# Convergence of Derivative-Free Iterative Methods with or without Memory in Banach Space

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## Abstract

**:**

## 1. Introduction

**Motivation**The convergence order is shown using the Taylor series expansion approach, which is based on derivatives up to order five (not on these methods), limiting their applicability. As a simple but motivational example:

## 2. Local Analysis

- (i)
- $${\xi}_{0}(t,at)-1$$
- (ii)
- $${\zeta}_{1}\left(t\right)-1$$$${\zeta}_{1}\left(t\right)=\frac{\xi (t,at)}{1-{\xi}_{0}(t,at)}.$$
- (iii)
- $${\xi}_{0}({\zeta}_{1}\left(t\right)t,0)-1,\phantom{\rule{0.166667em}{0ex}}{\xi}_{0}({\zeta}_{1}\left(t\right)t,at)-1$$
- (iv)
- $${\zeta}_{2}\left(t\right)-1$$$$\begin{array}{ccc}\hfill {\zeta}_{2}\left(t\right)& =& \left[\frac{\xi ({\zeta}_{1}\left(t\right)t,at){\xi}_{1}\left({\zeta}_{1}\left(t\right)t\right)}{(1-{\xi}_{0}({\zeta}_{1}\left(t\right)t,0))(1-{\xi}_{0}({\zeta}_{1}\left(t\right)t,at))}\right.\hfill \\ & & +\left.\frac{{\xi}_{2}(t,{\zeta}_{1}\left(t\right)t){\xi}_{1}\left({\zeta}_{1}\left(t\right)t\right)}{1-{\xi}_{0}({\zeta}_{1}\left(t\right)t,at)}\right]{\zeta}_{1}\left(t\right).\hfill \end{array}$$

- (h1)
- There exists an invertible operator L so that$$\parallel {L}^{-1}([x,y;F]-L)\parallel \le {\xi}_{0}(\parallel x-{x}_{*}\parallel ,\parallel y-{x}_{*}\parallel )$$$$\parallel I+\alpha [x,{x}_{*};F]\parallel \le a$$For each $x,y\in D.$Set ${D}_{0}=U({x}_{*},{R}_{0})\cap D.$
- (h2)
- $$\parallel {L}^{-1}([x,z;F]-[x,{x}_{*};F])\parallel \le \xi (\parallel x-{x}_{*}\parallel ,\parallel z-{x}_{*}\parallel ),$$$$\parallel {L}^{-1}{F}^{\prime}\left(x\right)\parallel \le {\xi}_{1}(\parallel x-{x}_{*}\parallel )$$$$\parallel I-A(x,y)\parallel \le {\xi}_{2}(\parallel x-{x}_{*}\parallel ,\parallel y-{x}_{*}\parallel )$$For each $x,y,z\in {D}_{0}.$
- (h3)
- $\overline{U}({x}_{*},{\tilde{d}}_{*})\subset D$ for ${\tilde{d}}_{*}=max\{a\tilde{d},\tilde{d}\}$ and $\tilde{d}$ to be given laterand
- (h4)
- There exists ${d}_{*}\ge {\tilde{d}}_{*}$, satisfying ${\xi}_{0}(0,{d}_{*})<1$ or ${\xi}_{0}({d}_{*},0)<1.$Let ${D}_{1}=\overline{U}({x}_{*},{d}_{*})\cap D.$

**Theorem**

**1.**

**Proof.**

**Remark**

**1.**

- (a)
- We can compute the computational order of convergence (COC), defined by$$\xi =ln\left(\frac{\parallel {x}_{n+1}-{x}^{*}\parallel}{\parallel {x}_{n}-{x}^{*}\parallel}\right)/ln\left(\frac{\parallel {x}_{n}-{x}^{*}\parallel}{\parallel {x}_{n-1}-{x}^{*}\parallel}\right)$$$${\xi}_{1}=ln\left(\frac{\parallel {x}_{n+1}-{x}_{n}\parallel}{\parallel {x}_{n}-{x}_{n-1}\parallel}\right)/ln\left(\frac{\parallel {x}_{n}-{x}_{n-1}\parallel}{\parallel {x}_{n-1}-{x}_{n-2}\parallel}\right).$$
- (b)
- The choice $A\left(t\right)=1+t+\beta {t}^{2},\phantom{\rule{0.166667em}{0ex}}t={F}^{\prime}{\left(x\right)}^{-1}F\left(y\right)$ satisfies the conditions $A\left(0\right)={A}^{\prime}\left(0\right)=1$ and ${A}^{\u2033}\left(0\right)<\infty $ required to show the fourth convergence order of Method (2). Next, we show how to choose function ${\xi}_{2}$ in this case. Notice that we have$$\begin{array}{ccc}& & \parallel {\left(L(x-{x}_{*})\right)}^{-1}(F\left(x\right)-F\left({x}_{*}\right)-L(x-{x}_{*}))\parallel \hfill \\ & \le & \frac{1}{\parallel x-{x}_{*}\parallel}\parallel {L}^{-1}([x,{x}_{*};F]-L)\parallel \parallel x-{x}_{*}\parallel \phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}for\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}x\ne {x}_{*}\hfill \\ & \le & {\xi}_{0}(\parallel x-{x}_{*}\parallel ,0),\hfill \end{array}$$$${\xi}_{2}(s,t)={\xi}_{0}(s,0).$$
- (c)
- The usual choice for $L={F}^{\prime}\left({x}_{*}\right)$ [8]. But this implies that the operator F is differentiable at $x={x}_{*}$ and ${x}_{*}$ is simple. This makes it unattractive for solving non-differentiable equations. However, if L is chosen to be different from ${F}^{\prime}\left({x}_{*}\right),$ then one can also solve non-differentiable equations.
- (d)
- The parameter a can be replaced by a real function as follows:$$\begin{array}{ccc}\hfill I+\alpha [x,{x}_{*};F]& =& I+\alpha L{L}^{-1}([x,{x}_{*};F]-L+L)\hfill \\ & =& I+\alpha L+\alpha L{L}^{-1}([x,{x}_{*};F]-L),\hfill \\ \hfill \parallel I+\alpha [x,{x}_{*};F]\parallel & \le & \parallel I+\alpha L\parallel \hfill \\ & & +\left|\alpha \right|\parallel L\parallel {\xi}_{0}(\parallel x-{x}_{*}\parallel ,\parallel {x}_{*}-{x}_{*}\parallel ).\hfill \end{array}$$Thus, we can set$$a\left(t\right)=\parallel I+\alpha L\parallel +\left|\alpha \right|\parallel L\parallel {\xi}_{0}(t,0),$$

**Theorem**

**2.**

## 3. Semi-Local Analysis

- (e1)
- There exist continuous and nondecreasing functions $f:M\u27f6\mathbb{R},\phantom{\rule{0.166667em}{0ex}}{p}_{0}:M\times M\u27f6\mathbb{R}$ so that the equation ${p}_{0}(t,f\left(t\right))-1=0$ has a smallest positive solution, denoted as $s.$ Set ${M}_{2}=[0,s).$
- (e2)
- There exists a continuous and nondecreasing function $p:{M}_{2}\times {M}_{2}\times {M}_{2}\u27f6\mathbb{R}.$ Define the sequence $\left\{{\alpha}_{n}\right\}$ for ${\alpha}_{0}=0,$ some ${\beta}_{0}\ge 0$, and each $n=0,1,2,\dots $ by$$\begin{array}{ccc}\hfill {c}_{n}& =& (1+{p}_{0}({\alpha}_{n},{\beta}_{n}))({\beta}_{n}-{\alpha}_{n})+(1+{p}_{0}({\alpha}_{n},f\left({\alpha}_{n}\right)))({\beta}_{n}-{\alpha}_{n}),\hfill \\ \hfill {\alpha}_{n+1}& =& {\beta}_{n}+\frac{p({\alpha}_{n},{\beta}_{n},f\left({\alpha}_{n}\right)){c}_{n}}{1-{p}_{0}({\beta}_{n},f\left({\alpha}_{n}\right))},\hfill \\ \hfill {b}_{n+1}& =& (1+{p}_{0}({\alpha}_{n},{\alpha}_{n+1}))({\alpha}_{n+1}-{\alpha}_{n})+(1+{p}_{0}({\alpha}_{n},f\left({\alpha}_{n}\right)))({\alpha}_{n+1}-{\alpha}_{n})\hfill \end{array}$$$${\beta}_{n+1}={\alpha}_{n+1}+\frac{{b}_{n+1}}{1-{p}_{0}({\alpha}_{n+1},f\left({\alpha}_{n+1}\right))}.$$A convergence criterion for this sequence is:
- (e3)
- There exists ${s}_{0}\in {M}_{2}$ such that for each $n=0,1,2,\dots $${p}_{0}({\beta}_{n},f\left({\alpha}_{n}\right))<1,\phantom{\rule{0.166667em}{0ex}}$${p}_{0}({\alpha}_{n},$$f\left({\alpha}_{n}\right))<1$, and ${\alpha}_{n}\le {s}_{0}.$ It follows by the definition of the sequence and this condition that $0\le {\alpha}_{n}\le {\beta}_{n}\le {\alpha}_{n+1}\le {s}_{0}$, and there exists ${\alpha}^{*}\in [0,{s}_{0}]$ such that ${lim}_{n\u27f6\infty}{\alpha}_{n}={\alpha}^{*}.$ These functions are connected to the operators of the method.
- (e4)
- There exists an invertible operator L so that for each $x,y\in D$ and some ${x}_{0}\in D$$$\parallel {L}^{-1}([x,y;F]-L)\parallel \le {p}_{0}(\parallel x-{x}_{0}\parallel ,\parallel y-{x}_{0}\parallel )$$$$z-{x}_{0}\parallel \le f(\parallel x-{x}_{0}\parallel )\le \parallel x-{x}_{0}\parallel .$$Set ${D}_{2}=D\cap U({x}_{0},s).$
- (e5)
- For $A=A(x,y,z)$ and each $x,y,z\in {D}_{2}$$$\parallel A\parallel \le p(\parallel x-{x}_{0}\parallel ,\parallel y-{x}_{0}\parallel ,\parallel z-{x}_{0}\parallel )$$
- (e6)
- $U[{x}_{0},{\alpha}^{*}]\subset D.$

**Theorem**

**3.**

**Proposition**

**1.**

- (i)
- There exists a solution $\overline{x}\in U({x}_{0},{s}_{1})$ of the equation $F\left(x\right)=0$ for some ${s}_{1}>0.$
- (ii)
- The first condition in (e4) holds in the ball $U({x}_{0},{s}_{1})$.
- (iii)
- There exists ${s}_{2}\ge {s}_{1}$ so that$${p}_{0}({s}_{1},{s}_{2})<1.$$Set ${D}_{3}=D\cap U[{x}_{0},{s}_{2}].$ Then, the equation $F\left(x\right)=0$ is uniquely solvable by $\overline{x}$ in the domain ${D}_{3}.$

**Proof.**

**Remark**

**2.**

- (i)
- The limit point ${\alpha}^{*}$ can be switched with s in the condition (e6).
- (ii)
- Under all the conditions of Theorem 3, we can take $\overline{x}={x}_{*}$ and ${s}_{1}={\alpha}^{*}.$
- (iii)
- As in the local case, a choice for the real function f can be provided, being motivated by the calculation:$$\begin{array}{ccc}\hfill z-{x}_{0}& =& x-{x}_{0}+\alpha (F\left(x\right)-F\left({x}_{0}\right))+\alpha F\left({x}_{0}\right)\hfill \\ & =& (I+\alpha [x,{x}_{0};F])(x-{x}_{0})+\alpha F\left({x}_{0}\right),\hfill \\ & =& [(I+\alpha L)+\alpha L{L}^{-1}([x,{x}_{0};F]-L)](x-{x}_{0})+\alpha F\left({x}_{0}\right).\hfill \end{array}$$Thus, we can take$$f\left(t\right)=[\parallel I+\alpha L\parallel +|\alpha |\parallel L\parallel {p}_{0}(t,0)]t+|\alpha |\parallel F\left({x}_{0}\right)\parallel .$$The semi-local analysis of convergence for Method (3) follows along the same lines.

## 4. Numerical Examples

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

**Example**

**4.**

**Example**

**5.**

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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n | ${\mathit{x}}_{\mathit{n}}$ by (2) | ${\mathit{x}}_{\mathit{n}}$ by (3) |
---|---|---|

−1 | (0.2000, 0.2000, 0.2000) | (0.2000, 0.2000, 0.2000) |

0 | (0.1000, 0.1000, 0.1000) | ( 0.1000, 0.1000, 0.1000) |

1 | ( 0.0044, 0.0526, 0) | ( 0.0000, 0.0457, 0) |

2 | (0.0000, 0.0325, 0) | (0.0000, 0.0276, 0) |

3 | (0.0000, 0.0215, 0) | (−0.0000, 0.0181, 0) |

4 | (0.0000, 0.0147, 0) | (−0.0000, 0.0124, 0) |

5 | (0.0000, 0.0103, 0) | (−0.0000, 0.0087, 0) |

6 | (0.0000, 0.0074, 0) | (−0.0000, 0.0062, 0) |

7 | (0.0000, 0.0053, 0) | (−0.0000, 0.0045, 0) |

8 | (0.0000, 0.0038, 0) | (−0.0000, 0.0032, 0) |

9 | (0.0000, 0.0028, 0) | (−0.0000, 0.0024, 0) |

10 | (0.0000, 0.0020, 0) | (−0.0000, 0.0017, 0) |

11 | (0.0000, 0.0015, 0) | (−0.0000, 0.0013, 0) |

12 | (0.0000, 0.0011, 0) | (−0.0000, 0.0009, 0) |

13 | (0.0000, 0.0008, 0) | (−0.0000, 0.0007, 0) |

14 | (0.0000, 0.0006, 0) | (−0.0000, 0.0005, 0) |

15 | (0.0000, 0.0004, 0) | (−0.0000, 0.0004, 0) |

16 | (0.0000, 0.0003, 0) | (−0.0000, 0.0003, 0) |

17 | (0.0000, 0.0002, 0) | (−0.0000, 0.0002, 0) |

18 | (0.0000, 0.0002, 0) | (−0.0000, 0.0001, 0) |

19 | (0.0000, 0.0001, 0) | (−0.0000, 0.0001, 0) |

20 | (0.0000, 0.0001, 0) | (−0.0000, 0.0001, 0) |

21 | (0.0000, 0.0001, 0) | (−0.0000, 0.0001, 0) |

22 | (0.0000, 0.0001, 0) | (0, 0, 0) |

n | ${\mathit{x}}_{\mathit{n}}$ by (2) | ${\mathit{x}}_{\mathit{n}}$ by (3) |
---|---|---|

−1 | — | (1.9, −0.9) |

0 | (2.000000, −1.000000) | (2.000000, −1.000000) |

1 | (1.953072, −0.962331) | (1.153994, 0.203527) |

2 | (1.903627, −0.920635) | (0.996799, 0.301846) |

3 | (1.851328, −0.874390) | (0.992780, 0.306440) |

4 | (1.795779, −0.822929) | (0.992780, 0.306440) |

5 | (1.736504, −0.765386) | |

6 | (1.672947, −0.700609) | |

7 | (1.604467, −0.627018) | |

8 | (1.530378, −0.542399) | |

9 | (1.450068, −0.443592) | |

10 | (1.363359, −0.326162) | |

11 | (1.271401, −0.184796) | |

12 | (1.178280, −0.018149) | |

13 | (1.091066, 0.152382) | |

14 | (1.020124, 0.270191) | |

15 | (0.993678, 0.305320) | |

16 | (0.992780, 0.306440) | |

17 | (0.992780, 0.306440) |

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## Share and Cite

**MDPI and ACS Style**

George, S.; Argyros, I.K.; Regmi, S.
Convergence of Derivative-Free Iterative Methods with or without Memory in Banach Space. *Foundations* **2023**, *3*, 589-601.
https://doi.org/10.3390/foundations3030035

**AMA Style**

George S, Argyros IK, Regmi S.
Convergence of Derivative-Free Iterative Methods with or without Memory in Banach Space. *Foundations*. 2023; 3(3):589-601.
https://doi.org/10.3390/foundations3030035

**Chicago/Turabian Style**

George, Santhosh, Ioannis K. Argyros, and Samundra Regmi.
2023. "Convergence of Derivative-Free Iterative Methods with or without Memory in Banach Space" *Foundations* 3, no. 3: 589-601.
https://doi.org/10.3390/foundations3030035