Extended Seventh Order Derivative Free Family of Methods for Solving Nonlinear Equations

Abstract

A plethora of applications from Computational Sciences can be identified for a system of nonlinear equations in an abstract space. These equations are mostly solved with an iterative method because an analytical method does not exist for such problems. The convergence of the method is established by sufficient conditions. Recently, there has been a surge in the development of high convergence order methods. Local convergence results reveal the degree of difficulty when choosing the initial points. However, these methods may converge even in cases not guaranteed by the conditions. Moreover, it is not known in advance how many iterations should be carried out to reach a certain error tolerance. Furthermore, no computable information is provided about the isolation of the solution in a certain region containing it. The aforementioned concerns constitute the motivation for writing this article. The novelty of the works is the expansion of the applicability of the method under $\omega -$continuity conditions considered for the involved operator. The technique is demonstrated using a derivative-free seventh convergence three-step method. However, it was found that it can be used with the same effectiveness as other methods containing inverses of linear operators. The technique also uses information about the operators appearing in this method. This is in contrast to earlier works utilizing derivatives or divided differences not on the method which may not even exist for the problem at hand. The numerical experiments complement the theory.

1. Introduction

Finding an iterative method for a locally unique solution $x^{*}$ of following a nonlinear system is one of the most challenging and difficult tasks

$$F\left(x\right)=0.$$

(1)

Here, the operator F is defined on subset $\mathbb{D}$ of a Banach space $\mathbb{X}$ with values in itself. Most of the iterative methods which generate iterations required certain initial hypotheses for convergence to $x^{*}$.

A popular iterative method is defined for all $\sigma =0,1,2,3,\cdots$ as

$$x_{\sigma +1}=x_{\sigma }-F^{\prime }{\left(x_{\sigma }\right)}^{-1}F\left(x_{\sigma }\right).$$

(2)

This is the so-called Newton’s method which is only quadratically convergent [1,2,3]. One of the biggest problems with this method is that it requires the inversion of the Fréchet derivative at each step. On the other hand, finding the derivative of nonlinear operators is quite a challenging and time-consuming task and some Fréchet derivatives do not even exist. Therefore, scholars focus on derivative free iterative methods.

If we replace $F^{\prime }$ in the scheme (2) by a divided difference of order one, an efficient derivative-free iterative method is defined for all $\sigma =0,1,2,3,\cdots$ as

$$x_{\sigma +1}=x_{\sigma }-F{[u_{\sigma },x_{\sigma }]}^{-1}F\left(x_{\sigma }\right),u_{\sigma }=x_{\sigma }+F\left(x_{\sigma }\right),$$

(3)

where $F[u_{\sigma },x_{\sigma }]:\mathbb{D}\times \mathbb{D}\to \ell \left(\mathbb{X}\right)$ is the first order divided difference for $\mathbb{D}$ and the notation of $\ell \left(\mathbb{X}\right)$ stands for the space of continuous linear operators from $\mathbb{X}$ into itself. This is the so-called Steffensen’s method which is also quadratically convergent [4,5,6].

Other single-step methods are the Secant [7,8] and the Kurchatov method [9] defined, respectively, by

$$\begin{array}{cc}\hfill & x_{\sigma +1}=x_{\sigma }-F{[x_{\sigma -1},x_{\sigma }]}^{-1}F\left(x_{\sigma }\right)\hfill \\ \hfill & \mathrm{and}\hfill \\ \hfill & x_{\sigma +1}=x_{\sigma }-F{[2x_{\sigma }-x_{\sigma -1},x_{\sigma -1}]}^{-1}F\left(x_{\sigma }\right).\hfill \end{array}$$

(4)

The convergence order of these methods is $\frac{1+\sqrt{5}}{2}$ and 2, respectively. To elevate the order convergence as well as the efficiency, a plethora of iterative methods have been developed [10,11,12,13]. Among them, special attention has been paid to the local convergence of the method studied in [14,15], which is defined for $x_0\in \mathbb{D}$ by

$$\begin{array}{cc}\hfill y_{\sigma }& =x_{\sigma }-F{[u_{\sigma },v_{\sigma }]}^{-1}F\left(x_{\sigma }\right),u_{\sigma }=x_{\sigma }+F\left(x_{\sigma }\right),v_{\sigma }=x_{\sigma }-F\left(x_{\sigma }\right),\sigma =0,1,2,\cdots \hfill \\ \hfill z_{\sigma }& =y_{\sigma }-\left(3I-2F{[u_{\sigma },v_{\sigma }]}^{-1}F[y_{\sigma },x_{\sigma }]\right)F{[u_{\sigma },v_{\sigma }]}^{-1}F\left(y_{\sigma }\right)\hfill \\ \hfill x_{\sigma +1}& =z_{\sigma }-\left[\frac{13}{4}I-F{[u_{\sigma },v_{\sigma }]}^{-1}F[z_{\sigma },y_{\sigma }]\left(\frac{7}{2}I-\frac{5}{4}F{[u_{\sigma },v_{\sigma }]}^{-1}F[z_{\sigma },y_{\sigma }]\right)\right]F{[u_{\sigma },v_{\sigma }]}^{-1}F\left(z_{\sigma }\right).\hfill \end{array}$$

(5)

The motivation, construction, validity and comparison to other methods using similar information are explained in [14,15]. Taylor series expansions and derivatives of the eighth-order are used in the convergence analysis, although such derivatives do not occur anywhere in the expression (5). This way, the order of convergence seven was determined. It is noticed that no computable error bounds or any information on the uniqueness of the solution are obtained using, for example, Lipschitz-type functions. Moreover, a uniqueness of the solution region is not specified either. These problems limit the applicability of the method.

A motivational example is considered. For example, define the following function: F on $\mathbb{X}=\mathbb{R}$ for $\mathbb{D}=[-0.4,1.3]$:

$$F\left(\tau \right)=\left\{\begin{array}{ccc}{\tau }^{2}ln{\tau }^2+{\tau }^4-{\tau }^3,\hfill & & \tau \ne 0\hfill \\ 0,\hfill & & \tau =0\hfill \end{array}.\right.$$

(6)

The second derivative of function $F\left(\tau \right)$ does not exist for $\mathbb{D}$. Therefore, results requiring the existence of $F^{″}\left(\tau \right)$ or higher cannot be applied when studying the convergence of (5) to ${\tau }_{*}=1$. Notice that there is a plethora of iteration functions [16,17,18,19,20,21,22] used for the solutions of equations that also raise the same concerns. It is mentioned in these articles that $x_0$ should be sufficiently close to $x_{*}$ for convergence to be realized. However, nothing is said about how close $x_0$ should be to $x_{*}$ for convergence. That is, the choice of the starting point is a shot in the “dark”. Hence, the radius of the convergence ball is required. The counter expression (6) can also be used for other methods proposed in [6,7,22,23]. Local convergence results are important, since they demonstrate how difficult it is to choose the starting point. In this article, only the hypothesis on the continuity of operator F is employed in method (5). Moreover, a method is developed that is based on Lipschitz constants to find the convergence radii as well as the error estimations and the uniqueness of the solution results. Furthermore, the range of the starting guess $x_0$ is given, which guarantees the convergence of (5).

2. Convergence

Parameters and functions are used in the local convergence of the method (5). Set $A=[0,\infty )$ and consider parameters $a\ge 0$, $b\ge 0$ and $d\ge 0$.

Suppose function:

(1)

$$p_0\left(\tau \right)-1$$

has a minimal zero $S_0\in A-\left\{0\right\}$ for some function $p_0:A\to A$ which is continuous and non decreasing. Set $A_0=[0,S_0).$

(2)

$$q_1\left(\tau \right)-1$$

has a minimal zero $R_1\in A_0-\left\{0\right\}$ for some function $p:A_0\to A$ which is a continuous and non decreasing function $q_1:A_0\to A$ defined by

$$q_1\left(\tau \right)=\frac{p\left(\tau \right)}{1-p_0\left(\tau \right)}.$$

(3)

$$q_2\left(\tau \right)-1$$

has a minimal zero $R_2\in A_0-\left\{0\right\}$ for some functions $p_1:A_0\to A,p_3:A_0\to A$ which are continuous and non decreasing and function $q_2:A_0\to A$ is defined by

$$q_2\left(\tau \right)=\left[\frac{p_3\left(\tau \right)}{1-p_0\left(\tau \right)}+\frac{2d\left(p_0\left(\tau \right)+p_1\left(\tau \right)\right)}{{(1-p_0\left(\tau \right))}^2}\right]q_1\left(\tau \right).$$

(4)

$$q_3\left(\tau \right)-1$$

has a minimal zero $R_3\in A_0-\left\{0\right\}$ for some functions $p_4:A_0\to A,p_5:A_0\to A$ which are continuous and non-decreasing and function $q_3:A_0\to A$ defined by

$$q_3\left(\tau \right)=\left[\frac{p_4\left(\tau \right)}{1-p_0\left(\tau \right)}+\frac{dh\left(\tau \right)\left(5h\left(\tau \right)+4\right)}{4\left(1-p_0\left(\tau \right)\right)}\right]q_2\left(\tau \right),$$

where

$$h\left(\tau \right)=\frac{p_0\left(\tau \right)+p_5\left(\tau \right)}{1-p_0\left(\tau \right)}.$$

The parameter R is defined by

$$R=min\left\{R_j\right\},j=1,2,3$$

(7)

is proven to be a convergence radius for method (5). Set $A_1=[0,R)$.

(5)

The definition of R implies that for all $\tau \in A_1$

$$0\le p_0\left(\tau \right)<1$$

(8)

and

$$0\le q_j\left(\tau \right)<1$$

(9)

hold.

The notations $U[x_{*},\rho )$ and $U[x_{*},\rho ]$ are used to denote open and closed balls in $\mathbb{X}$, respectively, of center $x_{*}$ and radius $\rho >0$. The functions “p” are as previously given and $x_{*}$ is a simple solution of Equation $\left(1\right)$. Moreover, the following conditions are considered.Suppose:

(H₁)

For all $x\in \mathbb{D}$

$$\begin{array}{cc}\hfill & ∥F^{\prime }{\left(x_{*}\right)}^{-1}\left(F\left[x+F\left(x\right),x-F\left(x\right)\right]-F^{\prime }\left(x_{*}\right)\right)∥\le p_0\left(\left|\right|x-x_{*}\left|\right|\right),\hfill \\ \hfill & ∥I+F\left[x,x_{*}\right]∥\le a\hfill \\ \hfill & \mathrm{and}\hfill \\ \hfill & ∥I-F\left[x,x_{*}\right]∥\le b.\hfill \end{array}$$

Set ${\mathbb{D}}_0=U[x_{*},R]\cap \mathbb{D}$.

(H₂)

For all $x\in {\mathbb{D}}_0$

$$\begin{array}{cc}\hfill & ∥F^{\prime }{\left(x_{*}\right)}^{-1}\left(F\left[x+F\left(x\right),x-F\left(x\right)\right]-\left[x,x_{*};F\right]\right)∥\le p\left(\right||x-x_{*}\left|\right|),\hfill \\ \hfill & ∥F^{\prime }{\left(x_{*}\right)}^{-1}\left(F\left[y,x\right]-F^{\prime }\left(x_{*}\right)\right)∥\le p_1\left(\left|\right|x-x_{*}\left|\right|\right),\hfill \\ \hfill & ∥F^{\prime }{\left(x_{*}\right)}^{-1}\left(F\left[x,x_{*}\right]-F^{\prime }\left(x_{*}\right)\right)∥\le p_2\left(\left|\right|x-x_{*}\left|\right|\right),\hfill \\ \hfill & ∥F^{\prime }{\left(x_{*}\right)}^{-1}\left(F\left[x+F\left(x\right),x-F\left(x\right)\right]-F\left[y,x_{*}\right]\right)∥\le p_3\left(\left|\right|x-x_{*}\left|\right|\right),\hfill \\ \hfill & ∥F^{\prime }{\left(x_{*}\right)}^{-1}\left(F\left[x+F\left(x\right),x-F\left(x\right)\right]-F\left[z,x_{*}\right]\right)∥\le p_4\left(\left|\right|x-x_{*}\left|\right|\right),\hfill \\ \hfill & ∥F^{\prime }{\left(x_{*}\right)}^{-1}\left(F\left[z,y\right]-F^{\prime }\left(x_{*}\right)\right)∥\le p_5\left(\left|\right|x-x_{*}\left|\right|\right),\hfill \\ \hfill & \mathrm{and}\hfill \\ \hfill & ∥F^{\prime }{\left(x_{*}\right)}^{-1}F[x,x_{*}]∥\le d\hfill \end{array}$$

and

(H₃)

$U\left[x_{*},R_{*}\right]\subseteq \mathbb{D},where{R}_{*}=max\{aR,bR,R\}$.

Next, the local convergence of method (5) is developed with the preceding notation.

Theorem 1. 

Suppose that the hypotheses is $\left(H_1\right)$–$\left(H_3\right)$ and hold and select a starting point $x_0\in U(x_{*},R)-\left\{x_{*}\right\}$. Then, the following error bounds hold for all $\sigma =0,1,2,...$

$$\left|\right|y_{\sigma }-x_{*}\left|\right|\le q_1\left(\left|\right|x_{\sigma }-x_{*}\left|\right|\right)\left|\right|x_{\sigma }-x_{*}\left|\right|\le \left|\right|x_{\sigma }-x_{*}\left|\right|<R,$$

(10)

$$\left|\right|z_{\sigma }-x_{*}\left|\right|\le q_2\left(\left|\right|x_{\sigma }-x_{*}\left|\right|\right)\left|\right|x_{\sigma }-x_{*}\left|\right|\le \left|\right|x_{\sigma }-x_{*}\left|\right|,$$

(11)

and

$$\left|\right|x_{\sigma +1}-x_{*}\left|\right|\le q_3\left(\left|\right|x_{\sigma }-x_{*}\left|\right|\right)\left|\right|x_{\sigma }-x_{*}\left|\right|\le \left|\right|x_{\sigma }-x_{*}\left|\right|,$$

(12)

with the functions $q_j(j=1,2,3)$ and the radius $R_{*}$ as previously given. Moreover, the sequence $\left\{x_{\sigma }\right\}$ produced by method (5) is convergent to $x_{*}$.

Proof. 

Items (10)–(12) are proven using induction. $Letx\in U(x_{*},R)-x_{*}$ be arbitrary. Then, it follows by $\left(H_1\right)$

$$\left|\right|x+F\left(x\right)-x_{*}\left|\right|=\left|\right|\left(I+F[x,x_{*}]\right)(x-x_{*})\left|\right|\le \left|\right|I+F[x,x_{*}]\left|\right|\left|\right|x-x_{*}\left|\right|\le aR\le R_{*}.$$

Similarly, the following is obtained:

$$\left|\right|x-F\left(x\right)-x_{*}\left|\right|=\left|\right|\left(I-F[x,x_{*}]\right)(x-x_{*})\left|\right|\le \left|\right|I-F[x,x_{*}]\left|\right|\left|\right|x-x_{*}\left|\right|\le bR\le R_{*}.$$

Thus, it follows the iterate $x+F\left(x\right),x-F\left(x\right)\in U[x_{*},R_{*}]\subseteq \mathbb{D}$ (by $\left(H_3\right)$). The application of $\left(H_1\right)$, (7) and (8) leads to

$$∥F^{\prime }{\left(x_{*}\right)}^{-1}\left(F[x+F\left(x\right),x-F\left(x\right)]-F^{\prime }\left(x_{*}\right)\right)∥\le p_0\left(\right||x-x_{*}\left|\right|)\le p_0\left(R\right)<1.$$

Hence, $F{\left[x+F\left(x\right),x-F\left(x\right)\right]}^{-1}\in \ell \left(\mathbb{X}\right)$ with

$$∥F{\left[x+F\left(x\right),x-F\left(x\right)\right]}^{-1}{F}^{\prime }\left(x_{*}\right)∥\le \frac{1}{1-p_0\left(\right||x-x_{*}\left|\right|)},$$

(13)

due to a lemma by Banach on invertible operators [1,2,8,23]. Notice also that for $x=x_0,$ iteration $y_0$ exits according to the first sub step of method (5) for $\sigma =0$, which can be written as follows:

$$\begin{array}{c}\hfill y_0-x_{*}=F{[u_0,v_0]}^{-1}\left(F[u_0,v_0]-F[x_0,x_{*}]\right)(x_0-x_{*}).\end{array}$$

(14)

The usage of (7), (9) (for $j=0$), $\left(H_2\right)$, (13) (for $x=x_0$) and (14) leads to

$$\begin{array}{c}\hfill \left|\right|y_0-x_{*}\left|\right|\le \frac{p\left(\right||x_0-x_{*}\left|\right|\left)\right||x_0-x_{*}\left|\right|}{1-p_0\left(\right||x_0-x_{*}\left|\right|)}\le q_1\left(\right||x_0-x_{*}\left|\right|\left)\right||x_0-x_{*}\left|\right|\le \left|\right|x_0-x_{*}\left|\right|<R,\end{array}$$

(15)

proving that the iterate $y_0\in U[x_{*},R]$ and (10) holds, if $\sigma =0.$ Iterates $z_0$ and $x_1$ also exist by (13) and the second and third substeps of method (5) for $\sigma =0.$ Then, the second substep is as follows:

$$\begin{array}{cc}\hfill z_0-x_{*}& =y_0-x_{*}-F{[u_0,v_0]}^{-1}F\left(y_0\right)\hfill \\ \hfill & +2F{[u_0,v_0]}^{-1}\left[\left(F[u_0,v_0]-F^{\prime }\left(x_{*}\right)\right)+\left(F^{\prime }\left(x_{*}\right)-F[y_0,x_0]\right)\right]F{[u_0,v_0]}^{-1}F\left(y_0\right).\hfill \end{array}$$

(16)

By (7), (9) (for $j=0$), ($H_2$), (13) (for $u=x_0$), (15), (16) and the triangle inequality, it follows

$$\begin{array}{cc}\hfill \left|\right|z_0-x_{*}\left|\right|& \le \left[\frac{p_3\left(\right||x_0-x_{*}\left|\right|)}{1-p_0\left(\right||x_0-x_{*}\left|\right|)}+\frac{2d\left(p_0\left(\right||x_0-x_{*}\left|\right|)+p_1\left(\right||x_0-x_{*}\left|\right|)\right)}{{\left(1-p_0\left(\right||x_0-x_{*}\left|\right|)\right)}^2}\right]\left|\right|y_0-x_{*}\left|\right|\hfill \\ \hfill & \le q_2\left(\right||x_0-x_{*}\left|\right|\left)\right||x_0-x_{*}\left|\right|\le \left|\right|x_0-x_{*}\left|\right|\hfill \end{array}$$

(17)

proving that the iterate $z_0\in U[x_{*},R]$ and (11) if $\sigma =0.$ An estimate is needed

$$\begin{array}{cc}\hfill ∥F{[u_0,v_0]}^{-1}F[z_0,y_0]-I∥& =∥F{[u_0,v_0]}^{-1}\left(\left(F[z_0,y_0]-F^{\prime }\left(x_{*}\right)\right)+\left(F^{\prime }\left(x_{*}\right)-F[u_0,v_0]\right)\right)∥\hfill \\ & \le \frac{p_0\left(\right||x_0-x_{*}\left|\right|)+p_5\left(\right||x_0-x_{*}\left|\right|)}{1-p\left(\right||x_0-x_{*}|\left|\right)}\le h\left(\right||x_0-x_{*}\left|\right|):=h_0\hfill \end{array}$$

(18)

Moreover, according to the third substep of method (5), it follows

$$\begin{array}{cc}\hfill x_1-x_{*}& =z_0-x_{*}-F{[u_0,v_0]}^{-1}F\left(z_0\right)\hfill \\ \hfill & -\frac{1}{4}\left[5{\left(F{[u_0,v_0]}^{-1}F[z_0,y_0]-I\right)}^2-4\left(F{[u_0,v_0]}^{-1}F[z_0,y_0]-I\right)\right]F{[u_0,v_0]}^{-1}F\left(z_0\right).\hfill \end{array}$$

(19)

Next, using (7), (9) (for $j=1$), ($H_2$), (13) (for $u=x_0$), (15), (17)–(19), and the triangle inequality, the following is obtained:

$$\begin{array}{cc}\hfill lll\left|\right|x_1-x_{*}\left|\right|& \le \left[\frac{p_4\left(\right||x_0-x_{*}\left|\right|)}{1-p_0\left(\right||x_0-x_{*}\left|\right|)}+\frac{d(5h_0^2+4h_0)}{4(1-p_0\left(\right||x_0-x_{*}\left|\right|)}\right]\left|\right|z_0-x_{*}\left|\right|\hfill \\ \hfill & \le q_3\left(\right||x_0-x_{*}\left|\right|\left)\right||x_0-x_{*}\left|\right|\hfill \\ \hfill & \le \left|\right|x_0-x_{*}\left|\right|,\hfill \end{array}$$

(20)

proving that the iterate $x_1\in U[x_{*},R]$ and (12) holds, if $\sigma =0.$ Hence, estimates (10)–(12) are shown for $\sigma =0$. Simply replace $x_0,y_0,z_0,x_1$ by $x_i,y_i,z_i,x_{i+1}$ in the preceding calculations to terminate the induction for items (10)–(12). It then follows by the estimation that

$$\begin{array}{c}\hfill \left|\right|x_{i+1}-x_{*}\left|\right|\le r\left|\right|x_L-x_{*}\left|\right|<R,\end{array}$$

(21)

proving that the iterate ${\displaystyle x_{i+1}\in U[x_{*},R]}$ and ${\displaystyle \underset{i\to \infty }{lim}{x}_i=x_{*}.}$ □

A uniqueness region for the solution $x_{*}$ is established in the next result.

Proposition 1. 

Suppose that the following conditions hold:

(H₄)

There exists a solution $z_{*}\in U(x_{*},r)$ of the equation $F\left(x\right)=0$ for some $r>0$.

(H₅)

The third condition in $\left(H_2\right)$ holds in the ball $U(x_{*},r)$

and

(H₆)

There exist $R_1>r$ such that

$$p_2\left(R_1\right)<1.$$

Define the region ${\mathbb{D}}_1=U[x_{*},R_1]\cap \mathbb{D}$. Then, the only solution of the equation $F\left(x\right)=0$ in the region ${\mathbb{D}}_1$ is $x_{*}.$

Proof. 

Define the linear operator $M=F[z_{*},x_{*}]$. By applying the conditions $\left(H_4\right)$–$\left(H_6\right)$, it follows that

$$∥F^{\prime }{\left(x_{*}\right)}^{-1}\left(M-F^{\prime }\left(x_{*}\right)\right)∥\le p_2(\parallel z_{*}-x_{*}\parallel )\le p_2\left(R_1\right)<1.$$

Thus, the linear operator is invertible. Then, it follows from

$$z_{*}-x_{*}=M^{-1}\left(F\left(z_{*}\right)-F\left(x_{*}\right)\right)=M^{-1}\left(0\right)=0.$$

Therefore, it is concluded that $z_{*}=x_{*}$. □

Remark 1. 

(a) 

The following choices are considered:

$$F[x,y]=\frac{1}{2}\left(F^{\prime }\left(x\right)+F^{\prime }\left(y\right)\right)$$

or

$$F[x,y]={\int }_0^1{F}^{\prime }\left(x+\theta (y-x)\right)d\theta$$

or the standard definition of the divided difference when $\mathbb{X}={\mathbb{R}}^i$ [6,7,13,23].

Moreover, suppose

$$∥F^{\prime }{\left(x_{*}\right)}^{-1}\left(F^{\prime }\left(x\right)-F^{\prime }\left(x_{*}\right)\right)∥\le {\phi }_0\left(\right||x-x_{*}\left|\right|)$$

and

$$∥F^{\prime }{\left(x_{*}\right)}^{-1}\left(F^{\prime }\left(x\right)-F^{\prime }\left(y\right)\right)∥\le \phi \left(\right||x-x_{*}\left|\right|),$$

where functions ${\phi }_0:\mathbb{A}\to \mathbb{A}$, $\phi :\mathbb{A}\to \mathbb{R}$ are continuous and nondecreasing (see also Examples 1 and 2).

(b) 

Conditions $\left(H_1\right)$–$\left(H_3\right)$ can be condensed using the classical condition for studying methods involving divided differences instead, as follows

$$\begin{array}{c}\hfill ∥F^{\prime }{\left(x_{*}\right)}^{-1}\left(F[{\theta }_1,{\theta }_2]-F[{\theta }_3,{\theta }_4]\right)∥\le p_6\left(\right||{\theta }_1,-{\theta }_2\left|\right|,\left|\right|{\theta }_2-{\theta }_4\left|\right|)\end{array}$$

for all ${\theta }_1,{\theta }_2,{\theta }_3,{\theta }_4\in \mathbb{D},$ where function $p_6:A\times A\to A$ is continuous and non decreasing in both variables.

(c) 

Clearly, under all the conditions $\left(H_1\right)$–$\left(H_3\right)$, set $r=R$ in the Proposition 1.

3. Numerical Applications

This section is dedicated to the application of the proposed theoretical results in the earlier sections. Some nonlinear problems are chosen for computational comparisons. The numerical results are listed in Table 1, Table 2, Table 3 and Table 4. Further, the Computational Order of Convergence $\left(COC\right)$ for an iterative method $\left\{x_{\sigma }\right\}$ in $\mathbb{X}$ by

$$\lambda =\frac{ln\frac{\parallel {\nu }_{\sigma +1}-{\nu }_{*}\parallel }{|{\nu }_{\sigma }-{\nu }_{*}\parallel }}{ln\frac{\parallel {\nu }_{\sigma }-{\nu }_{*}\parallel }{\parallel {\nu }_{\sigma -1}-{\nu }_{*}\parallel }},for\sigma =1,2,\dots$$

(22)

is used as well as the Approximated Computational Order of Convergence

$\left(ACOC\right)$ [22] as:

$${\lambda }^{*}=\frac{ln\frac{\parallel {\nu }_{\sigma +1}-{\nu }_{\sigma }\parallel }{\parallel {\nu }_{\sigma }-{\nu }_{\sigma -1}\parallel }}{ln\frac{\parallel {\nu }_{\sigma }-{\nu }_{\sigma -1}\parallel }{\parallel {\nu }_{\sigma -1}-{\nu }_{\sigma -2}\parallel }},for\sigma =2,3,\dots$$

(23)

The important thing that formulas (22) and (23) do not require any derivatives of the involved operator F for the determination of $\left(COC\right)$ and $\left(ACOC\right)$. The following terminating criteria for the solutions of nonlinear systems are used:

(i)

$\parallel {\nu }_{\sigma +1}-{\nu }_{\sigma }\parallel <ϵ,$ and

(ii)

$\parallel F\left({\nu }_{\sigma }\right)\parallel <ϵ$, where $ϵ={10}^{-100}$.

All the computational work is performed with the help of $Mathematica11$. In addition, multi precision arithmetic for better numerical results is employed.

Example 1. 

Let $\mathbb{X}={\mathbb{R}}^3$ and $\mathbb{D}=U[0,1]$. Then, for $w={(w_1,w_2,w_3)}^{tr}$, consider the operator F as

$$F\left(w\right)={\left(w_1,e^{w_2}-1,\frac{e-1}{2}{w}_3^2+w_3\right)}^{tr}.$$

(24)

The derivative $F^{\prime }$ is calculated as

$$F^{\prime }\left(w\right)=\left[\begin{array}{ccc}1& 0& 0\\ 0& e^{w_2}& 0\\ 0& 0& (e-1)w_3+1\end{array}\right].$$

The solution $x_{*}={(0,0,0)}^{tr}$ is verified using (24). Consequently, $F^{\prime }\left(x_{*}\right)=F^{\prime }{\left(x_{*}\right)}^{-1}=Diag\{1,1,1\}=I$. Then, under the first or the second choice listed in Remark 1, the conditions $\left(H_1\right)$–$\left(H_3\right)$ are verified, provided that

$$\begin{array}{cc}\hfill a=c& =d=\frac{l}{2}\left(1+e^{\frac{l}{e-1}}\right),b=\frac{l}{2}\left(3+e^{\frac{l}{e-1}}\right),{\phi }_0\left(\tau \right)=(e-1)\tau ,\phi \left(\tau \right)=e^{\frac{l}{e-1}}\tau ,\hfill \\ \hfill p_0\left(\tau \right)& =\frac{l}{2}\left({\phi }_0\left(at\right)+{\phi }_0\left(b\tau \right)\right),p\left(\tau \right)=\frac{l}{2}\left(\phi \left(c\tau \right)+{\phi }_0\left(b\tau \right)\right),\hfill \\ \hfill p_1\left(\tau \right)& =\frac{l}{2}\left({\phi }_0\left(q_1\left(\tau \right)\tau \right)+{\phi }_0\left(\tau \right)\right),p_2\left(\tau \right)=\frac{l}{2}{\phi }_0\left(\tau \right),\hfill \\ \hfill p_3\left(\tau \right)& =\frac{l}{2}\left[\phi \left((a+q_1\left(\tau \right))\tau \right)+{\phi }_0\left(b\tau \right)\right],p_4\left(\tau \right)=\frac{l}{2}\left[\phi \left((a+q_2\left(\tau \right))\tau \right)+{\phi }_0\left(b\tau \right)\right]\hfill \\ \hfill and& \\ \hfill p_5\left(\tau \right)& =\frac{l}{2}\left[{\phi }_0\left(q_2\left(\tau \right)\tau \right)+{\phi }_0\left(q_1\left(\tau \right)\tau \right)\right].\hfill \end{array}$$

Table 1 provides the radii, number of iterations, convergence order, CPU timing and initial approximation of method (5) for Example 1.

Table 1. Numerical results of Example 1.

Case	R	${\mathit{R}}_{*}$	${\mathit{x}}_0$	$\parallel \mathit{F}\left({\mathit{x}}_{\mathit{\sigma }}\right)\parallel$	$\parallel {\mathit{x}}_{\mathit{\sigma }+1}-\mathit{\sigma }\parallel$	$\mathit{\sigma }$	$\mathit{\lambda }$	CPU Timing
Method (5)	0.07531	0.180351	${\left(\frac{17}{100},\frac{17}{100},\frac{17}{100}\right)}^{tr}$	$8.39844\times {10}^{-222}$	$8.39844\times {10}^{-222}$	3	6.93750	0.20457

Table 1. Numerical results of Example 1.

Case	R	${\mathit{R}}_{*}$	${\mathit{x}}_0$	$\parallel \mathit{F}\left({\mathit{x}}_{\mathit{\sigma }}\right)\parallel$	$\parallel {\mathit{x}}_{\mathit{\sigma }+1}-\mathit{\sigma }\parallel$	$\mathit{\sigma }$	$\mathit{\lambda }$	CPU Timing
Method (5)	0.07531	0.180351	${\left(\frac{17}{100},\frac{17}{100},\frac{17}{100}\right)}^{tr}$	$8.39844\times {10}^{-222}$	$8.39844\times {10}^{-222}$	3	6.93750	0.20457

Example 2. 

Let $\mathbb{X}=K[0,1]$ and $\mathbb{D}=U(0,1)$. In addition, $\mathbb{X}$ is the space of operators continuous in $[0,1]$ having the max norm. An operator defined by:

$$\mathsf{\Psi }\left(\mathsf{\Omega }\right)\left(x\right)=\mathsf{\Psi }\left(x\right)-{\int }_0^{1}x\gamma \mathsf{\Theta }{\left(\gamma \right)}^{3}d\gamma .$$

(25)

Then, this definition gives

$${\mathsf{\Psi }}^{\prime }\left(\mathsf{\Omega }\left(\mu \right)\right)\left(x\right)=\mu \left(x\right)-3{\int }_0^{1}x\gamma \mathsf{\Theta }{\left(\gamma \right)}^2\mu \left(\gamma \right)d\gamma ,\mathrm{for}\mu \in \mathbb{D}.$$

The first or the second choice in Remark 1 the conditions $\left(H_1\right)$–$\left(H_3\right)$ are verified provided

$$\begin{array}{cc}\hfill a=b& =d=\frac{1}{2},{\phi }_0\left(\tau \right)=7.5\tau ,\phi \left(\tau \right)=15\tau ,p_0\left(\tau \right)=\frac{15}{4}\tau ,p\left(\tau \right)=\frac{15}{2}\tau ,\hfill \\ \hfill p_1\left(\tau \right)& =\frac{15}{2}\tau ,p_2\left(\tau \right)=\frac{15}{4}\tau ,p_3\left(\tau \right)=\frac{15}{2}\tau ,p_4\left(\tau \right)=\frac{15}{2}\tau \mathrm{and}{p}_5\left(\tau \right)=\frac{15}{4}\tau .\hfill \end{array}$$

For $x_{*}=0$, the radii of method (5), for Example 2, are recorded in Table 2.

Table 2. Radii of convergence for Example 2.

Case	R	${\mathit{R}}_{*}$
Method (5)	0.05744	0.02872

Table 2. Radii of convergence for Example 2.

Case	R	${\mathit{R}}_{*}$
Method (5)	0.05744	0.02872

Example 3. 

The kinematic synthesis for r automotive steering problem (for detail, please see in [10,21]), is defined by

$$\begin{array}{cc}\hfill & {\left[x_1\left(x_{2}sin\left({\alpha }_{\kappa }\right)-x_3\right)\left(x_{2}cos\left({\phi }_{\kappa }\right)+1\right)-x_1\left(x_{2}cos\left({\alpha }_{\kappa }\right)-x_3\right)\left(x_{2}sin\left({\phi }_{\kappa }\right)-x_3\right)\right]}^2\hfill \\ \hfill & -{\left[A_{\kappa }\left(x_{2}sin\left({\alpha }_{\kappa }\right)-x_3\right)-B_{\kappa }\left(x_{2}sin\left({\phi }_{\kappa }\right)-x_3\right)\right]}^2-[B_{\kappa }\left(x_{2}cos\left({\phi }_{\kappa }\right)+1\right)\hfill \\ \hfill & -B_{\kappa }\left(x_{2}cos\left({\alpha }_{\kappa }\right)-1\right){]}^2=0.\hfill \end{array}$$

The values for the angles $A_{\kappa }$ and $B_{\kappa }$ are given by

$$\begin{array}{cc}\hfill A_{\kappa }& =x_2\left(cos\left({\phi }_{\kappa }\right)-cos\left({\phi }_0\right)\right)-x_3{x}_2\left(sin\left({\phi }_{\kappa }\right)-sin\left({\phi }_0\right)\right)-x_1\left(x_{2}sin\left({\phi }_{\kappa }\right)-x_3\right),\hfill \\ \hfill B_{\kappa }& =(x_3-x_1)x_{2}sin\left({\eta }_0\right)-x_3{x}_{2}sin\left({\alpha }_{\kappa }\right)+x_{2}cos\left({\eta }_0\right)+x_1{x}_3-x_{2}cos\left({\alpha }_{\kappa }\right),\kappa =1,2,3.\hfill \end{array}$$

In Table 5, the values of ${\alpha }_{\kappa }$ and ${\phi }_{\kappa }$ appear (in radians).

The approximated solution is for $\mathbb{D}=U[x_{*},1]$

$$x_{*}={(0.9051567\dots ,0.6977417\dots ,0.6508335\dots )}^t.$$

The number of iterations, convergence order, CPU timing and initial approximation of method (5), for Example 3 are displayed in Table 3.

Table 3. Numerical results for Example 3.

Case	${\mathit{x}}_0$	$\parallel \mathit{F}\left({\mathit{x}}_{\mathit{\sigma }}\right)\parallel$	$\parallel {\mathit{x}}_{\mathit{\sigma }+1}-\mathit{\sigma }\parallel$	$\mathit{\sigma }$	$\mathit{\lambda }$	CPU Timing
Method (5)	${\left(\frac{7}{10},\frac{7}{10},\frac{7}{10}\right)}^{tr}$	$4.01003\times {10}^{-127}$	$3.66513\times {10}^{-125}$	4	5.03818	2.05587

Table 3. Numerical results for Example 3.

Case	${\mathit{x}}_0$	$\parallel \mathit{F}\left({\mathit{x}}_{\mathit{\sigma }}\right)\parallel$	$\parallel {\mathit{x}}_{\mathit{\sigma }+1}-\mathit{\sigma }\parallel$	$\mathit{\sigma }$	$\mathit{\lambda }$	CPU Timing
Method (5)	${\left(\frac{7}{10},\frac{7}{10},\frac{7}{10}\right)}^{tr}$	$4.01003\times {10}^{-127}$	$3.66513\times {10}^{-125}$	4	5.03818	2.05587

Example 4. 

Consider a system of nonlinear equations (details can be found in Grau-Sánchez et al. [13]), which is given by

$$F(x_1,x_2,\dots ,x_{\chi })=\sum _{j=1,j\ne k}^{\chi }{x}_j-e^{-x_k},1\le k\le \chi .$$

(26)

For particular value of $\chi =10$, a $10\times 10$ system of nonlinear equations is obtained. The convergence order, CPU timing and initial approximation of method (5), for this Example 4 appear in Table 4.

Table 4. Numerical results of Example 4.

Case	${\mathit{x}}_0$	$\parallel \mathit{F}\left({\mathit{x}}_{\mathit{\sigma }}\right)\parallel$	$\parallel {\mathit{x}}_{\mathit{\sigma }+1}-\mathit{\sigma }\parallel$	$\mathit{\sigma }$	$\mathit{\lambda }$	CPU Timing
Method (2)	${\left(\frac{1}{2},\frac{1}{2},\stackrel{10}{\cdots },\frac{1}{2}\right)}^{tr}$	$1.21348\times {10}^{-120}$	$8.60364\times {10}^{-122}$	6	2.01662	2.04416
Method (3)	${\left(\frac{1}{2},\frac{1}{2},\stackrel{10}{\cdots },\frac{1}{2}\right)}^{tr}$	$1.54853\times {10}^{-178}$	$1.09792\times {10}^{-179}$	8	2.0113	4.24586
Method (5)	${\left(\frac{1}{2},\frac{1}{2},\stackrel{10}{\cdots },\frac{1}{2}\right)}^{tr}$	$5.00670\times {10}^{-379}$	$3.54977\times {10}^{-380}$	3	7.06516	0.43371

Table 4. Numerical results of Example 4.

Case	${\mathit{x}}_0$	$\parallel \mathit{F}\left({\mathit{x}}_{\mathit{\sigma }}\right)\parallel$	$\parallel {\mathit{x}}_{\mathit{\sigma }+1}-\mathit{\sigma }\parallel$	$\mathit{\sigma }$	$\mathit{\lambda }$	CPU Timing
Method (2)	${\left(\frac{1}{2},\frac{1}{2},\stackrel{10}{\cdots },\frac{1}{2}\right)}^{tr}$	$1.21348\times {10}^{-120}$	$8.60364\times {10}^{-122}$	6	2.01662	2.04416
Method (3)	${\left(\frac{1}{2},\frac{1}{2},\stackrel{10}{\cdots },\frac{1}{2}\right)}^{tr}$	$1.54853\times {10}^{-178}$	$1.09792\times {10}^{-179}$	8	2.0113	4.24586
Method (5)	${\left(\frac{1}{2},\frac{1}{2},\stackrel{10}{\cdots },\frac{1}{2}\right)}^{tr}$	$5.00670\times {10}^{-379}$	$3.54977\times {10}^{-380}$	3	7.06516	0.43371

The Example 4 converges to the solution $x_{*}=(0.56714,0.56714,0.56714,$${-10times-,0.56714)}^{tr}$.

Table 5. Values of ${\alpha }_{\kappa }$ and ${\phi }_{\kappa }$ (in radians) for Example 3.

$\mathit{\kappa }$	${\mathit{\alpha }}_{\mathit{\kappa }}$	${\mathit{\phi }}_{\mathit{\kappa }}$
0	$1.39541$	$1.74617$
1	$1.74448$	$2.03646$
2	$2.06562$	$2.23909$
3	$2.46006$	$2.46006$

Table 5. Values of ${\alpha }_{\kappa }$ and ${\phi }_{\kappa }$ (in radians) for Example 3.

$\mathit{\kappa }$	${\mathit{\alpha }}_{\mathit{\kappa }}$	${\mathit{\phi }}_{\mathit{\kappa }}$
0	$1.39541$	$1.74617$
1	$1.74448$	$2.03646$
2	$2.06562$	$2.23909$
3	$2.46006$	$2.46006$

Remark 2. (1) It is clear from Table 4 that method (5) performs better as compared to method (2) and method (3) in terms of residual error, the difference between two consecutive iterations, CPU timing, $\left(COC\right)$ and number of iterations.

4. Concluding Remarks

The local convergence of high order methods has been established under weak $\omega -$ continuity conditions and by using conditions involving only operators for the method (5). The earlier convergence analysis [14,15,24,25] utilized Taylor series expansions under the existence of high order derivatives which are not included within the method (5). Moreover, they may not even exist for the problem at hand. The introduction of the $\omega -$continuity conditions provided a computable error analysis as well as the determination of the uniqueness ball for the solution. This way, the number of iterates that need to be computed to obtain a desired error tolerance is known a priori. A region is specified, inside which there is only one solution. The determination of the convergence ball provides a choice for initial points. It is also worth noting that in the previous approaches without the establishment of the convergence ball, the initial point was a shot in the dark. Hence, the applicability of method (5) is extended to cases where this was not possible before. The technique is applicable for similar methods using inverses which are single, multistep and multipoint [7,8,20,23]. This is the direction for future work.