_k^6+B_7{ϵ}_k^7+\cdots \right),$$

(3)

where $B_n={\displaystyle \frac{m!}{(m+n)!}}{\displaystyle \frac{f^{(m+n)}\left(r^{*}\right)}{f^{\left(m\right)}\left(r^{*}\right)}}$ for $n\in \mathbb{N}$.

Similarly, the expansion of $f^{\prime }\left(r_k\right)$ about $\alpha$ is:

$$f^{\prime }\left(r_k\right)={\displaystyle \frac{f^m\left(r^{*}\right)}{m!}}{ϵ}_k^{m-1}\left(m+(m+1)B_1{ϵ}_k+(m+2)B_2{ϵ}_k^2+(m+3)B_3{ϵ}_k^3+(m+4)B_4{ϵ}_k^4+\cdots \right).$$

(4)

Then, the first step of (2) yields

$$\begin{array}{cc}\hfill {ϵ}_{s_k}=& s_k-r^{*}\hfill \\ \hfill =& {\displaystyle \frac{B_1}{m}}{ϵ}_k^2+{\displaystyle \frac{-(1+m)B_1^2+2m{B}_2}{m^2}}{ϵ}_k^3+{\displaystyle \frac{1}{m^3}}({(1+m)}^2{B}_1^3-m(4+3m)B_1{B}_2\hfill \\ \hfill & +3m^2{B}_3){ϵ}_k^4-{\displaystyle \frac{1}{m^4}}({(1+m)}^3{B}_1^4-2m(3+5m+2m^2)B_1^2{B}_2\hfill \\ \hfill & +2m^2((2+m)B_2^2-2m{B}_4))+2m^2(3+2m)B_1{B}_3){ϵ}_k^5+{\displaystyle \frac{1}{m^5}}({(1+m)}^4{B}_1^5\hfill \\ \hfill & -m{(1+m)}^2(8+5m)B_1^3{B}_2+m^2(9+14m+5m^2)B_1^2{B}_3+m^2{B}_1((12+16m\hfill \\ \hfill & +5m^2)B_2^2-m(8+5m)B_4)+m^3(-(12+5m)B_2{B}_3+5m{B}_5)){ϵ}_k^6\hfill \\ \hfill & -{\displaystyle \frac{1}{m^6}}({(1+m)}^5{B}_1^6-2m{(1+m)}^3(5+3m)B_1^4{B}_2+6m^2{(1+m)}^2(2+m)B_1^3{B}_3\hfill \\ \hfill & +3m^2(2+3m+m^2)B_1^2((4+3m)B_2^2-2m{B}_4)-2m^3{B}_1(2(9+11m+3m^2)\hfill \\ \hfill & \times B_2{B}_3-m(5+3m)B_5)+m^3{(-2(2+m)}^2{B}_2^3+2m(8+3m)B_2{B}_4+m(3(3\hfill \\ \hfill & +m)B_3^2+m^2{B}_6))){ϵ}_k^7-{\displaystyle \frac{1}{m^7}}(-{(1+m)}^6{B}_1^7+m{(1+m)}^4(12+7m)B_1^5{B}_2\hfill \\ \hfill & -m^2{(1+m)}^3(15+7m)B_1^4{B}_3-m^2{(1+m)}^2{B}_1^3(2(20+24m+7m^2)B_2^2\hfill \\ \hfill & -m(16+7m)B_4)+m^3(1+m)B_1^2(3(24+27m+7m^2)B_2{B}_3\hfill \\ \hfill & -m(15+7m)C_5)-m^3{B}_1(-{(2+m)}^2(8+7m)B_2^3+2m(24+28m\hfill \\ \hfill & +7m^2)B_2{B}_4+m((27+30m+7m^2)B_3^2+m^2(1+m)B_6))\hfill \\ \hfill & +m^4(-(36+32m+7m^2)B_2^2{B}_3+m(20+7m)B_2{B}_5+m((24+7m)B_3{B}_4\hfill \\ \hfill & +m^2{B}_7))){ϵ}_k^8+O({ϵ}_k^9).\hfill \end{array}$$

(5)

Now, with the expansion $f(s_k)$ about $r^{*}$, we obtain

$$f\left(s_k\right)={\displaystyle \frac{f^m\left(r^{*}\right)}{m!}}{ϵ}_{s_k}^m\left(1+B_1{ϵ}_{s_k}+B_2{ϵ}_{s_k}^2+B_3{ϵ}_{s_k}^3+B_4{ϵ}_{s_k}^4+\cdots \right).$$

(6)

With the use of Equations (3) and (6), it follows that

$$\begin{array}{cc}\hfill u=& {\displaystyle \frac{B_1}{m}}{ϵ}_k+{\displaystyle \frac{-(2+m)B_1^2+2m{B}_2}{m^2}}{ϵ}_k^2+{\displaystyle \frac{1}{2m^3}}((7+7m+2m^2)B_1^3\hfill \\ \hfill & -2m(7+3m)B_1{B}_2+6m^2{B}_3){ϵ}_k^3-{\displaystyle \frac{1}{6m^4}}((34+51m+29m^2+6m^3)B_1^4\hfill \\ \hfill & -6m(17+16m+4m^2)B_1^2{B}_2+12m^2(5+2m)B_1{B}_3+12m^2((3+m)B_2^2\hfill \\ \hfill & -2m{B}_4)){ϵ}_k^4+{\displaystyle \frac{1}{24m^5}}((209+418m+355m^2+146m^3+24m^4)B_1^5\hfill \\ \hfill & -4m(209+306m+163m^2+30m^3)B_1^3{B}_2\hfill \\ \hfill & +12m^2(49+43m+10m^2)B_1^2{B}_3+12m^2{B}_1((53+47m+10m^2)B_2^2\hfill \\ \hfill & -2m(13+5m)B_4)-24m^3((17+5m)B_2{B}_3-5m{B}_5)){ϵ}_k^5+O({ϵ}_k^6),\hfill \end{array}$$

(7)

then

$$\begin{array}{cc}\hfill h=& {\displaystyle \frac{B_1}{m}}{ϵ}_k+{\displaystyle \frac{-(3+m)B_1^2+2m{B}_2}{m^2}}{ϵ}_k^2+{\displaystyle \frac{1}{2m^3}}((17+11m+2m^2)B_1^3\hfill \\ \hfill & -2m(11+3m)B_1{B}_2+6m^2{B}_3){ϵ}_k^3-{\displaystyle \frac{1}{6m^4}}((142+135m+47m^2\hfill \\ \hfill & +6m^3)B_1^4-6m(45+26m+4m^2)B_1^2{B}_2+24m^2(4+m)B_1{B}_3\hfill \\ \hfill & +12m^2((5+m)B_2^2-2m{B}_4)){ϵ}_k^4+{\displaystyle \frac{1}{24m^5}}((1573+1966m+995m^2\hfill \\ \hfill & +242m^3+24m^4)B_1^5-4m(983+864m+271m^2+30m^3)B_1^3{B}_2\hfill \\ \hfill & +12m^2(131+71m+10m^2)B_1^2{B}_3+12m^2{B}_1((157+79m+10m^2)B_2^2\hfill \\ \hfill & -2m(21+5m)B_4)-24m^3((29+5m)B_2{B}_3-5m{B}_5)){ϵ}_k^5+O({ϵ}_k^6).\hfill \end{array}$$

(8)

Inserting Equations (3)–(8) in the second step of (2), then some simple calculations yield

$$\begin{array}{cc}\hfill {ϵ}_{t_k}=& t_k-r^{*}\hfill \\ \hfill =& {\displaystyle \frac{(1-A_1)B_1}{m}}{ϵ}_k^2-{\displaystyle \frac{1}{m^2}}\left((1+A_2+m-A_1(4+m))B_1^2+2(-1+A_1)m{B}_2\right){ϵ}_k^3\hfill \\ \hfill & +{\displaystyle \frac{1}{2m^3}}((-A_1(25+15m+2m^2)+2({(1+m)}^2+A_2(7+2m)))B_1^3+2m(-4\hfill \\ \hfill & -4A_2-3m+3A_1(5+m))B_1{B}_2-6(-1+A_1)m^2{B}_3){ϵ}_k^4\hfill \\ \hfill & +\sum _{n=1}^4{\varphi }_n{ϵ}_k^{n+4}+O({ϵ}_k^9),\hfill \end{array}$$

(9)

where ${\varphi }_n={\varphi }_n(A_1,A_2,m,B_1,B_2,B_3,\dots ,B_7)$, $n=1,2,3,4$. As ${\varphi }_n$ is so long, its expression is not generated explicitly here.

Equation (9) makes it evident that, by concurrently setting the coefficients of ${ϵ}_k^2$ and ${ϵ}_k^3$ to zero, we will achieve at least fourth-order convergence. After resolving the ensuing equations,

$$A_1=1\mathrm{and}{A}_2=3$$

(10)

are obtained.

Consequently, the above error equation becomes

$$\begin{array}{cc}\hfill {ϵ}_{t_k}=& {\displaystyle \frac{(19+m)B_1^3-2m{B}_1{B}_2}{2m^3}}{ϵ}_k^4+\sum _{n=1}^4{\varphi }_n{ϵ}_k^{n+4}+O\left({ϵ}_k^9\right).\hfill \end{array}$$

(11)

Expanding $f\left(t_k\right)$ about $r^{*}$

$$f\left(t_k\right)={\displaystyle \frac{f^m\left(r^{*}\right)}{m!}}{ϵ}_{t_k}^m\left(1+B_1{ϵ}_{t_k}+B_2{ϵ}_{t_k}^2+B_3{ϵ}_{t_k}^3+B_4{ϵ}_{t_k}^4+\cdots \right).$$

(12)

Using Equations (6) and (12) in the expression of v, it follows that

$$\begin{array}{cc}\hfill v=& {\displaystyle \frac{(19+m)B_1^2-2m{B}_2}{2m^2}}{ϵ}_k^2-{\displaystyle \frac{1}{3m^3}}((163+57m+2m^2)B_1^3-6m(19+m)B_1{B}_2\hfill \\ \hfill & +6m^2{B}_3){ϵ}_k^3+{\displaystyle \frac{1}{24m^4}}((5279+3558m+673m^2+18m^3)B_1^4-12m(593\hfill \\ \hfill & +187m+6m^2)B_1^2{B}_2+24m^2(56+3m)B_1{B}_3+36m^2((25+m)B_2^2+2m{B}_4)){ϵ}_k^4\hfill \\ \hfill & -{\displaystyle \frac{1}{60m^5}}((47457+46810m+16635m^2+2210m^3+48m^4)B_1^5-20m(4681\hfill \\ \hfill & +2898m+497m^2+12m^3)B_1^3{B}_2+60m^2(429+129m+4m^2)B_1^2{B}_3\hfill \\ \hfill & +60m^2{B}_1((537+147m+4m^2)B_2^2-2m(37+2m)B_4)\hfill \\ \hfill & -120m^3((55+2m)B_2{B}_3-2m{B}_5)){ϵ}_k^5+O({ϵ}_k^6).\hfill \end{array}$$

(13)

Developing the weight function $H(h,v)$ in the vicinity of the origin $(0,0)$ using Taylor series up to third-order terms only,

$$\begin{array}{cc}\hfill H(h,v)=& H_{00}+H_{10}h+H_{01}v+{\displaystyle \frac{1}{2!}}(H_{20}{h}^2+2H_{11}hv+H_{02}{v}^2)+{\displaystyle \frac{1}{3!}}(H_{30}{h}^3\hfill \\ \hfill & +3H_{21}{h}^{2}v+3H_{12}h{v}^2+H_{03}{v}^3).\hfill \end{array}$$

(14)

By using (3), (4), (6), (7), (13), and (14) in third step of (2), we obtain that

$${ϵ}_{k+1}={\displaystyle \frac{(1-H_{00})B_1((19+m)B_1^2-2m{B}_2)}{2m^3}}{ϵ}_k^4+\sum _{n=1}^4{\psi }_n{ϵ}_k^{n+4}+O\left({ϵ}_k^9\right),$$

(15)

${\psi }_n={\psi }_n(m,B_1,B_2,\dots ,B_6)$, $n=1,2,3,4.$

Equation (15) shows that the eighth-order convergence is achieved if the coefficients of ${ϵ}_k^4$, ${ϵ}_k^5$, ${ϵ}_k^6$, and ${ϵ}_k^7$ are annihilated. Then, the resulting equations yield

$$H_{00}=1,H_{10}=2,H_{01}=1,H_{20}=-4,H_{11}=4,H_{30}=-72.$$

(16)

Then, the error Equation (15) reduces to

$$\begin{array}{cc}\hfill {ϵ}_{k+1}=& -{\displaystyle \frac{1}{48m^7}}(B_1((19+m)B_1^2-2m{B}_2)((3H_{02}{(19+m)}^2+2(-1121\hfill \\ \hfill & -156m-7m^2+3H_{21}(19+m)))B_1^4-12m(-52+H_{21}-4m\hfill \\ \hfill & +H_{02}(19+m))B_1^2{B}_2+12(-2+H_{02})m^2{B}_2^2-24m^2{B}_1{B}_3)){ϵ}_k^8\hfill \\ \hfill & +O({ϵ}_k^9).\hfill \end{array}$$

(17)

Thus, the result is established. □

Remark 1.

With just four functional evaluations (i.e., $f\left(r_k\right)$, $f\left(s_k\right)$, $f\left(t_k\right)$, and $f^{\prime }\left(r_k\right)$), the suggested scheme (2) achieves eighth-order convergence. Hence, according to the Kung–Traub theory [26] this scheme is optimal.

Remark 2.

According to [27], the computational efficiency $\left(E\right)$ is expressed as $E=p^{1/\theta }$, where p denotes the method’s order of convergence and θ denotes the number of function evaluations needed for each iteration. The one-step optimal technique ($E=2^{1/2}\approx 1.414$) and the two-step optimal fourth-order approach ($E=4^{1/3}\approx 1.587$) have substantially lower E-values than the new three-step optimal scheme, which has an E-value of $8^{1/4}\approx 1.682$.

Remark 3.

In order to apply the new method, the multiplicity m of the desired root should be known. If m is not known, then we can use the following formula to estimate it:

$$m\approx {\displaystyle \frac{r_{k+1}-r_k}{F\left(r_{k+1}\right)-F\left(r_k\right)}},$$

where $F\left(r_k\right)={\displaystyle \frac{f\left(r_k\right)}{f^{\prime }\left(r_k\right)}}$. This information can be found in [28].

Some Particular Methods

As specific cases of the family (2), we may produce a large number of iterative schemes based on the forms of the function $H(h,v)$ that meet the requirements of Theorem 1. We shall limit ourselves to selecting low-degree polynomials or straightforward rational functions, however. Consequently, consideration is given to the following basic forms:

$$\begin{array}{cc}\hfill \left(1\right)& H(h,v)=1+2h+v-2h^2+4hv-12h^3.\hfill \\ \hfill \left(2\right)& H(h,v)={\displaystyle \frac{1+2vs.-12h^3+h(2+6vs.-2h)}{1+v}}.\hfill \\ \hfill \left(3\right)& H(h,v)={\displaystyle \frac{1+3h+v+5hv-14h^3-12h^4}{1+h}}.\hfill \\ \hfill \left(4\right)& H(h,v)={\displaystyle \frac{1+3h+2v+8hv-14h^3}{(1+h)(1+v)}}.\hfill \\ \hfill \left(5\right)& H(h,v)={\displaystyle \frac{1+v-2h(2+v)-2h^2(6+11v)+h^3(4+8v)}{1-6h+2h^2}}.\hfill \end{array}$$

Each of the aforementioned forms has a corresponding procedure that is stated as follows:

•

Method 1 (M-1):

$$\begin{array}{cc}\hfill r_{k+1}=& t_k-muv\left[1+2h+v-2h^2+4hv-12h^3\right]{\displaystyle \frac{f\left(r_k\right)}{f^{\prime }\left(r_k\right)}}.\hfill \end{array}$$

•

Method 2 (M-2):

$$\begin{array}{cc}\hfill r_{k+1}=& t_k-muv\left[{\displaystyle \frac{1+2vs.-12h^3+h(2+6vs.-2h)}{1+v}}\right]{\displaystyle \frac{f\left(r_k\right)}{f^{\prime }\left(r_k\right)}}.\hfill \end{array}$$

•

Method 3 (M-3):

$$\begin{array}{cc}\hfill r_{k+1}=& t_k-muv\left[{\displaystyle \frac{1+3h+v+5hv-14h^3-12h^4}{1+h}}\right]{\displaystyle \frac{f\left(r_k\right)}{f^{\prime }\left(r_k\right)}}.\hfill \end{array}$$

•

Method 4 (M-4):

$$\begin{array}{cc}\hfill r_{k+1}=& t_k-muv\left[{\displaystyle \frac{1+3h+2v+8hv-14h^3}{(1+h)(1+v)}}\right]{\displaystyle \frac{f\left(r_k\right)}{f^{\prime }\left(r_k\right)}}.\hfill \end{array}$$

•

Method 5 (M-5):

$$\begin{array}{cc}\hfill r_{k+1}=& t_k-muv\left[{\displaystyle \frac{1+v-2h(2+v)-2h^2(6+11v)+h^3(4+8v)}{1-6h+2h^2}}\right]{\displaystyle \frac{f\left(r_k\right)}{f^{\prime }\left(r_k\right)}}.\hfill \end{array}$$

In above each, case $s_k=r_k-m{\displaystyle \frac{f\left(r_k\right)}{f^{\prime }\left(r_k\right)}}$ and $t_k=s_k-mh(1+3h){\displaystyle \frac{f\left(r_k\right)}{f^{\prime }\left(r_k\right)}}.$

3. Numerical Results

We use the above developed techniques to solve a nonlinear equation in this section. This helps verify the theoretical outcomes that have been deduced thus far and serves to practically demonstrate the techniques. We use the formula to generate the computational convergence order (CCO), which we use to verify the theoretical order of convergence (see [29]). The formula for CCO is

$$\mathrm{CCO}={\displaystyle \frac{ln|(r_{k+2}-r^{*})/(r_{k+1}-r^{*})|}{ln|(r_{k+1}-r^{*})/(r_k-r^{*})|}},\mathrm{for}\mathrm{each}k=1,2,\dots$$

(18)

Performance is compared with existing well-known eighth-order methods. For example, we choose the methods by Zafar et al. [18], Behl et al. [10], and Akram et al. [30]. Below, the methods are expressed for ready reference:

•

Methods by Zafar et al. (ZM-1 and ZM-2):

$$\begin{array}{cc}\hfill s_k=& r_k-m{\displaystyle \frac{f\left(r_k\right)}{f^{\prime }\left(r_k\right)}},\hfill \\ \hfill t_k=& s_k-m{u}_k\left({\displaystyle \frac{1-5u_k^2+8u_k^3}{1-2u_k}}\right){\displaystyle \frac{f\left(r_k\right)}{f^{\prime }\left(r_k\right)}},\hfill \\ \hfill r_{k+1}=& t_k-m{u}_k{v}_k(1+2u_k)(1+v_k)(1+2w_k){\displaystyle \frac{f\left(r_k\right)}{f^{\prime }\left(r_k\right)}}\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill s_k=& r_k-m{\displaystyle \frac{f\left(r_k\right)}{f^{\prime }\left(r_k\right)}},\hfill \\ \hfill t_k=& s_k-m{u}_k\left(1+2u_k-u_k^2+6u_k^3\right){\displaystyle \frac{f\left(r_k\right)}{f^{\prime }\left(r_k\right)}},\hfill \\ \hfill r_{k+1}=& t_k-m{u}_k{v}_k{e}^{v_k}{e}^{2w_k}(1+2u_k){\displaystyle \frac{f\left(r_k\right)}{f^{\prime }\left(r_k\right)}},\hfill \end{array}$$

where $u_k={\left({\displaystyle \frac{f\left(s_k\right)}{f\left(r_k\right)}}\right)}^{{\displaystyle \frac{1}{m}}}$, $v_k={\left({\displaystyle \frac{f\left(t_k\right)}{f\left(s_k\right)}}\right)}^{{\displaystyle \frac{1}{m}}}$ and $w_k={\left({\displaystyle \frac{f\left(t_k\right)}{f\left(r_k\right)}}\right)}^{{\displaystyle \frac{1}{m}}}.$

•

Methods by Behl et al. (BM-1 and BM-2):

$$\begin{array}{cc}\hfill s_k=& r_k-m{\displaystyle \frac{f\left(r_k\right)}{f^{\prime }\left(r_k\right)}},\hfill \\ \hfill t_k=& s_k-m{u}_k(1+2u_k-u_k^2){\displaystyle \frac{f\left(r_k\right)}{f^{\prime }\left(r_k\right)}},\hfill \\ \hfill r_{k+1}=& t_k+{\displaystyle \frac{m{u}_k{w}_k}{1-w_k}}\left[-1-2u_k+6u_k^3-{\displaystyle \frac{1}{6}}(85+21m+2m^2)u_k^4-2v_k\right]{\displaystyle \frac{f\left(r_k\right)}{f^{\prime }\left(r_k\right)}}\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill s_k=& r_k-m{\displaystyle \frac{f\left(r_k\right)}{f^{\prime }\left(r_k\right)}},\hfill \\ \hfill t_k=& s_k-m{u}_k(1+2u_k){\displaystyle \frac{f\left(r_k\right)}{f^{\prime }\left(r_k\right)}},\hfill \\ \hfill r_{k+1}=& t_k-{\displaystyle \frac{m{u}_k{w}_k}{1-w_k}}\left[{\displaystyle \frac{1+9u_k^2+2v_k+u_k(6+8v_k)}{1+4u_k}}\right]{\displaystyle \frac{f\left(r_k\right)}{f^{\prime }\left(r_k\right)}},\hfill \end{array}$$

where $u_k={\left({\displaystyle \frac{f\left(s_k\right)}{f\left(r_k\right)}}\right)}^{{\displaystyle \frac{1}{m}}}$, $v_k={\left({\displaystyle \frac{f\left(t_k\right)}{f\left(r_k\right)}}\right)}^{{\displaystyle \frac{1}{m}}}$, $w_k={\left({\displaystyle \frac{f\left(t_k\right)}{f\left(s_k\right)}}\right)}^{{\displaystyle \frac{1}{m}}}$.

•

Methods by Akram et al. (AM-1 and AM-2):

$$\begin{array}{cc}\hfill s_k=& r_k-m{\displaystyle \frac{f\left(r_k\right)}{f^{\prime }\left(r_k\right)}},\hfill \\ \hfill t_k=& s_k-mu(1+2u-u^2+6u^3){\displaystyle \frac{f\left(r_k\right)}{f^{\prime }\left(r_k\right)}},\hfill \\ \hfill r_{k+1}=& t_k-mu(v+v^2+2w+4vs.w){\displaystyle \frac{f\left(r_k\right)}{f^{\prime }\left(r_k\right)}}\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill s_k=& r_k-m{\displaystyle \frac{f\left(r_k\right)}{f^{\prime }\left(r_k\right)}},\hfill \\ \hfill t_k=& s_k-{\displaystyle \frac{mu(5+18u)}{5+8u-11u^2}}{\displaystyle \frac{f\left(r_k\right)}{f^{\prime }\left(r_k\right)}},\hfill \\ \hfill r_{k+1}=& t_k-mu(v+v^2+2w+4vs.w){\displaystyle \frac{f\left(r_k\right)}{f^{\prime }\left(r_k\right)}},\hfill \end{array}$$

where $u={\left({\displaystyle \frac{f\left(s_k\right)}{f\left(r_k\right)}}\right)}^{{\displaystyle \frac{1}{m}}}$, $v={\left({\displaystyle \frac{f\left(t_k\right)}{f\left(s_k\right)}}\right)}^{{\displaystyle \frac{1}{m}}}$ and $w={\left({\displaystyle \frac{f\left(t_k\right)}{f\left(r_k\right)}}\right)}^{{\displaystyle \frac{1}{m}}}$.

Computations are performed in the programming package of Mathematica software using multiple-precision arithmetic. Numerical results displayed in

Table 1. Numerical results for Example 1.

Methods	k	$|{\mathit{r}}_2-{\mathit{r}}_1|$	$|{\mathit{r}}_3-{\mathit{r}}_2|$	$|{\mathit{r}}_4-{\mathit{r}}_3|$	CCO	CPU-Time
ZM−1	3	$5.41\times {10}^{-7}$	$1.06\times {10}^{-58}$	0	8.000	0.4224
ZM−2	3	$3.74\times {10}^{-7}$	$3.73\times {10}^{-60}$	0	8.000	0.4683
BM−1	3	$2.05\times {10}^{-7}$	$1.93\times {10}^{-62}$	0	8.000	0.3925
BM−2	3	$2.05\times {10}^{-7}$	$1.82\times {10}^{-62}$	0	8.000	0.3914
AM−1	3	$5.48\times {10}^{-7}$	$1.28\times {10}^{-85}$	0	8.000	0.4217
AM−2	3	$5.47\times {10}^{-7}$	$1.27\times {10}^{-58}$	0	8.000	0.4052
M−1	3	$5.57\times {10}^{-7}$	$2.01\times {10}^{-58}$	0	8.000	0.3433
M−2	3	$8.91\times {10}^{-7}$	$1.40\times {10}^{-56}$	0	8.000	0.3902
M−3	3	$5.59\times {10}^{-7}$	$2.28\times {10}^{-58}$	0	8.000	0.3754
M−4	3	$8.94\times {10}^{-7}$	$1.56\times {10}^{-56}$	0	8.000	0.3585
M−5	3	$5.57\times {10}^{-7}$	$2.13\times {10}^{-58}$	0	8.000	0.3878

,

Table 2. Numerical results for Example 2.

Methods	k	$|{\mathit{r}}_2-{\mathit{r}}_1|$	$|{\mathit{r}}_3-{\mathit{r}}_2|$	$|{\mathit{r}}_4-{\mathit{r}}_3|$	CCO	CPU-Time
ZM−1	4	$7.86\times {10}^{-1}$	$3.66\times {10}^{-5}$	$2.31\times {10}^{-39}$	8.000	0.2802
ZM−2	5	$2.49$	$2.79\times {10}^{-2}$	$9.71\times {10}^{-17}$	8.000	0.4064
BM−1	6	$3.63\times {10}^{-2}$	$1.08\times {10}^{-5}$	$2.82\times {10}^{-16}$	3.000	0.3752
BM−2	6	$3.64\times {10}^{-2}$	$1.09\times {10}^{-5}$	$2.90\times {10}^{-16}$	3.000	0.3271
AM−1	4	$6.28\times {10}^{-1}$	$8.43\times {10}^{-6}$	$2.18\times {10}^{-44}$	8.000	0.3285
AM−2	4	$6.28\times {10}^{-1}$	$6.83\times {10}^{-6}$	$3.04\times {10}^{-45}$	8.000	0.3076
M−1	4	$5.99\times {10}^{-1}$	$1.60\times {10}^{-5}$	$1.14\times {10}^{-41}$	8.000	0.3124
M−2	4	$3.42\times {10}^{-1}$	$4.73\times {10}^{-7}$	$1.15\times {10}^{-53}$	8.000	0.2815
M−3	4	$5.92\times {10}^{-1}$	$1.91\times {10}^{-5}$	$6.62\times {10}^{-41}$	8.000	0.2976
M−4	4	$3.39\times {10}^{-1}$	$5.43\times {10}^{-7}$	$4.50\times {10}^{-53}$	8.000	0.2811
M−5	4	$5.99\times {10}^{-1}$	$3.38\times {10}^{-5}$	$8.13\times {10}^{-39}$	8.000	0.3120

,

Table 3. Numerical results for Example 3.

Methods	k	$|{\mathit{r}}_2-{\mathit{r}}_1|$	$|{\mathit{r}}_3-{\mathit{r}}_2|$	$|{\mathit{r}}_4-{\mathit{r}}_3|$	CCO	CPU-Time
ZM−1	23	$2.59$	$6.37$	$5.23$	2.000	1.5924
ZM−2	12	$7.80$	$4.30$	$3.25$	2.031	1.4243
BM−1	6	$6.27\times {10}^{-2}$	$1.50\times {10}^{-4}$	$1.52\times {10}^{-12}$	3.000	0.1566
BM−2	6	$6.26\times {10}^{-2}$	$1.49\times {10}^{-4}$	$1.49\times {10}^{-12}$	3.000	0.1405
AM−1	6	$9.78\times {10}^{-1}$	$2.15\times {10}^{-1}$	$7.16\times {10}^{-2}$	7.960	0.1413
AM−2	6	$9.69\times {10}^{-1}$	$2.01\times {10}^{-1}$	$6.37\times {10}^{-2}$	7.966	0.1567
M−1	4	$7.41\times {10}^{-1}$	$8.40\times {10}^{-7}$	$2.82\times {10}^{-50}$	8.000	0.1113
M−2	4	$3.95\times {10}^{-1}$	$1.77\times {10}^{-6}$	$1.89\times {10}^{-47}$	8.000	0.1094
M−3	4	$7.22\times {10}^{-1}$	$2.05\times {10}^{-6}$	$4.47\times {10}^{-47}$	8.000	0.1267
M−4	4	$3.88\times {10}^{-1}$	$1.75\times {10}^{-6}$	$2.01\times {10}^{-47}$	8.000	0.1242
M−5	4	$7.33\times {10}^{-1}$	$1.44\times {10}^{-6}$	$2.74\times {10}^{-48}$	8.000	0.1286

,

Table 4. Numerical results for Example 4.

Methods	k	$|{\mathit{r}}_2-{\mathit{r}}_1|$	$|{\mathit{r}}_3-{\mathit{r}}_2|$	$|{\mathit{r}}_4-{\mathit{r}}_3|$	CCO	CPU-Time
ZM−1	6	$3.49\times {10}^{-1}$	$2.37\times {10}^{-2}$	$1.05\times {10}^{-3}$	8.000	1.6694
ZM−2	fails	-	-	-	-	-
BM−1	6	$1.59\times {10}^{-2}$	$1.81\times {10}^{-6}$	$2.34\times {10}^{-18}$	3.000	1.5134
BM−2	6	$1.59\times {10}^{-2}$	$1.80\times {10}^{-6}$	$2.31\times {10}^{-18}$	3.000	1.4045
AM−1	4	$2.60\times {10}^{-1}$	$2.45\times {10}^{-8}$	$1.39\times {10}^{-62}$	8.000	1.2481
AM−2	4	$2.60\times {10}^{-1}$	$2.45\times {10}^{-8}$	$1.34\times {10}^{-62}$	8.000	1.2333
M−1	4	$2.47\times {10}^{-1}$	$2.53\times {10}^{-8}$	$3.17\times {10}^{-62}$	8.000	1.1556
M−2	4	$1.42\times {10}^{-1}$	$4.63\times {10}^{-9}$	$6.95\times {10}^{-68}$	8.000	1.1702
M−3	4	$2.44\times {10}^{-1}$	$2.52\times {10}^{-8}$	$3.65\times {10}^{-62}$	8.000	1.1396
M−4	4	$1.41\times {10}^{-1}$	$4.63\times {10}^{-9}$	$7.88\times {10}^{-68}$	8.000	1.1742
M−5	4	$2.47\times {10}^{-1}$	$2.53\times {10}^{-8}$	$3.69\times {10}^{-62}$	8.000	1.1864

,

Table 5. Numerical results for Example 5.

Methods	k	$|{\mathit{r}}_2-{\mathit{r}}_1|$	$|{\mathit{r}}_3-{\mathit{r}}_2|$	$|{\mathit{r}}_4-{\mathit{r}}_3|$	CCO	CPU-Time
ZM−1	4	$57.80$	$5.95\times {10}^{-4}$	$2.16\times {10}^{-43}$	8.000	1.4354
ZM−2	5	$178.00$	$3.60\times {10}^{-1}$	$1.55\times {10}^{-21}$	8.000	1.7940
BM−1	6	$2.62$	$4.80\times {10}^{-4}$	$2.88\times {10}^{-15}$	3.000	2.2154
BM−2	6	$2.62$	$4.79\times {10}^{-4}$	$2.87\times {10}^{-15}$	3.000	2.2315
AM−1	4	$47.10$	$1.60\times {10}^{-4}$	$7.27\times {10}^{-48}$	8.000	1.6857
AM−2	4	$47.00$	$1.31\times {10}^{-4}$	$1.09\times {10}^{-48}$	8.000	1.6963
M−1	4	$45.20$	$3.17\times {10}^{-4}$	$5.01\times {10}^{-45}$	8.000	1.3898
M−2	4	$26.00$	$1.01\times {10}^{-5}$	$9.75\times {10}^{-57}$	8.000	1.4204
M−3	4	$44.70$	$3.83\times {10}^{-4}$	$3.26\times {10}^{-44}$	8.000	1.4355
M−4	4	$25.80$	$1.17\times {10}^{-5}$	$4.09\times {10}^{-56}$	8.000	1.4674
M−5	4	$45.10$	$6.15\times {10}^{-4}$	$1.80\times {10}^{-42}$	8.000	1.4048

and

Table 6. Numerical results for Example 6.

Methods	k	$|{\mathit{r}}_2-{\mathit{r}}_1|$	$|{\mathit{r}}_3-{\mathit{r}}_2|$	$|{\mathit{r}}_4-{\mathit{r}}_3|$	CCO	CPU-Time
ZM−1	4	$3.19\times {10}^{-2}$	$2.77\times {10}^{-16}$	$1.01\times {10}^{-128}$	8.000	2.7613
ZM−2	4	$7.25\times {10}^{-2}$	$5.76\times {10}^{-14}$	$1.26\times {10}^{-110}$	8.000	2.7466
BM−1	5	$5.84\times {10}^{-4}$	$1.78\times {10}^{-11}$	$5.08\times {10}^{-34}$	3.000	2.7925
BM−2	5	$5.84\times {10}^{-4}$	$1.78\times {10}^{-11}$	$5.09\times {10}^{-34}$	3.000	2.8089
AM−1	4	$3.05\times {10}^{-2}$	$2.45\times {10}^{-16}$	$4.79\times {10}^{-129}$	8.000	2.7464
AM−2	4	$3.05\times {10}^{-2}$	$2.32\times {10}^{-16}$	$2.94\times {10}^{-129}$	8.000	2.7301
M−1	4	$3.05\times {10}^{-2}$	$4.52\times {10}^{-16}$	$1.26\times {10}^{-126}$	8.000	2.3873
M−2	4	$1.96\times {10}^{-2}$	$2.65\times {10}^{-17}$	$3.40\times {10}^{-136}$	8.000	2.4184
M−3	4	$3.04\times {10}^{-2}$	$5.46\times {10}^{-16}$	$7.15\times {10}^{-126}$	8.000	2.4185
M−4	4	$1.96\times {10}^{-2}$	$3.05\times {10}^{-17}$	$1.22\times {10}^{-135}$	8.000	2.4027
M−5	4	$3.05\times {10}^{-2}$	$5.43\times {10}^{-16}$	$6.88\times {10}^{-126}$	8.000	2.3873

include: (i) Number of iterations $\left(k\right)$ required to converge to the solution such that $|r_{k+1}-r_k|+|f\left(r_k\right)|<{10}^{-300}$; (ii) Estimated error $|r_{k+1}-r_k|$ in the first three iterations; (iii) Computational convergence order (CCO) using (18); and (iv) CPU-time utilized in the execution of program which is computed by the Mathematica command “TimeUsed[ ]”.

The following examples are chosen for numerical testing:

Example 1.

The energy density in an isothermal black body is determined by applying the Planck law of radiation [31], which gives

$$\varphi \left(\lambda \right)={\displaystyle \frac{8\pi ch{\lambda }^{-5}}{e^{ch/\lambda kT}-1}}.$$

(19)

Here, k is Boltzmann’s constant, h is Planck’s constant, T is the absolute temperature of the black body, c is the speed of light, and λ is the wavelength of the radiation. Finding the wavelength λ that corresponds to the highest energy density $\varphi \left(\lambda \right)$ is the difficulty. Thus,

$${\varphi }^{\prime }\left(\lambda \right)=\left({\displaystyle \frac{8\pi ch{\lambda }^{-6}}{e^{ch/\lambda kT}-1}}\right)\left({\displaystyle \frac{(ch/\lambda kT)e^{ch/\lambda kT}}{e^{ch/\lambda kT}-1}}-5\right)=A.B.$$

is the result of Equation (19). Keep in mind that a ϕ maxima will appear when $B=0$, or when

$${\displaystyle \frac{(ch/\lambda kT)e^{ch/\lambda kT}}{e^{ch/\lambda kT}-1}}=5.$$

The foregoing equation takes the form

$$1-{\displaystyle \frac{r}{5}}=e^{-r},$$

(20)

when $r=ch/\lambda kT$. Therefore, the necessary nonlinear function is

$$f_1\left(r\right)=e^{-r}-1+{\displaystyle \frac{r}{5}}.$$

(21)

Since the root $r=0$ is trivial, it is not discussed. Note that, in (20), the right-hand side is $e^{-5}\approx 6.74\times {10}^{-3}$, while the left-hand side is zero for $r=5$. Therefore, we hypothesize that another root may exist close to $r=5$. Actually, $r^{*}\approx 4.9651142317$ with $m=1$ gives the predicted root of Equation (21). Then, the wavelength of radiation (λ) corresponding to the maximum energy density is

$$\lambda \approx {\displaystyle \frac{ch}{4.9651142317\left(kT\right)}}.$$

The numerical results shown in Table 1 are computed with initial guess $r_0=10$.

Example 2.

Consider a nonlinear function of academic interest defined as

$$f_2\left(r\right)=e^r-r-1.$$

At $r^{*}=0$, this has multiple zeros of multiplicity 2. For this function, we choose the initial approximation $r_0=-0.5$ to obtain zero. Table 2 presents the numerical results.

Example 3.

Let us examine an another standard test function,

$$f_3\left(r\right)=r^9-29r^8+349r^7-2261r^6+8455r^5-17663r^4+15927r^3+6993r^2-24732r+12960.$$

At $r^{*}=3$ of multiplicity 4, there is one multiple zero in this function. We utilize the first approximation $r_0=3.3$ to find this zero. Table 3 displays the numerical results.

Example 4.

Next, we take a function

$$f_4\left(r\right)=r(r^2+1)(2e^{r^2+1}+r^2-1){cosh}^4\left({\displaystyle \frac{\pi r}{2}}\right),$$

which has imaginary zeros. There is a zero $r^{*}=i$ with multiplicity 6 in this function. To compute the zero, we use the first approximation, $r_0=0.8i$. Table 4 displays the acquired results.

Example 5.

The circuit L-C-R is governed by the equation

$$L{\displaystyle \frac{d^{2}q}{d{t}^2}}+R{\displaystyle \frac{dq}{dt}}+{\displaystyle \frac{q}{C}}=0,$$

whose solution $q\left(t\right)$ is

$$q\left(t\right)=q_0{e}^{-Rt/2L}cos\left(\sqrt{{\displaystyle \frac{1}{LC}}-{\left({\displaystyle \frac{R}{2L}}\right)}^2}t\right),$$

where $t=0,$ $q=q_0.$ Regarding a specific case study, the issue is stated as follows [32]: Assume that, with $L=5$ Henry and $C={10}^{-4}$ Farad, the charge dissipates to 1 percent of its initial value $(q/q_0=0.01)$ in $t=0.05$ seconds. Find a proper value of R? We examined this example six times using the numerical values, and each time, we were able to derive the necessary nonlinear function,

$$f_5\left(r\right)={\left[e^{-0.005r}cos\left(\sqrt{2000-0.01r^2}\left(0.05\right)\right)-0.01\right]}^6,$$

where $r=R.$ At $r^{*}\approx 328.15142908514817$, the function mentioned above has zero with multiplicity 6. We compute the required zero with the initial estimate $r_0=370$. Table 5 presents the numerical findings.

Example 6.

Assume isentropic supersonic flow around a pointed corner of expansion. The following formula (see [33]) describes the relationship between the Mach numbers before the corner (i.e., $M_1$) and after the corner (i.e., $M_2$):

$$\begin{array}{cc}\hfill \delta =b^{1/2}({tan}^{-1}{\left({\displaystyle \frac{M_2^2-1}{b}}\right)}^{1/2}& -{tan}^{-1}{\left({\displaystyle \frac{M_1^2-1}{b}}\right)}^{1/2})-({tan}^{-1}{(M_2^2-1)}^{1/2}\hfill \\ \hfill & -{tan}^{-1}{(M_1^2-1)}^{1/2}),\hfill \end{array}$$

where $b={\displaystyle \frac{\gamma +1}{\gamma -1}}$ and γ is the gas’s specific heat ratio.

We solve the equation for $M_2$ for a specific case study, where $M_1=1.5$, $\gamma =1.4$, and $\delta ={10}^0$. We have

$${tan}^{-1}\left({\displaystyle \frac{\sqrt{5}}{2}}\right)-{tan}^{-1}\left(\sqrt{r^2-1}\right)+\sqrt{6}\left({tan}^{-1}\left(\sqrt{{\displaystyle \frac{r^2-1}{6}}}\right)-{tan}^{-1}\left({\displaystyle \frac{1}{2}}\sqrt{{\displaystyle \frac{5}{6}}}\right)\right)-{\displaystyle \frac{11}{63}}=0,$$

where $r=M_2$.

Ten times over, we examined this situation and came up with the necessary nonlinear function,

$$\begin{array}{cc}\hfill f_6\left(r\right)=[{tan}^{-1}\left({\displaystyle \frac{\sqrt{5}}{2}}\right)-{tan}^{-1}\left(\sqrt{r^2-1}\right)& +\sqrt{6}({tan}^{-1}\left(\sqrt{{\displaystyle \frac{r^2-1}{6}}}\right)\hfill \\ \hfill & -{tan}^{-1}\left({\displaystyle \frac{1}{2}}\sqrt{{\displaystyle \frac{5}{6}}}\right))-{\displaystyle \frac{11}{63}}{]}^{10}.\hfill \end{array}$$

At $r^{*}\approx 1.8411027704926161$, the above function has zero with multiplicity 10. Using the initial approximation $r_0=2$, the needed zero is computed. Results in numbers are displayed in Table 6.

The presented methods demonstrate excellent convergence behavior, as evidenced by the data in Table 1, Table 2, Table 3, Table 4, Table 5 and Table 6. The numerical results clearly indicate that this good convergence is due to the increasing accuracy of successive approximations in each iteration, which also account for the stability of the procedures. Moreover, the new approaches yield approximations to the solutions that are either more accurate or comparable to those produced by existing methods. When the stopping criterion $|r_{k+1}-r_k|+|f\left(r_k\right)|<{10}^{-300}$ is met, the value of $|r_{k+1}-r_k|$ is shown as ‘0’. The computed order of convergence, displayed in the penultimate column of each table, confirms the theoretical eighth order of convergence. However, this eighth-order convergence is not maintained by the eighth-order techniques ZM-1, ZM-2, BM-1, and BM-2. The new methods also consume less CPU time compared to the existing ones, indicating their efficiency. Similar numerical tests on various other experiments corroborated these findings.

4. Basins of Attraction

In this section, we assess the complex geometry of the proposed methods by drawing the basins of attraction of the roots of a polynomial equation $F\left(z\right)=0$ in a complex plane. Basin of attraction is a helpful geometric tool for comparing the convergence zones of the iterative approaches [34,35,36,37]. By means of this tool, we can check visually the convergence properties and stability of an iterative method. Let us first review some fundamental concepts related to this graphical technique.

Consider a rational mapping on the Riemann sphere $R:\mathbb{C}\to \mathbb{C}.$ A point $z_0\in \mathbb{C}$ has an orbit defined as the set $\{z_0,R\left(z_0\right),R^2\left(z_0\right),\dots ,R^n\left(z_0\right),\dots \}$. If a point $z_0\in \mathbb{C}$ satisfies the equation $R\left(z_0\right)=z_0$, then it is a fixed point of the rational function R. If, for every n, the lowest such number, $R^n\left(z_0\right)=z_0$, then a point $z_0$ is said to be periodic with period $n>1$. If $|R^{\prime }\left(z_0\right)|<1$, $|R^{\prime }\left(z_0\right)|>1$, or $|R^{\prime }\left(z_0\right)|=1$, then a point $z_0$ is said to be attracting, repelling and neutral. Furthermore, the fixed point is extremely attractive if $|R^{\prime }\left(z_0\right)|=0$. Let $z^{*}$ be a attracting fixed point of the rational function R.

$$A\left(z^{*}\right)=\{z_0\in \mathbb{C}:R^n\left(z_0\right)\to z^{*},n\to \infty \}$$

defines the fixed-point $z^{*}$ basin of attraction.

The Fatou set is the collection of points whose orbits converge to an attractive fixed-point $z^{*}$. The closure of the set made up of repelling fixed points, known as the Julia set or complementary set, defines the boundaries separating the root basins of attraction. We can see which spots are good choices for starting points and which are not by using attraction basins to evaluate those starting points that, when we use an iterative procedure, converge to the relevant root of a polynomial.

To initialize D, a rectangular region in $\mathbb{C}$ that contains all the roots of the equation $F\left(z\right)=0,$ we select $z_0$. Starting from $z_0\in D$, an iterative process can either diverge or converge to the zero of $F\left(z\right)$. We use a stopping condition of ${10}^{-3}$ with a maximum of 25 iterations to evaluate the basins. If this tolerance is not met within the given number of iterations, the process is terminated, indicating divergence from the initial point $z_0$. To create the basins, each initial guess $z_0$ within the attraction basin of zero is assigned a color. If the iterative formula starting at $z_0$ converges, the corresponding area is colored accordingly; if it does not converge within the required number of iterations, the area is painted black.

To view complex geometry, we analyze the basins of attraction of the methods on different following polynomials:

Test problem 1. Let $F_1\left(z\right)={(z^2-1)}^2$ be a polynomial with multiplicities $m=2$ and two zeros $\{1,-1\}$. Figure 1 displays the polynomial’s basin of attractors. Every zero’s basin of attraction has a color allocated to it. Specifically, the attraction basins of the two zeros were assigned the colors red and green in order to obtain the basins.

Test problem 2. Let $F_2\left(z\right)={(z^3+z)}^3$ have multiplicities $m=3$ and three zeros $\{i,0,-i\}$. Figure 2 displays the polynomial’s basin of attractors. Every zero’s basin of attraction has a color allocated to it. Specifically, the colors red, blue, and green correspond to i, 0, and $-i$ attraction basins, respectively.

Test problem 3. Let $F_3\left(z\right)={\left(z^3+{\displaystyle \frac{1}{z}}\right)}^3$ have four zeros with multiplicities of $m=3$, namely $-0.707107+0.707107i$, $-0.707107-0.707107i$, $0.707107+0.707107i$, $0.707107-0.707107i$. Figure 3. displays the basin of attractors of these zeros as determined by the approaches under consideration. A color is assigned to a zero to identify its corresponding basin. For instance, the colors corresponding to $0.707107-0.707107i$, $0.707107+0.707107i$, $-0.707107-0.707107i$, and $-0.707107+0.707107i$ have been allocated, as have green, red, blue, and yellow.

Along with the basins of attraction, we also provide some other useful information of the performance of methods in

Table 7. Performance of methods for problems 1–3.

Methods	ZM-1	ZM-2	BM-1	BM-2	AM-1	AM-2	M-1	M-2	M-3	M-4	M-5
$\mathrm{Prob}.4.1$
IP	$4.59$	$4.60$	$3.20$	$3.21$	$4.85$	$4.00$	$4.54$	$4.21$	$4.54$	$4.22$	$4.89$
NC	$3.51$	$10.98$	$0.58$	$0.53$	$3.29$	$0.44$	$0.47$	$0.40$	$0.54$	$0.43$	$0.60$
IC	$3.85$	$3.51$	$3.08$	$3.10$	$4.16$	$3.91$	$4.44$	$4.13$	$4.43$	$4.73$	$4.77$
$\mathrm{Prob}.4.2$
IP	$7.47$	$7.27$	$5.21$	$5.10$	$7.85$	$6.42$	$6.26$	$5.83$	$6.25$	$5.77$	$6.62$
NC	$10.88$	$21.43$	$2.59$	$2.33$	$11.77$	$1.35$	$2.00$	$0.90$	$1.88$	$0.69$	$2.03$
IC	$5.33$	$5.36$	$4.69$	$4.63$	$5.57$	$6.16$	$5.88$	$5.65$	$5.89$	$5.64$	$6.24$
$\mathrm{Prob}.4.3$
IP	$5.62$	$5.61$	$3.49$	$3.53$	$5.63$	$4.68$	$4.68$	$4.29$	$4.70$	$4.24$	$4.82$
NC	$6.45$	$13.90$	$1.33$	$1.24$	$6.74$	$0.67$	$1.08$	$0.42$	$1.06$	$0.38$	$1.04$
IC	$4.29$	$4.16$	$3.20$	$3.26$	$4.23$	$4.54$	$4.46$	$4.20$	$4.48$	$4.16$	$4.61$

, which include:

• IP: Mean of iterations, measured in iterations/point.
• NC: Nonconvergent points, as a percentage of the total number of starting points evaluated.
• IC: Mean of iterations, measured in iterations/(point–non-convergent points).

From the above graphics, we can observe that, in general, the proposed methods M-1–M-5 along with the existing method AM-2 perform better than the other ones, since in all the examples, either these have no divergent points or very few divergent points. This feature can also be verified by the numerical results shown in Table 7. The methods ZM-2, ZM-1, and AM-1 have large divergent points shown by black zones followed by BM-1 and BM-2. The most visually attractive images appear when the basin boundaries are extremely complex. These images fall into the category of situations where the method is more demanding with respect to the starting point.

The convergence represented through the basin of attraction is a powerful tool, being the set of initial conditions leading to the long-iterated behavior that approaches the attractor [38]. Attractors can correspond to periodic, quasiperiodic, or chaotic behaviors of different types [39]. It has been found that a basin of attraction defined as a region in the state space may have a basic topological structure which can vary greatly from system to system [40,41]. Inspecting the basins of attraction for the test problems 4.1 to 4.3 one can see that all possess reflection symmetry.

A riddled basin of attraction [42,43] arises in some special systems that have a smooth invariant manifold (a smooth surface or hypersurface in the phase space where any initial condition on the surface produces an orbit that stays in the surface) due to symmetry or some other constraint.

5. Conclusions and Future Scope

In this paper, we presented a family of optimal eighth-order iterative methods for finding the multiple zeros of a nonlinear equation. The local convergence analysis has been studied, providing the eighth-order under standard assumptions, ensuring the methods’ theoretical robustness. Some particular cases have been explored and their performances have been checked by using two different ways viz. by numerical testing and by graphical tool of basins of attraction. Moreover, the comparison of performance of the methods with those of existing optimal eighth-order methods using these two ways has also been demonstrated. It has been observed that, unlike those of existing techniques, the proposed techniques have a consistent convergence behavior. Additionally, a comparison of the estimated CPU-time of the new methods with existing ones has been performed to rank the algorithms, demonstrating their computational efficiency. The results highlight that the ranking obtained through CPU-time estimation aligns well with the ranking derived from the efficiency results, confirming the superiority of the new methods. This dual validation through numerical and graphical analysis, coupled with efficiency rankings, underscores the practical applicability and advantages of the proposed methods in solving complex nonlinear equations with multiple roots.

The future scope of “An optimal family of eighth-order methods for multiple roots and their complex dynamics” is promising, with potential advancements in both theoretical and practical applications. These high-order iterative methods are crucial for solving nonlinear equations with multiple roots, offering significant improvements in convergence speed and computational efficiency. In the future, research could focus on the algorithm refinement, complex dynamics analysis, computational applications, software implementation, and interdisciplinary collaboration.