Unified Convergence Criteria of Derivative-Free Iterative Methods for Solving Nonlinear Equations

Abstract

A local and semi-local convergence is developed of a class of iterative methods without derivatives for solving nonlinear Banach space valued operator equations under the classical Lipschitz conditions for first-order divided differences. Special cases of this method are well-known iterative algorithms, in particular, the Secant, Kurchatov, and Steffensen methods as well as the Newton method. For the semi-local convergence analysis, we use a technique of recurrent functions and majorizing scalar sequences. First, the convergence of the scalar sequence is proved and its limit is determined. It is then shown that the sequence obtained by the proposed method is bounded by this scalar sequence. In the local convergence analysis, a computable radius of convergence is determined. Finally, the results of the numerical experiments are given that confirm obtained theoretical estimates.

1. Introduction

One of the greatest challenges numerical functional analysis and other computational disciplines the task of approximating a locally unique solution $x_{*}$ of the nonlinear equation

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

(1)

for $F:\mathsf{\Omega }\subseteq X\to X$, F is a continuous operator, acting between Banach space X and itself. The solution $x_{*}$ is needed in closed or analytical form but this is possible only in special cases. That is why iterative solution methods are used to generate a sequence approximating $x_{*}$ provided certain conditions are verified on the initial information.

Newton’s method (NM) defined for each $n=0,1,2,\dots$

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

(2)

has been used extensively to generate such a sequence converting quadratically to $x_{*}$ [1,2].

However, there are some difficulties with the implementation of it in case the inverse of linear operator $F^{\prime }\left(x_n\right)$ is very expensive to calculate or even does not exist.

This difficulty is handled by considering iterative methods of the form

$$x_{n+1}=x_n-T_n^{-1}F\left(x_n\right)foreachn=0,1,2,\dots ,$$

(3)

where $T_n=[G_n,H_n;F]$, $[·,·;F]:\mathsf{\Omega }\times \mathsf{\Omega }\to L(X,X)$, $G_n=G(x_n,x_{n-1})=a{x}_n+b{x}_{n-1}+cF\left(x_n\right)$, $H_n=H(x_n,x_{n-1})=d{x}_n+p{x}_{n-1}+qF\left(x_n\right)$, and $a,b,c,d,p$ an q are real numbers.

Motivation for writing this article. Some popular methods are special cases of (3):

Newton: set $a=d=1$, $b=c=p=q=0$ provided F is Fréchet-differentiable;

Secant [1,3,4]: set $a=p=1$, $b=c=d=q=0$;

Kurchatov [5,6,7,8]: pick $a=2$, $b=-1$, $p=1$, $c=d=q=0$;

Steffensen [1,9]: pick $a=d=1$, $c=1$ and $b=p=q=0$.

The convergence order of these iterative methods is 2, $1.6...$, 2 and 2, respectively, [1,2,7,9]. However, the convergence ctiteria differ, rendering the comparison between them difficult [10,11,12].

Other choices of the parameters lead to less well-known methods or new methods [1,7,8,13]. Iterative methods are constructed usually based on geometrical or algebraic considerations. Ours is the latter. The introduction of these parameters and function evaluations allow for a greater flexibility, tighter error accuracy, and the handling of equations not possible before (see also numerical section). The choice $qF\left(x_n\right)$ is not necessary more appropriate.

Semi-local and local constitute two types of convergence for iterative methods.

In the semi-local convergence analysis, information is used from the initial point $x_0$ to find usually sufficient convergence criteria for the method (3). A priori estimates on the norms $\parallel x_n-x^{*}\parallel$ are also obtained. In the local convergence analysis, data about the solution $x^{*}$ is taken into account to determine the radius of convergence for the method (3). Moreover, usually upper error bounds are calculated for the norms $\parallel x_n-x^{*}\parallel$. Generalized Lipschitz-type conditions are used for both types of convergence.

The novelty of the article. Therefore, it is important to study the convergence of method (3) in both the semi-local (Section 2 and Section 3) as well as the local convergence (Section 4) case. Our technique allows for a comparison between the convergence criteria of these methods. The new convergence criteria can be weaker than those ones given if the methods are studied separately. Section 5 contains the numerical examples, and Section 6 contains the conclusions.

2. Majorizing Sequence

It is convenient for the semi-local convergence analysis of method (3) to introduce some parameters, sequences, and functions. Let $L_0,L,\lambda ,h,{\eta }_0,\eta$, and $\overline{\eta }$ be given parameters. Define the parameters

$$A=\left|a\right|+\left|d\right|+\lambda \left|c\right|,B=\left|b\right|+\left|p\right|,$$

$$C=\left(\right|c|+|p\left|\right)h,\alpha =|1-a|+|1-d|,$$

$$\beta =\lambda \left(\right|c|+|q\left|\right),\gamma =|1-a-b|+|1-d-p|,$$

$$\delta =|1-a-b|\overline{\eta }+\left|c\right|{\eta }_0+|1-d-p|\overline{\eta }+\left|q\right|{\eta }_0,$$

$$t_{-1}=0,t_0=h,t_1=\eta +h,$$

sequences

$${\mu }_{n+1}=L_0(A(t_{n+1}-t_0)+B(t_n-t_0)+C),$$

$${\lambda }_{n+1}=L(t_{n+1}-t_n+\alpha (t_n-t_{n-1})+\beta (t_n-t_0)+\gamma (t_{n+1}-t_0)+\delta ),$$

$$t_{n+2}=t_{n+1}+\frac{{\lambda }_{n+1}}{1-{\mu }_{n+1}}(t_{n+1}-t_n).$$

(4)

We shall show that $\left\{t_n\right\}$ is a majorizing sequence for $\left\{x_n\right\}$ under certain conditions. Moreover, define parameters ${\theta }_i$, $i=1,2,\dots ,8$ by

$${\theta }_1=\frac{L_{0}A}{1-L_{0}C},{\theta }_2=\frac{L_{0}B}{1-L_{0}C},{\theta }_3=\frac{L}{1-L_{0}C},{\theta }_4=\frac{L\alpha }{1-L_{0}C},$$

$${\theta }_5=\frac{L\beta }{1-L_{0}C},{\theta }_6=\frac{L\gamma }{1-L_{0}C},{\theta }_7=\frac{L\delta }{1-L_{0}C},{\theta }_8=({\theta }_5+{\theta }_6)h+{\theta }_7,$$

$$s_{-1}=0,s_0=h,s_1=\eta +h,$$

and sequences

$$m_{n+1}={\theta }_1(t_{n+1}-t_0)+{\theta }_2(t_n-t_0),$$

$$l_{n+1}={\theta }_3(t_{n+1}-t_n)+{\theta }_4(t_n-t_{n-1})+{\theta }_5(t_n-t_0)+{\theta }_6(t_{n-1}-t_0)+{\theta }_7,$$

$$s_{n+2}=s_{n+1}+\frac{l_{n+1}(s_{n+1}-s_n)}{1-m_{n+1}}.$$

(5)

We shall study the simplified version $\left\{s_n\right\}$ of sequence $\left\{t_n\right\}$.

Furthermore, define the interval $[0,1)$ quadratic polynomial

$$g\left(t\right)=({\theta }_1+{\theta }_3)t^2+({\theta }_2+{\theta }_4-{\theta }_3+{\theta }_5)t+{\theta }_6-{\theta }_4,$$

function

$$Q_{\infty }\left(t\right)=\frac{{\theta }_5\eta }{1-t}+\frac{{\theta }_6\eta }{1-t}+\frac{{\theta }_1\eta t}{1-t}+\frac{{\theta }_2\eta t}{1-t}+{\theta }_{1}ht+{\theta }_{1}ht-t+{\theta }_8$$

and sequence

$$\begin{array}{ccc}\hfill Q_n\left(t\right)& =& {\theta }_3\eta t^n+{\theta }_4\eta t^{n-1}+{\theta }_5\left(h+\frac{1-t^n}{1-t}\eta \right)+{\theta }_6\left(h+\frac{1-t^{n-1}}{1-t}\eta \right)+{\theta }_7\hfill \\ & & +t{\theta }_1\left(h+\frac{1-t^n}{1-t}\eta \right)+t{\theta }_2\left(h+\frac{1-t^{n-1}}{1-t}\eta \right)-t+{\theta }_8\hfill \\ & =& {\theta }_3\eta t^n+{\theta }_4\eta t^{n-1}+{\theta }_5(1+t+\dots +t^{n-1})\eta +{\theta }_6(1+t+\dots +t^{n-2})\eta \hfill \\ & & +{\theta }_7+t{\theta }_1(1+t+\dots +t^{n-1})\eta +t{\theta }_2(1+t+\dots +t^{n-2})\eta \hfill \\ & & +t{\theta }_{1}h+t{\theta }_{2}h-t+{\theta }_8.\hfill \end{array}$$

Suppose that either of the following conditions hold:

(I)

$$L_{0}C<1$$

equation $Q_{\infty }\left(t\right)=0$ has a minimal solution $w\in (0,1)$ satisfying

$$0\le \frac{l_1}{1-m_1}\le w$$

and

$$g\left(w\right)\ge 0.$$

(II)

$$L_{0}C<1$$

and w exists satisfying

$$0\le \frac{l_1}{1-m_1}\le w,$$

$$Q_1\left(w\right)\le 0$$

and

$$g\left(w\right)\ge 0.$$

Then, we can show the following result on majorizing sequences for method (3).

Lemma 1.

Under conditions (I) or (II), sequence $\left\{s_n\right\}$ generated by (5) is nondecreasing, bounded from above by ${\displaystyle s_{**}=h+\frac{\eta }{1-w}}$ and converges to its unique least upper bound $s_{*}\in [h+\eta ,s_{**}]$.

Proof. 

Induction is used to show

$$0\le \frac{l_{k+1}}{1-m_{k+1}}\le w$$

(6)

and

$$m_{k+1}<1.$$

(7)

These estimates are true for $k=0$ by (I) (or II). It then follows from (5) that

$$0\le s_2-s_1\le w(s_1-s_0)=w\eta$$

and

$$s_2\le h+(1+w)\eta =h+\frac{1-w^2}{1-w}\eta \le s_{**}.$$

Assume

$$0\le s_{k+1}-s_k\le w^k\eta .$$

(8)

Then, we also have

$$\begin{array}{ccc}\hfill s_{k+1}& \le & s_k+w^k\eta \le s_{k-1}+w^{k-1}\eta +w^k\eta \le \dots \hfill \\ \hfill & \le & s_1+\eta +w\eta +\dots +w^k\eta =h+\frac{1-w^{k+1}}{1-w}\eta \le s_{**}.\hfill \end{array}$$

(9)

Evidently, if we use (5), (8), and (9) estimates (6) and (7) are true if

$${\theta }_3\eta w^k+{\theta }_4\eta w^{k-1}+{\theta }_5\left(h+\frac{1-w^k}{1-w}\eta \right)+{\theta }_6\left(h+\frac{1-w^{k-1}}{1-w}\eta \right)+{\theta }_7$$

$$+t{\theta }_1\left(h+\frac{1-w^k}{1-w}\eta \right)+w{\theta }_2\left(h+\frac{1-w^{n-1}}{1-w}\eta \right)-w\le 0.$$

(10)

Define recurrent functions $Q_k$ on the interval $[0,1)$ by

$$\begin{array}{ccc}\hfill Q_k\left(t\right)& =& {\theta }_3\eta t^k+{\theta }_4\eta t^{k-1}+{\theta }_5\eta (1+t+\dots +t^{k-1})+{\theta }_6\eta (1+t+\dots +t^{k-2})\hfill \\ \hfill & & +t\eta {\theta }_1(1+t+\dots +t^{k-2})+t\eta {\theta }_2(1+t+\dots +t^{k-2})\hfill \\ \hfill & & +t{\theta }_{1}h+t{\theta }_{2}h-t+{\theta }_8.\hfill \end{array}$$

(11)

Then, we can show instead of (10) that

$$Q_k\left(w\right)\le 0.$$

(12)

Next, we relate two consecutive functions $Q_k$. By the definition of these functions we have

$$Q_{k+1}\left(t\right)=Q_{k+1}\left(t\right)-Q_k\left(t\right)+Q_k\left(t\right)=Q_k\left(t\right)+g\left(t\right)t^{k-1}\eta .$$

(13)

Case I. We have by (13) $Q_{k+1}\left(w\right)=Q_k\left(w\right)$ since $g\left(w\right)=0$. Define function $Q_{\infty }$ by

$$Q_{\infty }\left(t\right)=\underset{k\to \infty }{lim}{Q}_k\left(t\right).$$

(14)

Then, we have by (11) and (14) that

$$Q_{\infty }\left(t\right)=\frac{{\theta }_5\eta }{1-t}+\frac{{\theta }_6\eta }{1-t}+\frac{{\theta }_1\eta t}{1-t}+\frac{{\theta }_2\eta t}{1-t}+{\theta }_{1}ht+{\theta }_{2}ht-t+{\theta }_8.$$

(15)

It follows by $Q_{\infty }\left(w\right)=Q_{\infty }\left(w\right)$, (12) and (15) that we can show instead that

$$Q_{\infty }\left(w\right)\le 0,$$

(16)

which is true by the choice of w.

Case II. By $g\left(w\right)\le 0$ and (13), we have

$$Q_{k+1}\left(w\right)\le Q_k\left(w\right).$$

(17)

Thus, we can show instead of (12) that

$$Q_1\left(w\right)\le 0.$$

which is true by the definition of w. The induction for (6) and (7) is completed. Hence, in either case (I) or (II) sequence $\left\{s_n\right\}$ is nondecreasing and bounded from above by $s_{**}$ and as such it converges to its unique least upper bound $s_{*}$. □

Remark 1.

(a) Clearly sequence $\left\{t_n\right\}$ can replace $\left\{s_n\right\}$ in Lemma 1 (since they are equivalent).

(b) It follows from the proof on the Theorem 1 that the convergence of the method (3) depends on the majorizing sequence (4). Sufficient convergence criteria for the majorizing sequence are given in Lemma 1.

Next, more general sufficient convergence criteria are developed so that the conditions of the Lemma 1 imply those of the Lemma 2 but not necessarily vice versa.

Lemma 2.

Suppose that there exists $\rho >0$ such that for each $n=0,1,2,...$

$${\mu }_{n+1}<1and{t}_n<\rho .$$

(18)

Then, the following assertion holds

$$0\le t_n\le t_{n+1}<\rho and{t}_n<\rho .$$

(19)

and ${\rho }_{*}\in [0,\rho ]$ exists such that

$$\underset{n\to \infty }{lim}{t}_n={\rho }_{*}.$$

(20)

Proof. 

The definition of the sequence $\left\{t_n\right\}$ given by the formula (4) and the condition (18) imply the assertion (19) from which the item (20) is implied. □

Remark 2.

A possibly choice for ρ under the conditions of the Lemma 1 is $s_{*}$.

3. Semi-Local Convergence

The following condition (R) shall be used in the semi-local convergence.

($R_1$)

$x_{-1},x_0\in \mathsf{\Omega }$, $h\ge 0$, $\eta \ge 0$, ${\eta }_0\ge 0$, and $\overline{\eta }\ge 0$ exist such that

$$\parallel x_{-1}-x_0\parallel \le h,\parallel T_0^{-1}F\left(x_0\right)\parallel \le \eta ,\parallel F\left(x_0\right)\parallel \le {\eta }_{0}and\parallel x_0\parallel \le \overline{\eta }.$$

($R_2$)

$L_0\ge 0$, $L\ge 0$, and $\lambda \ge 0$ exist such that for all $x,y,z\in \mathsf{\Omega }$

$$\parallel T_0^{-1}([G(y,x),H(y,x);F]-T_0)\parallel \le L_0(\parallel G(y,x)-G_0\parallel +\parallel H(y,x)-H_0\parallel ),$$

$$\parallel T_0^{-1}([z,y;F]-[G(y,x),H(y,x);F])\parallel \le L(\parallel z-G(y,x)\parallel +\parallel y-H(y,x)\parallel )$$

and

$$\parallel F\left(y\right)-F\left(x_0\right)\parallel \le \lambda \parallel y-x_0\parallel .$$

($R_3$)

Conditions of Lemma 1 hold with $s_{*}$ also satisfying

$$\parallel (a+b-1)x_0+cF\left(x_0\right)\parallel \le s_{*}(1-|a|-|b|-\lambda |c\left|\right)$$

and

$$\parallel (d+p-1)x_0+qF\left(x_0\right)\parallel \le s_{*}(1-|d|-|p|-\lambda |q\left|\right).$$

($R_4$)

$U[x_0,s_{*}]\subset \mathsf{\Omega }$.

Next, we show the semi-local convergence analysis of method (3) using conditions (R) and the preceding notation.

Theorem 1.

Suppose that conditions (R) hold. Then, sequence $\left\{x_n\right\}$ starting with $x_{-1},$$x_0\in U[x_0,s_{*}]$ and generated by method (3) is well-defined in $U[x_0,s_{*}]$, remains in $U[x_0,s_{*}]$, and converges to a solution $x_{*}\in U[x_0,s_{*}]$ of equation $F\left(x\right)=0$.

Proof. 

We shall show that $\left\{t_k\right\}$ is a majorizing sequence for $\left\{x_k\right\}$ using induction. Notice that $\parallel x_0-x_{-1}\parallel \le t_0-t_{-1}$ and $\parallel x_1-x_0\parallel \le t_1-t_0$. Suppose $\parallel x_{k+1}-x_k\parallel \le t_{k+1}-t_k$.

First, we show that linear operator $T_{k+1}^{-1}\in L(X,X)$ exists. We have by the first condition in ($R_2$) that

$$\begin{array}{ccc}\hfill \parallel T_0^{-1}(T_{k+1}-T_0)\parallel & =& \parallel T_0^{-1}([G_{k+1},H_{k+1};F]-T_0)\parallel \hfill \\ \hfill & \le & L_0(\parallel G_{k+1}-G_0\parallel +\parallel H_{k+1}-H_0\parallel ).\hfill \end{array}$$

(21)

However, we have by ($R_2$) and ($R_3$)

$$\parallel a{x}_{k+1}+b{x}_k+cF\left(x_{k+1}\right)-x_0\parallel \le \parallel a(x_{k+1}-x_0)+b(x_k-x_0)+c(F\left(x_{k+1}\right)-F\left(x_0\right))$$

$$+a{x}_0+b{x}_0+cF\left(x_0\right)-x_0\parallel \le \parallel (a+b-1)x_0+cF\left(x_0\right)\parallel +\left|a\right|s_{*}+\left|b\right|s_{*}+\left|c\right|\lambda s_{*}\le s_{*},$$

and similarly

$$\parallel d{x}_{k+1}+p{x}_k+qF\left(x_{k+1}\right)-x_0\parallel \le \parallel (d+p-1)x_0+qF\left(x_0\right)\parallel$$

$$+\left|d\right|s_{*}+\left|p\right|s_{*}+\left|q\right|\lambda s_{*}\le s_{*},$$

thus, the iteration $a{x}_{k+1}+b{x}_k+cF\left(x_{k+1}\right)$, $d{x}_{k+1}+p{x}_k+qF\left(x_{k+1}\right)$ belong in $U[x_0,s_{*}]$.

Moreover, we have

$$\begin{array}{ccc}\hfill \parallel G_{k+1}-G_0\parallel & =& \parallel a{x}_{k+1}+b{x}_k+cF\left(x_{k+1}\right)-a{x}_0+b{x}_{-1}+cF\left(x_0\right)\parallel \hfill \\ \hfill & \le & \parallel a(x_{k+1}-x_0)+b(x_k-x_0)+c(F\left(x_{k+1}\right)-F\left(x_0\right))\parallel \hfill \\ \hfill & \le & \left|a\right|\parallel x_{k+1}-x_0\parallel +\left|b\right|\parallel x_k-x_0+x_0-x_{-1}\parallel \hfill \\ \hfill & & +\left|c\right|\parallel F\left(x_{k+1}\right)-F\left(x_0\right)\parallel \hfill \\ \hfill & \le & \left|a\right|(t_{k+1}-t_0)+\left|b\right|(t_k-t_0)+\left|b\right|h+\left|c\right|\lambda (t_{k+1}-t_0).\hfill \end{array}$$

(22)

Similarly, it follows

$$\parallel H_{k+1}-H_0\parallel \le \left|d\right|(t_{k+1}-t_0)+\left|p\right|(t_k-t_0)+\left|p\right|h+\left|q\right|\lambda (t_{k+1}-t_0),$$

(23)

hence (21) gives by summing up

$$\parallel T_0^{-1}(T_{k+1}-T_0)\parallel \le {\mu }_{k+1}<1$$

by Lemma 1, so $T_{k+1}^{-1}$ exists and

$$\parallel T_{k+1}^{-1}{T}_0\parallel \le \frac{1}{1-{\mu }_{k+1}}.$$

(24)

Furthermore, we can write

$$\begin{array}{ccc}\hfill F\left(x_{k+1}\right)& =& F\left(x_{k+1}\right)-F\left(x_k\right)-T_k(x_{k+1}-x_k)\hfill \\ \hfill & =& ([x_{k+1},x_k;F]-T_k)(x_{k+1}-x_k)\hfill \\ \hfill & =& ([x_{k+1},x_k;F]-[G_k,H_k;F])(x_{k+1}-x_k).\hfill \end{array}$$

(25)

Using $\left(R_2\right)$ and (25), we obtain

$$\parallel T_0^{-1}F\left(x_{k+1}\right)\parallel \le L(\parallel x_{k+1}-G_k\parallel +\parallel x_k-H_k\parallel )\parallel x_{k+1}-x_k\parallel .$$

(26)

However, we also have

$$\begin{array}{ccc}\hfill x_{k+1}-G_k& =& x_{k+1}-a{x}_k-b{x}_{k-1}-cF\left(x_k\right)=x_{k+1}-x_k\hfill \\ & & +(1-a)(x_k-x_{k-1})+(1-a)x_{k-1}-b{x}_{k-1}-cF\left(x_k\right)\hfill \\ & =& x_{k+1}-x_k+(1-a)(x_k-x_{k-1})+(1-a-b)(x_{k-1}-x_0)\hfill \\ & & +(1-a-b)x_0-c(F\left(x_k\right)-F\left(x_0\right))-cF\left(x_0\right),\hfill \end{array}$$

thus

$$\begin{array}{ccc}\hfill \parallel x_{k+1}-G_k\parallel & \le & \parallel x_{k+1}-x_k\parallel +|1-a|\parallel x_k-x_{k-1}\parallel +|1-a-b|\parallel x_{k-1}-x_0\parallel \hfill \\ & & +|1-a-b|\parallel x_0\parallel +\lambda \left|c\right|\parallel x_k-x_0\parallel +\left|c\right|\parallel F\left(x_0\right)\parallel \hfill \\ & \le & t_{k+1}-t_k+|1-a|(t_k-t_{k-1})+|1-a-b|(t_{k+1}-t_0)\hfill \\ & & +|1-a-b|\overline{\eta }+\lambda \left|c\right|(t_k-t_0)+\left|c\right|{\eta }_0,\hfill \end{array}$$

Similarly,

$$\begin{array}{ccc}\hfill x_k-H_k& =& x_k-d{x}_k-p{x}_{k-1}-qF\left(x_k\right)\hfill \\ & =& (1-d)(x_k-x_{k-1})+(1-d-p)(x_{k-1}-x_0)\hfill \\ & & +(1-d-p)x_0-qF\left(x_k\right),\hfill \end{array}$$

so

$$\begin{array}{ccc}\hfill \parallel x_k-H_k\parallel & \le & |1-d|(t_k-t_{k-1})+|1-d-p|(t_{k-1}-t_0)+|1-d-p|\overline{\eta }\hfill \\ & & +\left|q\right|\lambda (t_k-t_0)+\left|q\right|{\eta }_0,\hfill \end{array}$$

hence,

$$\begin{array}{c}\parallel x_{k+1}-G_k\parallel +\parallel x_k-H_k\parallel \le t_{k+1}-t_k+|1-a|(t_k-t_{k-1})\\ +|1-a-b|(t_{k+1}-t_0)+|1-a-b|\overline{\eta }+\lambda \left|c\right|(t_k-t_0)+\left|c\right|{\eta }_0\\ +|1-d|(t_k-t_{k-1})+|1-d-p|(t_{k-1}-t_0)+|1-d-p|\overline{\eta }\\ +\left|q\right|\lambda (t_k-t_0)+\left|q\right|{\eta }_0\\ \le t_{k+1}-t_k+\alpha (t_k-t_{k-1})+\beta (t_k-t_0)+\gamma (t_{k-1}-t_0)+\delta .\end{array}$$

(27)

Therefore, by (26), (27) and the definition of sequence ${\lambda }_{k+1}$, we obtain

$$\parallel T_0^{-1}F\left(x_{k+1}\right)\parallel \le {\lambda }_{k+1}(t_{k+1}-t_k).$$

(28)

It then follows from (3), (24), and (28) that

$$\parallel x_{k+2}-x_{k+1}\parallel \le \parallel T_{k+1}^{-1}{T}_0\parallel \parallel T_0^{-1}F\left(x_{k+1}\right)\parallel \le \frac{{\lambda }_{k+1}(t_{k+1}-t_k)}{1-{\mu }_{k+1}}=t_{k+2}-t_{k+1}$$

and

$$\begin{array}{ccc}\hfill \parallel x_{k+2}-x_0\parallel & \le & \parallel x_{k+2}-x_{k+1}\parallel +\parallel x_{k+1}-x_k\parallel +\dots +\parallel x_1-x_0\parallel \hfill \\ & \le & t_{k+2}-t_0\le t_{k+2}\le t_{**}.\hfill \end{array}$$

It follows that sequence $\left\{x_k\right\}$ is Cauchy (since $\left\{t_k\right\}$ is Cauchy as convergence by Lemma 4) and as such it converges to some $x_{*}\in U[x_0,s_{*}]$. By letting $k\to \infty$ in (28), we conclude $F\left(x_{*}\right)=0$. □

Remark 3.

Clearly, the conditions of Lemma 2 and ρ can replace Lemma 1 and $s_{*}$ in Theorem 1.

4. Local Convergence

Suppose:

(C₁)

There exists a simple solution $x_{*}\in \mathsf{\Omega }$ of equation $F\left(x\right)=0$.

(C₂)

For each $x,y\in \mathsf{\Omega }$

$$\parallel F^{\prime }{\left(x_{*}\right)}^{-1}([G(y,x),H(y,x);F]-F^{\prime }\left(x_{*}\right))\parallel \le l_0(\parallel G(y,x)-x_{*}\parallel +\parallel H(y,x)-x_{*}\parallel ),$$

$$\parallel F^{\prime }{\left(x_{*}\right)}^{-1}([G(y,x),H(y,x);F]-[y,x_{*};F])\parallel \le l(\parallel G(y,x)-y\parallel +\parallel H(y,x)-x_{*}\parallel ),$$

$$\parallel F\left(y\right)\parallel \le \lambda \parallel y-x_{*}\parallel .$$

(C₃)

The parameter $r_{*}$ satisfies the conditions

$$\parallel (a+b-1)x_{*}\parallel \le r_{*}(1-|a|-|b|-\lambda |c\left|\right)$$

and

$$\parallel (d+p-1)x_{*}\parallel \le r_{*}(1-|d|-|p|-\lambda |q\left|\right).$$

(C₄)

$U(x_{*},r_{*})\subset \mathsf{\Omega }$, where ${\displaystyle r_{*}=\frac{1}{2l_0+3l}}$.

Theorem 2.

Suppose that conditions (C) hold. Then, sequence $\left\{x_n\right\}$ starting with $x_{-1},x_0\in U(x_{*},r_{*})$ and generated by method (3) is well-defined in $U(x_{*},r_{*})$, remains in $U(x_{*},r_{*})$ and converges to a solution $x_{*}$.

Proof. 

We have by $\left(C_2\right)$ and $\left(C_3\right)$ that

$$\parallel a{x}_k+b{x}_{k-1}+cF\left(x_k\right)-x_{*}\parallel \le \parallel a(x_k-x_{*})+b(x_{k-1}-x_{*})+cF\left(x_k\right)$$

$$+a{x}_{*}+b{x}_{*}-x_{*}\parallel \le \parallel (a+b-1)x_{*}\parallel +\left|a\right|r_{*}+\left|b\right|r_{*}+\left|c\right|\lambda r_{*}\le r_{*},$$

$$\parallel d{x}_k+p{x}_{k-1}+qF\left(x_k\right)-x_{*}\parallel \le \parallel (d+p-1)x_{*}\parallel +\left|d\right|r_{*}+\left|p\right|r_{*}+\left|q\right|\lambda r_{*}\le r_{*},$$

$$\parallel a{x}_k+b{x}_{k-1}+cF\left(x_k\right)-x_k\parallel \le \parallel a{x}_k+b{x}_{k-1}+cF\left(x_k\right)-x_{*}\parallel +\parallel x_{*}-x_k\parallel \le 2r_{*},$$

$$\parallel F^{\prime }{\left(x_{*}\right)}^{-1}(T_k-F^{\prime }\left(x_{*}\right))\parallel \le l_0(\parallel G_k-x_{*}\parallel +\parallel H_k-x_{*}\parallel )\le 2l_0{r}_{*}<1,$$

so

$$\parallel T_k^{-1}{F}^{\prime }\left(x_{*}\right)\parallel \le \frac{1}{1-l_0(\parallel G_k-x_{*}\parallel +\parallel H_k-x_{*}\parallel )}.$$

We also get by ($C_2$)

$$\parallel F^{\prime }{\left(x_{*}\right)}^{-1}(T_k-[x_k,x_{*};F])\parallel \le l(\parallel H_k-x_k\parallel +\parallel G_k-x_{*}\parallel ),$$

thus

$$\begin{array}{ccc}\hfill \parallel x_{k+1}-x_{*}\parallel & =& \parallel x_k-x_{*}-T_k^{-1}F\left(x_k\right)\parallel \hfill \\ & \le & \parallel T_k^{-1}{F}^{\prime }\left(x_{*}\right)\parallel \parallel F^{\prime }{\left(x_{*}\right)}^{-1}(T_k-[x_k,x_{*};F])(x_k-x_{*})\parallel \hfill \\ & \le & \parallel T_k^{-1}{F}^{\prime }\left(x_{*}\right)\parallel \parallel F^{\prime }{\left(x_{*}\right)}^{-1}(T_k-[x_k,x_{*};F])\parallel \parallel x_k-x_{*}\parallel \hfill \\ & \le & \frac{l(\parallel H_k-x_k\parallel +\parallel G_k-x_{*}\parallel )}{1-l_0(\parallel G_k-x_{*}\parallel +\parallel H_k-x_{*}\parallel )}<\parallel x_k-x_{*}\parallel <r_{*},\hfill \end{array}$$

hence, the iterate $x_{k+1}\in U(x_{*},r_{*})$ and $\underset{k\to \infty }{lim}{x}_k=x_{*}$. □

A uniqueness of the solution domain can be specified.

Proposition 1.

Suppose that there exists a solution $x_{*}\in \mathsf{\Omega }$ of the equation $F\left(x\right)=0$ such that for each $x\in U(x_{*},{\rho }_1)$

$$\parallel F^{\prime }{\left(x_{*}\right)}^{-1}([x_{*},x;F]-F^{\prime }\left(x_{*}\right))\parallel \le l_1\parallel x-x_{*}\parallel forsome{\rho }_1,l_1>0;$$

(29)

$$l_1{\rho }_1<1.$$

(30)

Then, the point $x_{*}$ is the only solution of the equation $F\left(x\right)=0$ in the domain $U_0=U(x_{*},{\rho }_1)\cap U[x_{*},\frac{1}{l_1}]$.

Proof. 

Let $y_{*}\in U_0$ with $F\left(x\right)=0$. Define the linear operator $S=[x_{*},y_{*};F]$. By applying the condition (29) and (30), it follows that

$$\parallel F^{\prime }{\left(x_{*}\right)}^{-1}(S-F^{\prime }\left(x_{*}\right))\parallel \le l_1\parallel x_{*}-y_{*}\parallel \le l_1{\rho }_1<1,$$

thus, $S^{-1}$ exists. Then, from the identity $x_{*}-y_{*}=S^{-1}(F\left(x_{*}\right)-F\left(y_{*}\right))=S^{-1}\left(0\right)$, we conclude that $y_{*}=x_{*}$. □

5. Numerical Examples

In this section, we present numerical examples that confirm obtained semi-local theoretical results.

Firstly, we consider a nonlinear equation. Let $X=\mathbb{R}$, $\mathsf{\Omega }=(0.8,1.3)$ and

$$F\left(x\right)=x^3-1=0.$$

Let us determine the Lipschitz constants from conditions $\left(R_2\right)$. We can write

$$|F\left(y\right)-F\left(x_0\right)|=|y^3-x_0^3|=|y^2+y{x}_0+x_0^2||y-x_0|.$$

It follows that $\lambda =\underset{y\in \mathsf{\Omega }}{max}|y^2+y{x}_0+x_0^2|$. For divided difference $[x,y;F]$, we have

$$[x,y;F]=x^2+xy+y^2$$

and

$$[x,y;F]-[u,v;F]=(x-u)(x+y+u)+(y-v)(y+u+v).$$

We obtain from the last equality that

$$\left|T_0^{-1}([x,y;F]-[u,y;F])\right|\le \frac{1}{\left|T_0\right|}\underset{x,y,u,v\in \mathsf{\Omega }}{max}\{|x+y+u|,|y+u+v|\}(|x-u|+|y-v|).$$

If $a=d$, $b=p$, $c=q$, and F is Fréchet-differentiable, then we obtain methods with derivatives. In this case, $[x,x;F]=F^{\prime }\left(x\right)$ and

$$F^{\prime }\left(u\right)-F^{\prime }\left(u_0\right)=3(u+u_0)(u-u_0)\Rightarrow L_0=\frac{1.5}{\left|T_0\right|}\underset{u\in \mathsf{\Omega }}{max}|u+u_0|.$$

In Table 1, there are Lipschitz constants from conditions $\left(R_2\right)$ and the value $s_{*}$ to which the sequence $\left\{t_n\right\}$ converges. We see that in both cases sequences $\left\{x_n\right\}$ is contained in $U(x_0,s_{*})\subset \mathsf{\Omega }$.

Table 1. Lipschitz constants and radii.

Method	${\mathit{L}}_0$	L	$\mathit{\lambda }$	${\mathit{s}}_{*}$
Newton	0.9917	1.0744	4.3300	0.1023
Secant	1.0101	1.0647	4.3300	0.1129

In Table 2, there are values of the error at each step. The calculations were performed for initial approximation $x_0=1.1$ and an accuracy $\epsilon ={10}^{-10}$. For the Secant method, $x_{-1}=1.11$. We see from the obtained results that

$$|x_n-x_{n-1}|\le t_n-t_{n-1}$$

is performed for each $n\ge 1$.

Table 2. Results for Newton and Secant method.

n	Newton Method			Secant Method
	${\mathit{x}}_{\mathit{n}}$	$|{\mathit{x}}_{\mathit{n}}-{\mathit{x}}_{\mathit{n}-\mathbf{1}}|$	${\mathit{t}}_{\mathit{n}}-{\mathit{t}}_{\mathit{n}-\mathbf{1}}$	${\mathit{x}}_{\mathit{n}}$	$|{\mathit{x}}_{\mathit{n}}-{\mathit{x}}_{\mathit{n}-\mathbf{1}}|$	${\mathit{t}}_{\mathit{n}}-{\mathit{t}}_{\mathit{n}-\mathbf{1}}$
1	1.0088	9.1185 × 10${}^{-2}$	9.1185 × 10${}^{-2}$	1.0096	9.0361 × 10${}^{-2}$	9.0361 × 10${}^{-2}$
2	1.0001	8.7386 × 10${}^{-3}$	1.0905 × 10${}^{-2}$	1.0009	8.7419 × 10${}^{-3}$	1.0991 × 10${}^{-2}$
3	1.0000	7.6802 × 10${}^{-5}$	1.6022 × 10${}^{-4}$	1.0000	8.8887 × 10${}^{-4}$	1.5283 × 10${}^{-3}$
4	1.0000	5.8989 × 10${}^{-9}$	3.4595 × 10${}^{-8}$	1.0000	8.5828 × 10${}^{-6}$	2.6685 × 10${}^{-5}$
5	1.0000	0	1.6098 × 10${}^{-15}$	1.0000	7.7050 × 10${}^{-9}$	5.7989 × 10${}^{-8}$
6				1.0000	6.6169 × 10${}^{-14}$	2.1673 × 10${}^{-12}$

Then, we consider a system of nonlinear equations. Let $X={\mathbb{R}}^3$, $\mathsf{\Omega }=U(0,1)$ and

$$F\left(x\right)=\left(\begin{array}{c}{e}^{x_1}-1\\ \frac{e-1}{2}{x}_2^3+x_2\\ x_3\end{array}\right)=0.$$

Since $|e^{t_1}-e^{t_2}|\le e|t_1-t_2|$, then

$$\lambda =max\left\{e,\frac{e-1}{2}max|y_2^2+y_2{\tau }_0+{\tau }_0^2|+1,1\right\},x_0={({\xi }_0,{\tau }_0,{\rho }_0)}^T$$

and

$$L_0=\parallel T_0^{-1}\parallel max\left\{\frac{e}{2},\frac{e-1}{2}{M}_0\right\},L=\parallel T_0^{-1}\parallel max\left\{\frac{e}{2},\frac{e-1}{2}M\right\}.$$

The constants $M_0$ and M are calculated similarly to the previous example.

Table 3 and Table 4 show results for system of nonlinear equations. The calculations were performed for initial approximations $x_0={(0.07,0.07,0.07)}^T$, $x_{-1}={(0.08,0.08,0.08)}^T$ and an accuracy $\epsilon ={10}^{-10}$. From the obtained results we see that

$$\parallel x_n-x_{n-1}\parallel \le t_n-t_{n-1}$$

is satisfied for each $n\ge 1$.

Table 3. Lipschitz constants and radii.

Method	${\mathit{L}}_0$	L	$\mathit{\lambda }$	${\mathit{s}}_{*}$
Newton	1.3789	2.5774	2.7183	0.0864
Secant	1.7784	2.5774	2.7183	0.1041

Table 4. Results for Newton and Secant method.

n	Newton Method		Secant Method
	$\parallel {\mathit{x}}_{\mathit{n}}-{\mathit{x}}_{\mathit{n}-\mathbf{1}}\parallel$	${\mathit{t}}_{\mathit{n}}-{\mathit{t}}_{\mathit{n}-\mathbf{1}}$	$\parallel {\mathit{x}}_{\mathit{n}}-{\mathit{x}}_{\mathit{n}-\mathbf{1}}\parallel$	${\mathbf{t}}_{\mathbf{n}}-{\mathbf{t}}_{\mathbf{n}-\mathbf{1}}$
1	7.0000 × 10${}^{-2}$	7.0000 × 10${}^{-2}$	7.0000 × 10${}^{-2}$	7.0000 × 10${}^{-2}$
2	2.3910 × 10${}^{-3}$	1.5651 × 10${}^{-2}$	2.6368 × 10${}^{-3}$	1.7556 × 10${}^{-2}$
3	2.8629 × 10${}^{-6}$	8.2657 × 10${}^{-4}$	9.4309 × 10${}^{-5}$	5.9446 × 10${}^{-3}$
4	4.0981 × 10${}^{-12}$	2.3125 × 10${}^{-6}$	1.2890 × 10${}^{-7}$	5.7643 × 10${}^{-4}$
5			6.0867 × 10${}^{-12}$	1.5803 × 10${}^{-5}$

6. Conclusions

A unified convergence analysis of the method without derivatives is provided under the classical Lipschitz conditions for first-order divided differences. The current convergence analysis allows for a comparison between specialized methods that was not possible before under the same set of conditions. The results of the numerical experiment that confirmed the theoretical one are given. The developed technique can also be employed on multipoint as well as multi-step iterative methods [13,14]. This is a possible direction for future areas of research.