# Local Convergence Analysis of an Eighth Order Scheme Using Hypothesis Only on the First Derivative

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Local Convergence: One Dimensional Case

**Theorem**

**1.**

**Proof**

**of Theorem 1.**

**Remark**

**1**

- (a)
- In view of Equation (13) and the estimate$$\begin{array}{ccc}\u2225{F}^{\prime}{({x}^{*})}^{-1}{F}^{\prime}(x)\u2225\hfill & =& \u2225{F}^{\prime}{({x}^{*})}^{-1}({F}^{\prime}(x)-{F}^{\prime}({x}^{*}))+{F}^{\prime}({x}^{*})\u2225\hfill \\ & \le & 1+\u2225{F}^{\prime}{({x}^{*})}^{-1}({F}^{\prime}(x)-{F}^{\prime}({x}^{*}))\u2225\hfill \\ & \le & 1+{L}_{0}\parallel x-{x}^{*}\parallel \hfill \end{array}$$$$M=M(t)=1+{L}_{0}t$$
- (b)
- The results obtained here can be used for operators F satisfying the autonomous differential equation [4,5] of the form$${F}^{\prime}(x)=P(F(x)),$$
- (c)
- The radius ${r}_{1}$ was shown in [4,5] to be the convergence radius for Newton’s method under conditions (11) and (12). It follows from Equation (4) and the definition of ${r}_{1}$ that the convergence radius r of the method Equation (2) cannot be larger than the convergence radius ${r}_{1}$ of the second order Newton’s method. As already noted in [4,5] ${r}_{1}$ is at least the convergence radius given by Rheinboldt [12]$$\begin{array}{c}{r}_{R}={\displaystyle \frac{2}{3L}}.\hfill \end{array}$$In particular, for ${L}_{0}<L$ we have that$${r}_{R}<{r}_{1}$$$$\frac{{r}_{R}}{{r}_{1}}}\to {\displaystyle \frac{1}{3}}\phantom{\rule{1.em}{0ex}}\mathrm{as}\phantom{\rule{1.em}{0ex}}{\displaystyle \frac{{L}_{0}}{L}}\to 0.$$That is our convergence radius r is at most three times larger than Rheinboldt’s. The same value for ${r}_{R}$ given by Traub [14].
- (d)
- We shall show how to define function ${g}_{2}$ and l appearing in condition Equation (3) for the method$$\begin{array}{cc}\hfill {y}_{n}& ={x}_{n}-{F}^{\prime}{({x}_{n})}^{-1}F({x}_{n})\hfill \\ \hfill {z}_{n}& ={\varphi}_{4}({x}_{n},\phantom{\rule{0.277778em}{0ex}}{y}_{n}):={y}_{n}-{[{y}_{n},\phantom{\rule{0.277778em}{0ex}}{x}_{n};\phantom{\rule{0.277778em}{0ex}}F]}^{-1}{F}^{\prime}({x}_{n}){[{y}_{n},\phantom{\rule{0.277778em}{0ex}}{x}_{n};\phantom{\rule{0.277778em}{0ex}}F]}^{-1}F({y}_{n})\hfill \\ \hfill {x}_{n+1}& ={\varphi}_{8}({x}_{n},\phantom{\rule{0.277778em}{0ex}}{y}_{n},\phantom{\rule{0.277778em}{0ex}}{z}_{n}).\hfill \end{array}$$Clearly method Equation (33) is a special case of method Equation (2). If $\mathbb{X}=\mathbb{Y}=\mathbb{R},$ then method Equation (33) reduces to Kung-Traub method [14]. We shall follow the proof of Theorem 1 but first we need to show that ${[{y}_{n},\phantom{\rule{0.277778em}{0ex}}{x}_{n};\phantom{\rule{0.277778em}{0ex}}F]}^{-1}\in L(\mathbb{Y},\phantom{\rule{0.277778em}{0ex}}\mathbb{X})$. By assuming the hypotheses of Theorem 1, we get that$$\begin{array}{cc}\hfill \u2225{F}^{\prime}{({x}^{*})}^{-1}\left([{y}_{n},\phantom{\rule{0.277778em}{0ex}}{x}_{n};F]-{F}^{\prime}({x}^{*})\right)\u2225& \le {K}_{0}\parallel {y}_{n}-{x}^{*}\parallel +{K}_{1}\parallel {x}_{n}-{x}^{*}\parallel \hfill \\ & \le \left({K}_{0}{g}_{1}(\parallel {x}_{n}-{x}^{*}\parallel )+{K}_{1}\right)\parallel {x}_{n}-{x}^{*}\parallel \hfill \\ & ={p}_{0}(\parallel {x}_{n}-{x}^{*}\parallel ),\hfill \end{array}$$The function ${h}_{{p}_{0}}(t)={p}_{0}(t)-1$, where ${p}_{0}(t)=({K}_{0}{g}_{1}(t)+{K}_{1})t$, has a smallest zero denoted by ${r}_{{p}_{0}}$ in the interval $\left(0,\phantom{\rule{0.277778em}{0ex}}\frac{1}{{L}_{0}}\right)$. Set $l={r}_{{p}_{0}}$. Then, we have from the last substep of method Equation (33) that$$\begin{array}{cc}\hfill \parallel {z}_{n}-{x}^{*}\parallel \le & \parallel {y}_{n}-{x}^{*}\parallel +\u2225{[{y}_{n},\phantom{\rule{0.277778em}{0ex}}{x}_{n};\phantom{\rule{0.277778em}{0ex}}F]}^{-1}{F}^{\prime}({x}^{*})\u2225\u2225{F}^{\prime}{({x}^{*})}^{-1}{F}^{\prime}({x}^{*})\u2225\hfill \\ & \times \u2225{[{y}_{n},\phantom{\rule{0.277778em}{0ex}}{x}_{n};\phantom{\rule{0.277778em}{0ex}}F]}^{-1}{F}^{\prime}({x}^{*})\u2225\u2225{F}^{\prime}{({x}^{*})}^{-1}F({y}_{n})\u2225\hfill \\ & \le \parallel {y}_{n}-{x}^{*}\parallel +\frac{{M}^{2}}{{\left(1-{p}_{0}(\parallel {x}_{n}-{x}^{*}\parallel )\right)}^{2}}\parallel {y}_{n}-{x}^{*}\parallel \hfill \\ & \le \left(1+\frac{{M}^{2}}{{\left(1-{p}_{0}(\parallel {x}_{n}-{x}^{*}\parallel )\right)}^{2}}\right){g}_{1}(\parallel {x}_{n}-{x}^{*}\parallel )\parallel {x}_{n}-{x}^{*}\parallel \hfill \\ & =\frac{1}{2}\left(1+\frac{{M}^{2}}{{\left(1-{p}_{0}(\parallel {x}_{n}-{x}^{*}\parallel )\right)}^{2}}\right)\frac{L\parallel {x}_{n}-{x}^{*}{\parallel}^{2}}{1-{L}_{0}\parallel {x}_{n}-{x}^{*}\parallel}.\hfill \end{array}$$It follows from Equation (34) that $\lambda =2$ and$${g}_{2}(t)=\frac{1}{2}\frac{L}{1-{L}_{0}t}\left(1+\frac{{M}^{2}}{{\left(1-{p}_{0}(t)\right)}^{2}}\right).$$Then, the convergence radius is given by$$r=min\{{r}_{1},\phantom{\rule{0.277778em}{0ex}}{r}_{2},\phantom{\rule{0.277778em}{0ex}}{r}_{{p}_{0}},\phantom{\rule{0.277778em}{0ex}}{r}_{3}\}.$$

## 3. Numerical Example and Applications

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

**Example**

**4.**

#### 3.1. Results and Discussion

## 4. Basin of Attractions

**Test Problem**

**1.**

**Test Problem**

**2.**

**Test Problem**

**3.**

**Let**${p}_{3}(z)=({z}^{6}+z),$ having simple zeros $\{-0.809017-0.587785i,\phantom{\rule{0.277778em}{0ex}}-0.809017+0.587785i,\phantom{\rule{0.277778em}{0ex}}0,\phantom{\rule{0.277778em}{0ex}}0.309017-0.951057i,\phantom{\rule{0.277778em}{0ex}}0.309017+0.951057i,\phantom{\rule{0.277778em}{0ex}}1\}$. It is concluded on the basis of Figure 3, that the method ${M}_{1}$ has a much lower number of non convergent points as compared to ${M}_{2}$ and ${M}_{3}$ (in fact we can say that method ${M}_{1}$ has almost zero non convergent points in this region). Further, the dynamics behaviors of the methods ${M}_{2}$ and ${M}_{3}$ are shown to be very chaotic on the boundary points.

## 5. Conclusions

## Author Contributions

## Conflicts of Interest

## References

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${\mathit{r}}_{\mathit{R}}$ | ${\mathit{r}}_{1}$ | ${\mathit{r}}_{2}$ | ${\mathit{r}}_{3}$ | ${\mathit{r}}_{\mathit{q}}$ | ${\mathit{r}}_{{\mathit{p}}_{0}}$ | r |
---|---|---|---|---|---|---|

$0.017098$ | $0.017098$ | $0.0061188$ | $0.0077833$ | $0.0080573$ | $0.019592$ | $0.0061188$ |

${\mathit{r}}_{\mathit{R}}$ | ${\mathit{r}}_{1}$ | ${\mathit{r}}_{2}$ | ${\mathit{r}}_{3}$ | ${\mathit{r}}_{\mathit{q}}$ | ${\mathit{r}}_{{\mathit{p}}_{0}}$ | r |
---|---|---|---|---|---|---|

$0.22856$ | $0.22856$ | $0.089459$ | $0.086166$ | $0.11255$ | $0.26191$ | $0.086166$ |

${\mathit{r}}_{\mathit{R}}$ | ${\mathit{r}}_{1}$ | ${\mathit{r}}_{2}$ | ${\mathit{r}}_{3}$ | ${\mathit{r}}_{\mathit{q}}$ | ${\mathit{r}}_{{\mathit{p}}_{0}}$ | r |
---|---|---|---|---|---|---|

$0.37258$ | $0.38269$ | $0.15120$ | $0.13166$ | $0.19038$ | $0.44149$ | $0.13166$ |

${\mathit{r}}_{\mathit{R}}$ | ${\mathit{r}}_{1}$ | ${\mathit{r}}_{2}$ | ${\mathit{r}}_{3}$ | ${\mathit{r}}_{\mathit{q}}$ | ${\mathit{r}}_{{\mathit{p}}_{0}}$ | r |
---|---|---|---|---|---|---|

$0.0075648$ | $0.0075648$ | $0.0027072$ | $0.0035090$ | $0.0035649$ | $0.0086685$ | $0.0027072$ |

Methods | n | $\parallel \mathit{F}({\mathit{x}}_{\mathit{n}})\parallel $ | $\parallel {\mathit{e}}_{\mathit{n}}\parallel $ | ξ | $\u2225\frac{{\mathit{e}}_{\mathit{n}}}{{\mathit{e}}_{\mathit{n}-1}^{\mathit{p}}}\u2225$ | η |
---|---|---|---|---|---|---|

${M}_{1}$ | 0 | 6.1e−2 | 5.6e−3 | |||

1 | 1.6e−17 | 1.8e−18 | $0.00006065014094$ | 2.153729894e+49 | ||

2 | 1.0e−93 | 9.1e−95 | $4.9289$ | 2.314984786e+12 | ||

3 | 1.4e−514 | 1.3e−515 | $5.5155$ | 2.153729894e+49 | ||

${M}_{2}$ | 0 | 6.1e−2 | 5.6e−3 | |||

1 | 2.3e−13 | 2.1e−14 | $0.00002153200773$ | 4.146304252e−6 | ||

2 | 2.3e−59 | 2.1e−60 | $4.0264$ | $0.00001076045277$ | ||

3 | 8.6e−244 | 7.9e−245 | $4.0090$ | 4.146304252e−6 | ||

${M}_{3}$ | 0 | 6.1e−2 | 5.6e−3 | |||

1 | 3.6e−13 | 3.3e−14 | $0.00003395540062$ | 9.787194795e−6 | ||

2 | 2.2e−58 | 2.0e−59 | $4.0278$ | $0.00001654426326$ | ||

3 | 1.7e−239 | 1.5e−240 | $4.0050$ | 9.787194795e−6 |

Methods | n | ${\mathit{x}}_{\mathit{n}}$ | $|\mathit{F}({\mathit{x}}_{\mathit{n}})|$ | $|{\mathit{e}}_{\mathit{n}}|$ | ξ | $\left|\frac{{\mathit{e}}_{\mathit{n}}}{{\mathit{e}}_{\mathit{n}-1}^{\mathit{p}}}\right|$ | η |
---|---|---|---|---|---|---|---|

${M}_{1}$ | 0 | $0.65$ | 2.3e−1 | 6.5e−2 | |||

1 | $0.714805912187027$ | 6.5e−10 | 1.8e−10 | $0.5649113510$ | $0.3156709071$ | ||

2 | $0.714805912362778$ | 1.1e−78 | 2.9e−79 | $8.0295$ | $0.3156709076$ | ||

3 | $0.714805912362778$ | 1.1e−629 | 1.5e−629 | $8.0000$ | $0.3156709071$ | ||

${M}_{2}$ | 0 | $0.65$ | 2.3e−1 | 6.5e−2 | |||

1 | $0.714805910701679$ | 6.1e−9 | 1.7e−9 | $5.339230124$ | $2.311637753$ | ||

2 | $0.714805912362778$ | 4.9e−70 | 1.3e−70 | $8.0479$ | $2.311637800$ | ||

3 | $0.714805912362778$ | 8.8e−559 | 2.4e−559 | $8.0000$ | $2.311637753$ | ||

${M}_{3}$ | 0 | $0.65$ | 2.3e−1 | 6.5e−2 | |||

1 | $0.714805907218054$ | 1.9e−8 | 5.1e−9 | $16.53656565$ | $6.144786331$ | ||

2 | $0.714805912362778$ | 1.1e−65 | 3.0e−66 | $8.0606$ | $6.144786786$ | ||

3 | $0.714805912362778$ | 1.5e−523 | 4.2e−524 | $8.0000$ | $6.144786331$ |

Methods | n | $\parallel \mathit{F}({\mathit{x}}_{\mathit{n}})\parallel $ | $\parallel {\mathit{e}}_{\mathit{n}}\parallel $ | ξ | $\u2225\frac{{\mathit{e}}_{\mathit{n}}}{{\mathit{e}}_{\mathit{n}-1}^{\mathit{p}}}\u2225$ | η |
---|---|---|---|---|---|---|

${M}_{1}$ | 0 | 1.7e−1 | 1.6e−1 | |||

1 | 7.2e−12 | 7.2 e−12 | $0.00001412834175$ | $0.001157325063$ | ||

2 | 3.0e−93 | 3.0e−93 | $7.8569$ | $0.0004283379739$ | ||

3 | 7.9e−744 | 7.9e−744 | $7.9947$ | $0.001157325063$ | ||

${M}_{2}$ | 0 | 1.7e−1 | 1.6e−1 | |||

1 | 1.9e−5 | 1.9e−5 | $0.02642235158$ | $0.2812313492$ | ||

2 | 2.8e−20 | 2.8e−20 | $3.7651$ | $0.2223712525$ | ||

3 | 1.7e−79 | 1.7e−79 | $3.9931$ | $0.2812313492$ | ||

${M}_{3}$ | 0 | 1.7e−1 | 1.6e−1 | |||

1 | 2.8e−5 | 2.5e−5 | $0.03967215686$ | $0.4999499551$ | ||

2 | 2.5e−19 | 2.5e−19 | $3.7373$ | $0.3860243040$ | ||

3 | 1.9e−75 | 1.9e−75 | $3.9920$ | $0.4999499551$ |

Methods | n | ${\mathit{x}}_{\mathit{n}}$ | $|\mathit{F}({\mathit{x}}_{\mathit{n}})|$ | $|{\mathit{e}}_{\mathit{n}}|$ | ξ | $\left|\frac{{\mathit{e}}_{\mathit{n}}}{{\mathit{e}}_{\mathit{n}-1}^{\mathit{p}}}\right|$ | η |
---|---|---|---|---|---|---|---|

${M}_{1}$ | 0 | $0.317$ | 3.0e−4 | 1.3e−3 | |||

1 | $0.318309886183791$ | 3.8e−18 | 1.6e−17 | 1.881774410e+6 | 1.746669349e+6 | ||

2 | $0.318309886183791$ | 2.1e−129 | 8.7e−129 | $8.0023$ | 1.746669349e+6 | ||

3 | $0.318309886183791$ | 1.4e−1019 | 6.0e−1019 | $8.0000$ | 1.746669349e+6 | ||

${M}_{2}$ | 0 | $0.317$ | 3.0e−4 | 1.3e−3 | |||

1 | $0.318309886183790$ | 5.5e−17 | 2.3e−16 | 2.711175817e+7 | 2.406103772e+7 | ||

2 | $0.318309886183791$ | 5.3e−119 | 2.2e−118 | $8.0041$ | 2.406103772e+7 | ||

3 | $0.318309886183791$ | 3.5e−935 | 1.5e−934 | $8.0000$ | 2.406103772e+7 | ||

${M}_{3}$ | 0 | $0.317$ | 3.0e−4 | 1.3e−3 | |||

1 | $0.318309886183790$ | 1.7e−16 | 7.2 e−16 | 8.333410395e+7 | 7.204280059e+7 | ||

2 | $0.318309886183791$ | 1.3e−114 | 5.3e−114 | $8.0052$ | 7.204280059e+7 | ||

3 | $0.318309886183791$ | 1.1e−899 | 4.7e−899 | $8.0000$ | 7.204280059e+7 |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Argyros, I.K.; Behl, R.; Motsa, S.S.
Local Convergence Analysis of an Eighth Order Scheme Using Hypothesis Only on the First Derivative. *Algorithms* **2016**, *9*, 65.
https://doi.org/10.3390/a9040065

**AMA Style**

Argyros IK, Behl R, Motsa SS.
Local Convergence Analysis of an Eighth Order Scheme Using Hypothesis Only on the First Derivative. *Algorithms*. 2016; 9(4):65.
https://doi.org/10.3390/a9040065

**Chicago/Turabian Style**

Argyros, Ioannis K., Ramandeep Behl, and Sandile S. Motsa.
2016. "Local Convergence Analysis of an Eighth Order Scheme Using Hypothesis Only on the First Derivative" *Algorithms* 9, no. 4: 65.
https://doi.org/10.3390/a9040065