Multi-Step Quantum Numerical Techniques for Finding the Solutions of Nonlinear Equations

Abstract

In this paper, we analyze the $\mathrm{q}$-iterative schemes to determine the roots of nonlinear equations by applying the decomposition technique with Simpson’s $\frac{1}{3}$-rule in the setting of q-calculus. We discuss the convergence analysis of our suggested iterative methods. To check the efficiency and performance, we also compare our main outcomes with some well known techniques existing in the literature.

1. Introduction

In recent years, many numerical analysts have shown their keen interest in developing new different order iterative schemes for finding the solutions of nonlinear equations using various techniques, such as variational iterative methods and decomposition techniques; for details, see [1,2,3,4,5,6,7,8,9,10,11,12,13,14]. A famous decomposition technique is the Adomian decomposition [15] in which a solution is considered in terms of an in infinite series, which converges towards the exact solution. Using this technique, Chun [2] and Abbasbandy [16] constructed and investigated different higher order iterative methods. Using different modifications of the Adomian decomposition method, Daftardar-Gejji and Jafari [17] have used different modifications and suggested a technique that does not need derivative evaluation of the Adomian polynomial. Saqib and Iqbal [18] and Ali et al. [19,20] have used this decomposition technique and developed a family of iterative methods having better efficiency and convergence order.

In this paper, we introduce new two and three-step iterative methods for solving the nonlinear equations. We also prove the convergence of the proposed method. Some numerical examples are presented to make a comparative study of the newly constructed methods with known iterative algorithms.

2. Construction of q-Iterative Methods

In this section, some new different order multi-step $\mathrm{q}$-iterative methods are constructed by considering the Simpson’s one third rule in the setting of $\mathrm{q}$-calculus and using the technique of decomposition [17,21]. However, before that, we recall some basic preliminaries from quantum calculus.

Quantum calculus is the branch of mathematics in which we obtain q-analogues of mathematical objects that can be recaptured by taking $\mathrm{q}\to 1^{-}$. It is also known as calculus without limits. Although the history of quantum calculus is old, but during the last century it has experienced rapid development due to its many applications in various fields of pure and applied sciences. Quantum calculus also serves as the bridge between mathematics and physics, see [22,23,24,25].

In quantum calculus [24] (often known as q-calculus), a q-integer is defined as ${\left[n\right]}_{\mathrm{q}}=\frac{1-{\mathrm{q}}^n}{1-\mathrm{q}}$, where $0<\mathrm{q}<1$. The $\mathrm{q}$-factorial is defined as:

$${\left[n\right]}_{\mathrm{q}}!={\left[n\right]}_{\mathrm{q}}{[n-1]}_{\mathrm{q}}\dots {\left[1\right]}_{\mathrm{q}},$$

where ${\left[0\right]}_{\mathrm{q}}!=1$.

For $0\le k\le n$, the $\mathrm{q}$-binomials are defined as:

$${\left(\genfrac{}{}{0pt}{}{n}{k}\right)}_{\mathrm{q}}=\frac{{\left[n\right]}_{\mathrm{q}}!}{{\left[k\right]}_{\mathrm{q}}!{[n-k]}_{\mathrm{q}}!}.$$

The $\mathrm{q}$-derivative is defined as:

Definition 1.

The $\mathrm{q}$-derivative of a real valued and continuous function Ψ is defined as:

$$\begin{array}{c}\hfill {\mathrm{D}}_q\mathsf{\Psi }\left(\varpi \right)=\frac{\mathsf{\Psi }\left(\varpi \right)-\mathsf{\Psi }\left(\mathrm{q}\varpi \right)}{(1-\mathrm{q})\varpi }\end{array}$$

We can recapture the classical derivative by taking $\mathrm{q}\to 1^{-}$.

The q-derivatives of product and quotient of functions f and g are defined, respectively, as:

$$\begin{array}{c}\hfill {\mathrm{D}}_{\mathrm{q}}\left(\mathsf{\Psi }\left(\varpi \right)\mathsf{\Phi }\left(\varpi \right)\right)=\mathsf{\Phi }\left(\varpi \right){\mathrm{D}}_{\mathrm{q}}\mathsf{\Psi }\left(\varpi \right)+\mathsf{\Psi }\left(\mathrm{q}\varpi \right){\mathrm{D}}_q\mathsf{\Phi }\left(\varpi \right),\end{array}$$

and

$$\begin{array}{c}\hfill {\mathrm{D}}_{\mathrm{q}}\left(\frac{\mathsf{\Psi }\left(\varpi \right)}{\mathsf{\Phi }\left(\varpi \right)}\right)=\frac{\mathsf{\Phi }\left(\varpi \right){\mathrm{D}}_{\mathrm{q}}\mathsf{\Psi }\left(\varpi \right)-\mathsf{\Psi }\left(\varpi \right){\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\varpi \right)}{\mathsf{\Phi }\left(\mathrm{q}\varpi \right)\mathsf{\Phi }\left(\varpi \right)},\mathsf{\Phi }\left(\mathrm{q}\varpi \right)\mathsf{\Phi }\left(\varpi \right)\ne 0.\end{array}$$

Definition 2.

Let Ψ be a continuous function; then, for any c the $\mathrm{q}$-Taylor’s formula is given by:

$$\begin{array}{c}\hfill \mathsf{\Psi }\left(\varpi \right)=\sum _{k=1}^{\infty }\frac{{\mathrm{D}}_{\mathrm{q}}^k\mathsf{\Psi }\left(c\right){(\varpi -c)}^k}{\left[k\right]!}.\end{array}$$

Jackson [26] introduced the $\mathrm{q}$-Taylor’s formula. Then, Jing and Fan [27] derived $\mathrm{q}$-Taylor’s formula with its $\mathrm{q}$-remainder by using the $\mathrm{q}$-differentiation approach. For further details, see [24].

Simpson’s One Third Rule

Consider the nonlinear equation

$$\mathsf{\Psi }\left(\varpi \right)=0,$$

(1)

which is equivalent to

$$\varpi =\mathsf{\Phi }\left(\varpi \right).$$

(2)

Assume that $\alpha$ is a simple root of nonlinear Equation (1), and $\gamma$ is an initial guess sufficiently close to the root. Using the fundamental theorem of calculus and Simpson’s one third quadrature formula in the $\mathrm{q}$-calculus, we have

$$\varpi =\mathsf{\Phi }\left(\gamma \right)+\frac{1}{6}\left(\varpi -\gamma \right)\left({\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\varpi \right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{\varpi +\gamma }{2}\right)+{\mathrm{D}}_{\mathrm{q}}\left(\gamma \right)\right).$$

Now, using the technique of He [28], the nonlinear Equation (2) can be written as an equivalent coupled system of equations

$$\varpi =\mathsf{\Phi }\left(\gamma \right)+\frac{1}{6}\left(\varpi -\gamma \right)\left({\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\varpi \right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{\varpi +\gamma }{2}\right)+{\mathrm{D}}_{\mathrm{q}}\left(\gamma \right)\right)+H\left(\varpi \right)$$

$$\begin{array}{ccc}\hfill H\left(\varpi \right)& =& \mathsf{\Phi }\left(\varpi \right)-\mathsf{\Phi }\left(\gamma \right)-\frac{1}{6}\left(\varpi -\gamma \right)\left({\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\varpi \right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{\varpi +\gamma }{2}\right)+{\mathrm{D}}_{\mathrm{q}}\left(\gamma \right)\right)\hfill \\ & =& \varpi \left(1-\frac{1}{6}\left({\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\varpi \right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{\varpi +\gamma }{2}\right)+{\mathrm{D}}_{\mathrm{q}}\left(\gamma \right)\right)\right)-\mathsf{\Phi }\left(\gamma \right)\hfill \\ & & +\frac{\gamma }{6}\left({\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\varpi \right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{\varpi +\gamma }{2}\right)+{\mathrm{D}}_{\mathrm{q}}\left(\gamma \right)\right),\hfill \end{array}$$

(3)

from which it follows that

$$\begin{array}{ccc}\hfill \varpi & =& \frac{H\left(\varpi \right)}{1-\frac{1}{6}\left({\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\varpi \right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{\varpi +\gamma }{2}\right)+{\mathrm{D}}_{\mathrm{q}}\left(\gamma \right)\right)}\hfill \\ & & +\frac{\mathsf{\Phi }\left(\gamma \right)-\frac{\gamma }{6}\left({\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\varpi \right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{\varpi +\gamma }{2}\right)+{\mathrm{D}}_{\mathrm{q}}\left(\gamma \right)\right)}{1-\frac{1}{6}\left({\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\varpi \right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{\varpi +\gamma }{2}\right)+{\mathrm{D}}_{\mathrm{q}}\left(\gamma \right)\right)}\hfill \\ & =& \mu +M_{\mathrm{q}}\left(\varpi \right),\hfill \end{array}$$

(4)

where

$$\mu =\gamma$$

(5)

$$\begin{array}{ccc}\hfill M_{\mathrm{q}}\left(\varpi \right)& =& \frac{H\left(\varpi \right)}{1-\frac{1}{6}\left({\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\varpi \right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{\varpi +\gamma }{2}\right)+{\mathrm{D}}_{\mathrm{q}}\left(\gamma \right)\right)}\hfill \\ & & +\frac{\mathsf{\Phi }\left(\gamma \right)-\gamma }{1-\frac{1}{6}\left({\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\varpi \right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{\varpi +\gamma }{2}\right)+{\mathrm{D}}_{\mathrm{q}}\left(\gamma \right)\right)}.\hfill \end{array}$$

(6)

Now, by applying the decomposition technique of Daftardar-Gejji and Jafari [17], we have

For $n=0,$

$$\varpi \approx U_0={\varpi }_0=\mu =\gamma .$$

(7)

From (3), it can easily be computed as

$$H\left({\varpi }_0\right)=0.$$

Using (6), we get

$$\begin{array}{ccc}\hfill {\varpi }_1& =& M_{\mathrm{q}}\left({\varpi }_0\right)=\frac{H\left({\varpi }_0\right)}{1-\frac{1}{6}\left({\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left({\varpi }_0\right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{{\varpi }_0+\gamma }{2}\right)+{\mathrm{D}}_{\mathrm{q}}\left(\gamma \right)\right)}\hfill \\ & & +\frac{\mathsf{\Phi }\left(\gamma \right)-\gamma }{1-\frac{1}{6}\left({\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left({\varpi }_0\right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{{\varpi }_0+\gamma }{2}\right)+{\mathrm{D}}_{\mathrm{q}}\left(\gamma \right)\right)}\hfill \\ & =& \frac{\mathsf{\Phi }\left(\gamma \right)-\gamma }{1-\frac{1}{6}\left({\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left({\varpi }_0\right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{{\varpi }_0+\gamma }{2}\right)+{\mathrm{D}}_{\mathrm{q}}\left(\gamma \right)\right)}.\hfill \end{array}$$

For $n=1,$

$$\varpi \approx U_1={\varpi }_0+{\varpi }_1={\varpi }_0+M_{\mathrm{q}}\left({\varpi }_0\right)$$

(8)

$$\varpi \approx U_1={\varpi }_0+{\varpi }_1=\gamma +\frac{\mathsf{\Phi }\left(\gamma \right)-\gamma }{1-\frac{1}{6}\left({\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left({\varpi }_0\right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{{\varpi }_0+\gamma }{2}\right)+{\mathrm{D}}_{\mathrm{q}}\left(\gamma \right)\right)}.$$

Using (7), we have

$$\varpi =\frac{\mathsf{\Phi }\left(\gamma \right)-\gamma {\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\gamma \right)}{1-{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\gamma \right)}.$$

(9)

From (3), we have

$$\begin{array}{ccc}\hfill H\left({\varpi }_0+{\varpi }_1\right)& =& \mathsf{\Phi }\left({\varpi }_0+{\varpi }_1\right)-\mathsf{\Phi }\left(\gamma \right)\hfill \\ & & -\frac{1}{6}\left({\varpi }_0+{\varpi }_1-\gamma \right)\left({\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left({\varpi }_0+{\varpi }_1\right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{{\varpi }_0+{\varpi }_1+\gamma }{2}\right)+{\mathrm{D}}_{\mathrm{q}}\left(\gamma \right)\right).\hfill \end{array}$$

and

$$\begin{array}{ccc}& & {\varpi }_1+{\varpi }_2=M_{\mathrm{q}}\left({\varpi }_0+{\varpi }_1\right)\\ & =& \frac{H\left({\varpi }_0+{\varpi }_1\right)}{1-\frac{1}{6}\left({\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left({\varpi }_0+{\varpi }_1\right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{{\varpi }_0+{\varpi }_1+\gamma }{2}\right)+{\mathrm{D}}_{\mathrm{q}}\left(\gamma \right)\right)}\hfill \\ & & +\frac{\mathsf{\Phi }\left(\gamma \right)-\gamma }{1-\frac{1}{6}\left({\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left({\varpi }_0+{\varpi }_1\right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{{\varpi }_0+{\varpi }_1+\gamma }{2}\right)+{\mathrm{D}}_{\mathrm{q}}\left(\gamma \right)\right)}\hfill \\ & =& \frac{\mathsf{\Phi }\left({\varpi }_0+{\varpi }_1\right)-\mathsf{\Phi }\left(\gamma \right)-\frac{1}{6}\left({\varpi }_0+{\varpi }_1-\gamma \right)\left({\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left({\varpi }_0+{\varpi }_1\right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{{\varpi }_0+{\varpi }_1+\gamma }{2}\right)+{\mathrm{D}}_{\mathrm{q}}\left(\gamma \right)\right)}{1-\frac{1}{6}\left({\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left({\varpi }_0+{\varpi }_1\right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{{\varpi }_0+{\varpi }_1+\gamma }{2}\right)+{\mathrm{D}}_{\mathrm{q}}\left(\gamma \right)\right)}\hfill \\ & & +\frac{\mathsf{\Phi }\left(\gamma \right)-\gamma }{1-\frac{1}{6}\left({\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left({\varpi }_0+{\varpi }_1\right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{{\varpi }_0+{\varpi }_1+\gamma }{2}\right)+{\mathrm{D}}_{\mathrm{q}}\left(\gamma \right)\right)}\hfill \\ & =& \frac{\mathsf{\Phi }\left({\varpi }_0+{\varpi }_1\right)}{1-\frac{1}{6}\left({\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left({\varpi }_0+{\varpi }_1\right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{{\varpi }_0+{\varpi }_1+\gamma }{2}\right)+{\mathrm{D}}_{\mathrm{q}}\left(\gamma \right)\right)}\hfill \\ & & -\frac{\frac{1}{6}\left({\varpi }_0+{\varpi }_1-\gamma \right)\left({\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left({\varpi }_0+{\varpi }_1\right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{{\varpi }_0+{\varpi }_1+\gamma }{2}\right)+{\mathrm{D}}_{\mathrm{q}}\left(\gamma \right)\right)}{1-\frac{1}{6}\left({\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left({\varpi }_0+{\varpi }_1\right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{{\varpi }_0+{\varpi }_1+\gamma }{2}\right)+{\mathrm{D}}_{\mathrm{q}}\left(\gamma \right)\right)}\hfill \\ & & -\frac{\gamma }{1-\frac{1}{6}\left({\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left({\varpi }_0+{\varpi }_1\right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{{\varpi }_0+{\varpi }_1+\gamma }{2}\right)+{\mathrm{D}}_{\mathrm{q}}\left(\gamma \right)\right)}.\hfill \end{array}$$

For $n=2,$

$$\begin{array}{ccc}\hfill \varpi & \approx & U_2={\varpi }_0+{\varpi }_1+{\varpi }_2=\mu +M_{\mathrm{q}}\left({\varpi }_0+{\varpi }_1\right).\hfill \\ & =& \gamma +\frac{\mathsf{\Phi }\left({\varpi }_0+{\varpi }_1\right)}{1-\frac{1}{6}\left({\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left({\varpi }_0+{\varpi }_1\right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{{\varpi }_0+{\varpi }_1+\gamma }{2}\right)+{\mathrm{D}}_{\mathrm{q}}\left(\gamma \right)\right)}\hfill \\ & & -\frac{\frac{1}{6}\left({\varpi }_0+{\varpi }_1-\gamma \right)\left({\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left({\varpi }_0+{\varpi }_1\right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{{\varpi }_0+{\varpi }_1+\gamma }{2}\right)+{\mathrm{D}}_{\mathrm{q}}\left(\gamma \right)\right)}{1-\frac{1}{6}\left({\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left({\varpi }_0+{\varpi }_1\right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{{\varpi }_0+{\varpi }_1+\gamma }{2}\right)+{\mathrm{D}}_{\mathrm{q}}\left(\gamma \right)\right)}\hfill \\ & & -\frac{\gamma }{1-\frac{1}{6}\left({\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left({\varpi }_0+{\varpi }_1\right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{{\varpi }_0+{\varpi }_1+\gamma }{2}\right)+{\mathrm{D}}_{\mathrm{q}}\left(\gamma \right)\right)}.\hfill \end{array}$$

Take

$$\begin{array}{cc}{\varpi }_0+{\varpi }_1=v=& \frac{\mathsf{\Phi }\left(\gamma \right)-\gamma {\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\gamma \right)}{1-{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\gamma \right)}\hfill \\ & =\gamma +\frac{\mathsf{\Phi }\left(v\right)}{1-\frac{1}{6}\left({\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(v\right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{v+\gamma }{2}\right)+{\mathrm{D}}_{\mathrm{q}}\left(\gamma \right)\right)}\hfill \\ & \hspace{1em}-\frac{\frac{1}{6}\left(v-\gamma \right)\left({\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(v\right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{v+\gamma }{2}\right)+{\mathrm{D}}_{\mathrm{q}}\left(\gamma \right)\right)}{1-\frac{1}{6}\left({\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(v\right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{v+\gamma }{2}\right)+{\mathrm{D}}_{\mathrm{q}}\left(\gamma \right)\right)}\hfill \\ & \hspace{1em}-\frac{\gamma }{1-\frac{1}{6}\left({\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(v\right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{v+\gamma }{2}\right)+{\mathrm{D}}_{\mathrm{q}}\left(\gamma \right)\right)}\hfill \\ & =\frac{\mathsf{\Phi }\left(v\right)}{1-\frac{1}{6}\left({\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(v\right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{v+\gamma }{2}\right)+{\mathrm{D}}_{\mathrm{q}}\left(\gamma \right)\right)}\hfill \\ & \hspace{1em}-\frac{\frac{1}{6}v\left({\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(v\right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{v+\gamma }{2}\right)+{\mathrm{D}}_{\mathrm{q}}\left(\gamma \right)\right)}{1-\frac{1}{6}\left({\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(v\right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{v+\gamma }{2}\right)+{\mathrm{D}}_{\mathrm{q}}\left(\gamma \right)\right)}.\hfill \end{array}$$

This relation yields the following two-step method for solving a nonlinear equation.

It is noted that

$$\begin{array}{cc}{\varpi }_0+{\varpi }_1& +{\varpi }_2=w\hfill \\ & =\frac{\mathsf{\Phi }\left({\varpi }_0+{\varpi }_1\right)}{1-\frac{1}{6}\left({\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left({\varpi }_0+{\varpi }_1\right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{{\varpi }_0+{\varpi }_1+\gamma }{2}\right)+{\mathrm{D}}_{\mathrm{q}}\left(\gamma \right)\right)}\hfill \\ & \hspace{1em}-\frac{\frac{1}{6}\left({\varpi }_0+{\varpi }_1\right)\left({\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left({\varpi }_0+{\varpi }_1\right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{{\varpi }_0+{\varpi }_1+\gamma }{2}\right)+{\mathrm{D}}_{\mathrm{q}}\left(\gamma \right)\right)}{1-\frac{1}{6}\left({\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left({\varpi }_0+{\varpi }_1\right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{{\varpi }_0+{\varpi }_1+\gamma }{2}\right)+{\mathrm{D}}_{\mathrm{q}}\left(\gamma \right)\right)},\hfill \end{array}$$

(10)

From (3), we can write

$$\begin{array}{ccc}\hfill H\left({\varpi }_0+{\varpi }_1+{\varpi }_2\right)& =& \mathsf{\Phi }\left({\varpi }_0+{\varpi }_1+{\varpi }_2\right)-\mathsf{\Phi }\left(\gamma \right)\hfill \\ & & -\frac{1}{6}\left({\varpi }_0+{\varpi }_1+{\varpi }_2-\gamma \right)\left({\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left({\varpi }_0+{\varpi }_1+{\varpi }_2\right)\right.\hfill \\ & & \left.+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{{\varpi }_0+{\varpi }_1+{\varpi }_2+\gamma }{2}\right)+{\mathrm{D}}_{\mathrm{q}}\left(\gamma \right)\right),\hfill \end{array}$$

and

$$\begin{array}{c}{\varpi }_1+{\varpi }_2+{\varpi }_3=M_{\mathrm{q}}\left({\varpi }_0+{\varpi }_1+{\varpi }_2\right)\hfill \\ =\frac{H\left({\varpi }_0+{\varpi }_1+{\varpi }_2\right)}{1-\frac{1}{6}\left({\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left({\varpi }_0+{\varpi }_1+{\varpi }_2\right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{{\varpi }_0+{\varpi }_1+{\varpi }_2+\gamma }{2}\right)+{\mathrm{D}}_{\mathrm{q}}\left(\gamma \right)\right)}\hfill \\ \hspace{1em}+\frac{\mathsf{\Phi }\left(\gamma \right)-\gamma }{1-\frac{1}{6}\left({\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left({\varpi }_0+{\varpi }_1+{\varpi }_2\right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{{\varpi }_0+{\varpi }_1+{\varpi }_2+\gamma }{2}\right)+{\mathrm{D}}_{\mathrm{q}}\left(\gamma \right)\right)}\hfill \\ =\frac{1}{1-\frac{1}{6}\left({\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left({\varpi }_0+{\varpi }_1+{\varpi }_2\right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{{\varpi }_0+{\varpi }_1+{\varpi }_2+\gamma }{2}\right)+{\mathrm{D}}_{\mathrm{q}}\left(\gamma \right)\right)}\left(\mathsf{\Phi }\left({\varpi }_0+{\varpi }_1+{\varpi }_2\right)\right.\hfill \\ \hspace{1em}-\mathsf{\Phi }\left(\gamma \right)-\frac{1}{6}\left({\varpi }_0+{\varpi }_1+{\varpi }_2-\gamma \right)\left({\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left({\varpi }_0+{\varpi }_1+{\varpi }_2\right)\right.\hfill \\ \left.\hspace{1em}+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{{\varpi }_0+{\varpi }_1+{\varpi }_2+\gamma }{2}\right)+{\mathrm{D}}_{\mathrm{q}}\left(\gamma \right)\right)\hfill \\ \hspace{1em}+\frac{\mathsf{\Phi }\left(\gamma \right)-\gamma }{1-\frac{1}{6}\left({\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left({\varpi }_0+{\varpi }_1+{\varpi }_2\right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{{\varpi }_0+{\varpi }_1+{\varpi }_2+\gamma }{2}\right)+{\mathrm{D}}_{\mathrm{q}}\left(\gamma \right)\right)}\hfill \\ =\frac{\mathsf{\Phi }\left({\varpi }_0+{\varpi }_1+{\varpi }_2\right)}{1-\frac{1}{6}\left({\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left({\varpi }_0+{\varpi }_1+{\varpi }_2\right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{{\varpi }_0+{\varpi }_1+{\varpi }_2+\gamma }{2}\right)+{\mathrm{D}}_{\mathrm{q}}\left(\gamma \right)\right)}\hfill \\ \hspace{1em}-\frac{\frac{1}{6}\left({\varpi }_0+{\varpi }_1+{\varpi }_2-\gamma \right)\left({\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left({\varpi }_0+{\varpi }_1+{\varpi }_2\right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{{\varpi }_0+{\varpi }_1+{\varpi }_2+\gamma }{2}\right)+{\mathrm{D}}_{\mathrm{q}}\left(\gamma \right)\right)}{1-\frac{1}{6}\left({\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left({\varpi }_0+{\varpi }_1+{\varpi }_2\right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{{\varpi }_0+{\varpi }_1+{\varpi }_2+\gamma }{2}\right)+{\mathrm{D}}_{\mathrm{q}}\left(\gamma \right)\right)}\hfill \\ \hspace{1em}-\frac{\gamma }{1-\frac{1}{6}\left({\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left({\varpi }_0+{\varpi }_1+{\varpi }_2\right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{{\varpi }_0+{\varpi }_1+{\varpi }_2+\gamma }{2}\right)+{\mathrm{D}}_{\mathrm{q}}\left(\gamma \right)\right)}.\hfill \end{array}$$

For $n=3$

$$\begin{array}{ccc}\hfill \varpi & \approx & U_3={\varpi }_0+{\varpi }_1+{\varpi }_2+{\varpi }_3={\varpi }_0+M_{\mathrm{q}}\left({\varpi }_0+{\varpi }_1+{\varpi }_2\right)\hfill \\ & =& \gamma +\frac{\mathsf{\Phi }\left({\varpi }_0+{\varpi }_1+{\varpi }_2\right)}{1-\frac{1}{6}\left({\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left({\varpi }_0+{\varpi }_1+{\varpi }_2\right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{{\varpi }_0+{\varpi }_1+{\varpi }_2+\gamma }{2}\right)+{\mathrm{D}}_{\mathrm{q}}\left(\gamma \right)\right)}\hfill \\ & & -\frac{\frac{1}{6}\left({\varpi }_0+{\varpi }_1+{\varpi }_2-\gamma \right)\left({\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left({\varpi }_0+{\varpi }_1+{\varpi }_2\right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{{\varpi }_0+{\varpi }_1+{\varpi }_2+\gamma }{2}\right)+{\mathrm{D}}_{\mathrm{q}}\left(\gamma \right)\right)}{1-\frac{1}{6}\left({\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left({\varpi }_0+{\varpi }_1+{\varpi }_2\right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{{\varpi }_0+{\varpi }_1+{\varpi }_2+\gamma }{2}\right)+{\mathrm{D}}_{\mathrm{q}}\left(\gamma \right)\right)}\hfill \\ & & -\frac{\gamma }{1-\frac{1}{6}\left({\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left({\varpi }_0+{\varpi }_1+{\varpi }_2\right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{{\varpi }_0+{\varpi }_1+{\varpi }_2+\gamma }{2}\right)+{\mathrm{D}}_{\mathrm{q}}\left(\gamma \right)\right)}.\hfill \end{array}$$

Using (10), we obtain

$$\begin{array}{ccc}& =& \gamma +\frac{\mathsf{\Phi }\left(w\right)}{1-\frac{1}{6}\left({\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(w\right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{w+\gamma }{2}\right)+{\mathrm{D}}_{\mathrm{q}}\left(\gamma \right)\right)}\hfill \\ & & -\frac{\frac{1}{6}\left(w-\gamma \right)\left({\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(w\right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{w+\gamma }{2}\right)+{\mathrm{D}}_{\mathrm{q}}\left(\gamma \right)\right)}{1-\frac{1}{6}\left({\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(w\right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{w+\gamma }{2}\right)+{\mathrm{D}}_{\mathrm{q}}\left(\gamma \right)\right)}\hfill \\ & & -\frac{\gamma }{1-\frac{1}{6}\left({\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(w\right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{w+\gamma }{2}\right)+{\mathrm{D}}_{\mathrm{q}}\left(\gamma \right)\right)}\hfill \\ & =& \frac{\mathsf{\Phi }\left(w\right)}{1-\frac{1}{6}\left({\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(w\right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{w+\gamma }{2}\right)+{\mathrm{D}}_{\mathrm{q}}\left(\gamma \right)\right)}\hfill \\ & & -\frac{\frac{1}{6}w\left({\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(w\right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{w+\gamma }{2}\right)+{\mathrm{D}}_{\mathrm{q}}\left(\gamma \right)\right)}{1-\frac{1}{6}\left({\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(w\right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{w+\gamma }{2}\right)+{\mathrm{D}}_{\mathrm{q}}\left(\gamma \right)\right)},\hfill \end{array}$$

This formulation yields the following three step method for solving nonlinear Equation (1).

3. Order of Convergence

This section is comprised of the convergence analysis of the $\mathrm{q}$-iterative methods determined by Algorithms 1 and 2 in the previous section.

Algorithm 1: q-iterative Scheme of Order 3

For a given initial guess ${\varpi }_0$, the approximate solution ${\varpi }_{n+1}$ can be computed by the following iterative schemes:

$$v_n=\frac{\mathsf{\Phi }\left({\varpi }_n\right)-{\varpi }_n{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left({\varpi }_n\right)}{1-{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left({\varpi }_n\right)}$$ (11) $${\varpi }_{n+1}=\frac{\mathsf{\Phi }\left(v_n\right)-\frac{1}{6}{v}_n\left({\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(v_n\right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{v_n+{\varpi }_n}{2}\right)+{\mathrm{D}}_{\mathrm{q}}\left({\varpi }_n\right)\right)}{1-\frac{1}{6}\left({\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(v_n\right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{v_n+{\varpi }_n}{2}\right)+{\mathrm{D}}_{\mathrm{q}}\left({\varpi }_n\right)\right)},$$ (12)

where $n=0,1,2,\dots$

Algorithm 2: q-iterative Scheme of Order 4

For a given initial guess ${\varpi }_0$, the approximated solution ${\varpi }_{n+1}$ is computed by the following iterative schemes.

$$v_n=\frac{\mathsf{\Phi }\left({\varpi }_n\right)-{\varpi }_n{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left({\varpi }_n\right)}{1-{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left({\varpi }_n\right)}$$ $$w_n=\frac{\mathsf{\Phi }\left(v_n\right)-\frac{1}{6}{v}_n\left({\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(v_n\right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{v_n+{\varpi }_n}{2}\right)+{\mathrm{D}}_{\mathrm{q}}\left({\varpi }_n\right)\right)}{1-\frac{1}{6}\left({\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(v_n\right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{v_n+{\varpi }_n}{2}\right)+{\mathrm{D}}_{\mathrm{q}}\left({\varpi }_n\right)\right)}$$ (13) $${\varpi }_{n+1}=\frac{\mathsf{\Phi }\left(w_n\right)-\frac{1}{6}{w}_n\left({\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(w_n\right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{w_n+{\varpi }_n}{2}\right)+{\mathrm{D}}_{\mathrm{q}}\left({\varpi }_n\right)\right)}{1-\frac{1}{6}\left({\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(w_n\right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{w_n+{\varpi }_n}{2}\right)+{\mathrm{D}}_{\mathrm{q}}\left({\varpi }_n\right)\right)},$$ (14)

where $n=0,1,2,\dots$

Theorem 1.

Let $I\subset$R be an open interval, and Ψ$:I$→R is differential function. If $\beta \in I$ is root of $\mathsf{\Psi }\left(\varpi \right)=0$ and ${\varpi }_0$ is sufficiently close to β, then the multi-step method defined by Algorithm 1 has the third order of convergence.

Proof. 

Let $\beta$ be the root of nonlinear equation $\mathsf{\Psi }\left(\varpi \right)=0$, or equivalently, $\varpi =\mathsf{\Phi }\left(\varpi \right)$. Let $e_n$ and $e_{n+1}$ be the errors at $nth$ and $\left(n+1\right)$ iterations, respectively.

Now, expanding $\mathsf{\Phi }\left({\varpi }_n\right)$ and ${\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left({\varpi }_n\right)$ in terms of $\mathrm{q}$-Taylor’s series about $\beta$ gets

$$\begin{array}{ccc}\hfill \mathsf{\Phi }\left({\varpi }_n\right)& =& {\mathrm{D}}_{\mathrm{q}}^0\mathsf{\Phi }\left(\beta \right)+{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\beta \right)e_n+\frac{1}{\left[2\right]!}{\mathrm{D}}_{\mathrm{q}}^2\mathsf{\Phi }\left(\beta \right)e_n^2+\frac{1}{\left[3\right]!}{\mathrm{D}}_{\mathrm{q}}^3\mathsf{\Phi }\left(\beta \right)e_n^3+\frac{1}{\left[4\right]!}{\mathrm{D}}_{\mathrm{q}}^4\mathsf{\Phi }\left(\beta \right)e_n^4+O\left(e_n^5\right).\hfill \\ & =& \beta +{\mu }_1{e}_n+{\mu }_2{e}_n^2+{\mu }_3{e}_n^3+{\mu }_4{e}_n^4+O\left(e_n^5\right).\hfill \end{array}$$

(15)

$$\begin{array}{ccc}\hfill {\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left({\varpi }_n\right)& =& {\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\beta \right)+{\mathrm{D}}_{\mathrm{q}}^2\mathsf{\Phi }\left(\beta \right)e_n+\frac{1}{\left[2\right]!}{\mathrm{D}}_{\mathrm{q}}^3\mathsf{\Phi }\left(\beta \right)e_n^2+\frac{1}{\left[3\right]!}{\mathrm{D}}_{\mathrm{q}}^4\mathsf{\Phi }\left(\beta \right)e_n^3+\frac{1}{\left[4\right]!}{\mathrm{D}}_{\mathrm{q}}^5\mathsf{\Phi }\left(\beta \right)e_n^4+O\left(e_n^5\right).\hfill \\ & =& {\mu }_1+2{\mu }_2{e}_n+3{\mu }_3{e}_n^2+4{\mu }_4{e}_n^3+5{\mu }_5{e}_n^4+O\left(e_n^5\right),\hfill \end{array}$$

(16)

where

$${\mu }_m=\frac{{\mathrm{D}}_{\mathrm{q}}^m\mathsf{\Phi }\left(\beta \right)}{\left[m\right]!}\mathrm{for}m=1,2,3,\dots$$

and

$$e_n={\varpi }_n-\beta$$

$$\begin{array}{ccc}\hfill {\varpi }_n{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left({\varpi }_n\right)& =& \beta {\mu }_1+2\beta {\mu }_2{e}_n+3\beta {\mu }_3{e}_n^2+4\beta {\mu }_4{e}_n^3+5\beta {\mu }_5{e}_n^4\hfill \\ & & +{\mu }_1{e}_n+2{\mu }_2{e}_n^2+3{\mu }_3{e}_n^3+4{\mu }_4{e}_n^4+O\left(e_n^5\right).\hfill \end{array}$$

(17)

Subtract (15) from (17). Then, we have

$$\begin{array}{ccc}\hfill \mathsf{\Phi }\left({\varpi }_n\right)-{\varpi }_n{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left({\varpi }_n\right)& =& \beta -\beta {\mu }_1-2\beta {\mu }_2{e}_n-3\beta {\mu }_3{e}_n^2-4\beta {\mu }_4{e}_n^3-5\beta {\mu }_5{e}_n^4\hfill \\ & & -{\mu }_2{e}_n^2-2{\mu }_3{e}_n^3-3{\mu }_4{e}_n^4+O\left(e_n^5\right).\hfill \end{array}$$

(18)

$$1-{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left({\varpi }_n\right)=1-{\mu }_1-2{\mu }_2{e}_n-3{\mu }_3{e}_n^2-4{\mu }_4{e}_n^3-5{\mu }_5{e}_n^4+O\left(e_n^5\right).$$

(19)

Dividing (18) and (19), we have

$$\begin{array}{c}\frac{\mathsf{\Phi }\left({\varpi }_n\right)-{\varpi }_n{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left({\varpi }_n\right)}{1-{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left({\varpi }_n\right)}\hfill \\ \hspace{1em}=\left[\beta -\beta {\mu }_1-2\beta {\mu }_2{e}_n-3\beta {\mu }_3{e}_n^2-4\beta {\mu }_4{e}_n^3-5\beta {\mu }_5{e}_n^4-{\mu }_2{e}_n^2-2{\mu }_3{e}_n^3\right.\hfill \\ \hspace{1em}\hspace{1em}\left.-3{\mu }_4{e}_n^4+O\left(e_n^5\right)\right]{\left[1-\left({\mu }_1+2{\mu }_2{e}_n+3{\mu }_3{e}_n^2+4{\mu }_4{e}_n^3+O\left(e_n^4\right)\right)\right]}^{-1}\hfill \end{array}$$

From Equation (11), we have

$$\begin{array}{ccc}\hfill v_n& =& \beta +\frac{1}{\left(-1+{\mu }_1\right)}{\mu }_2{e}_n^2+\frac{1}{{\left(-1+{\mu }_1\right)}^2}\left(2{\mu }_1{\mu }_3-2{\mu }_3-2{\mu }_2^2\right)e_n^3\hfill \\ & & +\frac{1}{2{\left(-1+{\mu }_1\right)}^2}{\mu }_1\left(4{\mu }_1{\mu }_2{\mu }_3-{\mu }_2{\mu }_3-4{\mu }_2^3\right)e_n^4+O\left(e_n^5\right).\hfill \end{array}$$

Expanding $\mathsf{\Phi }\left(v_n\right)$ in terms of $\mathrm{q}$-Taylor’s series about $\beta ,$ we get

$$\begin{array}{ccc}\hfill \mathsf{\Phi }\left(v_n\right)& =& \beta +\frac{1}{\left(-1+{\mu }_1\right)}{\mu }_1{\mu }_2{e}_n^2+\frac{1}{{\left(-1+{\mu }_1\right)}^2}{\mu }_1\left(2{\mu }_1{\mu }_3-2{\mu }_3-2{\mu }_2^2\right)e_n^3\hfill \\ & & +\frac{1}{{\left(-1+{\mu }_1\right)}^3}{\mu }_1\left(3{\mu }_4+4{\mu }_2^3-6{\mu }_1{\mu }_4+3{\mu }_1^2{\mu }_4+7{\mu }_2{\mu }_3-7{\mu }_1{\mu }_2{\mu }_3\right)e_n^4\hfill \\ & & +O\left(e_n^5\right).\hfill \end{array}$$

(20)

By expanding ${\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(v_n\right),$ in terms of $\mathrm{q}$-Taylor’s series, we get

$$\begin{array}{ccc}\hfill {\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(v_n\right)& =& {\mu }_1+\frac{1}{\left(-1+{\mu }_1\right)}2{\mu }_2^2{e}_n^2+\frac{1}{{\left(-1+{\mu }_1\right)}^2}2{\mu }_2\left(2{\mu }_1{\mu }_3-2{\mu }_3-2{\mu }_2^2\right)e_n^3\hfill \\ & & +\frac{1}{{\left(-1+{\mu }_1\right)}^3}2{\mu }_2\left(3{\mu }_4+4{\mu }_2^3-6{\mu }_1{\mu }_4+3{\mu }_1^2{\mu }_4+7{\mu }_2{\mu }_3-7{\mu }_1{\mu }_2{\mu }_3\right)e_n^4\hfill \\ & & +O\left(e_n^5\right).\hfill \end{array}$$

(21)

Expanding ${\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{v_{n+{\varpi }_n}}{2}\right)$ in terms of $\mathrm{q}$-Taylor’s series about $\beta ,$ we get

$$\begin{array}{ccc}\hfill {\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{v_{n+{\varpi }_n}}{2}\right)& =& {\mu }_1+{\mu }_2{e}_n+\frac{1}{\left(-1+{\mu }_1\right)}2{\mu }_2^2{e}_n^2+\frac{1}{{\left(-1+{\mu }_1\right)}^2}2{\mu }_2\left(2{\mu }_1{\mu }_3-2{\mu }_3-2{\mu }_2^2\right)e_n^3\hfill \\ & & +\frac{1}{{\left(-1+{\mu }_1\right)}^3}2{\mu }_2\left(3{\mu }_4+4{\mu }_2^3-6{\mu }_1{\mu }_4+3{\mu }_1^2{\mu }_4+7{\mu }_2{\mu }_3-7{\mu }_1{\mu }_2{\mu }_3\right)e_n^4\hfill \\ & & +O\left(e_n^5\right).\hfill \end{array}$$

(22)

Using (16), (21) and (22), we have

$$\begin{array}{cc}\frac{1}{6}& \left[{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(v_n\right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{v_n+{\varpi }_n}{2}\right)+{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left({\varpi }_n\right)\right]\hfill \\ & ={\mu }_1+{\mu }_2{e}_n+\frac{1}{2}{\mu }_3{e}_n^2+\frac{1}{3\left(-1+{\mu }_1\right)}5{\mu }_2^2{e}_n^2+\frac{2}{3}{\mu }_4{e}_n^3+\frac{1}{3{\left(-1+{\mu }_1\right)}^2}5{\mu }_2\left(2{\mu }_1{\mu }_3\right.\hfill \\ & \left.\hspace{1em}-2{\mu }_3-2{\mu }_2^2\right)e_n^3+\frac{5}{6}{\mu }_5{e}_n^4+\frac{1}{3{\left(-1+{\mu }_1\right)}^3}5{\mu }_2\left(3{\mu }_4+4{\mu }_2^3-6{\mu }_1{\mu }_4+3{\mu }_1^2{\mu }_4\right.\hfill \\ & \left.\hspace{1em}+7{\mu }_2{\mu }_3-7{\mu }_1{\mu }_2{\mu }_3\right)e_n^4+O\left(e_n^5\right).\hfill \end{array}$$

Now

$$\begin{array}{cc}\hfill \frac{v_n}{6}[{\mathrm{D}}_{\mathrm{q}}& \mathsf{\Phi }\left(v_n\right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{v_n+{\varpi }_n}{2}\right)+{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left({\varpi }_n\right)]\hfill \\ & =\beta {\mu }_1+\beta {\mu }_2{e}_n+\frac{1}{2}\beta {\mu }_3{e}_n^2+\frac{1}{3\left(-1+{\mu }_1\right)}5\beta {\mu }_2^2{e}_n^2+\frac{2}{3}\beta {\mu }_4{e}_n^3\hfill \\ & \hspace{1em}+\frac{1}{3{\left(-1+{\mu }_1\right)}^2}5\beta {\mu }_2\left(2{\mu }_1{\mu }_3-2{\mu }_3-2{\mu }_2^2\right)e_n^3+\frac{5}{6}\beta {\mu }_5{e}_n^4\hfill \\ & \hspace{1em}+\frac{1}{3{\left(-1+{\mu }_1\right)}^3}5\beta {\mu }_2\left(3{\mu }_4+4{\mu }_2^3-6{\mu }_1{\mu }_4+3{\mu }_1^2{\mu }_4\right.\hfill \\ & \left.\hspace{1em}+7{\mu }_2{\mu }_3-7{\mu }_1{\mu }_2{\mu }_3\right)e_n^4+\frac{1}{\left(-1+{\mu }_1\right)}{\mu }_1{\mu }_2{e}_n^2+\frac{1}{2{\left(-1+{\mu }_1\right)}^2}{\mu }_1\left(2{\mu }_1{\mu }_3\right.\hfill \\ & \left.\hspace{1em}-2{\mu }_3-2{\mu }_2^2\right)e_n^3+\frac{1}{{\left(-1+{\mu }_1\right)}^3}{\mu }_1\left(3{\mu }_4+4{\mu }_2^3-6{\mu }_1{\mu }_4+3{\mu }_1^2{\mu }_4\right.\hfill \\ & \left.\hspace{1em}+7{\mu }_2{\mu }_3-7{\mu }_1{\mu }_2{\mu }_3\right)e_n^4+\frac{1}{\left(-1+{\mu }_1\right)}{\mu }_2^2{e}_n^3+\frac{1}{{\left(-1+{\mu }_1\right)}^2}{\mu }_2\left(2{\mu }_1{\mu }_3\right.\hfill \\ & \left.\hspace{1em}-2{\mu }_3-2{\mu }_2^2\right)e_n^4+\frac{1}{2\left(-1+{\mu }_1\right)}{\mu }_2{\mu }_3{e}_n^4+\frac{1}{3{\left(-1+{\mu }_1\right)}^2}5{\mu }_2^3{e}_n^4+O\left(e_n^5\right).\hfill \end{array}$$

(23)

Using (21) and (23), we have

$$\begin{array}{cc}\hfill \mathsf{\Phi }\left(v_n\right)& -\frac{v_n}{6}\left[{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(v_n\right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{v_n+{\varpi }_n}{2}\right)+{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left({\varpi }_n\right)\right]\hfill \\ & =\beta -\beta \left(-1+{\mu }_1\right)-\beta {\mu }_2{e}_n-\frac{1}{2}\beta {\mu }_3{e}_n^2-\frac{1}{3\left(-1+{\mu }_1\right)}5\beta {\mu }_2^2{e}_n^2-\frac{2}{3}\beta {\mu }_4{e}_n^3\hfill \\ & \hspace{1em}-\frac{1}{3{\left(-1+{\mu }_1\right)}^2}5\beta {\mu }_2\left(2{\mu }_1{\mu }_3-2{\mu }_3-2{\mu }_2^2\right)e_n^3-\frac{5}{6}\beta {\mu }_5{e}_n^4\hfill \\ & \hspace{1em}-\frac{1}{3{\left(-1+{\mu }_1\right)}^3}5\beta {\mu }_2\left(3{\mu }_4+4{\mu }_2^3-6{\mu }_1{\mu }_4\right.\hfill \\ & \left.\hspace{1em}+3{\mu }_1^2{\mu }_4+7{\mu }_2{\mu }_3-7{\mu }_1{\mu }_2{\mu }_3\right)e_n^4-\frac{1}{{\left(-1+{\mu }_1\right)}^2}{\mu }_2^2{e}_n^3\hfill \\ & \hspace{1em}-\frac{1}{6{\left(-1+{\mu }_1\right)}^3}\left(15{\mu }_1{\mu }_2{\mu }_3-15{\mu }_2{\mu }_3-8{\mu }_2^3\right)e_n^4+O\left(e_n^5\right).\hfill \end{array}$$

(24)

Now

$$\begin{array}{c}1-\frac{1}{6}\left[{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(v_n\right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{v_n+{\varpi }_n}{2}\right)+{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left({\varpi }_n\right)\right]\hfill \\ \hfill =1-{\mu }_1-{\mu }_2{e}_n-\frac{1}{2}{\mu }_3{e}_n^2-\frac{1}{3\left(-1+{\mu }_1\right)}5{\mu }_2^2{e}_n^2-\frac{2}{3}{\mu }_4{e}_n^3+O\left(e_n^4\right).\end{array}$$

(25)

Dividing (24) by (25), we have

$$\begin{array}{c}\frac{\mathsf{\Phi }\left(v_n\right)-\frac{v_n}{6}\left[{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(v_n\right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{v_n+{\varpi }_n}{2}\right)+{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left({\varpi }_n\right)\right]}{1-\frac{1}{6}\left[{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(v_n\right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{v_n+{\varpi }_n}{2}\right)+{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left({\varpi }_n\right)\right]}\hfill \\ =\beta +\frac{1}{{\left(-1+{\mu }_1\right)}^2}{\mu }_2^2{e}_n^3+\frac{1}{6{\left(-1+{\mu }_1\right)}^3}\left(15{\mu }_1{\mu }_2{\mu }_3-15{\mu }_2{\mu }_3-8{\mu }_2^3\right)e_n^4+O\left(e_n^5\right).\hfill \end{array}$$

Using (12), we have

$$\begin{array}{ccc}\hfill {\varpi }_{n+1}& =& \beta +\frac{1}{{\left(-1+{\mu }_1\right)}^2}{\mu }_2^2{e}_n^3+\frac{1}{6{\left(-1+{\mu }_1\right)}^3}\left(15{\mu }_1{\mu }_2{\mu }_3\right.\hfill \\ & & \left.-15{\mu }_2{\mu }_3-8{\mu }_2^3\right)e_n^4+O\left(e_n^5\right).\hfill \end{array}$$

(26)

Hence,

$$\begin{array}{c}{e}_{n+1}\hfill \\ \hspace{1em}=\frac{1}{{\left(-1+{\mu }_1\right)}^2}{\mu }_2^2{e}_n^3+\frac{1}{6{\left(-1+{\mu }_1\right)}^3}\left(15{\mu }_1{\mu }_2{\mu }_3-15{\mu }_2{\mu }_3-8{\mu }_2^3\right)e_n^4+O\left(e_n^5\right).\hfill \end{array}$$

This completes the proof. □

Theorem 2.

Let $I\subset$R be an open interval, and Ψ$:I$→R is a differential function. If $\beta \in I$ is a root of $\mathsf{\Psi }\left(\varpi \right)=0$ and ${\varpi }_0$ is sufficiently close to β, then the multi-step method defined by Algorithm 2 has the fourth order of convergence.

Proof. 

From equations (13) and (26), we have

$$\begin{array}{c}{w}_n\hfill \\ =\beta +\frac{1}{{\left(-1+{\mu }_1\right)}^2}{\mu }_2^2{e}_n^3+\frac{1}{6{\left(-1+{\mu }_1\right)}^3}\left(15{\mu }_1{\mu }_2{\mu }_3-15{\mu }_2{\mu }_3-8{\mu }_2^3\right)e_n^4+O\left(e_n^5\right).\hfill \end{array}$$

Expanding ${\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(w_n\right),$ in terms of $\mathrm{q}$-Taylor’s series gives

$$\begin{array}{c}{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(w_n\right)\hfill \\ \hspace{1em}={\mu }_1+\frac{1}{{\left(-1+{\mu }_1\right)}^2}2{\mu }_2^3{e}_n^3+\frac{1}{3{\left(-1+{\mu }_1\right)}^3}{\mu }_2\left(15{\mu }_1{\mu }_2{\mu }_3-15{\mu }_2{\mu }_3-8{\mu }_2^3\right)e_n^4\hfill \\ \hspace{1em}\hspace{1em}+O\left(e_n^5\right).\hfill \end{array}$$

(27)

Expanding ${\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{w_n+{\varpi }_n}{2}\right)$ in terms of $\mathrm{q}$-Taylor’s series gives

$$\begin{array}{c}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{w_n+{\varpi }_n}{2}\right)\hfill \\ ={\mu }_1+{\mu }_2{e}_n+\frac{1}{{\left(-1+{\mu }_1\right)}^2}2{\mu }_2^3{e}_n^3+\frac{1}{3{\left(-1+{\mu }_1\right)}^3}{\mu }_2\left(15{\mu }_1{\mu }_2{\mu }_3-15{\mu }_2{\mu }_3-8{\mu }_2^3\right)e_n^4\hfill \\ \hspace{1em}+O\left(e_n^5\right).\hfill \end{array}$$

(28)

Using (16), (27) and (28), we have

$$\begin{array}{c}\frac{1}{6}\left[{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(w_n\right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{w_n+{\varpi }_n}{2}\right)+{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left({\varpi }_n\right)\right]\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}={\mu }_1+{\mu }_2{e}_n+\frac{1}{2}{\mu }_3{e}_n^2+\frac{2}{3}{\mu }_4{e}_n^3+\frac{1}{3{\left(-1+{\mu }_1\right)}^2}5{\mu }_2^3{e}_n^3+\frac{5}{6}{\mu }_5{e}_n^4\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+\frac{1}{18{\left(-1+{\mu }_1\right)}^3}5{\mu }_2\left(15{\mu }_1{\mu }_2{\mu }_3-15{\mu }_2{\mu }_3-8{\mu }_2^3\right)e_n^4+O\left(e_n^5\right).\hfill \end{array}$$

(29)

Now,

$$\begin{array}{c}\frac{w_n}{6}\left[{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(w_n\right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{w_n+{\varpi }_n}{2}\right)+{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left({\varpi }_n\right)\right]\hfill \\ \hspace{1em}=\beta {\mu }_1+\beta {\mu }_2{e}_n+\frac{1}{2}\beta {\mu }_3{e}_n^2+\frac{2}{3}\beta {\mu }_4{e}_n^3+\frac{1}{2}\beta {\mu }_3{e}_n^2+\frac{1}{3{\left(-1+{\mu }_1\right)}^2}5\beta {\mu }_2^3{e}_n^3\hfill \\ \hspace{1em}\hspace{1em}+\frac{5}{6}\beta {\mu }_5{e}_n^4+\frac{1}{18{\left(-1+{\mu }_1\right)}^3}5\beta {\mu }_2\left(15{\mu }_1{\mu }_2{\mu }_3-15{\mu }_2{\mu }_3-8{\mu }_2^3\right)e_n^4\hfill \\ \hspace{1em}\hspace{1em}+\frac{1}{{\left(-1+{\mu }_1\right)}^2}{\mu }_1{\mu }_2^2{e}_n^3+\frac{1}{6{\left(-1+{\mu }_1\right)}^3}{\mu }_1\left(15{\mu }_1{\mu }_2{\mu }_3-15{\mu }_2{\mu }_3-8{\mu }_2^3\right)e_n^4\hfill \\ \hspace{1em}\hspace{1em}+\frac{1}{{\left(-1+{\mu }_1\right)}^2}{\mu }_2^3{e}_n^4+O\left(e_n^5\right).\hfill \end{array}$$

(30)

Now,

$$\begin{array}{cc}& \mathsf{\Phi }\left(w_n\right)\hfill \\ =& \beta +\frac{1}{{\left(-1+{\mu }_1\right)}^2}{\mu }_1{\mu }_2^2{e}_n^3+\frac{1}{6{\left(-1+{\mu }_1\right)}^3}{\mu }_1\left(15{\mu }_1{\mu }_2{\mu }_3-15{\mu }_2{\mu }_3-8{\mu }_2^3\right)e_n^4+O\left(e_n^5\right).\hfill \end{array}$$

(31)

Subtract (30) from (31), we have

$$\begin{array}{cc}\hfill \mathsf{\Phi }\left(w_n\right)& -\frac{w_n}{6}\left[{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(w_n\right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{w_n+{\varpi }_n}{2}\right)+{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left({\varpi }_n\right)\right]\hfill \\ =& -\beta \left(-1+{\mu }_1\right)-\beta {\mu }_2{e}_n-\frac{1}{2}\beta {\mu }_3{e}_n^2-\frac{2}{3}\beta {\mu }_4{e}_n^3-\frac{1}{2}\beta {\mu }_3{e}_n^2-\frac{1}{3{\left(-1+{\mu }_1\right)}^2}5\beta {\mu }_2^3{e}_n^3\hfill \\ & -\frac{5}{6}\beta {\mu }_5{e}_n^4-\frac{1}{18{\left(-1+{\mu }_1\right)}^3}5\beta {\mu }_2\left(-8{\mu }_2^3-15{\mu }_2{\mu }_3+15{\mu }_1{\mu }_2{\mu }_3\right)e_n^4\hfill \\ & -\frac{1}{{\left(-1+{\mu }_1\right)}^2}{\mu }_2^3{e}_n^4+O\left(e_n^5\right).\hfill \end{array}$$

(32)

Now,

$$\begin{array}{cc}\hfill 1-& \frac{1}{6}\left[{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(w_n\right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{w_n+{\varpi }_n}{2}\right)+{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left({\varpi }_n\right)\right]\hfill \\ & =1-{\mu }_1-{\mu }_2{e}_n-\frac{1}{2}{\mu }_3{e}_n^2-\frac{2}{3}{\mu }_4{e}_n^3-\frac{1}{2}{\mu }_3{e}_n^2-\frac{1}{3{\left(-1+{\mu }_1\right)}^2}5{\mu }_2^3{e}_n^3-\frac{5}{6}{\mu }_5{e}_n^4\hfill \\ & \hspace{1em}-\frac{1}{18{\left(-1+{\mu }_1\right)}^3}5{\mu }_2\left(-8{\mu }_2^3-15{\mu }_2{\mu }_3+15{\mu }_1{\mu }_2{\mu }_3\right)e_n^4+O\left(e_n^5\right).\hfill \end{array}$$

(33)

Dividing (32) and (33), we have

$$\begin{array}{c}\frac{\mathsf{\Phi }\left(w_n\right)-\frac{w_n}{6}\left[{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(w_n\right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{w_n+{\varpi }_n}{2}\right)+{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left({\varpi }_n\right)\right]}{1-\frac{1}{6}\left[{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(w_n\right)+4{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left(\frac{w_n+{\varpi }_n}{2}\right)+{\mathrm{D}}_{\mathrm{q}}\mathsf{\Phi }\left({\varpi }_n\right)\right]}\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=\beta +\frac{1}{{\left(-1+{\mu }_1\right)}^2}{\mu }_2^3{e}_n^4+O\left(e_n^5\right).\hfill \end{array}$$

Using (14), we have

$${\varpi }_{n+1}=\beta +\frac{1}{{\left(-1+{\mu }_1\right)}^2}{\mu }_2^3{e}_n^4+O\left(e_n^5\right).$$

(34)

Hence,

$$e_{n+1}=\frac{1}{{\left(-1+{\mu }_1\right)}^2}{\mu }_2^3{e}_n^4+O\left(e_n^5\right).$$

This completes the proof. □

4. Numerical Examples and Comparison Results

We now give some numerical examples to check the efficiency of the proposed algorithms. The numerical experiments were performed on the Intel (R) core (TM) $2\times 2.1$ GHz, 12 GB of RAM, and the code was written in Maple. Using $\epsilon ={10}^{-5}$, we obtained an approximated simple root instead of exact.

For the sake of simplicity, we use abbreviation CAGs for classical iterative methods and QAG for their $\mathrm{q}$-analogues.

Now recall the classical List 2.2 in [29] (CAG1), defined by

$$y_n=x_n-\frac{f\left(x_n\right)}{f^{\prime }\left(x_n\right)}\mathrm{and}{x}_{n+1}=y_n-\frac{f\left(y_n\right)}{f^{\prime }\left(y_n\right)}(\forall n=0,1,2,\cdots ).$$

and the classical List 2.3 in [30] (CAG2), we defined by

$$y_n=x_n-\frac{f\left(x_n\right)}{f^{\prime }\left(x_n\right)},$$

$$z_n=y_n-\frac{f\left(y_n\right)}{f^{\prime }\left(y_n\right)},$$

$$x_{n+1}=z_n-\frac{f\left(z_n\right)}{f^{\prime }\left(z_n\right)},\mathrm{for}n=0,1,2,\cdots$$

For brevity, we denote the Algorithms 1 and 2 by QAG1 and QAG2, respectively. The computational results presented in the Tables elaborate the performance and efficacy of our newly developed $\mathrm{q}$ iterative methods.

The first two examples are concerned with the performance of $\mathrm{q}$-iterative methods in comparison with the classical methods for different values of $\mathrm{q}$ for up to three iterations. Additionally, we can check the performances of these methods for different values of $\mathrm{q}$.

Example 1.

$\mathsf{\Psi }\left(\varpi \right)={\varpi }^3-10,\mathsf{\Phi }\left(\varpi \right)=\sqrt{\frac{10}{\varpi }},{\varpi }_0=1.5$

Table 1. Calculation of ${\varpi }_i$ and $\mathsf{\Psi }\left({\varpi }_i\right)$ for $i=1,2,3$ and different values of $\mathrm{q}$ by using QAG1.

$\mathbf{q}$	${\mathit{\varpi }}_1$	$\mathbf{\Psi }\left({\mathit{\varpi }}_1\right)$	${\mathit{\varpi }}_2$	$\mathbf{\Psi }\left({\mathit{\varpi }}_2\right)$	${\mathit{\varpi }}_3$	$\mathbf{\Psi }\left({\mathit{\varpi }}_3\right)$
$1.02$	$2.1515084$	$-0.406926\times {10}^{-1}$	$2.1544338$	$-0.124\times {10}^{-4}$	$2.1544347$	0
$1.01$	$2.1513563$	$-0.428046\times {10}^{-1}$	$2.1544343$	$-0.54\times {10}^{-5}$	$2.1544347$	0
$0.9999$	$2.1511829$	$-0.452121\times {10}^{-1}$	$2.1544347$	0	$2.1544349$	$0.3\times {10}^{-5}$
$0.98$	$2.1508648$	$-0.496276\times {10}^{-1}$	$2.1544358$	$0.15\times {10}^{-4}$	$2.1544346$	$-0.13\times {10}^{-5}$
$0.97$	$2.1506881$	$-0.520797\times {10}^{-1}$	$2.1544365$	$0.25\times {10}^{-4}$	$2.1544347$	0
$0.99$	$2.1510350$	$-0.472652\times {10}^{-1}$	$2.1544354$	$0.10\times {10}^{-4}$	$2.1544347$	0
$0.96$	$2.1505046$	$-0.546258\times {10}^{-1}$	$2.1544373$	$0.36\times {10}^{-4}$	$2.1544349$	$0.3\times {10}^{-5}$
$0.95$	$2.1503135$	$-0.572769\times {10}^{-1}$	$2.1544381$	$0.47\times {10}^{-4}$	$2.1544347$	0
$0.9$	$2.1492442$	$-0.721024\times {10}^{-1}$	$2.1544438$	$0.127\times {10}^{-3}$	$2.1544347$	0
$0.8$	$2.1463948$	$-0.1115363$	$2.1544670$	$0.450\times {10}^{-3}$	$2.1544345$	$-0.26\times {10}^{-5}$
$0.7$	$2.1422283$	$-0.1690099$	$2.1545204$	$0.1194\times {10}^{-2}$	$2.1544340$	$-0.96\times {10}^{-5}$
$0.6$	$2.1360665$	$-0.2535983$	$2.1546349$	$0.2788\times {10}^{-2}$	$2.1544324$	$-0.319\times {10}^{-4}$
$0.5$	$2.1267640$	$-0.3803803$	$2.1548704$	$0.6068\times {10}^{-2}$	$2.1544273$	$-0.1029\times {10}^{-3}$
$0.4$	$2.1122542$	$-0.5759291$	$2.1553307$	$0.12482\times {10}^{-1}$	$2.1544122$	$-0.3132\times {10}^{-1}$

shows the computation of ${\varpi }_i$ and $\mathsf{\Psi }\left({\varpi }_i\right)$ for $i=1,2,3$ and different values of $\mathrm{q}$ by using QAG1.

We can observe from Table 1 that we can get more accurate values of ${\varpi }_1,{\varpi }_2,{\varpi }_3$ when $\mathrm{q}\to 1^{-}$ and for which $\mathsf{\Psi }\left({\varpi }_1\right),\mathsf{\Psi }\left({\varpi }_2\right),\mathsf{\Psi }\left({\varpi }_3\right)$ tend towards zero. The values of $\mathsf{\Psi }\left({\varpi }_1\right)=-0.452121\times {10}^{-1},$$\mathsf{\Psi }\left({\varpi }_2\right)=0,$$\mathsf{\Psi }\left({\varpi }_3\right)=0.3\times {10}^{-5}$ calculated by QAG1 at $\mathrm{q}=0.9999$ are closer to zero as compared to the values $\mathsf{\Psi }\left({\varpi }_1\right)=0.584879,$$\mathsf{\Psi }\left({\varpi }_2\right)=0.3\times {10}^{-6},$$\mathsf{\Psi }\left({\varpi }_3\right)=0$ calculated by CAG1.

Table 2. Calculation of ${\varpi }_i$ and $\mathsf{\Psi }\left({\varpi }_i\right)$ for $i=1,2,3$ and different values of $\mathrm{q}$ by using QAG2.

$\mathbf{q}$	${\mathit{\varpi }}_1$	$\mathbf{\Psi }\left({\mathit{\varpi }}_1\right)$	${\mathit{\varpi }}_2$	$\mathbf{\Psi }\left({\mathit{\varpi }}_2\right)$	${\mathit{\varpi }}_3$	$\mathbf{\Psi }\left({\mathit{\varpi }}_3\right)$
$1.02$	$2.1543094$	$-0.17445\times {10}^{-2}$	$2.1544347$	0	$2.1544348$	$0.2\times {10}^{-5}$
$1.01$	$2.1543000$	$-0.18754\times {10}^{-2}$	$2.1544347$	0	$2.1544346$	$-0.13\times {10}^{-5}$
$0.9999$	$2.1542886$	$-0.20341\times {10}^{-2}$	$2.1544348$	$0.2\times {10}^{-5}$	$2.1544348$	$0.2\times {10}^{-5}$
$0.98$	$2.1542681$	$-0.23195\times {10}^{-2}$	$2.1544348$	$0.2\times {10}^{-5}$	$2.1544346$	$-0.13\times {10}^{-5}$
$0.97$	$2.1542562$	$-0.24852\times {10}^{-2}$	$2.1544346$	$-0.13\times {10}^{-5}$	$2.1544346$	$-0.13\times {10}^{-5}$
$0.99$	$2.1542794$	$-0.21622\times {10}^{-2}$	$2.1544347$	0	$2.1544347$	0
$0.96$	$2.1542433$	$-0.26648\times {10}^{-2}$	$2.1544346$	$-0.13\times {10}^{-5}$	$2.1544347$	0
$0.95$	$2.1542297$	$-0.28542\times {10}^{-2}$	$2.1544347$	0	$2.1544347$	0
$0.9$	$2.1541480$	$-0.39916\times {10}^{-2}$	$2.1544348$	$0.2\times {10}^{-5}$	$2.1544347$	0
$0.8$	$2.1538876$	$-0.76162\times {10}^{-2}$	$2.1544345$	$-0.26\times {10}^{-5}$	$2.1544348$	$0.2\times {10}^{-5}$
$0.7$	$2.1534120$	$-0.142340\times {10}^{-1}$	$2.1544342$	$-0.68\times {10}^{-5}$	$2.1544347$	0
$0.6$	$2.1525319$	$-0.264725\times {10}^{-1}$	$2.1544329$	$-0.249\times {10}^{-4}$	$2.1544347$	0
$0.5$	$2.1508570$	$-0.497358\times {10}^{-1}$	$2.1544290$	$-0.792\times {10}^{-4}$	$2.1544347$	0
$0.4$	$2.1475211$	$-0.959615\times {10}^{-1}$	$2.1544175$	$-0.2394\times {10}^{-3}$	$2.1544348$	$0.2\times {10}^{-5}$

shows the computation of ${\varpi }_i$ and $\mathsf{\Psi }\left({\varpi }_i\right)$ for $i=1,2,3$ and different values of $\mathrm{q}$ by using QAG2.

We can observe from Table 2 that we can get more accurate values of ${\varpi }_1,{\varpi }_2,{\varpi }_3$ when $\mathrm{q}\to 1^{-}$ and for which $\mathsf{\Psi }\left({\varpi }_1\right),\mathsf{\Psi }\left({\varpi }_2\right),\mathsf{\Psi }\left({\varpi }_3\right)$ tend towards zero. The values of $\mathsf{\Psi }\left({\varpi }_1\right)=-0.20341\times {10}^{-2},$$\mathsf{\Psi }\left({\varpi }_2\right)=0.2\times {10}^{-5},$$\mathsf{\Psi }\left({\varpi }_3\right)=0.2\times {10}^{-5}$ calculated by QAG2 at $\mathrm{q}=0.9999$ are closer to zero as compared to the values $\mathsf{\Psi }\left({\varpi }_1\right)=-0.64313\times {10}^{-4},$$\mathsf{\Psi }\left({\varpi }_2\right)=1.0\times {10}^{-8},$$\mathsf{\Psi }\left({\varpi }_3\right)=1.0\times {10}^{-8}$ calculated by CAG2.

Example 2.

$\mathsf{\Psi }\left(\varpi \right)=cos\varpi -\varpi ,$   $\mathsf{\Phi }\left(\varpi \right)=cos\varpi ,$   ${\varpi }_0=1.5$

Table 3. Calculation of ${\varpi }_i$ and $\mathsf{\Psi }\left({\varpi }_i\right)$ for $i=1,2,3$ and different values of $\mathrm{q}$ by using QAG1.

$\mathbf{q}$	${\mathit{\varpi }}_1$	$\mathbf{\Psi }\left({\mathit{\varpi }}_1\right)$	${\mathit{\varpi }}_2$	$\mathbf{\Psi }\left({\mathit{\varpi }}_2\right)$	${\mathit{\varpi }}_3$	$\mathbf{\Psi }\left({\mathit{\varpi }}_3\right)$
$1.02$	$0.74168017$	$-0.434557\times {10}^{-2}$	$0.73908448$	$0.109\times {10}^{-5}$	$0.73908512$	$2.0\times {10}^{-8}$
$1.01$	$0.74165397$	$-0.430167\times {10}^{-2}$	$0.73908470$	$7.3\times {10}^{-7}$	$0.73908517$	$-6.0\times {10}^{-8}$
$0.9999$	$0.74162250$	$-0.424894\times {10}^{-2}$	$0.73908503$	$1.7\times {10}^{-7}$	$0.73908514$	$-1.0\times {10}^{-8}$
$0.98$	$0.74154549$	$-0.411992\times {10}^{-2}$	$0.73908564$	$-8.5\times {10}^{-7}$	$0.73908518$	$-8.0\times {10}^{-8}$
$0.97$	$0.74149993$	$-0.404359\times {10}^{-2}$	$0.73908598$	$-0.142\times {10}^{-5}$	$0.73908508$	$9.0\times {10}^{-8}$
$0.99$	$0.74158649$	$-0.418861\times {10}^{-2}$	$0.73908534$	$-3.5\times {10}^{-7}$	$0.73908511$	$4.0\times {10}^{-8}$
$0.96$	$0.74145003$	$-0.395998\times {10}^{-2}$	$0.73908627$	$-0.190\times {10}^{-5}$	$0.73908518$	$-8.0\times {10}^{-8}$
$0.95$	$0.74139590$	$-0.386930\times {10}^{-2}$	$0.73908656$	$-0.239\times {10}^{-5}$	$0.73908509$	$7.0\times {10}^{-8}$
$0.9$	$0.74106976$	$-0.332295\times {10}^{-2}$	$0.73908818$	$-0.510\times {10}^{-5}$	$0.73908513$	$1.0\times {10}^{-8}$
$0.8$	$0.74022788$	$-0.191300\times {10}^{-2}$	$0.73908968$	$-0.761\times {10}^{-5}$	$0.73908519$	$-1.0\times {10}^{-7}$
$0.7$	$0.73934263$	$-0.43097\times {10}^{-3}$	$0.73908703$	$-0.317\times {10}^{-5}$	$0.73908511$	$4.0\times {10}^{-8}$
$0.6$	$0.73870344$	$0.00063875\times {10}^{-3}$	$0.73908049$	$0.777\times {10}^{-5}$	$0.73908509$	$7.0\times {10}^{-8}$
$0.5$	$0.73864134$	$0.00074266\times {10}^{-3}$	$0.73907702$	$0.1358\times {10}^{-4}$	$0.73908500$	$2.2\times {10}^{-7}$
$0.4$	$0.73952233$	$-0.73177\times {10}^{-3}$	$0.73909643$	$-0.1891\times {10}^{-4}$	$0.73908541$	$-4.6\times {10}^{-7}$

shows the computation of ${\varpi }_i$ and $\mathsf{\Psi }\left({\varpi }_i\right)$ for $i=1,2,3$ and different values of $\mathrm{q}$ by using QAG1.

We can observe from Table 1 that we can get more accurate values of ${\varpi }_1,{\varpi }_2,{\varpi }_3$ when $\mathrm{q}\to 1^{-}$ and for which $\mathsf{\Psi }\left({\varpi }_1\right),\mathsf{\Psi }\left({\varpi }_2\right),\mathsf{\Psi }\left({\varpi }_3\right)$ tend towards zero. The values of $\mathsf{\Psi }\left({\varpi }_1\right)=-0.424894\times {10}^{-2},$$\mathsf{\Psi }\left({\varpi }_2\right)=1.7\times {10}^{-7},$$\mathsf{\Psi }\left({\varpi }_3\right)=-1.0\times {10}^{-8}$ calculated by QAG1 at $\mathrm{q}=0.9999$ are closer to zero as compared to the values $\mathsf{\Psi }\left({\varpi }_1\right)=0.10707\times {10}^{-1},$$\mathsf{\Psi }\left({\varpi }_2\right)=0,$$\mathsf{\Psi }\left({\varpi }_3\right)=0$ calculated by CAG1.

Table 4. Calculation of ${\varpi }_i$ and $\mathsf{\Psi }\left({\varpi }_i\right)$ for $i=1,2,3$ and different values of $\mathrm{q}$ by using QAG2.

$\mathbf{q}$	${\mathit{\varpi }}_1$	$\mathbf{\Psi }\left({\mathit{\varpi }}_1\right)$	${\mathit{\varpi }}_2$	$\mathbf{\Psi }\left({\mathit{\varpi }}_2\right)$	${\mathit{\varpi }}_3$	$\mathbf{\Psi }\left({\mathit{\varpi }}_3\right)$
$1.02$	$0.73926160$	$-0.29535\times {10}^{-3}$	$0.73908518$	$-8.0\times {10}^{-8}$	$0.73908517$	$-6.0\times {10}^{-8}$
$1.01$	$0.73925632$	$-0.28651\times {10}^{-3}$	$0.73908515$	$-3.0\times {10}^{-8}$	$0.73908513$	$1.0\times {10}^{-8}$
$0.9999$	$0.73925085$	$-0.27736\times {10}^{-3}$	$0.73908515$	$-3.0\times {10}^{-8}$	$0.73908516$	$-4.0\times {10}^{-8}$
$0.98$	$0.73923896$	$-0.25746\times {10}^{-3}$	$0.73908511$	$4.0\times {10}^{-8}$	$0.73908513$	$1.0\times {10}^{-8}$
$0.97$	$0.73923266$	$-0.24691\times {10}^{-3}$	$0.73908512$	$2.0\times {10}^{-8}$	$0.73908514$	$-1.0\times {10}^{-8}$
$0.99$	$0.73924499$	$-0.26755\times {10}^{-3}$	$0.73908515$	$-3.0\times {10}^{-8}$	$0.73908517$	$-6.0\times {10}^{-8}$
$0.96$	$0.73922623$	$-0.23615\times {10}^{-3}$	$0.73908513$	$1.0\times {10}^{-8}$	$0.73908517$	$-6.0\times {10}^{-8}$
$0.95$	$0.73921962$	$-0.22509\times {10}^{-3}$	$0.73908514$	$-1.0\times {10}^{-8}$	$0.73908514$	$-1.0\times {10}^{-8}$
$0.9$	$0.73918540$	$-0.16781\times {10}^{-3}$	$0.73908511$	$4.0\times {10}^{-8}$	$0.73908513$	$1.0\times {10}^{-8}$
$0.8$	$0.73912331$	$-0.6389\times {10}^{-4}$	$0.73908512$	$2.0\times {10}^{-8}$	$0.73908508$	$9.0\times {10}^{-8}$
$0.7$	$0.73908870$	$-0.597\times {10}^{-4}$	$0.73908513$	$1.0\times {10}^{-8}$	$0.73908509$	$1.0\times {10}^{-8}$
$0.6$	$0.73908827$	$-0.525\times {10}^{-5}$	$0.73908511$	$4.0\times {10}^{-8}$	$0.73908508$	$4.0\times {10}^{-8}$
$0.5$	$0.73909977$	$-0.2450\times {10}^{-4}$	$0.73908510$	$6.0\times {10}^{-8}$	$0.73908517$	$-6.0\times {10}^{-8}$
$0.4$	$0.73905847$	$0.4462\times {10}^{-4}$	$0.73908529$	$-2.6\times {10}^{-7}$	$0.73908512$	$2.0\times {10}^{-8}$

shows the computation of ${\varpi }_i$ and $\mathsf{\Psi }\left({\varpi }_i\right)$ for $i=1,2,3$ and different values of $\mathrm{q}$ by using QAG2.

We can observe from Table 2 that we can get more accurate values of ${\varpi }_1,{\varpi }_2,{\varpi }_3$ when $\mathrm{q}\to 1^{-}$ and for which $\mathsf{\Psi }\left({\varpi }_1\right),\mathsf{\Psi }\left({\varpi }_2\right),\mathsf{\Psi }\left({\varpi }_3\right)$ tend towards zero. The values of $\mathsf{\Psi }\left({\varpi }_1\right)=-0.27736\times {10}^{-3},$$\mathsf{\Psi }\left({\varpi }_2\right)=-3.0\times {10}^{-8},$$\mathsf{\Psi }\left({\varpi }_3\right)=-4.0\times {10}^{-8}$ calculated by QAG2 at $\mathrm{q}=0.9999$ are closer to zero as compared to the values $\mathsf{\Psi }\left({\varpi }_1\right)=-6.0\times {10}^{-8},$$\mathsf{\Psi }\left({\varpi }_2\right)=1.0\times {10}^{-8},$$\mathsf{\Psi }\left({\varpi }_3\right)=0$ calculated by CAG2.

Comparison of the Classical and q-Analogue of Iterative Methods

Here, we check the efficiency of our new iterative methods by considering some of the nonlinear equations. Furthermore, we compare the standard Newton method (NM), Halley method (HM), Algorithm 2B [8] and Algorithm 2C [8] with our new Algorithms 1 and 2. In following tables, we display the number of iterations (IT), the approximate root $x_n$, the value $\mathsf{\Psi }\left(x_n\right)$ and $\delta$, the distance between two successive estimations. It is important to mention that in order to get better computational results of the $\mathrm{q}$-iterative methods, we take the value of $\mathrm{q}=0.9999.$

Example 3.

$\mathsf{\Psi }\left(\varpi \right)={\varpi }^3-10,\mathsf{\Phi }\left(\varpi \right)=\sqrt{\frac{10}{\varpi }},{\varpi }_0=1.5$.

$Methods$	$IT$	${\varpi }_n$	$\mathsf{\Psi }\left({\varpi }_n\right)$	$\delta =\mid {\varpi }_n-{\varpi }_{n-1}\mid$
NM [8]	5	$2.1544346900319185890$	$4.85518\times {10}^{-13}$	$2.740789522304\times {10}^{-7}$
HM [8]	5	$2.1544346900319185890$	$4.85518\times {10}^{-13}$	$2.740789522304\times {10}^{-7}$
Algorithm 2B [8]	3	$2.1544346900318837218$	$1.0\times {10}^{-13}$	$3.6257326037\times {10}^{-9}$
Algorithm 2C [8]	3	$2.1544346900318837217$	$-8.0\times {10}^{-19}$	$1.457\times {10}^{-16}$
QAG1	3	$2.1544346900317721381$	$-1.5537764\times {10}^{-12}$	$7.58921442268\times {10}^{-8}$
QAG2	3	$2.1544346900318837199$	$2.59\times {10}^{-17}$	$2.09269700\times {10}^{-11}$

Example 4.

$\mathsf{\Psi }\left(\varpi \right)=cos\left(\varpi \right)-\varpi ,$  $\mathsf{\Phi }\left(\varpi \right)=cos\left(\varpi \right),$  ${\varpi }_0=1.7$.

$Methods$	$IT$	${\varpi }_n$	$\mathsf{\Psi }\left({\varpi }_n\right)$	$\delta =\mid {\varpi }_n-{\varpi }_{n-1}\mid$
NM [8]	4	$0.73908513321516087614$	$-3.9244\times {10}^{-16}$	$3.258805388731\times {10}^{-8}$
HM [8]	4	$0.73908513321516087614$	$3.9244\times {10}^{-16}$	$3.258805388731\times {10}^{-8}$
Algorithm 2B [8]	3	$0.73908513321516064166$	$-1.0\times {10}^{-20}$	$8.63747112426\times {10}^{-9}$
Algorithm 2C [8]	3	$0.73908513321516064166$	$-1.0\times {10}^{-20}$	$3.72234\times {10}^{-15}$
QAG1	3	$0.73908513321504533240$	$1.9298296\times {10}^{-13}$	$9.810393747542\times {10}^{-8}$
QAG2	3	$0.73908513321516064015$	$2.52\times {10}^{-18}$	$1.732950960\times {10}^{-11}$

Example 5.

$\mathsf{\Psi }\left(\varpi \right)={sin}^2\left(\varpi \right)-{\varpi }^2+1,$  $\mathsf{\Phi }\left(\varpi \right)=sin\left(\varpi \right)+\frac{1}{sin\left(\varpi \right)+\varpi },$  ${\varpi }_0=1$.

$Methods$	$IT$	${\varpi }_n$	$\mathsf{\Psi }\left({\varpi }_n\right)$	$\delta =\mid {\varpi }_n-{\varpi }_{n-1}\mid$
NM [8]	5	$1.4044916482156470349$	$7.591622\times {10}^{-13}$	$6.247205954873\times {10}^{-7}$
HM [8]	5	$1.4044916482156470349$	$7.591622\times {10}^{-13}$	$6.247205954873\times {10}^{-7}$
Algorithm 2B [8]	3	$1.4044916482153412261$	$-2.0\times {10}^{-19}$	$4.830530998\times {10}^{-10}$
Algorithm 2C [8]	3	$1.4044916482153412261$	$-2.0\times {10}^{-19}$	$3.27\times {10}^{-17}$
QAG1	3	$1.4044916482153396720$	$3.85782\times {10}^{-15}$	$5.9936291807\times {10}^{-9}$
QAG2	3	$1.4044916482153412261$	$-2.0\times {10}^{-19}$	$4.64231\times {10}^{-14}$

Example 6.

$\mathsf{\Psi }\left(\varpi \right)=e^{\varpi }-3{\varpi }^2,$   $\mathsf{\Phi }\left(\varpi \right)=\sqrt{\frac{e^{\varpi }}{3}},$   ${\varpi }_0=0.8$

$Methods$	$IT$	${\varpi }_n$	$\mathsf{\Psi }\left({\varpi }_n\right)$	$\delta =\mid {\varpi }_n-{\varpi }_{n-1}\mid$
NM [8]	4	$0.91000757248870906142$	$-2.3\times {10}^{-18}$	$1.13400774851\times {10}^{-9}$
HM [8]	4	$0.91000757248870906142$	$-2.3\times {10}^{-18}$	$1.13400774848\times {10}^{-9}$
Algorithm 2B [8]	3	$0.91000757248870906069$	$-1.0\times {10}^{-19}$	$5.97372\times {10}^{-15}$
Algorithm 2C [8]	2	$0.91000757248870906058$	$2.0\times {10}^{-19}$	$0.114201199652787\times {10}^{-5}$
QAG1	3	$0.91000757248870109207$	$2.37121\times {10}^{-14}$	$3.43403658042\times {10}^{-9}$
QAG2	3	$0.91000757248870905516$	$1.64\times {10}^{-17}$	$1.914022857\times {10}^{-11}$

Tables compare the solutions obtained by using the classical and our $\mathrm{q}$-iterative methods. The results show that our $\mathrm{q}$-analogue iterative methods QAG1 and QAG2 give the same results as the classical methods NM, HM, Algorithm 2B [8] and Algorithm 2C [8].

Remark 1.

The efficiency index is considered as $E=P^{\frac{1}{m}}$, where P represents the order of the method and m is the total number of function evaluations per iteration necessary by the method. The computational efficiency of QAG1 is $1.245730$, and that of QAG2 is $1.189207$, which is better than many of the existing methods.

5. Conclusions

Using the quantum calculus approach and essentially using the decomposition technique, we have derived a new family of two-step and three-step $\mathrm{q}$-iterative methods for solving nonlinear equations. We have discussed the convergence analysis of these newly derived methods. The computational results concluded that the $\mathrm{q}$-analogues of the iterative methods for solving the nonlinear equations generate the same results as the classical methods, but the convergence rate towards approaching the root is faster than those of the classical methods. It is worth mentioning here that one can extend the methods developed in this paper by using the techniques of post quantum calculus. It will be an interesting problem for future research.