On Iterative Methods for Solving Nonlinear Equations in Quantum Calculus

Abstract

Quantum calculus (also known as the q-calculus) is a technique that is similar to traditional calculus, but focuses on the concept of deriving q-analogous results without the use of the limits. In this paper, we suggest and analyze some new q-iterative methods by using the q-analogue of the Taylor’s series and the coupled system technique. In the domain of q-calculus, we determine the convergence of our proposed q-algorithms. Numerical examples demonstrate that the new q-iterative methods can generate solutions to the nonlinear equations with acceptable accuracy. These newly established methods also exhibit predictability. Furthermore, an analogy is settled between the well known classical methods and our proposed q-Iterative methods.

1. Introduction

In most scientific and engineering applications, the problem of finding the solution of nonlinear equations have become an active area of research. Many researchers have explored various order iterative methods to find solutions of the nonlinear equations using various techniques such as homotopy perturbation technique, variational iterative methods and decomposition technique, for details, see [1,2,3,4,5,6,7,8,9,10,11]. Firstly, Traub [12] initiated the study of the iterative methods for the solution of the nonlinear equations and introduced a basic quadratic convergent Newton iterative method for the solution of the nonlinear equations, which have much significance in the literature. Later on, in order to improve efficiency and local order of convergence of the Newtons method, Cordero and Torregrosa [13], Frontini and Sormani [14], Hasanov [15], Weerakoon and Fernando [16] and Ozban [17] have presented different modifications of the Newton’s method using quadrature rules. Daftardar-Gejji and Jafari [18] have used different modifications of Adomian decomposition method [19] and suggested a simple technique that does not need derivative evaluation of the Adomian polynomial, which is major advantage of using this technique over Adomian decomposition method. Saqib and Iqbal [20] and Ali et al. [21,22] have used this decomposition technique and developed a family of iterative methods with better efficiency and convergence order for solving the nonlinear equations. This study shifts the paradigm of determining higher order iterative methods for solving the nonlinear equations towards the q-analogue of the iterative methods in the q-calculus.

In the last quarter of the 20th century, q-calculus appeared as an amalgamation of mathematics and physics (see [23,24,25,26,27,28,29]), and much consideration has been given by many researchers because of its wide range of applications in many fields of mathematics such as combinatorics, theory of relativity, mechanics, number theory and orthogonal polynomials. Firstly, Jackson [30] introduced the q-Taylor’s formula. Then, Jing and Fan [31] derived q-Taylor’s formula with its q-remainder by using the q-differentiation approach and established results on the q-remainder in the q-Taylor’s formula. Ernest [32] presented the four different q-Taylor’s formulas along with q integral remainder. Prashant et al. [33] have used the q-Taylor’s formula and investigated the q-analogue of the iterative methods, particularly the q-analogue of generalized Newton Raphson method and the q-analogue of the Newton Raphson method for the solution of algebraic transcendental equations and compared the accuracy of the results obtained by the classical methods. Many linear and nonlinear models appearing in science and engineering problems can be modeled by using the q differential equations. Jafari et al. [34] have adopted Daftardar decomposition technique for solving the q difference equations and also determined the convergence of the method.

In this study, we determine the q-analogue of the iterative methods proposed and suggested by Noor and Noor [35] and Ullah et al. [36] with the help of the q-Taylor’s series and decomposition technique [14].

Now, we recall some of the basic results in the area of the q-calculus [37] for $0<q<1$ that will support the development of our proposed q-iterative methods for the solution of the nonlinear equations.

Let q∈(0,1) the q-integer be defined as:

$$\begin{array}{cc}\hfill & {\left[n\right]}_q=1+q+q^2+\cdots +q^n-1=\frac{1-q^n}{1-q}\mathrm{for}n=1,2,\dots \hfill \end{array}$$

(1)

$$\begin{array}{cc}\hfill & {\left[n\right]}_q=n\mathrm{for}q=1.\hfill \end{array}$$

(2)

For q factorial and for $0\le k\le n$, the q binomials are defined as:

$$\begin{array}{cc}\hfill & {\left[n\right]}_q!={\left[n\right]}_q{[n-1]}_q\cdots {\left[1\right]}_q{\left[0\right]}_q!=1,{\left[\begin{array}{c}n\\ k\end{array}\right]}_q=\frac{{\left[n\right]}_q!}{{\left[k\right]}_q!{[n-k]}_q!}.\hfill \end{array}$$

(3)

Definition 1

(see [37]). The q-derivative for real valued continuous function $f\left(x\right)$ is defined as follows

$$\begin{array}{c}\hfill D_q\left(x\right)={\left(\frac{d}{dx}\right)}_{q}f\left(x\right)=\frac{f\left(qx\right)-f\left(x\right)}{qx-x},q\ne 1.\end{array}$$

(4)

When $q\to 1$, then the q-derivative is reduced to the standard derivative. Furthermore, the q-derivative can be represented as $\left(D_{q}f\right)\left(x\right)$, and it is known as the Jackson Derivative. The higher order q-derivative for the function $f\left(x\right)$ is given as

$$\begin{array}{c}\hfill D_q^{0}f=f,D_q^{n}f=D_q\left(D_q^{n-1}\right)\mathit{for}n=1,2,3,\dots .\end{array}$$

(5)

Definition 2

(see [37]). The q-derivative of product and quotient of function $f\left(x\right)$ and $g\left(x\right)$ is defined as follows

$$\begin{array}{cc}\hfill & D_q\left(f\left(x\right)g\left(x\right)\right)=g\left(x\right)D_{q}f\left(x\right)+f\left(qx\right)D_{q}g\left(x\right)=g\left(qx\right)D_{q}f\left(x\right)+f\left(x\right)D_{q}g\left(x\right),\hfill \\ \hfill & D_q\left(\frac{f\left(x\right)}{g\left(x\right)}\right)=\frac{g\left(x\right)D_{q}f\left(x\right)-f\left(x\right)D_{q}g\left(x\right)}{g\left(qx\right)g\left(x\right)}\mathit{such}\mathit{that}g\left(x\right)g\left(qx\right)\ne 0.\hfill \end{array}$$

Definition 3

(see [30,31,32]). Let $f\left(x\right)$ be a continuous function on some interval $(a,b)$ and $c\in$[a,b] then Jackson q-Taylor’s formula is given as:

$$\begin{array}{c}\hfill f\left(x\right)=\sum _{k=1}^{\infty }\frac{D_q^k{(x-c)}^k}{\left[k\right]!}\left(\forall x\in (a,b)\right),\end{array}$$

(6)

where,

$$\begin{array}{c}\hfill {(x-c)}^0=1,{(x-c)}^k=\prod _{i=0}^{k-1}{(x-cq)}^i,k\in N,\end{array}$$

(7)

where $0<q<1$ and $D_q,D_q^2,\dots$ are all q-derivatives.

The rest of this article is organized as follows: in Section 2, the structures of the q-iterative methods will be designed by proposing the Lists A–C. In Section 3, we deal with the convergence analysis of the proposed q-iterative methods, and it is established that these methods have the same order of convergence as the classical methods for $q=1$. In Section 4, we present some of the examples to check the efficacy and performance of these methods. Furthermore, the comparisons of the results obtained by q-iterative methods with the previously known iterative methods will be discussed in the same section. Section 5 explores a general form of the q-iterative method based on the proposed iterative methods. Finally, the findings of our article are given in Section 6.

2. Construction of the $\mathit{q}$-Iterative Methods

In this section, some new different order multi-step q-iterative methods are constructed by considering the Taylor’s series in the q-calculus. Here, we consider the nonlinear equation

$$\begin{array}{c}\hfill f\left(x\right)=0(\forall x\in R).\end{array}$$

(8)

Suppose that $\alpha$ is the root of Equation (8) and $\lambda$ is an initial guess in the neighborhood of $\alpha$. By the same technique used in [28], we rewrite the nonlinear Equation (8) as the coupled system of equations by using the Taylor’s series in the neighborhood of $\lambda$ in the q-calculus:

$$\begin{array}{cc}\hfill & f\left(\lambda \right)+(x-\lambda )D_{q}f\left(\lambda \right)+G\left(x\right)=0,\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill & G\left(x\right)=f\left(x\right)-f\left(\lambda \right)-(x-\lambda )D_{q}f\left(\lambda \right).\hfill \end{array}$$

(10)

Since $f\left(x\right)=0$, the relation (10) can be written as:

$$\begin{array}{cc}\hfill & x=\lambda -\frac{f\left(\lambda \right)}{D_{q}f\left(\lambda \right)}-\frac{G\left(x\right)}{D_{q}f\left(\lambda \right)},\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill & x=c+M_q\left(x\right),\hfill \end{array}$$

(12)

where

$$\begin{array}{c}\hfill c≔\lambda -\frac{f\left(\lambda \right)}{D_{q}f\left(\lambda \right)},\end{array}$$

(13)

and

$$\begin{array}{c}\hfill M_q\left(x\right)≔-\frac{G\left(x\right)}{D_{q}f\left(\lambda \right)}\end{array}$$

(14)

is a nonlinear operator and c is treated as constant.

It is noted that if we consider $x_0$ as an initial guess, then from (10), we have

$$\begin{array}{c}\hfill f\left(x_0\right)=G\left(x_0\right).\end{array}$$

(15)

It is wort mentioning that Equation (15) plays a very significant role in the development of new multi-step q-iterative methods. Now, we establish a sequence of higher order iterative methods implementing the decomposition technique presented by Daftardar-Gejji and Jafari [18]. The main idea behind the implementation of this technique is to find out the solution of q-type functional Equation (12) in terms of infinite series:

$$\begin{array}{c}\hfill x=\sum _{i=0}^{\infty }{x}_i.\end{array}$$

(16)

Now, we decompose the operator $M_q\left(x\right)$ defined in (14), such as:

$$\begin{array}{c}\hfill M_q\left(x\right)=M_q\left(x_0\right)+\sum _{i=1}^{\infty }\left\{M_q\left(\sum _{j=0}^i{x}_j\right)-M_q\left(\sum _{j=0}^{i-1}{x}_j\right)\right\}.\end{array}$$

(17)

From the Equations (12), (16) and (17), we have

$$\begin{array}{c}\hfill \sum _{i=1}^{\infty }{x}_i=c+M_q\left(x_0\right)+\sum _{i=1}^{\infty }\left\{M_q\left(\sum _{j=0}^i{x}_j\right)-M_q\left(\sum _{j=0}^{i-1}{x}_j\right)\right\},\end{array}$$

(18)

which generates the following iterative scheme

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \left\{\begin{array}{cc}{x}_0\hfill & =c,\hfill \\ x_1\hfill & =M_q\left(x_0\right),\hfill \\ x_2\hfill & =M_q(x_0+x_1)-M_q\left(x_0\right),\hfill \\ \hfill & ⋮\hfill \\ x_{n+1}\hfill & =M_q\left(\sum _{j=0}^n{x}_j\right)-M_q\left(\sum _{j=0}^{n-1}{x}_j\right),n=1,2,\dots .\hfill \end{array}\right.\end{array}\end{array}$$

(19)

It follows that

$$\begin{array}{c}\hfill x_1+x_2+\cdots +x_{n+1}=M_q\left(x_0+x_1+x_2+\cdots +x_n\right),\end{array}$$

and

$$\begin{array}{c}\hfill x=c+\sum _{i=1}^{\infty }{x}_i.\end{array}$$

(20)

Theorem 1

(see [34]). If M is a contraction mapping, the defined series in (16) is absolutely convergent.

Proof. 

If M is a contraction mapping, we see that

$$\begin{array}{c}\hfill ∥M_q\left(x\right)-M_q\left(y\right)∥\le \alpha ∥x-y∥0<\alpha <1,\end{array}$$

(21)

then in view of (19), we have

$$\begin{array}{c}\hfill ∥x_{n+1}∥=∥M_q(x_1+x_2+\dots +x_n)-M_q(x_1+x_2+\dots +x_{n-1})∥\le \alpha ∥x_n∥\le {\alpha }^n∥x_0∥\\ \hfill n=0,1,2,\dots .\end{array}$$

So, the series x=${\sum }_{i=0}^{\infty }{x}_i$ converges uniformly and absolutely to the solution of Equation (12) (see [38]). It is noted that x is approximated by

$$\begin{array}{cc}\hfill & X_n=x_0+x_1+x_2+\cdots +x_n,\hfill \end{array}$$

and thus $\underset{x\to \infty }{lim}{X}_n=x$.

This completes the proof. □

Our iterative techniques proceed with the following algorithms:

List A: 

From (19), we have for $n=0:$

$$\begin{array}{c}\hfill x\approx X_0=x_0=c=\lambda -\frac{f\left(\lambda \right)}{D_{q}f\left(\lambda \right)}.\end{array}$$

(22)

This formulation suggests the following iterative scheme for solving the nonlinear Equation (8). Now, for the given initial guess $x_0$, the approximate solution is computed by the iterative representation

$$\begin{array}{cc}\hfill x_{n+1}& =x_n-\frac{f\left(x_n\right)}{D_{q}f\left(x_n\right)}\hfill \end{array}$$

(23)

This represents the Newton’s method in the q-calculus which has quadratic convergence. It is proved by Singh et al. [33].

By replacing the value of the q-derivative in (23), we have

$$\begin{array}{cc}\hfill x_{n+1}=& x_n-\frac{f\left(x_n\right)}{\frac{f\left(q{x}_n\right)-f\left(x_n\right)}{(q-1)x_n}}\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill =& x_n-\frac{(q-1)x_{n}f\left(x_n\right)}{f\left(q{x}_n\right)-f\left(x_n\right)},n=0,1,\dots .\hfill \end{array}$$

(25)

This method resembles the method of secants (chords).

Now, with the help of (10) and (19), we get

$$\begin{array}{cc}\hfill x_1=M_q\left(x_0\right)& =-\frac{G\left(x_0\right)}{D_{q}f\left(\lambda \right)}=-\frac{f\left(x_0\right)}{D_{q}f\left(\lambda \right)}.\hfill \end{array}$$

(26)

List B: 

From (19), we have for $n=1:$

$$\begin{array}{c}\hfill x\approx X_1=x_0+x_1=x_0+M_q\left(x_0\right).\end{array}$$

(27)

By using (22) and (26), we have

$$\begin{array}{cc}\hfill x& =\lambda -\frac{f\left(\lambda \right)}{D_{q}f\left(\lambda \right))}-\frac{f\left(x_0\right)}{D_{q}f\left(\lambda \right)}.\hfill \end{array}$$

(28)

This formulation suggests the following iterative scheme for solving the nonlinear Equation (8).

For the given initial guess $x_0$, the approximate solution is computed by the following iterative method:

$$\begin{array}{cc}\hfill & y_n=x_n-\frac{f\left(x_n\right)}{D_{q}f\left(x_n\right)},\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill & x_{n+1}=y_n-\frac{f\left(y_n\right)}{D_{q}f\left(x_n\right)},n=0,1,2,\dots .\hfill \end{array}$$

(30)

This is q-analogue of Chun method [3], which has cubic convergence for $q=1$. The error term for this algorithm is computed in Theorem (2).

By using (10), (19) and (26), we can obtain

$$\begin{array}{cc}\hfill x_1+x_2& =M_q(x_0+x_1)=-\frac{G(x_0+x_1)}{D_{q}f\left(\lambda \right)}\hfill \\ \hfill & =-\frac{f(x_0+x_1)-f\left(\lambda \right)-(x_0+x_1-\lambda )D_{q}f\left(\lambda \right)}{D_{q}f\left(\lambda \right)}\hfill \\ \hfill & =-\frac{f(x_0+x_1)}{D_{q}f\left(\lambda \right)}-\frac{f\left(x_0\right)}{D_{q}f\left(\lambda \right)}.\hfill \end{array}$$

(31)

List C: 

Again, from (19), we have for $n=2:$

$$\begin{array}{cc}\hfill x\approx X_2& =x_0+x_1+x_2=x_0+M_q\left(x_0\right)+M_q(x_0+x_1)-M_q\left(x_0\right)\hfill \\ \hfill & =c+M_q(x_0+x_1).\hfill \end{array}$$

(32)

By using (22) and (31), we get

$$\begin{array}{cc}\hfill x& =\lambda -\frac{f\left(\lambda \right)}{D_{q}f\left(\lambda \right)}-\frac{f\left(x_0\right)}{D_{q}f\left(\lambda \right)}-\frac{f(x_0+x_1)}{D_{q}f\left(\lambda \right)}.\hfill \end{array}$$

(33)

This formulation allows us to suggest the following iterative method for solving the nonlinear Equation (8).

For the given initial guess $x_0$, the approximate solution is computed by the following iterative method:

$$\begin{array}{cc}\hfill & y_n=x_n-\frac{f\left(x_n\right)}{D_{q}f\left(x_n\right)},\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill & z_n=y_n-\frac{f\left(y_n\right)}{D_{q}f\left(x_n\right)},\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill & x_{n+1}=z_n-\frac{f\left(z_n\right)}{D_{q}f\left(x_n\right)},n=0,1,2,\dots .\hfill \end{array}$$

(36)

This is q-analogue of convergent iterative method was investigated by Ullah et al. [36]. Furthermore, it has fourth order convergence for q = 1. The Error equation for this algorithm is computed in Theorem (2).

3. Convergence Analysis

This section is comprised of the convergence analysis of the q-iterative methods determined by Lists B and C in the previous section.

Theorem 2.

Let $I\subset R$ be an open interval and $f:I\to R$ be a differentiable function. If $\alpha \in I$ is a simple root of $f\left(x\right)=0$ and $x_0$ is sufficiently close to α, then the convergence order of multi-step methods determined byLists BandChave convergence of the order at least three and four, respectively, and we write it as $[3;q]$ and $[4;q]$, where q represents q-calculus. Furthermore, it satisfies the error equations

$$\begin{array}{cc}\hfill & e_{n+1}=2b_2^2{e}_n^3+(7b_2{b}_3-9b_2^3)e_n^4+O\left(e_n^5\right),\hfill \\ \hfill & e_{n+1}=4b_2^3{e}_n^4+(20b_3{b}_2^2-26b_2^4)e_n^5+O\left(e_n^6\right).\hfill \end{array}$$

Proof. 

Expanding $f\left(x_n\right)$ and $D_{q}f\left(x_n\right)$ in terms of q-Taylor’s series about $\alpha$ to get

$$\begin{array}{cc}\hfill f\left(x_n\right)=& D_{q}f\left(\alpha \right)e_n+\frac{1}{\left[2\right]!}{D}_q^{2}f\left(\alpha \right)e_n^2+\frac{1}{\left[3\right]!}{D}_q^{3}f\left(\alpha \right)e_n^3+\cdots ,\hfill \end{array}$$

(37)

$$\begin{array}{cc}\hfill f\left(x_n\right)=& D_{q}f\left(\alpha \right)\left\{e_n+b_2{e}_n^2+b_3{e}_n^3+b_4{e}_n^4+\cdots \right\},\hfill \end{array}$$

(38)

$$\begin{array}{cc}\hfill D_{q}f\left(x_n\right)& =D_{q}f\left(\alpha \right)\left\{1+2b_2{e}_n+3b_3{e}_n^2+4b_4{e}_n^3+\cdots \right\},\hfill \end{array}$$

(39)

where

$$\begin{array}{c}\hfill b_m=\frac{D_q^{m}f\left(\alpha \right)}{\left[m\right]!D_{q}f\left(\alpha \right)}\mathrm{for}m=2,3,\dots \mathrm{and}{e}_n=x_n-\alpha .\end{array}$$

(40)

In view of the Equations (38) and (39), we get

$$\begin{array}{cc}\hfill & \frac{f\left(x_n\right)}{D_{q}f\left(x_n\right)}=e_n-b_2{e}_n^2-\left(2b_3-2b_2^2\right)e_n^3+O\left(e_n^4\right).\hfill \end{array}$$

(41)

Now, by substituting (41) into (29), we get

$$\begin{array}{cc}\hfill y_n& =x_n-\frac{f_q\left(x_n\right)}{D_{q}f\left(x_n\right)}=\alpha +b_2{e}_n^2+\left(2b_3-2b_2^2\right)e_n^3+O\left(e_n^4\right).\hfill \end{array}$$

(42)

Expanding $f\left(y_n\right)$ in terms of Taylor’s series about $\alpha$ to get

$$\begin{array}{cc}\hfill f\left(y_n\right)& =D_{q}f\left(\alpha \right)\left\{b_2{e}_n^2+\left(2b_3-2b_2^2\right)e_n^3+\left(3b_4-7b_2{b}_3+5b_2^3\right)e_n^4+O\left(e_n^5\right)\right\}.\hfill \end{array}$$

(43)

From (39) and (43), we have

$$\begin{array}{c}\hfill \frac{f\left(y_n\right)}{D_{q}f\left(x_n\right)}=b_2{e}_n^2+(2b_3-4b_2^2)e_n^3+(3b_4-14b_2{b}_3+13b_2^3)e_n^4+O\left(e_n^5\right).\end{array}$$

(44)

By applying (42) and (44) into (30), we get the error term of the List B:

$$\begin{array}{cc}\hfill x_{n+1}& =\alpha +2b_2^2{e}_n^3+\left(7b_2{b}_3-9b_2^3\right)e_n^4+O\left(e_n^5\right),\hfill \\ \hfill e_{n+1}& =2b_2^2{e}_n^3+\left(7b_2{b}_3-9b_2^3\right)e_n^4+O\left(e_n^5\right),\hfill \\ \hfill z_n& =\alpha +2b_2^2{e}_n^3+\left(7b_2{b}_3-9b_2^3\right)e_n^4+O\left(e_n^5\right).\hfill \end{array}$$

(45)

Expanding $f\left(z_n\right)$ in terms of Taylor’s series about $\alpha$ to get

$$\begin{array}{c}f\left(z_n\right)=D_{q}f\left(\alpha \right)\{2b_2^2{e}_n^3+\left(7b_2{b}_3-9b_2^3\right)e_n^4\hfill \\ \hfill +\left(10b_2{b}_4+6c_3^2-44b_3{b}_2^2+30b_2^4\right)e_n^5+O\left(e_n^6\right)\}.\end{array}$$

(46)

By the help of (39) and (46), we can deduce

$$\begin{array}{c}\frac{f\left(z_n\right)}{D_{q}f\left(x_n\right)}=2b_2^2{e}_n^3+\left(7b_2{b}_3-13b_2^3\right)e_n^4\hfill \\ \hfill +\left(10b_2{b}_4+6b_3^2-64b_3{b}_2^2+56b_2^4\right)e_n^5+O\left(e_n^6\right).\end{array}$$

(47)

By applying (45) and (47) into (34), we get the error term of the List C:

$$\begin{array}{c}\hfill x_{n+1}=\alpha +4b_2^3{e}_n^4+\left(20b_3{b}_2^2-26b_2^4\right)e_n^5+O\left(e_n^6\right),\end{array}$$

(48)

$$\begin{array}{c}\hfill e_{n+1}=4b_2^3{e}_n^4+\left(20b_3{b}_2^2-26b_2^4\right)e_n^5+O\left(e_n^6\right).\end{array}$$

(49)

This completes the proof. □

4. Numerical Examples and Comparison Results

This section elaborates on the efficacy of algorithms introduced in this paper with the support of examples. All the numerical experiments are performed with Intel (R) Core [TM] 2 × 2.1 GHz, 12 GB of RAM, and all the codes are written in Maple. We use $\epsilon ={10}^{-15}$ and obtain an approximated simple root rather than the exact based on the exactness $\epsilon$ of the computer.

For the computational work, we use the following stopping criteria:

$$\begin{array}{c}\hfill \left(i\right)|x_{n+1}-x_n|<\epsilon and\left(ii\right)|f\left(x_{n+1}\right)|<\epsilon \end{array}$$

(50)

Abbreviation $\mathsf{CAG}$ is used for classical iterative method and $\mathsf{QAG}$ for the q-analogue of classical iterative method, and term div is used for divergence of method.

Recall the classical List 2.2 in [35] ($\mathsf{CAG}1$), defined by

$$\begin{array}{cc}\hfill & y_n=x_n-\frac{f\left(x_n\right)}{f^{\prime }\left(x_n\right)}\mathrm{and}{x}_{n+1}=y_n-\frac{f\left(y_n\right)}{f^{\prime }\left(x_n\right)}(\forall n=0,1,2,\dots ).\hfill \end{array}$$

and the classical List 2.3 in [36] ($\mathsf{CAG}2$), defined by

$$\begin{array}{cc}\hfill & y_n=x_n-\frac{f\left(x_n\right)}{f^{\prime }\left(x_n\right)},\hfill \\ \hfill & z_n=y_n-\frac{f\left(y_n\right)}{f^{\prime }\left(x_n\right)},\hfill \\ \hfill & x_{n+1}=z_n-\frac{f\left(z_n\right)}{f^{\prime }\left(x_n\right)},\mathrm{for}n=0,1,2,\dots .\hfill \end{array}$$

For simplicity, we denote the iterative Lists B and C by $\mathsf{QAG}1$ and $\mathsf{QAG}2$, respectively. The computational results are presented in Table 1, Table 2, Table 3, Table 4, Table 5 and Table 6 to elaborate the performance and efficacy of our q iterative methods that is the main motivation of transformation of the classical methods towards the q-iterative methods.

Table 1. Calculation of $x_i$ and $f\left(x_i\right)$ for $i=1,2,3$ and different values of q by using $\mathsf{QAG}1$.

q	${\mathit{x}}_1$	$\mathit{f}\left({\mathit{x}}_1\right)$	${\mathit{x}}_2$	$\mathit{f}\left({\mathit{x}}_2\right)$	${\mathit{x}}_3$	$\mathit{f}\left({\mathit{x}}_3\right)$
1.02	1.7250206931	3.025329e+01	1.4474835253	7.379764e+00	1.2518646781	9.612675e-01
1.01	1.7146118692	2.883931e+01	1.4305859139	6.635881e+00	1.2420681659	7.368529e-01
0.9999	1.7039972727	2.746138e+01	1.4138444432	5.942719e+00	1.2336242968	5.488326e-01
0.98	1.6827973761	2.489016e+01	1.3819710496	4.732768e+00	1.2210048207	2.767866e-01
0.97	1.6720053179	2.366777e+01	1.3665877297	4.195734e+00	1.2164975903	1.821333e-01
0.99	1.6934971518	2.615875e+01	1.3977906174	5.316156e+00	1.2267061933	3.983936e-01
0.96	1.6611219970	2.249026e+01	1.3516779288	3.701888e+00	1.2130682867	1.109783e-01
0.95	1.6501477370	2.135624e+01	1.3372782267	3.248313e+00	1.2105873872	5.995948e-02
0.9	1.5938858602	1.628822e+01	1.2741571956	1.498487e+00	1.2074940697	3.121695e-03
0.8	1.4737314129	8.631573e+00	1.2037818654	7.805411e-02	1.2070782899	1.155594e-02
0.7	1.3426598641	3.415231e+00	1.2098745341	4.537035e-02	1.2083400493	1.407187e-02
0.6	1.2021358529	1.110139e-01	1.2043272420	6.709762e-02	1.2056858306	3.972595e-02
0.5	1.0551639918	2.509831e+00	0.8000270036	5.058134e+00	5.3921025287	2.284390e+13
0.4	1.2618215234	1.196531e+00	1.2449990477	8.032748e-01	1.2395189569	6.795706e-01

Table 2. Calculation of $x_i$ and $f\left(x_i\right)$ for $i=1,2,3$ and different values of q by using $\mathsf{QAG}2$.

q	${\mathit{x}}_1$	$\mathit{f}\left({\mathit{x}}_1\right)$	${\mathit{x}}_2$	$\mathit{f}\left({\mathit{x}}_2\right)$	${\mathit{x}}_3$	$\mathit{f}\left({\mathit{x}}_3\right)$
1.02	1.6694991316	2.339181e+01	1.3531960626	3.751021e+00	1.2127490200	1.043912e-01
1.01	1.6589604001	2.226275e+01	1.3391400100	3.305714e+00	1.2107771567	6.384855e-02
0.9999	1.6483076255	2.117115e+01	1.3256085221	2.896627e+00	1.2094270195	3.622758e-02
0.99	1.6378605159	2.014678e+01	1.3130192725	2.532342e+00	1.2085807901	1.897265e-02
0.98	1.6273047549	1.915594e+01	1.3010027394	2.198535e+00	1.2080737484	8.654896e-03
0.97	1.6167470984	1.820720e+01	1.2897082127	1.896564e+00	1.2078048652	3.189788e-03
0.96	1.6061879189	1.729859e+01	1.2791506509	1.624142e+00	1.2076848052	7.509703e-04
0.95	1.5956267820	1.642815e+01	1.2693411891	1.379186e+00	1.2076465297	2.634743e-05
0.9	1.5427300899	1.258434e+01	1.2316217651	5.049568e-01	1.2076068661	8.317613e-04
0.8	1.4351491676	6.832250e+00	1.2061294870	3.076338e-02	1.2077295521	1.659826e-03
0.7	1.3223454750	2.800732e+00	1.2066348770	2.053914e-02	1.2078327767	3.756890e-03
0.6	1.2029703471	9.432425e-02	1.2097943109	4.373048e-02	1.2067390111	1.843054e-02
0.5	1.0780919174	2.195871e+00	1.4172230345	6.079244e+00	1.2081162362	9.518872e-03
0.4	0.9495337533	3.745734e+00	0.5641997450	6.473432e+00	div	div

Table 3. The computed values of $x_i$ and $f\left(x_i\right)$ for $i=1,2,3$ and different values of q by using $\mathsf{QAG}1$.

q	${\mathit{x}}_1$	$\mathit{f}\left({\mathit{x}}_1\right)$	${\mathit{x}}_2$	$\mathit{f}\left({\mathit{x}}_2\right)$	${\mathit{x}}_3$	$\mathit{f}\left({\mathit{x}}_3\right)$
1.02	0.5142791696	6.772056e+05	0.1232018298	3.004429e+04	0.1010035073	7.468324e+00
1.01	0.5097137812	6.681808e+05	0.1222121457	2.869170e+04	0.1010024505	6.053356e+00
0.9999	0.5051056237	6.591112e+05	0.1212441701	2.736999e+04	0.1010015437	4.839200e+00
0.99	0.5005916534	6.502654e+05	0.1203256645	2.611695e+04	0.1010007932	3.834194e+00
0.98	0.4960351599	6.413748e+05	0.1194278075	2.489313e+04	0.1010001573	2.982748e+00
0.97	0.4914818733	6.325288e+05	0.1185594952	2.371057e+04	0.1009996292	2.275633e+00
0.96	0.4869319106	6.237276e+05	0.1177201915	2.256843e+04	0.1009991952	1.694573e+00
0.95	0.4823853861	6.149710e+05	0.1169093583	2.146590e+04	0.1009988432	1.223128e+00
0.9	0.4597081677	5.718559e+05	0.1132633228	1.651858e+04	0.1009979633	4.506777e-02
0.8	0.4146701230	4.889359e+05	0.1077950521	9.130505e+03	0.1009981060	2.360760e-01
0.7	0.3701214518	4.103351e+05	0.1043132261	4.446092e+03	0.1009984405	6.839589e-01
0.6	0.3261190990	3.358939e+05	0.1023101377	1.758120e+03	0.1009983768	5.987359e-01
0.5	0.2826923050	2.654100e+05	0.1013283547	4.425044e+02	0.1009981173	2.511882e-01
0.4	0.2398433266	1.986490e+05	0.1009777753	2.698620e+01	0.1009979129	2.244417e-02

Table 4. The computed values of $x_i$ and $f\left(x_i\right)$ for $i=1,2,3$ and different values of q by using $\mathsf{QAG}2$.

q	${\mathit{x}}_1$	$\mathit{f}\left({\mathit{x}}_1\right)$	${\mathit{x}}_2$	$\mathit{f}\left({\mathit{x}}_2\right)$	${\mathit{x}}_3$	$\mathit{f}\left({\mathit{x}}_3\right)$
1.02	0.3836245813	4.338095e+05	0.1028722679	2.511938e+03	0.1009979297	3.473304e-05
1.01	0.3798572173	4.272299e+05	0.1027414335	2.336453e+03	0.1009979297	1.377622e-05
0.9999	0.3760675536	4.206350e+05	0.1026168764	2.169407e+03	0.1009979297	3.799508e-06
0.99	0.3723680428	4.142198e+05	0.1025018597	2.015174e+03	0.1009979297	3.969817e-07
0.98	0.3686464634	4.077891e+05	0.1023924737	1.868507e+03	0.1009979297	7.033782e-08
0.97	0.3649403724	4.014077e+05	0.1022896062	1.730594e+03	0.1009979297	7.550798e-08
0.96	0.3612498787	3.950754e+05	0.1021929573	1.601031e+03	0.1009979297	1.396345e-06
0.95	0.3575750888	3.887920e+05	0.1021022363	1.479425e+03	0.1009979297	5.200839e-06
0.9	0.3394402376	3.581029e+05	0.1017280112	9.779077e+02	0.1009979296	7.367781e-05
0.8	0.3044020340	3.002834e+05	0.1012821950	3.806792e+02	0.1009979295	3.115139e-04
0.7	0.2710667500	2.470282e+05	0.1010864659	1.185540e+02	0.1009979294	3.461354e-04
0.6	0.2394852751	1.981025e+05	0.1010160263	2.423142e+01	0.1009979296	1.702203e-04
0.5	0.2096843622	1.532571e+05	0.1009984548	7.031754e-01	0.1009979297	9.707395e-06

Table 5. The calculated values of $x_i$ and $f\left(x_i\right)$ for $i=1,2,3$ and different values of q by using $\mathsf{QAG}1$.

q	${\mathit{x}}_1$	$\mathit{f}\left({\mathit{x}}_1\right)$	${\mathit{x}}_2$	$\mathit{f}\left({\mathit{x}}_2\right)$	${\mathit{x}}_3$	$\mathit{f}\left({\mathit{x}}_3\right)$
0.9999	2.5202589147	4.378340e-01	2.1949546968	7.835843e-02	2.0201748257	1.283785e-02
0.99	2.5115105320	4.227815e-01	2.1831373107	7.175670e-02	2.0087578377	1.058966e-02
0.98	2.5025058715	4.076474e-01	2.1709804383	6.533731e-02	1.9971609647	8.509070e-03
0.97	2.4933259775	3.925895e-01	2.1585882787	5.917107e-02	1.9855208132	6.617240e-03
0.95	2.4744109780	3.627241e-01	2.1330269816	4.760752e-02	1.9622425575	3.387207e-03
0.9	2.4234483030	2.897601e-01	2.0632626947	2.326221e-02	1.9080108794	1.646041e-03
0.8	2.2993595203	1.537076e-01	1.8634563691	3.701777e-03	1.9049539739	1.839852e-03
0.7	2.1233417817	4.361666e-02	1.7302041526	4.077219e-03	1.7584545862	4.282942e-03
0.6	1.8638867517	3.689049e-03	1.8851091208	2.899285e-03	1.9024486639	1.992358e-03
0.5	1.5633390425	9.364898e-03	1.5791722007	7.962836e-03	1.5931768979	1.567623e-03
0.4	1.3893319818	4.831340e-02	1.4342000910	3.331323e-02	1.4679207997	2.455640e-02

Table 6. The calculated values of $x_i$ and $f\left(x_i\right)$ for $i=1,2,3$ and different values of q by using $\mathsf{QAG}2$.

q	${\mathit{x}}_1$	$\mathit{f}\left({\mathit{x}}_1\right)$	${\mathit{x}}_2$	$\mathit{f}\left({\mathit{x}}_2\right)$	${\mathit{x}}_3$	$\mathit{f}\left({\mathit{x}}_3\right)$
1.02	2.4541906439	3.324888e-01	2.1294971623	4.612880e-02	1.9793132558	5.685628e-03
1.01	2.4465460254	3.215032e-01	2.1199949026	4.228580e-02	1.9716511617	4.607177e-03
0.9999	2.4387247327	3.105124e-01	2.1103029536	3.856881e-02	1.9641322014	3.623214e-03
0.99	2.4309573266	2.998435e-01	2.1007067436	3.508499e-02	1.9570428610	2.760648e-03
0.98	2.4230062579	2.891735e-01	2.0909118772	3.172520e-02	1.9502501877	1.991531e-03
0.97	2.4149452314	2.786122e-01	2.0810081426	2.852389e-02	1.9439396580	1.325652e-03
0.96	2.4067694211	2.681611e-01	2.0709868209	2.547916e-02	1.9382576782	7.649241e-04
0.95	2.3984733326	2.578211e-01	2.0608364625	2.258868e-02	1.9333949804	3.134112e-04
0.9	2.3549374981	2.077991e-01	2.0074163967	1.033870e-02	1.9308693710	8.898415e-05
0.8	2.2524565779	1.158895e-01	1.8514430978	4.009256e-03	1.9336672846	3.380165e-04
0.7	2.1059949469	3.698096e-02	1.8046133598	4.487988e-03	1.8679147777	3.563896e-03
0.6	1.8657872051	3.631368e-03	1.8953688005	2.393342e-03	1.9150164412	1.168978e-03
0.5	1.5693521850	8.800824e-03	1.5911971423	7.070040e-03	1.6098711773	5.951690e-03
0.4	1.4247737346	3.613315e-02	1.4748490103	2.300092e-02	1.5105345877	1.619046e-02

For simplicity, initially in Examples 1–3, we check the performance of q-iterative methods with the classical methods for different values of q up to three iterations. Similarly, we can check the performance of the q-iterative methods for the different values of q for the rest of the iterations until we achieve the desired accuracy.

Example 1

(see [3]). We consider the nonlinear equation:

$$\begin{array}{c}\hfill f\left(x\right)=x{e}^{x^2}-{sin}^{2}x+3cosx+5.\end{array}$$

(51)

The exact solution for this example is $x=1.207647827130919$. We take $x_0=-2$ as an initial guess. Table 1 shows the computation of $x_i$ and $f\left(x_i\right)$ for $i=1,2,3$ and different values of q by using $\mathsf{QAG}1$.

Proceeding in the way of Table 1, we get $x=1.2076478271$ for different values of q, which is the required solution. One can observe from Table 1 that more accurate values of $x_i$s can be obtained when q approaches towards one and for which $f\left(x_i\right)$ tend towards zero. The values of $f\left(x_1\right)$ = 2.746138e + 01, $f\left(x_2\right)$ = 5.942719e + 00, $f\left(x_3\right)$ = 5.488326e − 01 calculated by $\mathsf{QAG}1$ at q = 0.9999 are closer to zero as compared to the values $f\left(x_1\right)$ = 2.747477e + 01, $f\left(x_2\right)$ = 5.949309e + 00, $f\left(x_3\right)$ = 5.505169e − 01 calculated by $\mathsf{CAG}1$. Furthermore, Equation (51) converges towards the root $x_5$ = 1.2076478271 for q = 0.9999 and $f\left(x\right)$ = 4.435401e − 12. Table 2 shows the computation of $x_i$ and $f\left(x_i\right)$ for i = 1, 2, 3 for different values of q by using the List C $\left(\mathsf{QAG}2\right)$.

Proceeding in the way of Table 2, we get the required solution x = 1.2076478271. From Table 2, we see that if q approaches one, we can obtain more accurate values of $x_i$s for which $f\left(x_i\right)$s tends towards zero, where $i=1,2,3$. It is also observe that the values of $f\left(x_1\right)$ = 2.117115e + 01, $f\left(x_2\right)$ = 2.896627e + 00, $f\left(x_3\right)$ = 3.622758e − 02 at q = 0.9999 calculated by $\mathsf{QAG}2$ are closer to zero as compared to the values of $f\left(x_1\right)$ = 2.118172e + 01, $f\left(x_2\right)$ = 2.900487e + 00, $f\left(x_3\right)$ = 3.644647e − 02 calculated by $\mathsf{CAG}2$. Furthermore, Equation (51) converges to the root $x_5$ = 1.2076478271 for q = 0.9999 and $f\left(x\right)$ = 1.424874e − 20.

Example 2

(see [39] (Population growth model)). Consider the nonlinear equation

$$\begin{array}{c}\hfill 1564000=1000000e^{\lambda }+\frac{435000}{\lambda }\left(e^{\lambda }-1\right).\end{array}$$

(52)

This equation appears in the mathematical modeling of the growth of population over short periods of time, where λ denotes the constant birth rate of population whose value needs to determined.

For computational work, we take $x_0$ = 1.5 as an initial estimate. The solution of this example approximated to 16 decimal digits is 0.1009979296857498. In Table 3, we compute the values of $x_1,x_2,x_3$ and $f\left(x_1\right),f\left(x_2\right),f\left(x_3\right)$ for different values of q by using $\mathsf{QAG}1$.

Proceeding as the way of Table 3, we get the required solution $x=0.1009979296857498$. From Table 3, It can easily observe that we obtain more accurate values of $x_1,x_2,x_3$ when q approaches one and for which $f\left(x_1\right),f\left(x_2\right),f\left(x_3\right)$ tend towards zero. The values of $f\left(x_1\right)$ = 6.591112e + 05, $f\left(x_2\right)$ = 2.736999e + 04, $f\left(x_3\right)$ = 4.839200e + 00 at q = 0.9999 computed by $\mathsf{QAG}1$ are closer to zero as compared to the other values of $f\left(x_1\right)$ = 6.592007e + 05, $f\left(x_2\right)$ = 2.738286e + 04, $f\left(x_3\right)$ = 4.850247e + 00 computed by $\mathsf{CAG}1$. Furthermore, Equation (52) converges towards the root $x_5$ = 0.1009979297 in the fifth iteration for q = 0.9999 and $f\left(x_5\right)$ = 4.545278e − 22.

Meantime, in Table 4, we compute the values of $x_1,x_2,x_3$ and $f\left(x_1\right),f\left(x_2\right),f\left(x_3\right)$ for different values of q by using $\mathsf{QAG}2$.

We can observe from Table 4 that we get more accurate values of $x_1,x_2,x_3$ when q approaches one and for which $f\left(x_1\right),f\left(x_2\right),f\left(x_3\right)$ tend towards zero. In addition, the values of $f\left(x_1\right)$ = 4.206350e + 05, $f\left(x_2\right)$ = 2.169407e + 03, $f\left(x_3\right)$ = 3.799508e − 06 calculated by $\mathsf{QAG}2$ at $q=0.9999$ are closer to zero as compared to the values $f\left(x_1\right)$ = 4.207000e + 05, $f\left(x_2\right)$ = 2.171013e + 03, $f\left(x_3\right)$ = 3.859455e − 06 calculated by $\mathsf{CAG}2$. Furthermore, Equation (52) converges towards the root $x_4=0.1009979296857498$ in the fourth iteration for q = 0.9999 and $f\left(x\right)$ = 4.103556e − 22.

Example 3

(see [40]). Consider the van der Waal’s equation

$$f\left(x\right)=0.986x^3-5.181x^2+9.067x-5.289.$$

(53)

This equation is used to interpret the real and ideal gas behavior that has been converted to the non-linear form after choosing the appropriate values of the parameters. Its exact solution is x = 1.92984624284786221849. Here, we take $x_0$ = 3.10.

By proceeding as the above table, we can obtain the required solution x = 1.9298462428. From Table 5, we can observe that we get more accurate values of $x_1,x_2,x_3$ for which $f\left(x_1\right),f\left(x_2\right),f\left(x_3\right)$ tend towards zero when q approaches one. The values of $f\left(x_1\right)$ = 4.378340e − 01, $f\left(x_2\right)$ = 7.835843e − 02, $f\left(x_3\right)$ = 1.283785e − 02 at q = 0.9999 computed by $\mathsf{QAG}1$ are closer to zero as compared to the values of $f\left(x_1\right)$ = 4.379863e − 01, $f\left(x_2\right)$ = 7.842636e − 0, $f\left(x_3\right)$ = 1.286153e − 02 computed by $\mathsf{CAG}1$. Furthermore, Equation (53) converges towards the root $x_7$ = 1.92984624284786221850 in the seventh iteration for q = 0.9999 and $f\left(x\right)$ = 3.081547e − 18.

The computational results obtained from Table 6 illustrate the accuracy of the values of $x_1,x_2,x_3$ when q approaches one and for which $f\left(x_1\right),f\left(x_2\right),f\left(x_3\right)$ tend towards zero. Moreover, the values of $f\left(x_1\right)$ = 3.105124e − 01, $f\left(x_2\right)$ = 3.856881e − 02, $f\left(x_3\right)$ = 3.623214e − 03 at q = 0.9999 computed by $\mathsf{QAG}2$, are closer to zero as compared to $f\left(x_1\right)$ = 3.106207e − 01, $f\left(x_2\right)$ = 3.860481e − 02, $f\left(x_3\right)$ = 3.632441e − 03 computed by $\mathsf{CAG}2$. Furthermore, Equation (53) converges towards the root $x_6$ = 1.92984624284786221849 in the sixth iteration for q = 0.9999 and $f\left(x\right)$ = 2.935505e − 21.

4.1. Error Analysis and Application of the q-Iterative Methods

Error is considered as the difference between a true value and an estimate (see [39]), or an approximation, it can easily be observed from the numerical values that if we compute the errors of q-iterative methods than they fluctuate for different values of q. The error decreases when q approaches to the extreme values between 0 and 1. In view of this result, the q-iterative methods are calculated for a large value of $q=0.99999$, which will approximate the ordinary iterative methods.

List B$\left(\mathsf{QAG}1\right)$
Nonlinear equation	True Solution	Approximate solution at q = 0.9999	Error
Equation (51)	1.20764782713091892701	1.20764782713091892700	7.187741e-21
Equation (52)	0.10099792968574978895	0.10099792968574978895	3.394397e-28
Equation (53)	1.92984624284786221849	1.92984624284786218283	3.565288e-17
List C$\left(\mathsf{QAG}2\right)$
Nonlinear equation	True Solution	Approximate solution at q = 0.9999	Error
Equation (51)	1.20764782713091892701	1.20764782713091892701	7.016521e-22
Equation (52)	0.10099792968574978895	0.10099792968574978895	3.064831e-28
Equation (53)	1.92984624284786221849	1.92984624284786221845	3.396321e-20

4.2. Comparison of the Classical and q-Analogue of Iterative Methods

Here, we check the robustness and efficiency of our new iterative methods by considering some of the nonlinear equations. Furthermore, we compare the standard Newton’s method ($\mathsf{NM}$), fourth order Chun method ($\mathsf{CM}$) (see [3]), $\mathsf{CAG}1$ and $\mathsf{CAG}2$ with our new iterative methods $\mathsf{QAG}1$ and $\mathsf{QAG}2$. In Table 7, we display the number of iterations ($\mathsf{IT}$), the approximate root $x_n$, the value $f\left(x_n\right)$ and $\delta$ be the distance between two successive estimations. It is important to mention that in order to get better computational results of the q-iterative methods, we take the value of $q=0.9999$.

$$\begin{array}{cc}\hfill f_1\left(x\right)& =x{e}^{x^2}-{sin}^{2}x+3cos\left(x\right)+5,\hfill \\ \hfill f_2\left(x\right)& =0.1056885297x^2+\left(\frac{e^x-e^{-x}}{2}\right)-sin\left(x\right),\hfill \\ \hfill f_3\left(x\right)& =0.986x^3-5.181x^2+9.067x-5.289,\hfill \\ \hfill f_4\left(x\right)& =\mathrm{1,000,000}{e}^x+\mathrm{435,000}\left(\frac{e^x-1}{x}\right)-\mathrm{1,564,000},\hfill \\ \hfill f_5\left(x\right)& =x^3+4x^2-10,\hfill \\ \hfill f_6\left(x\right)& =x^2-e^x-3x+2,\hfill \\ \hfill f_7\left(x\right)& =x^2-{(1-x)}^5,\hfill \\ \hfill f_8\left(x\right)& ={\left(x^3-9x^2+24x-20\right)}^{1/3}+e^{x/2},\hfill \\ \hfill f_9\left(x\right)& =\sqrt[3]{x^3-3x^2}+log(x+1).\hfill \end{array}$$

Table 7. The methods and their $\mathsf{IT}$, $x_n$, $f\left(x_n\right)$ and $\delta$.

$f_1\left(x\right)=x{e}^{x^2}-{sin}^{2}x+3cosx+5$, $x_0=-2$
Methods	$\mathsf{IT}$	$x_n$	$f\left(x_n\right)$	$\delta =|x_n-x_{n-1}|$
$\mathsf{NM}$	8	1.2076478271309189	5.538944e-20	4.261204e-11
CM	6	1.2076478271309189	1.063857e-17	4.175854e-10
$\mathsf{CAG}1$	6	1.2076478271309189	6.780624e-34	1.948602e-12
$\mathsf{QAG}1$	5	1.2076478271309189	4.435401e-12	7.441078e-05
$\mathsf{CAG}2$	5	1.2076478271309189	1.009799e-37	1.383883e-10
$\mathsf{QAG}2$	5	1.2076478271309189	1.424874e-20	1.175311e-10
$f_2\left(x\right)=0.1056885297x^2+\left(\frac{e^x-e^{-x}}{2}\right)-sin\left(x\right)$, $x_0=-1.2$
$\mathsf{NM}$	9	0.3170617745729571	5.220965e-21	1.571531e-10
CM	7	0.3170617745729571	2.444123e-46	4.508430e-16
$\mathsf{CAG}1$	6	0.3170617745729575	1.510838e-17	1.782637e-06
$\mathsf{QAG}1$	6	0.3170617745730157	1.964445e-15	1.741348e-06
$\mathsf{CAG}2$	5	0.3170617745729571	3.494619e-19	1.009576e-05
$\mathsf{QAG}2$	5	0.3170617745729571	2.502232e-19	1.000084e-05
$f_3\left(x\right)=0.986x^3-5.181x^2+9.067x-5.289$, $x_0=3.10$
$\mathsf{NM}$	10	1.9298462428478622	1.815641e-25	5.866916e-13
CM	7	1.9298462428478621	6.121465e-18	2.408838e-09
$\mathsf{CAG}1$	7	1.9298462428478622	1.803102e-28	3.036789e-10
$\mathsf{QAG}1$	7	1.9298462428478622	3.081547e-18	2.566058e-11
$\mathsf{CAG}2$	6	1.9298462428478622	7.761545e-40	5.605978e-11
$\mathsf{QAG}2$	6	1.9298462428478622	2.935505e-21	2.073804e-11
$f_4\left(x\right)=\mathrm{1,000,000}{e}^x+\mathrm{435,000}\left(\frac{e^x-1}{x}\right)-\mathrm{1,564,000}$, $x_0=1.5$
$\mathsf{NM}$	7	0.1009979296857498	7.697779e-31	1.104193e-18
CM	6	0.1009979296857498	2.569791e-33	4.511244e-20
$\mathsf{CAG}1$	5	0.1009979296857498	5.620211e-45	2.113407e-17
$\mathsf{QAG}1$	5	0.1009979296857498	4.545278e-22	1.496790e-17
$\mathsf{CAG}2$	4	0.1009979296857498	3.875494e-41	2.882365e-12
$\mathsf{QAG}2$	4	0.1009979296857498	4.103556e-22	2.837594e-12
$f_5\left(x\right)=x^3+4x^2-10$, $x_0=1$
$\mathsf{NM}$	5	1.2076478271309189	3.662513e-21	2.126976e-11
CM	41	1.2076478271309189	5.538944e-20	4.261204e-11
$\mathsf{CAG}1$	4	1.3652300134140968	3.000665e-38	1.557777e-13
$\mathsf{QAG}1$	4	1.3652300134140968	6.937438e-20	9.377190e-13
$\mathsf{CAG}2$	3	1.3652300134140968	2.672414e-27	1.361253e-07
$\mathsf{QAG}2$	3	1.3652300134140968	6.648073e-19	1.349188e-07
$f_6\left(x\right)=x^2-e^x-3x+2$, $x_0=2$
$\mathsf{NM}$	5	0.2575302854398608	3.439576e-27	9.869210e-14
CM	5	0.2575302854398608	9.348485e-20	3.638191e-10
$\mathsf{CAG}1$	4	0.2575302854398608	1.218871e-41	5.694586e-14
$\mathsf{QAG}1$	4	0.2575302854398608	9.378610e-25	4.284785e-14
$\mathsf{CAG}2$	3	0.2575302854398608	9.720474e-29	2.979348e-07
$\mathsf{QAG}2$	3	0.2575302854398608	1.658292e-23	2.973073e-07
$f_7\left(x\right)=x^2-{(1-x)}^5$, $x_0=0.9$
$\mathsf{NM}$	5	0.3459548158482420	6.819645e-17	6.158927e-09
CM	6	0.3459548158482420	4.839061e-22	1.160085e-11
$\mathsf{CAG}1$	4	0.3459548158482420	9.410608e-42	1.327460e-14
$\mathsf{QAG}1$	3	0.3459548158482420	7.943736e-17	1.736512e-05
$\mathsf{CAG}2$	3	0.3459548158482420	1.606415e-38	2.055302e-10
$\mathsf{QAG}2$	3	0.3459548158482420	1.855349e-23	1.990689e-10
$f_8\left(x\right)={\left(x^3-9x^2+24x-20\right)}^{1/3}+e^{x/2}$, $x_0=2$
$\mathsf{NM}$		Fail
CM		Fail
$\mathsf{CAG}1$		Fail
$\mathsf{QAG}1$	4	0.9694264485832326	2.465462e-15	7.651474e-06
$\mathsf{CAG}2$		Fail
$\mathsf{QAG}2$	4	0.9694264485832314	7.378707e-25	1.231459e-10
$f_9\left(x\right)=\sqrt[3]{x^3-3x^2}+log(x+1)$, $x_0=3$
$\mathsf{NM}$		Fail
CM		Fail
$\mathsf{CAG}1$		Fail
$\mathsf{QAG}1$	6	2.6925176762621718	1.218786e-16	6.562094e-10
$\mathsf{CAG}2$		Fail
$\mathsf{QAG}2$	5	2.6925176762621717	3.027817e-17	4.453735e-07

Table 7 compares the solutions obtained by using the classical and our the q-iterative methods. The results show that our q-analogue iterative methods $\mathsf{QAG}1$ and $\mathsf{QAG}2$ give the same results as the classical methods $\mathsf{CAG}1$ and $\mathsf{CAG}2$. The functions $f_8\left(x\right),f_9\left(x\right)$ are not differentiable at $x=2$ and $x=3$, respectively, when we choose $x=2$ and $x=3$ as initial guesses for $f_8\left(x\right)$ and $f_9\left(x\right)$, respectively. Then, the iterative methods: $\mathsf{NM},\mathsf{CM},\mathsf{CAG}1$ and $\mathsf{CAG}2$ fail, while the new iterative methods $\mathsf{QAG}1$ and $\mathsf{QAG}2$ are applicable and give rapid convergent results.

Remark 1

(see [12]). The efficiency index is considered as $P^{\frac{1}{m}}$, where P represents the order of the method and m is the total number of function evaluations per iteration necessary by the method.

• Efficiency index of $\mathsf{CAG}1$ is $3^{\frac{1}{3}}=1.442250$.
• Efficiency index of $\mathsf{CAG}2$ is $4^{\frac{1}{4}}=1.414214$.
• Efficiency index of $\mathsf{QAG}1$ is $3^{\frac{1}{2}}=1.442250$.
• Efficiency index of $\mathsf{QAG}2$ is $4^{\frac{1}{4}}=1.414214$.

We conclude that the efficiency indexes calculated by $\mathsf{QAG}1$ and $\mathsf{QAG}2$ are the same as those calculated by $\mathsf{CAG}1$ and $\mathsf{CAG}2$.

5. Generalization of the Iterative Scheme in $\mathit{q}$-Calculus

In this section, based on our previous results, we determine the generalized q-iterative scheme. By adding the values of $x_i{s}^{\prime }$$(\forall i=1,2,\dots ,n)$ in (19), we obtain

$$\begin{array}{c}\hfill x_1+x_2+\cdots +x_n=x_0+M_q\left(x_0+x_1+\cdots +x_{n-1}\right).\end{array}$$

From (22), we have

$$\begin{array}{cc}\hfill & x_0=\lambda -\frac{f\left(\lambda \right)}{D_{q}f\left(\lambda \right)},\hfill \\ \hfill & x_0+x_1=x_0-\frac{f\left(x_0\right)}{D_{q}f\left(\lambda \right)}=x_0+M_q\left(x_0\right),\hfill \\ \hfill & x=x_0+x_1+x_2=x_0+x_1-\frac{f(x_0+x_1)}{D_{q}f\left(\lambda \right)}=x_0+M_q(x_0+x_1).\hfill \end{array}$$

Now, if x is approximated by

$$\begin{array}{cc}\hfill & x=x_0+x_1+\cdots +x_n=x_0+M_q(x_0+x_1+\cdots +x_{n-1})\hfill \\ \hfill & x=x_0+x_1+\cdots +x_{n-1}-\frac{f(x_0+x_1+\cdots +x_{n-1})}{D_{q}f\left(\lambda \right)}\hfill \\ \hfill & x=x_0+x_1+\cdots +x_n=x_0+x_1+\cdots +x_{n-1}-\frac{f(x_0+x_1+\cdots +x_{n-1})}{D_{q}f\left(\lambda \right)}.\hfill \end{array}$$

(54)

Therefore, (54) gives the following iterative scheme:

$$\begin{array}{cc}\hfill & x_n=-\frac{f(x_0+x_1+\cdots +x_{n-1})}{D_{q}f\left(\lambda \right)}(\forall n=1,2,3,\dots );\hfill \end{array}$$

(55)

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill & x_0=\lambda -\frac{f\left(\lambda \right)}{D_{q}f\left(\lambda \right)},\hfill \\ \hfill & x_0+x_1=x_0-\frac{f\left(x_0\right)}{D_{q}f\left(\lambda \right)},\hfill \\ \hfill & x_0+x_1+x_2=x_0+x_1-\frac{f(x_0+x_1)}{D_{q}f\left(\lambda \right)},\hfill \\ \hfill & ⋮\hfill \\ \hfill & x_0+x_1+x_2+\dots +x_{n-1}+x_n=x_0+x_1+x_2+\dots +x_{n-1}-\frac{f(x_0+x_1+x_2+\dots +x_{n-1})}{D_{q}f\left(\lambda \right)}.\hfill \end{array}\right.\end{array}$$

(56)

This formulation allows us to suggest the following generalized q iterative method:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill & y_0=x_m,\hfill \\ \hfill & y_1=y_0-\frac{f\left(y_0\right)}{D_{q}f\left(y_0\right)},\hfill \\ \hfill & y_2=y_1-\frac{f\left(y_1\right)}{D_{q}f\left(y_0\right)},\hfill \\ \hfill & y_3=y_2-\frac{f\left(y_2\right)}{D_{q}f\left(y_0\right)},\hfill \\ \hfill & ⋮\hfill \\ \hfill & y_{n+1}=y_n-\frac{f\left(y_n\right)}{D_{q}f\left(y_0\right)},\hfill \\ \hfill & x_{m+1}=y_{n+1}.\hfill \end{array}\right.\end{array}$$

The order of convergence of the iterative scheme is $n+2$ for $n=0,1,2,\dots$ and the number of functional evaluations is also $n+2$ as well.

6. Conclusions

The main target of this article is to introduce novel algorithms for solution of the nonlinear equations in the context of the q-calculus. The new algorithms are introduced by using the Daftardar–Jafari decomposition technique. The comparison of these newly established algorithms with the classical methods reflects that the proposed q-iterative methods are reliable and best alternatives to the already known algorithms. The computational results conclude that the q-analogue of the iterative methods for solving the algebraic nonlinear equations generate the same results as the classical methods, but convergence rate towards approaching the root is higher than convergence rate suggested by the classical methods. Furthermore, the errors associated with the proposed methods are comparatively lesser by the appropriately chosen value of q being close to one. The difficulty in this method, which needs future investigation, is that we need to estimate the value of $q\in (0,1)$.