# Semi-Local Convergence of a Seventh Order Method with One Parameter for Solving Non-Linear Equations

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Convergence for the Majorizing Sequence

- (i)
- There exists a function ${\omega}_{0}$ with $Dom\left({\omega}_{0}\right)={M}_{0}$ and $Range\left({\omega}_{0}\right)=\mathbb{R}$ which is continuous and non-decreasing such that the equation$${\omega}_{0}\left(u\right)-1=0$$
- (ii)
- There exist functions $\omega ,\psi $ with $Dom\left(\omega \right)=M,Dom\left(\psi \right)=M\times M\times M$ whose range is $\mathbb{R}$ which are continuous and non-decreasing.

**Lemma**

**1.**

**Proof.**

**Remark**

**1.**

## 3. Convergence for the Method (3)

- (${A}_{1}$)
- There exists a point ${x}_{0}\in B$ and a parameter $\Delta \ge 0$ such that ${F}^{\prime}{\left({x}_{0}\right)}^{-1}\in L(V,U)$ and $\parallel {F}^{\prime}{\left({x}_{0}\right)}^{-1}F\left({x}_{0}\right)\parallel \le \Delta $.
- (${A}_{2}$)
- $\parallel {F}^{\prime}{\left({x}_{0}\right)}^{-1}({F}^{\prime}\left(u\right)-{F}^{\prime}\left({x}_{0}\right))\parallel \le {\omega}_{0}(\parallel u-{x}_{0}\parallel )$ for all $u\in B$. Set ${B}_{0}=S({x}_{0},{\rho}_{0})\cap B$.
- (${A}_{3}$)
- $\parallel {F}^{\prime}{\left({x}_{0}\right)}^{-1}({F}^{\prime}\left({u}_{2}\right)-{F}^{\prime}\left({u}_{1}\right))\parallel \le \omega (\parallel {u}_{2}-{u}_{1}\parallel )$ and$\parallel {F}^{\prime}{\left({x}_{0}\right)}^{-1}([{u}_{1},{u}_{2};F]-{F}^{\prime}\left({u}_{3}\right))\parallel \le \psi (\parallel {u}_{1}-{x}_{0}\parallel ,\parallel {u}_{2}-{x}_{0}\parallel ,\parallel {u}_{3}-{x}_{0}\parallel )$ for all ${u}_{1},{u}_{2},{u}_{3}\in {B}_{0}$.
- (${A}_{4}$)
- (${A}_{5}$)
- $S[{x}_{0},{a}^{*}]\subset B$.

**Theorem**

**1.**

**Proof.**

**Proposition**

**1.**

**Proof.**

**Remark**

**2.**

- (b)
- Proposition 1 is not using all the conditions of the Theorem 1. But if this is the case, then set ${\rho}_{1}={a}^{*}$ or ${\rho}_{1}={\rho}_{0}$.
- (c)
- The second condition in (${A}_{3}$) involving the function ψ can be dropped as follows: Suppose that there exists a function ${\psi}_{0}:M\times M\to \mathbb{R}$ continuous and non-decreasing such that for all $y,z\in {B}_{0}$$$\parallel {F}^{\prime}{\left({x}_{0}\right)}^{-1}([y,z;F]-{F}^{\prime}\left({x}_{0}\right))\parallel \le {\psi}_{0}(\parallel y-{x}_{0}\parallel ,\parallel z-{x}_{0}\parallel ).$$Then, in view of the estimate$$\parallel {F}^{\prime}{\left({x}_{0}\right)}^{-1}({F}^{\prime}\left(x\right)-[y,z;F]\parallel \le \parallel {F}^{\prime}{\left({x}_{0}\right)}^{-1}({F}^{\prime}\left(x\right)-{F}^{\prime}\left({x}_{0}\right))\parallel +\parallel {F}^{\prime}{\left({x}_{0}\right)}^{-1}([y,z;F]-{F}^{\prime}\left({x}_{0}\right))\parallel ,$$$$\psi ({t}_{1},{t}_{2},{t}_{3})={\omega}_{0}\left({t}_{1}\right)+{\psi}_{0}({t}_{2},{t}_{3})\phantom{\rule{1.em}{0ex}}for\phantom{\rule{4.pt}{0ex}}all\phantom{\rule{1.em}{0ex}}{t}_{1},{t}_{2},{t}_{3}\in \left[0,{\rho}_{0}\right).$$Moreover, if the divided difference is defined by$$[y,z;F]={\int}_{0}^{1}{F}^{\prime}(z+\theta (y-z))d\theta ,$$$$\psi ({t}_{1},{t}_{2},{t}_{3})={\omega}_{0}\left({t}_{1}\right)+\frac{1}{2}({\psi}_{0}\left({t}_{2}\right)+{\psi}_{0}\left({t}_{3}\right)),$$

## 4. Numerical Examples

**Example**

**1.**

**Estimates for ${\omega}_{\mathbf{0}}\left({\mathit{a}}_{\mathit{n}}\right)$ and ${\mathit{a}}_{\mathit{n}}$**.

**Example**

**2.**

**Estimates for ${\omega}_{\mathbf{0}}\left({\mathit{a}}_{\mathit{n}}\right)$ and ${\mathit{a}}_{\mathit{n}}$**.

**Example**

**3.**

**Estimates for ${\omega}_{\mathbf{0}}\left({\mathit{a}}_{\mathit{n}}\right)$ and ${\mathit{a}}_{\mathit{n}}$**.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

$\omega ,{\omega}_{0}$ | Lipschitz constants |

$L(U,V)$ | Set of Linear operators from U to V |

$\left\{{a}_{n}\right\}$ | Scalar sequence |

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n | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|---|---|

${\omega}_{\mathbf{0}}\left({\mathit{a}}_{\mathit{n}}\right)$ | 0 | 0.0380825 | 0.0438734 | 0.0446644 | 0.04477 | 0.0447841 | 0.044786 | 0.0447862 | 0.0447862 |

${\mathit{a}}_{\mathit{n}}$ | 0 | 0.0185768 | 0.0214017 | 0.0217875 | 0.021839 | 0.0218459 | 0.0218468 | 0.0218469 | 0.0218469 |

n | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|---|

${\omega}_{\mathbf{0}}\left({\mathit{a}}_{\mathit{n}}\right)$ | 0 | 0.0368743 | 0.0409929 | 0.0413345 | 0.0413615 | 0.0413636 | 0.0413638 | 0.0413638 |

${\mathit{a}}_{\mathit{n}}$ | 0 | 0.0179875 | 0.0199966 | 0.0201632 | 0.0201764 | 0.0201774 | 0.0201775 | 0.0201775 |

n | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|---|

${\omega}_{\mathbf{0}}\left({\mathit{a}}_{\mathit{n}}\right)$ | 0 | 0.0376078 | 0.042654 | 0.0431496 | 0.0431953 | 0.0431994 | 0.0431998 | 0.0431998 |

${\mathit{a}}_{\mathit{n}}$ | 0 | 0.0183453 | 0.0208068 | 0.0210486 | 0.0210709 | 0.0210729 | 0.0210731 | 0.0210731 |

n | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|---|

${\omega}_{\mathbf{0}}\left({\mathit{a}}_{\mathit{n}}\right)$ | 0 | 0.0366798 | 0.0404137 | 0.040727 | 0.0407527 | 0.0407548 | 0.0407549 | 0.0407549 |

${\mathit{a}}_{\mathit{n}}$ | 0 | 0.0235073 | 0.0259002 | 0.026101 | 0.0261175 | 0.0261188 | 0.0261189 | 0.0261189 |

n | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|---|

${\omega}_{\mathbf{0}}\left({\mathit{a}}_{\mathit{n}}\right)$ | 0 | 0.0358561 | 0.0385603 | 0.0387014 | 0.0387085 | 0.0387089 | 0.0387089 | 0.0387089 |

${\mathit{a}}_{\mathit{n}}$ | 0 | 0.0229794 | 0.0247124 | 0.0248029 | 0.0248074 | 0.0248076 | 0.0248077 | 0.0248077 |

n | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|---|

${\omega}_{\mathbf{0}}\left({\mathit{a}}_{\mathit{n}}\right)$ | 0 | 0.0363564 | 0.0396474 | 0.0398455 | 0.0398568 | 0.0398574 | 0.0398575 | 0.0398575 |

${\mathit{a}}_{\mathit{n}}$ | 0 | 0.0233 | 0.0254091 | 0.0255361 | 0.0255433 | 0.0255437 | 0.0255438 | 0.0255438 |

n | 0 | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|---|

${\omega}_{\mathbf{0}}\left({\mathit{a}}_{\mathit{n}}\right)$ | 0 | 0.0591554 | 0.0600971 | 0.0601084 | 0.0601086 | 0.0601086 |

${\mathit{a}}_{\mathit{n}}$ | 0 | 0.0046183 | 0.00469182 | 0.00469271 | 0.00469272 | 0.00469272 |

n | 0 | 1 | 2 | 3 | 4 |
---|---|---|---|---|---|

${\omega}_{\mathbf{0}}\left({\mathit{a}}_{\mathit{n}}\right)$ | 0 | 0.0589339 | 0.0596338 | 0.0596393 | 0.0596394 |

${\mathit{a}}_{\mathit{n}}$ | 0 | 0.00460101 | 0.00465565 | 0.00465609 | 0.00465609 |

n | 0 | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|---|

${\omega}_{\mathbf{0}}\left({\mathit{a}}_{\mathit{n}}\right)$ | 0 | 0.059072 | 0.0599212 | 0.0599287 | 0.0599287 | 0.0599287 |

${\mathit{a}}_{\mathit{n}}$ | 0 | 0.00461179 | 0.00467809 | 0.00467867 | 0.00467868 | 0.00467868 |

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**MDPI and ACS Style**

Argyros, C.I.; Argyros, I.K.; Regmi, S.; John, J.A.; Jayaraman, J.
Semi-Local Convergence of a Seventh Order Method with One Parameter for Solving Non-Linear Equations. *Foundations* **2022**, *2*, 827-838.
https://doi.org/10.3390/foundations2040056

**AMA Style**

Argyros CI, Argyros IK, Regmi S, John JA, Jayaraman J.
Semi-Local Convergence of a Seventh Order Method with One Parameter for Solving Non-Linear Equations. *Foundations*. 2022; 2(4):827-838.
https://doi.org/10.3390/foundations2040056

**Chicago/Turabian Style**

Argyros, Christopher I., Ioannis K. Argyros, Samundra Regmi, Jinny Ann John, and Jayakumar Jayaraman.
2022. "Semi-Local Convergence of a Seventh Order Method with One Parameter for Solving Non-Linear Equations" *Foundations* 2, no. 4: 827-838.
https://doi.org/10.3390/foundations2040056