# Convergence Analysis and Complex Geometry of an Efficient Derivative-Free Iterative Method

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## Abstract

**:**

## 1. Introduction

## 2. Local Convergence Analysis

- (a1)
- $F:\Omega \subseteq {B}_{1}\to {B}_{2}$ is a continuously differentiable operator and $[\xb7,\xb7\phantom{\rule{0.166667em}{0ex}};\phantom{\rule{0.166667em}{0ex}}F]:\Omega \times \Omega \to \mathcal{L}({B}_{1},{B}_{2})$ is a first divided difference operator of F.
- (a2)
- There exists ${x}^{\ast}\in \Omega $ so that $F\left({x}^{\ast}\right)=0$ and ${F}^{\prime}{\left({x}^{\ast}\right)}^{-1}\in \mathcal{L}({B}_{2},{B}_{1}).$
- (a3)
- There exists a continuous and nondecreasing function ${w}_{0}\phantom{\rule{0.166667em}{0ex}}:{\mathbb{R}}_{+}\cup \left\{0\right\}\to {\mathbb{R}}_{+}\cup \left\{0\right\}$ with ${w}_{0}\left(0\right)=0$ such that, for each $x\in \Omega $,$$\parallel {F}^{\prime}{\left({x}^{\ast}\right)}^{-1}([x,y\phantom{\rule{0.166667em}{0ex}};\phantom{\rule{0.166667em}{0ex}}F]-{F}^{\prime}\left({x}^{\ast}\right))\parallel \le {w}_{0}(\parallel x-{x}^{\ast}\parallel ,\parallel y-{x}^{\ast}\parallel ).$$
- (a4)
- Let ${\Omega}_{0}=\Omega \cap U({x}^{\ast},r)$, where r has been defined before. There exists continuous and nondecreasing function ${v}_{0}:[0,r)\to {\mathbb{R}}_{+}\cup \left\{0\right\}$ such that, for each $x,y\in {\Omega}_{0}$,$$\parallel \beta [x,{x}^{\ast}\phantom{\rule{0.166667em}{0ex}};\phantom{\rule{0.166667em}{0ex}}F]\parallel \le {v}_{0}(\parallel {x}_{0}-{x}^{\ast}\parallel ),$$$$\overline{U}({x}^{\ast},r)\subset \Omega ,$$$$\parallel I+\beta [x,{x}^{\ast}\phantom{\rule{0.166667em}{0ex}};\phantom{\rule{0.166667em}{0ex}}F]\parallel \le p.$$
- (a5)
- $\overline{U}({x}^{\ast},{r}_{3})\subseteq \Omega $ and $\parallel {F}^{\prime}{\left({x}^{\ast}\right)}^{-1}{F}^{\prime}\left(x\right)\parallel \le M$.
- (a6)
- Let $R\ge {r}_{3}$ and set ${\Omega}_{1}=\Omega \cap \overline{U}({x}^{\ast},R)$, ${\int}_{0}^{1}{w}_{0}\left(\theta R\right)d\theta <1.$

**Theorem**

**1.**

**Proof.**

## 3. Numerical Examples

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

## 4. Basins of Attraction

**Test problem 1.**Consider the polynomial ${p}_{1}\left(z\right)={z}^{4}-6{z}^{2}+8$ that has four simple zeros $\{\pm 2,\pm 1.414\dots \}$. We use a grid of $400\times 400$ points in a rectangle $R\in \mathbb{C}$ of size $[-3,3]\times [-3,3]$ and allocate the red, blue, green and yellow colors to the basins of attraction of these four zeros. Basins obtained for the method (3) are shown in Figure 1(i)–(iii) corresponding to $\beta ={10}^{-2},\phantom{\rule{0.166667em}{0ex}}{10}^{-4},\phantom{\rule{0.166667em}{0ex}}{10}^{-9}$. Observing the behavior of the method, we say that the divergent zones (black zones) are becoming smaller with the decreasing value of $\beta $.

**Problem 2**. Let us take the polynomial ${p}_{2}\left(z\right)={z}^{3}-z$ having zeros $\{0,\pm 1\}$. In this case, we also consider a rectangle $R=[-3,3]\times [-3,3]\in \mathbb{C}$ with $400\times 400$ grid points and allocate the colors red, green and blue to each point in the basin of attraction of $-1$, 0 and 1, respectively. Basins obtained for the method (3) are displayed in Figure 2(i)–(iii) for the parameter values $\beta ={10}^{-2},\phantom{\rule{0.166667em}{0ex}}{10}^{-4},\phantom{\rule{0.166667em}{0ex}}{10}^{-9}$. Notice that the divergent zones are becoming smaller in size as parameter $\beta $ assumes smaller values.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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Kumar, D.; Sharma, J.R.; Jäntschi, L.
Convergence Analysis and Complex Geometry of an Efficient Derivative-Free Iterative Method. *Mathematics* **2019**, *7*, 919.
https://doi.org/10.3390/math7100919

**AMA Style**

Kumar D, Sharma JR, Jäntschi L.
Convergence Analysis and Complex Geometry of an Efficient Derivative-Free Iterative Method. *Mathematics*. 2019; 7(10):919.
https://doi.org/10.3390/math7100919

**Chicago/Turabian Style**

Kumar, Deepak, Janak Raj Sharma, and Lorentz Jäntschi.
2019. "Convergence Analysis and Complex Geometry of an Efficient Derivative-Free Iterative Method" *Mathematics* 7, no. 10: 919.
https://doi.org/10.3390/math7100919