# A Convex Combination Approach for Mean-Based Variants of Newton’s Method

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^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Convex Combination

**Theorem**

**1.**

**Proof.**

**Remark**

**1.**

**Remark**

**2.**

**Corollary**

**1**

**(θ-test)**

#### Re-Proving Known Results for MBN

- (i)
- Arithmetic mean:$$\begin{array}{cc}\hfill {M}_{A}({f}^{\prime}({x}_{n}),{f}^{\prime}({z}_{n}))& =\frac{{f}^{\prime}({x}_{n})+{f}^{\prime}({z}_{n})}{2}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\frac{1}{2}\left({f}^{\prime}(\alpha )[1+2{c}_{2}\phantom{\rule{0.166667em}{0ex}}{e}_{n}+\mathcal{O}({e}_{n}^{2})]+{f}^{\prime}(\alpha )[1+\mathcal{O}({e}_{n}^{2})]\right)\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& ={f}^{\prime}(\alpha )[1+{c}_{2}\phantom{\rule{0.166667em}{0ex}}{e}_{n}+\mathcal{O}({e}_{n}^{2})].\hfill \end{array}$$
- (ii)
- Heronian mean: In this case, the associated $\theta $-test is:$$\begin{array}{cc}\hfill {M}_{He}{f}^{\prime}({x}_{n}),{f}^{\prime}({z}_{n})& =\frac{1}{3}\left({f}^{\prime}(\alpha )[1+2{c}_{2}{e}_{n}+\mathcal{O}({e}_{n}^{2})]+{f}^{\prime}(\alpha )[1+{c}_{2}{e}_{n}+\mathcal{O}({e}_{n}^{2})]+{f}^{\prime}(\alpha )[1+\mathcal{O}({e}_{n}^{2})]\right)\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\frac{{f}^{\prime}(\alpha )}{3}[3+2{c}_{2}{e}_{n}+{c}_{2}{e}_{n}+\mathcal{O}({e}_{n}^{2})].\hfill \end{array}$$
- (iii)
- Generalized mean:$$\begin{array}{cc}\hfill {M}_{G}({f}^{\prime}({x}_{n}),{f}^{\prime}({z}_{n}))& ={\left(\frac{{f}^{\prime}{({x}_{n})}^{m}+{f}^{\prime}{({z}_{n})}^{m})}{2}\right)}^{1/m}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& ={\left(\frac{{f}^{\prime}{(\alpha )}^{m}{[1+2{c}_{2}\phantom{\rule{0.166667em}{0ex}}{e}_{n}+\mathcal{O}({e}_{n}^{2})]}^{m}+{f}^{\prime}{(\alpha )}^{m}{[1+\mathcal{O}({e}_{n}^{2})]}^{m}}{2}\right)}^{1/m}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& ={f}^{\prime}(\alpha ){\left({[1+{c}_{2}\phantom{\rule{0.166667em}{0ex}}{e}_{n}+\mathcal{O}({e}_{n}^{2})]}^{m}\right)}^{1/m}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& ={f}^{\prime}(\alpha )[1+{c}_{2}\phantom{\rule{0.166667em}{0ex}}{e}_{n}+\mathcal{O}({e}_{n}^{2})].\hfill \end{array}$$
- (iv)
- Centroidal mean:$$\begin{array}{cc}\hfill {M}_{Ce}({f}^{\prime}({x}_{n}),{f}^{\prime}({z}_{n}))& =\frac{2({f}^{\prime}{({x}_{n})}^{2}+{f}^{\prime}({x}_{n}){f}^{\prime}({z}_{n})+{f}^{\prime}({z}_{n}))}{3({f}^{\prime}({x}_{n})+{f}^{\prime}({z}_{n}))}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\frac{2({f}^{\prime}{(\alpha )}^{2}[1+2{c}_{2}{e}_{n}+\mathcal{O}({e}_{n}^{2})]+{f}^{\prime}{(\alpha )}^{2}[2+4{c}_{2}{e}_{n}+\mathcal{O}({e}_{n}^{2})])}{3({f}^{\prime}(\alpha )[2+2{c}_{2}{e}_{n}+\mathcal{O}({e}_{n}^{2})])}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\frac{2({f}^{\prime}{(\alpha )}^{2}[3+6{c}_{2}{e}_{n}+\mathcal{O}({e}_{n}^{2})])}{3({f}^{\prime}(\alpha )[2+2{c}_{2}{e}_{n}+\mathcal{O}({e}_{n}^{2})])}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& ={f}^{\prime}(\alpha )[1+2{c}_{2}{e}_{n}+\mathcal{O}({e}_{n}^{2})][1+{c}_{2}{e}_{n}+\mathcal{O}({e}_{n}^{2})]\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& ={f}^{\prime}(\alpha )[1+{c}_{2}{e}_{n}+\mathcal{O}({e}_{n}^{2})].\hfill \end{array}$$

## 3. New MBN by Using the Lehmer Mean and Its Generalization

#### $\sigma $-Means

**Definition**

**1**

**(σ-mean)**

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

## 4. Numerical Results and Dependence on Initial Estimations

- (a)
- ${f}_{1}(x)={x}^{3}+4{x}^{2}-10$,
- (b)
- ${f}_{2}(x)=sin{(x)}^{2}-{x}^{2}+1$,
- (c)
- ${f}_{3}(x)={x}^{2}-{\mathrm{e}}^{x}-3x+2$,
- (d)
- ${f}_{4}(x)=cos(x)-x$,
- (e)
- ${f}_{5}(x)={(x-1)}^{3}-1$.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Table 1.**Numerical results. HN, the harmonic mean Newton method; CHN, the contraharmonic mean Newton method; LN, the Lehmer–Newton method; CN, the classic Newton method.

Function | x${}_{0}$ | Number of Iterations | ACOC | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

HN | CHN | LN(−7) | 1/3 N | CN | HN | CHN | LN(−7) | 1/3 N | CN | ||

(a) | −0.5 | 50 | 18 | 55 | 6 | 132 | 3.10 | 3.03 | 2.97 | 1.99 | 2.00 |

1 | 4 | 5 | 5 | 5 | 6 | 2.94 | 3.01 | 2.96 | 2.02 | 2.00 | |

2 | 4 | 5 | 5 | 5 | 6 | 3.10 | 2.99 | 3.02 | 2.00 | 2.00 | |

(b) | 1 | 4 | 5 | 6 | 6 | 7 | 3.06 | 3.16 | 3.01 | 2.01 | 2.00 |

3 | 4 | 5 | 7 | 6 | 7 | 3.01 | 2.95 | 3.02 | 2.01 | 2.00 | |

(c) | 2 | 5 | 5 | 5 | 5 | 6 | 3.01 | 2.99 | 3.11 | 2.01 | 2.00 |

3 | 5 | 6 | 5 | 6 | 7 | 3.10 | 3.00 | 3.10 | 2.01 | 2.00 | |

(d) | −0.3 | 5 | 5 | 5 | 6 | 6 | 2.99 | 3.14 | 3.02 | 2.01 | 1.99 |

1 | 4 | 4 | 4 | 5 | 5 | 2.99 | 2.87 | 2.88 | 2.01 | 2.00 | |

1.7 | 4 | 4 | 5 | 5 | 5 | 3.00 | 2.72 | 3.02 | 2.01 | 1.99 | |

(e) | 0 | 6 | $>1000$ | 7 | 7 | 10 | 3.06 | 3.00 | 3.02 | 2.01 | 2.00 |

1.5 | 5 | 7 | 7 | 7 | 8 | 3.04 | 3.01 | 2.99 | 2.01 | 2.00 | |

2.5 | 4 | 5 | 5 | 5 | 7 | 3.07 | 2.96 | 3.01 | 1.99 | 2.00 | |

3.0 | 5 | 6 | 6 | 6 | 7 | 3.04 | 2.99 | 2.98 | 2.00 | 2.00 | |

3.5 | 5 | 6 | 6 | 6 | 8 | 3.07 | 2.95 | 2.99 | 2.00 | 2.00 |

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**MDPI and ACS Style**

Cordero, A.; Franceschi, J.; Torregrosa, J.R.; Zagati, A.C.
A Convex Combination Approach for Mean-Based Variants of Newton’s Method. *Symmetry* **2019**, *11*, 1106.
https://doi.org/10.3390/sym11091106

**AMA Style**

Cordero A, Franceschi J, Torregrosa JR, Zagati AC.
A Convex Combination Approach for Mean-Based Variants of Newton’s Method. *Symmetry*. 2019; 11(9):1106.
https://doi.org/10.3390/sym11091106

**Chicago/Turabian Style**

Cordero, Alicia, Jonathan Franceschi, Juan R. Torregrosa, and Anna C. Zagati.
2019. "A Convex Combination Approach for Mean-Based Variants of Newton’s Method" *Symmetry* 11, no. 9: 1106.
https://doi.org/10.3390/sym11091106