# Stability Analysis of Jacobian-Free Newton’s Iterative Method

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

**Theorem**

**1.**

**Corollary**

**1.**

**Lemma**

**1.**

**Definition**

**1.**

**Theorem**

**2.**

- (a)
- If all eigenvalues ${\lambda}_{j}$ have $|{\lambda}_{j}|<1$, then ${x}^{*}$ is attracting.
- (b)
- If one eigenvalue $|{\lambda}_{{j}_{0}}|>1$, ${j}_{0}\in \{1,2,\cdots ,n\}$ then ${x}^{*}$ is unstable, that is, a repelling or a saddle point.
- (c)
- If all eigenvalues ${\lambda}_{j}$ have $|{\lambda}_{j}|>1$, then ${x}^{*}$ is repelling.

**Proposition**

**1.**

- (a)
- If $\mid \frac{\partial {g}_{i}\left({x}^{*}\right)}{\partial {x}_{j}}\mid <\frac{1}{n}$ for all $i,j\in \{1,\dots ,n\}$, then ${x}^{*}\in {\mathbb{R}}^{n}$ is attracting.
- (b)
- If $\mid \frac{\partial {g}_{i}\left({x}^{*}\right)}{\partial {x}_{j}}\mid =0$ for all $i,j\in \{1,\dots ,n\}$, then ${x}^{*}\in {\mathbb{R}}^{n}$ is superattracting.
- (c)
- If $\mid \frac{\partial {g}_{i}\left({x}^{*}\right)}{\partial {x}_{j}}\mid >\frac{1}{n}$ for all $i,j\in \{1,\dots ,n\}$, then ${x}^{*}\in {\mathbb{R}}^{n}$${x}^{*}$ is unstable and lies at the Julia set.

**Definition**

**2.**

## 2. Jacobian-Free Variants of Newton’s Method

_{m}, for $m=1,2,\dots $. In a similar way, when central divided difference (6) is used to replace Jacobian matrix in Newton’s procedure, the resulting modified schemes are denoted by CMN

_{m}, for $m=1,2,\dots $. Also, the modified Newton’s method obtained by using divided difference (7) is denoted by RMN

_{m}, for $m=1,2,\dots $.

_{m}, CMN

_{m}and RMN

_{m}on the polynomial systems $q\left(x\right)=0$, $r\left(x\right)=0$ and $p\left(x\right)=0$. In the following sections, the coordinate functions of the different classes of iterative methods, joint with their fixed and critical points are summarized.

#### 2.1. Second-Degree Polynomial System $q\left(x\right)=0$

**Proposition**

**2.**

_{m}for $m=1,2,\dots $ on polynomial system $q\left(x\right)=0$ are

- (a)
- For $m=1,2,\dots $, the only fixed points are the roots of $q\left(x\right)$, $(-1,-1)$, $(-1,1)$, $(1,-1)$ and $(1,1)$, that are also superattracting. There is no strange fixed point in this case.
- (b)
- The components of free critical points ${c}_{{\lambda}^{1,m}}^{n}=(k,l)$ are roots of $2{q}_{j}\left(x\right)+(2+2m){x}_{j}{q}_{j}{\left(x\right)}^{m}+{q}_{j}{\left(x\right)}^{\left(2m\right)}=0$, provided that k and l are not equal to 1 and $-1$, simultaneously.

**Remark**

**1.**

_{m}. In particular, free critical points for the fixed point function ${\lambda}^{1,m}\left(x\right)$, $m=1,2,\dots ,6$ are

**Proposition**

**3.**

_{m}, for $m=1,2,\dots $, and RMN

_{m}, for $m=1,2,\dots $, on polynomial system $q\left(x\right)=0$ are

_{m}and RMN

_{m}methods coincide with that of Newton’s method. In this case, $(-1,-1)$, $(-1,1)$, $(1,-1)$ and $(1,1)$ are superattracting fixed points and there are no strange fixed points or free critical points. Figure 2 shows the resulting dynamical plane, that coincides with that of Newton’s method.

#### 2.2. Sixth-Degree Polynomial System $r\left(x\right)=0$

_{m}, CMN

_{m}and RMN

_{m}classes applied on sixth-degree polynomial system $r\left(x\right)=0$ are summarized in the following propositions. They resume, in each case, the iteration function, the fixed and critical points.

**Proposition**

**4.**

_{m}on $r\left(x\right)$, for $m=1,2,\dots $, are

- (a)
- For $m\ge 1$ the fixed points are $(-1,-1)$, $(-1,1)$, $(1,-1)$ and $(1,1)$ that are also superattracting. There are no strange fixed points in this case.
- (b)
- The coordinates of free critical points ${c}_{{h}^{1,m}}^{n}=(k,l)$ are the roots of the polynomial$$\begin{array}{ccc}\hfill {({x}_{j}^{6}-{({x}_{j}+{r}_{j}{\left(x\right)}^{m})}^{6})}^{2}& +& {r}_{j}{\left(x\right)}^{1+m}(-6{x}_{j}^{5}+6(1+6m{x}_{j}^{5}{r}_{j}{\left(x\right)}^{-1+m}){({x}_{j}+{r}_{j}{\left(x\right)}^{m})}^{5})\hfill \\ & +& 6(1+m){x}_{j}^{5}{r}_{j}{\left(x\right)}^{m}({x}_{j}^{6}-{({x}_{j}+{r}_{j}{\left(x\right)}^{m})}^{6}),\hfill \end{array}$$

**Remark**

**2.**

_{m}, for $m=2,3,\dots $, has 32 free critical points. The free critical points in this case are

_{2}and FMN

_{3}methods. These figures show the difference between the basins of attraction of an odd member with an even member of the family FMN

_{m}. Because of the symmetry, the transpose of the basin of attraction of $(1,-1)$ coincide with that of $(-1,1)$, so we only depicted one of them.

**Proposition**

**5.**

_{m}, for $m=1,2,\dots $, on $r\left(x\right)$ are

- (a)
- The fixed points are $(-1,-1)$, $(1,-1)$, $(-1,1)$ and $(1,1)$ that are also superattracting and there are not strange fixed points.
- (b)
- The components of free critical points ${c}_{{h}^{2,m}}^{n}=(k,l)$ in this case are the roots of the polynomial ${P}_{{h}^{2,m}}^{j}\left(x\right)$, provided that ${Q}_{{h}^{2,m}}^{j}\ne 0$, being$$\begin{array}{ccc}{P}_{{h}^{2,m}}^{j}\left(x\right)\hfill & =& 15{x}_{j}^{10}+30(3+4m){x}_{j}^{8}{r}_{j}{\left(x\right)}^{2m}-3{r}_{j}{\left(x\right)}^{4m}+(221+72m){x}_{j}^{6}{r}_{j}{\left(x\right)}^{4m}\hfill \\ & & +6{x}_{j}^{2}{r}_{j}{\left(x\right)}^{2m}(-5+3{r}_{j}{\left(x\right)}^{6m})+15{x}_{j}^{4}(-1+8{r}_{j}{\left(x\right)}^{6m}),\phantom{\rule{4pt}{0ex}}j=1,2,\hfill \\ \hfill {Q}_{{h}^{2,m}}^{j}& =& 2{(3{x}_{j}^{5}+10{x}_{j}^{3}{r}_{j}{\left(x\right)}^{2m}+3{x}_{j}{r}_{j}{\left(x\right)}^{4m})}^{2},\phantom{\rule{4pt}{0ex}}j=1,2,\hfill \end{array}$$

**Remark**

**3.**

**Proposition**

**6.**

_{m}on $r\left(x\right)$, for $m=1,2,\dots $, are

- (a)
- For $m=1,2,\dots $, the fixed points are $(-1,-1)$, $(-1,1)$, $(1,-1)$ and $(1,1)$ that are also superattracting and there are not strange fixed points in this case.
- (b)
- The components of free critical points ${c}_{{h}^{3,m}}^{n}=(k,l)$ are the roots of polynomial$$\begin{array}{c}\hfill {P}_{{h}^{3,m}}^{j}\left(x\right)=-40{x}_{j}^{4}+40{x}_{j}^{10}+2{r}_{j}{\left(x\right)}^{4m}-2(7+24m){x}_{j}^{6}{r}_{j}{\left(x\right)}^{4m}+3{x}_{j}^{2}{r}_{j}{\left(x\right)}^{8m},\end{array}$$

**Remark**

**4.**

#### 2.3. Second-Degree Polynomial System $p\left(x\right)=0$

_{m}, CMN

_{m}and RMN

_{m}classes applied on second-degree polynomial system $p\left(x\right)=0$ are stated in the following propositions. They resume, in each case, the iteration function, the fixed and critical points.

**Proposition**

**7.**

_{m}on $p\left(x\right)$, for $m=1,2,\dots $, are

- (a)
- For $m\ge 1$ the only fixed points are the roots $(-1,0)$, $(0,-1)$, $\left(\frac{1}{2}\left(1-\sqrt{5}\right),\frac{1}{2}\left(1-\sqrt{5}\right)\right)$ and $\left(\frac{1}{2}\left(1+\sqrt{5}\right),\frac{1}{2}\left(1+\sqrt{5}\right)\right)$, that are superattracting.
- (b)
- For $m=1$, the free critical points are $(-0.632026,-0.460233)$, $(-0.460233,-0.632026)$, $(-0.203287,0.294659)$ and $(0.294659,-0.203287)$.

_{m}, for $m=1,2,\dots $, and RMN

_{m}, for $m=1,2,\dots $, on polynomial system $p\left(x\right)=0$, we obtain, as was the case for the system $q\left(x\right)=0$, the same fixed point function as the classical Newton’s method.

**Proposition**

**8.**

_{m}for $m=1,2,\dots $ and RMN

_{m}for $m=1,2,\dots $ on polynomial system $p\left(x\right)$ are

_{m}and RMN

_{m}for any $m=1,2,\dots $ as well as their original partner, Newton’s scheme.

## 3. Conclusions

_{m}. This central differences method does not need to calculate and evaluate the Jacobian matrix as Newton’s method and provides similar basins of attraction. Although Richardson’s method reaches good results of convergence, has a computational cost that discourages its use.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Dynamical plane for FMN

_{m}, $m=1,2,\dots ,6$ on $q\left(x\right)$. Green, orange, red and blue areas are the basins of attraction of the roots of $q\left(x\right)$ (marked as white stars); black area denotes divergence and white squares are free critical points.

**Figure 2.**Dynamical plane of CMN

_{m}, RMN

_{m}and Newton’s method on $q\left(x\right)$. Green, orange, red and blue areas are the basins of attraction of the roots of $q\left(x\right)$ (marked as white stars).

**Figure 3.**Dynamical plane of FMN

_{m}method for $m=1,2,\dots ,6$ on $r\left(x\right)$. Green, orange, red and blue areas are the basins of attraction of the roots of $q\left(x\right)$ (marked as white stars); black area denotes divergence and white squares are free critical points.

**Figure 4.**Basins of attraction of $(-1,-1)$, $(-1,1)$ and $(1,1)$ for FMN

_{2}method on $r\left(x\right)$. Green, orange, red and blue areas are the basins of attraction of the roots of $q\left(x\right)$ (marked as white stars); black area denotes divergence and white squares are free critical points.

**Figure 5.**Basins of attraction of $(-1,-1)$, $(1,-1)$ and $(1,1)$ for FMN

_{3}method on $r\left(x\right)$. Green, orange, red and blue areas are the basins of attraction of the roots of $q\left(x\right)$ (marked as white stars); black area denotes divergence and white squares are free critical points.

**Figure 6.**Dynamical plane for CMN

_{m}method, $m=1,2,3,4$ on $r\left(x\right)$. Green, orange, red and blue areas are the basins of attraction of the roots of $q\left(x\right)$ (marked as white stars); black area denotes divergence and white squares are free critical points.

**Figure 7.**Dynamical plane for RMN

_{m}method, $m=1,2,3,4$ on $r\left(x\right)$. Green, orange, red and blue areas are the basins of attraction of the roots of $q\left(x\right)$ (marked as white stars); black area denotes divergence and white squares are free critical points.

**Figure 8.**Dynamical plane for FMN

_{m}method, $m=1,2,3,4$ on $p\left(x\right)$. Green, orange, red and blue areas are the basins of attraction of the roots of $q\left(x\right)$ (marked as white stars); black area denotes divergence and white squares are free critical points.

**Figure 9.**Dynamical plane of CMN

_{m}, RMN

_{m}and Newton’s method on $p\left(x\right)$. Green, orange, red and blue areas are the basins of attraction of the roots of $q\left(x\right)$ (marked as white stars); black area denotes divergence and white squares are free critical points.

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**MDPI and ACS Style**

Amiri, A.; Cordero, A.; Darvishi, M.T.; Torregrosa, J.R.
Stability Analysis of Jacobian-Free Newton’s Iterative Method. *Algorithms* **2019**, *12*, 236.
https://doi.org/10.3390/a12110236

**AMA Style**

Amiri A, Cordero A, Darvishi MT, Torregrosa JR.
Stability Analysis of Jacobian-Free Newton’s Iterative Method. *Algorithms*. 2019; 12(11):236.
https://doi.org/10.3390/a12110236

**Chicago/Turabian Style**

Amiri, Abdolreza, Alicia Cordero, Mohammad Taghi Darvishi, and Juan R. Torregrosa.
2019. "Stability Analysis of Jacobian-Free Newton’s Iterative Method" *Algorithms* 12, no. 11: 236.
https://doi.org/10.3390/a12110236