# Geometrically Constructed Family of the Simple Fixed Point Iteration Method

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## Abstract

**:**

## 1. Introduction

## 2. Geometric Derivation of the Family

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

#### Special Cases

- 1.
- For $m=0$, Formula (7) corresponds to the classical fixed point method ${x}_{n+1}=\varphi \left({x}_{n}\right)$.
- 2.
- 3.
- For $m={\displaystyle \frac{1-{\gamma}_{n}}{{\gamma}_{n}}}$, where $\left\{{\gamma}_{n}\right\}$ is a real sequence in $(0,1]$, Formula (7) corresponds to the following well-known Mann’s iteration [19]$${x}_{n+1}=(1-{\gamma}_{n}){x}_{n}+{\gamma}_{n}\phantom{\rule{0.277778em}{0ex}}\varphi \left({x}_{n}\right).$$
- 4.
- By inserting $m=1$, in scheme (7), one achieves the following well-known Kranselski’s iteration [20]$${x}_{n+1}=\frac{{x}_{n}+\varphi \left({x}_{n}\right)}{2},$$Similarly, we can derive several other formulas by taking different specific values of m. Furthermore, we proposed the following new schemes on the basis of some standard means of two quantities ${x}_{n}$ and $\varphi \left({x}_{n}\right)$ of same signs:
- 5.
- Geometric mean-based fixed point formula is given by$${x}_{n+1}=\sqrt{{x}_{n}\varphi \left({x}_{n}\right)},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{where}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{x}_{0}\ne 0.$$
- 6.
- Harmonic mean-based fixed point formula is defined by$${x}_{n+1}=\frac{2{x}_{n}\varphi \left({x}_{n}\right)}{{x}_{n}+\varphi \left({x}_{n}\right)},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{where}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{x}_{0}\ne 0.$$
- 7.
- Centroidal mean-based fixed point formula is mentioned as follows:$${x}_{n+1}=\frac{2\left(\right)open="("\; close=")">{x}_{n}^{2}+{x}_{n}\varphi \left({x}_{n}\right)+{\varphi}^{2}\left({x}_{n}\right)}{}3\left(\right)open="("\; close=")">{x}_{n}+\varphi \left({x}_{n}\right).$$
- 8.
- The following fixed point formula based on the Heronian mean is defined as$${x}_{n+1}=\frac{{x}_{n}+\sqrt{{x}_{n}\varphi \left({x}_{n}\right)}+\varphi \left({x}_{n}\right)}{3}.$$
- 9.
- The fixed point formula based on Contra-harmonic is depicted as follows:$${x}_{n+1}=\frac{{x}_{n}^{2}+{\varphi}^{2}\left({x}_{n}\right)}{{x}_{n}+\varphi \left({x}_{n}\right)}.$$

**Remark**

**1.**

## 3. Two-Step Iterative Schemes

- 1.
- Ishikawa [22] has proposed the following iterative scheme:$$\begin{array}{cc}\hfill {x}_{n+1}& =(1-{\beta}_{n}){x}_{n}+{\beta}_{n}\phantom{\rule{0.277778em}{0ex}}\varphi \left({y}_{n}\right),\hfill \\ \hfill {y}_{n}& =(1-{\gamma}_{n}){x}_{n}+{\gamma}_{n}\phantom{\rule{0.222222em}{0ex}}\varphi \left({x}_{n}\right),\hfill \end{array}$$
- 2.
- Agarwal et al. [1] have proposed the following iteration scheme defined as$$\begin{array}{cc}\hfill {x}_{n+1}& =(1-{\beta}_{n})\phantom{\rule{0.277778em}{0ex}}\varphi \left({x}_{n}\right)+{\beta}_{n}\phantom{\rule{0.277778em}{0ex}}\varphi \left({y}_{n}\right),\hfill \\ \hfill {y}_{n}& =(1-{\gamma}_{n}){x}_{n}+{\gamma}_{n}\phantom{\rule{0.222222em}{0ex}}\varphi \left({x}_{n}\right),\hfill \end{array}$$
- 3.
- Thianwan [23] defined the following two-step iteration scheme as$$\begin{array}{c}\hfill {x}_{n+1}=(1-{\beta}_{n}){y}_{n}+{\beta}_{n}\phantom{\rule{0.277778em}{0ex}}\varphi \left({y}_{n}\right),\\ \hfill {y}_{n}=(1-{\gamma}_{n}){x}_{n}+{\gamma}_{n}\phantom{\rule{0.222222em}{0ex}}\varphi \left({x}_{n}\right),\end{array}$$

#### Modified Schemes

## 4. Numerical Examples

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

**Example**

**4.**

**Example**

**5.**

## 5. Role of the Parameter ‘m’

- 1.
- Since $a\le x\le b$ implies that $a\le \varphi \left(x\right)\le b$. Therefore, the parameter ‘$m\ge 0$’ ensures that the fixed point divides the interval between ${x}_{0}$ and $\varphi \left({x}_{0}\right)$ internally in the ratio $m:1$ or $1:m$, otherwise, there will be an external division and hence, $h\left(x\right)\notin [a,b]$.
- 2.
- Since $h\left(x\right)={\displaystyle \frac{mx+\varphi \left(x\right)}{m+1}}$. As $|{h}^{\prime}\left({x}_{n}\right)|<1$ is the sufficient condition for the convergence of modified fixed point method, then we have$$\left(\right)open="|"\; close="|">\frac{m+{\varphi}^{\prime}\left({x}_{n}\right)}{m+1}$$This further implies that$$-(2m+1)<{\varphi}^{\prime}\left({x}_{n}\right)<1.$$This is the interval of convergence of our proposed scheme (7). As $m\ge 0$, so (26) represents a wider domain of convergence in contrast to the classical fixed point method $x=\varphi \left(x\right)$. In particular for $m=1$ (arithmetic mean), (26) gives the following interval of convergence as$$-3<{\varphi}^{\prime}\left({x}_{n}\right)<1.$$Therefore, the arithmetic mean formula has a bigger interval of convergence as compared to simple fixed point method.

**Remark**

**2.**

**Example**

**6.**

**Example**

**7.**

- 1.
- ${x}_{n+1}=\varphi \left({x}_{n}\right)={\displaystyle \frac{4}{{x}_{n}}},\phantom{\rule{0.277778em}{0ex}}n=0,1,2,\cdots ,$
- 2.
- ${x}_{n+1}=\varphi \left({x}_{n}\right)={\displaystyle \frac{{x}_{n}}{2}}+{\displaystyle \frac{2}{{x}_{n}}},\phantom{\rule{0.277778em}{0ex}}n=0,1,2,\cdots .$

**Remark**

**3.**

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Agawal, R.P.; Regan, D.O.; Sahu, D.R. Iterative constructions of the fixed point of nearly asymptotically nonexpansive mapping. J. Nonlinear Convex Anal.
**2012**, 27, 145–156. [Google Scholar] - Traub, J.F. Iterative Methods for the Solution of Equations; Prentice-Hall: Englewood Cliffs, NJ, USA, 1964. [Google Scholar]
- Behl, R.; Salimi, M.; Ferrara, M.; Sharifi, S.; Samaher, K.A. Some real life applications of a newly constructed derivative free iterative scheme. Symmetry
**2019**, 11, 239. [Google Scholar] [CrossRef][Green Version] - Salimi, M.; Nik Long, N.M.A.; Sharifi, S.; Pansera, B.A. A multi-point iterative method for solving nonlinear equations with optimal order of convergence. Jpn. J. Ind. Appl. Math.
**2018**, 35, 497–509. [Google Scholar] [CrossRef] - Sharifi, S.; Salimi, M.; Siegmund, S.; Lotfi, T. A new class of optimal four-point methods with convergence order 16 for solving nonlinear equations. Math. Comput. Simul.
**2016**, 119, 69–90. [Google Scholar] [CrossRef][Green Version] - Salimi, M.; Lotfi, T.; Sharifi, S.; Siegmund, S. Optimal Newton-Secant like methods without memory for solving nonlinear equations with its dynamics. Int. J. Comput. Math.
**2017**, 94, 1759–1777. [Google Scholar] [CrossRef][Green Version] - Matthies, G.; Salimi, M.; Sharifi, S.; Varona, J.L. An optimal eighth-order iterative method with its dynamics. Jpn. J. Ind. Appl. Math.
**2016**, 33, 751–766. [Google Scholar] [CrossRef][Green Version] - Sharifi, S.; Ferrara, M.; Salimi, M.; Siegmund, S. New modification of Maheshwari method with optimal eighth order of convergence for solving nonlinear equations. Open Math. (Former. Cent. Eur. J. Math.)
**2016**, 14, 443–451. [Google Scholar] - Lotfi, T.; Sharifi, S.; Salimi, M.; Siegmund, S. A new class of three point methods with optimal convergence order eight and its dynamics. Numer. Algor.
**2016**, 68, 261–288. [Google Scholar] [CrossRef] - Jamaludin, N.A.A.; Nik Long, N.M.A.; Salimi, M.; Sharifi, S. Review of some iterative methods for solving nonlinear equations with multiple zeros. Afr. Mat.
**2019**, 30, 355–369. [Google Scholar] [CrossRef] - Nik Long, N.M.A.; Salimi, M.; Sharifi, S.; Ferrara, M. Developing a new family of Newton–Secant method with memory based on a weight function. SeMA J.
**2017**, 74, 503–512. [Google Scholar] [CrossRef] - Ferrara, M.; Sharifi, S.; Salimi, M. Computing multiple zeros by using a parameter in Newton-Secant method. SeMA J.
**2017**, 74, 361–369. [Google Scholar] [CrossRef][Green Version] - Magreñán, A.A.; Argyros, I.K. A Contemporary Study of Iterative Methods: Convergence, Dynamics and Applications; Academic Press: Cambridge, MA, USA; Elsevier: Amsterdam, The Netherlands, 2019. [Google Scholar]
- Argyros, I.K.; Magreñán, A.A. Iterative Methods and Their Dynamics with Applications; CRC Press: New York, NY, USA; Taylor & Francis: Abingdon, UK, 2021. [Google Scholar]
- Burden, R.L.; Faires, J.D. Numerical Analysis; PWS Publishing Company: Boston, MA, USA, 2001. [Google Scholar]
- Ostrowski, A.M. Solution of Equations and Systems of Equation; Pure and Applied Mathematics; Academic Press: New York, NY, USA; London, UK, 1960; Volume IX. [Google Scholar]
- Petkovic, M.S.; Neta, B.; Petkovic, L.; Džunič, J. Multipoint Methods for Solving Nonlinear Equation; Elsevier: Amsterdam, The Netherlands, 2013. [Google Scholar]
- Schaefer, H. Über die methods sukzessiver approximationen. Jahreber Deutsch. Math. Verein
**1957**, 59, 131–140. [Google Scholar] - Mann, W.R. Mean Value Methods in Iteration. Proc. Am. Math. Soc.
**1953**, 4, 506–510. [Google Scholar] [CrossRef] - Kranselski, M.A. Two remarks on the method of successive approximation (Russian). Uspei Nauk.
**1955**, 10, 123–127. [Google Scholar] - Berinde, V. Iterative Approximation of Fixed Points; Springer: Berlin/Heidelberg, Germany; New York, NY, USA, 2002. [Google Scholar] [CrossRef]
- Ishikawa, S. Fixed Point by a New Iteration Method. Proc. Am. Math. Soc.
**1974**, 44, 147–150. [Google Scholar] [CrossRef] - Thianwan, S. Common fixed Points of new iterations for two asymptotically nonexpansive nonself mappings in Banach spaces. J. Comput. Appl. Math.
**2009**, 224, 688–695. [Google Scholar] [CrossRef][Green Version] - Noor, M.A. New approximation schemes for general variational inequalities. J. Math. Anal. Appl.
**2000**, 251, 217–229. [Google Scholar] [CrossRef][Green Version] - Phuengrattana, W.; Suantai, S. On the rate of convergence of Mann Ishikawa, Noor and SP iterations for continuous functions on an arbitrary interval. J. Comput. Appl. Math.
**2011**, 235, 3006–3014. [Google Scholar] [CrossRef][Green Version] - Cordero, A.; Torregrosa, J.R. Variants of Newtons method using fith-order quadrature formulas. Appl. Math. Comput.
**2007**, 190, 686–698. [Google Scholar]

**Figure 1.**The graph of approximate nonlinear function $y=\varphi \left(x\right)$ by a linear approximation.

**Table 1.**Some modified schemes based on Ishikawa’s, Agarwal and Thianwan as corrector, respectively.

Predictor | Ishikawa’s | Agarwal | Thianwan |
---|---|---|---|

Corrector | Corrector | Corrector | |

${y}_{n}=\sqrt{{x}_{n}\varphi \left({x}_{n}\right)}$ | ${x}_{n+1}=(1-{\beta}_{n}){x}_{n}+{\beta}_{n}\phantom{\rule{0.277778em}{0ex}}\varphi \left({y}_{n}\right),$ | ${x}_{n+1}=(1-{\gamma}_{n})\phantom{\rule{0.277778em}{0ex}}\varphi \left({x}_{n}\right)+{\gamma}_{n}\phantom{\rule{0.277778em}{0ex}}\varphi \left({y}_{n}\right),$ | ${x}_{n+1}=(1-{\gamma}_{n}){y}_{n}+{\gamma}_{n}\phantom{\rule{0.277778em}{0ex}}\varphi \left({y}_{n}\right),$ |

called by | $\left(IGM\right)$ | $\left(AGM\right)$ | $\left(TGM\right)$ |

${y}_{n}=\frac{2{x}_{n}\varphi \left({x}_{n}\right)}{{x}_{n}+\varphi \left({x}_{n}\right)}$ | ${x}_{n+1}=(1-{\beta}_{n}){x}_{n}+{\beta}_{n}\phantom{\rule{0.277778em}{0ex}}\varphi \left({y}_{n}\right),$ | ${x}_{n+1}=(1-{\gamma}_{n})\phantom{\rule{0.277778em}{0ex}}\varphi \left({x}_{n}\right)+{\gamma}_{n}\phantom{\rule{0.277778em}{0ex}}\varphi \left({y}_{n}\right),$ | ${x}_{n+1}=(1-{\gamma}_{n}){y}_{n}+{\gamma}_{n}\phantom{\rule{0.277778em}{0ex}}\varphi \left({y}_{n}\right),$ |

known by | $\left(IHM\right)$ | $\left(AHM\right)$ | $\left(THM\right)$ |

${y}_{n}=\frac{{x}_{n}+2\varphi \left({x}_{n}\right)}{3}$ | ${x}_{n+1}=(1-{\beta}_{n}){x}_{n}+{\beta}_{n}\phantom{\rule{0.277778em}{0ex}}\varphi \left({y}_{n}\right),$ | ${x}_{n+1}=(1-{\gamma}_{n})\phantom{\rule{0.277778em}{0ex}}\varphi \left({x}_{n}\right)+{\gamma}_{n}\phantom{\rule{0.277778em}{0ex}}\varphi \left({y}_{n}\right),$ | ${x}_{n+1}=(1-{\gamma}_{n}){y}_{n}+{\gamma}_{n}\phantom{\rule{0.277778em}{0ex}}\varphi \left({y}_{n}\right),$ |

denoted by | $\left(IOM1\right)$ | $\left(AOM1\right)$ | $\left(TOM1\right)$ |

${y}_{n}=\frac{{x}_{n}+4\varphi \left({x}_{n}\right)}{5}$ | |||

called by | $\left(IOM2\right)$ | $\left(AOM2\right)$ | $\left(TOM2\right)$ |

${y}_{n}=\frac{{x}_{n}+10\varphi \left({x}_{n}\right)}{11}$ | |||

known by | $\left(IOM3\right)$ | $\left(AOM3\right)$ | $\left(TOM3\right)$ |

Examples | E.C. | FIM | KM | GM | HM | OM1 | OM2 | OM3 | OM4 |
---|---|---|---|---|---|---|---|---|---|

R.E. | |||||||||

$\mathbf{\rho}$ | |||||||||

(1) | $|{x}_{k+1}-{x}_{k}|$ | $3.4(-4)$ | $9.3(-16)$ | $9.1(-16)$ | $8.9(-16)$ | $8.1(-11)$ | $2.2(-7)$ | $2.0(-5)$ | $8.2(-14,080)$ |

$f\left({x}_{k}\right)$ | $5.7(-4)$ | $3.1(-15)$ | $3.1(-15)$ | $3.0(-15)$ | $4.6(-11)$ | $4.7(-7)$ | $3.7(-5)$ | $2.5(-14,079)$ | |

$\rho $ | $0.9998$ | $1.000$ | $1.000$ | $1.000$ | $1.000$ | $1.000$ | $1.000$ | $2.000$ | |

(2) | $|{x}_{k+1}-{x}_{k}|$ | $5.6(-5)$ | $2.8(-10)$ | $2.5(-10)$ | $2.3(-10)$ | $1.9(-18)$ | $2.9(-9)$ | $1.6(-6)$ | $6.5(-9126)$ |

$f\left({x}_{k}\right)$ | $5.6(-5)$ | $5.6(-10)$ | $5.0(-10)$ | $4.5(-10)$ | $2.9(-18)$ | $3.6(-9)$ | $1.7(-6)$ | $1.0(-9125)$ | |

$\rho $ | $1.000$ | $1.000$ | $1.000$ | $1.000$ | $1.000$ | $1.000$ | $1.000$ | $2.000$ | |

(3) | $|{x}_{k+1}-{x}_{k}|$ | $1.1(-18)$ | $2.3(-6)$ | $2.3(-6)$ | $2.3(-6)$ | $3.9(-8)$ | $2.7(-10)$ | $3.0(-13)$ | $2.6(-13,415)$ |

$f\left({x}_{k}\right)$ | $1.1(-17)$ | $4.6(-5)$ | $4.5(-5)$ | $4.5(-5)$ | $5.9(-7)$ | $3.4(-9)$ | $3.3(-12)$ | $2.5(-13,414)$ | |

$\rho $ | $1.000$ | $1.000$ | $1.000$ | $1.000$ | $1.000$ | $1.000$ | $1.000$ | $2.000$ | |

(4) | $|{x}_{k+1}-{x}_{k}|$ | $5.3(-10)$ | $3.7(-7)$ | $3.5(-7)$ | $3.4(-7)$ | $8.2(-11)$ | $1.1(-20)$ | $8.3(-14)$ | $2.6(-10,135)$ |

$f\left({x}_{k}\right)$ | $1.4(-9)$ | $2.0(-6)$ | $1.9(-6)$ | $1.8(-6)$ | $3.3(-10)$ | $3.7(-20)$ | $2.4(-13)$ | $8.3(-10,135)$ | |

$\rho $ | $1.000$ | $1.000$ | $1.000$ | $1.000$ | $1.000$ | $1.000$ | $1.000$ | $2.000$ | |

(5) | $|{x}_{k+1}-{x}_{k}|$ | $5.3(-7)$ | $4.1(-9)$ | $4.0(-9)$ | $3.9(-9)$ | $1.7(-17)$ | $4.4(-13)$ | $5.5(-9)$ | $6.0(-1,102,284)$ |

$f\left({x}_{k}\right)$ | $5.3(-7)$ | $8.2(-9)$ | $8.0(-9)$ | $7.9(-9)$ | $2.5(-17)$ | $5.5(-13)$ | $6.1(-9)$ | $8.5(-1,102,284)$ | |

$\rho $ | $1.000$ | $1.000$ | $1.000$ | $1.000$ | $1.000$ | $1.000$ | $1.000$ | $2.000$ |

**Table 3.**Comparison of our methods with existing Ishikawa method based on $k=12$ number of iterations.

Examples | E.C. | $\mathit{I}\mathit{S}$ | IGM | IHM | IOM1 | IOM2 | IOM3 |
---|---|---|---|---|---|---|---|

R.E. | |||||||

(1) | $|{x}_{k+1}-{x}_{k}|$ | $5.1(-7)$ | $3.9(-8)$ | $3.9(-18)$ | $9.1(-8)$ | $1.8(-7)$ | $2.3(-7)$ |

$f\left({x}_{k}\right)$ | $1.5(-3)$ | $1.9(-4)$ | $1.9(-4)$ | $5.8(-4)$ | $1.5(-3)$ | $2.5(-3)$ | |

(2) | $|{x}_{k+1}-{x}_{k}|$ | $3.1(-6)$ | $8.7(-7)$ | $7.9(-7)$ | $1.7(-7)$ | $1.1(-6)$ | $1.8(-6)$ |

$f\left({x}_{k}\right)$ | $5.4(-3)$ | $2.1(-3)$ | $1.9(-3)$ | $4.8(-4)$ | $3.5(-3)$ | $6.6(-3)$ | |

(1) | $|{x}_{k+1}-{x}_{k}|$ | $5.7(-9)$ | $6.6(-8)$ | $6.6(-8)$ | $4.6(-8)$ | $3.0(-8)$ | $1.7(-8)$ |

$f\left({x}_{k}\right)$ | $9.9(-5)$ | $1.1(-3)$ | $1.1(-3)$ | $7.7(-4)$ | $5.0(-4)$ | $2.8(-4)$ | |

(4) | $|{x}_{k+1}-{x}_{k}|$ | $5.3(-7)$ | $8.6(-7)$ | $8.4(-7)$ | $4.4(-7)$ | $7.2(-8)$ | $2.4(-7)$ |

$f\left({x}_{k}\right)$ | $2.4(-3)$ | $4.4(-3)$ | $4.3(-3)$ | $2.4(-3)$ | $4.0(-4)$ | $1.3(-3)$ | |

(5) | $|{x}_{k+1}-{x}_{k}|$ | $6.9(-7)$ | $4.0(-7)$ | $5.1(-7)$ | $8.2(-8)$ | $1.9(-7)$ | $4.2(-7)$ |

$f\left({x}_{k}\right)$ | $1.2(-3)$ | $8.7(-4)$ | $8.6(-4)$ | $2.0(-4)$ | $4.9(-4)$ | $1.2(-3)$ |

Examples | E.C. | AS | AGM | AHM | AOM1 | AOM2 | AOM3 |
---|---|---|---|---|---|---|---|

R.E. | |||||||

(1) | $|{x}_{k+1}-{x}_{k}|$ | $1.5(-4)$ | $1.4(-5)$ | $1.4(-5)$ | $2.2(-5)$ | $3.1(-5)$ | $3.4(-5)$ |

$f\left({x}_{k}\right)$ | $2.5(-4)$ | $2.4(-5)$ | $2.4(-5)$ | $3.7(-5)$ | $5.3(-5)$ | $5.8(-5)$ | |

(2) | $|{x}_{k+1}-{x}_{k}|$ | $1.9(-5)$ | $6.1(-6)$ | $5.6(-6)$ | $8.6(-7)$ | $4.1(-6)$ | $5.2(-6)$ |

$f\left({x}_{k}\right)$ | $1.9(-5)$ | $6.1(-6)$ | $5.6(-6)$ | $8.6(-7)$ | $4.1(-6)$ | $5.2(-6)$ | |

(3) | $|{x}_{k+1}-{x}_{k}|$ | $3.7(-20)$ | $3.9(-15)$ | $3.9(-19)$ | $5.1(-18)$ | $1.4(-19)$ | $6.8(-20)$ |

$f\left({x}_{k}\right)$ | $3.7(-19)$ | $3.9(-14)$ | $3.9(-18)$ | $5.1(-17)$ | $1.4(-18)$ | $6.8(-19)$ | |

(4) | $|{x}_{k+1}-{x}_{k}|$ | $7.4(-11)$ | $1.3(-10)$ | $1.2(-10)$ | $5.3(-11)$ | $7.2(-12)$ | $2.0(-11)$ |

$f\left({x}_{k}\right)$ | $2.0(-10)$ | $3.4(-10)$ | $3.3(-10)$ | $1.4(-10)$ | $1.9(-11)$ | $5.2(-11)$ | |

(5) | $|{x}_{k+1}-{x}_{k}|$ | $1.4(-7)$ | $8.6(-8)$ | $8.5(-8)$ | $1.3(-8)$ | $2.4(-8)$ | $4.2(-8)$ |

$f\left({x}_{k}\right)$ | $1.4(-7)$ | $3.6(-8)$ | $8.5(-8)$ | $1.3(-8)$ | $2.4(-8)$ | $4.2(-8)$ |

**Table 5.**Comparison of our methods with classical Mann’s and Thianwan method after $k=12$ number of iterations.

Examples | E.C. | MS | TS | TGM | THM | TOM1 | TOM2 | TOM3 |
---|---|---|---|---|---|---|---|---|

R.E. | ||||||||

(1) | $|{x}_{k+1}-{x}_{k}|$ | $1.2(-7)$ | $1.3(-8)$ | $4.9(-17)$ | $4.8(-17)$ | $9.7(-13)$ | $1.2(-8)$ | $1.1(-6)$ |

$f\left({x}_{k}\right)$ | $3.6(-4)$ | $1.9(-5)$ | $1.6(-16)$ | $1.6(-16)$ | $2.4(-12)$ | $2.5(-8)$ | $2.0(-6)$ | |

(2) | $|{x}_{k+1}-{x}_{k}|$ | $2.6(-6)$ | $4.9(-7)$ | $2.3(-11)$ | $2.1(-11)$ | $1.7(-19)$ | $2.6(-10)$ | $3.4(-7)$ |

$f\left({x}_{k}\right)$ | $4.5(-3)$ | $4.2(-4)$ | $4.6(-11)$ | $4.3(-11)$ | $2.6(-19)$ | $3.2(-10)$ | $3.7(-7)$ | |

(3) | $|{x}_{k+1}-{x}_{k}|$ | $1.3(-7)$ | $5.1(-9)$ | $4.6(-8)$ | $4.6(-8)$ | $8.0(-10)$ | $5.5(-12)$ | $6.8(-20)$ |

$f\left({x}_{k}\right)$ | $2.2(-3)$ | $4.4(-5)$ | $4.8(-7)$ | $9.2(-7)$ | $1.2(-8)$ | $6.9(-11)$ | $6.8(-19)$ | |

(4) | $|{x}_{k+1}-{x}_{k}|$ | $2.7(-6)$ | $2.8(-7)$ | $2.4(-8)$ | $2.3(-8)$ | $5.4(-12)$ | $7.4(-22)$ | $5.5(-15)$ |

$f\left({x}_{k}\right)$ | $9.5(-3)$ | $6.4(-4)$ | $1.3(-7)$ | $1.2(-7)$ | $2.2(-11)$ | $2.4(-21)$ | $1.6(-14)$ | |

(5) | $|{x}_{k+1}-{x}_{k}|$ | $1.1(-6)$ | $2.0(-7)$ | $3.7(-10)$ | $3.7(-10)$ | $1.6(-18)$ | $4.0(-14)$ | $5.1(-10)$ |

$f\left({x}_{k}\right)$ | $1.9(-3)$ | $1.8(-4)$ | $7.5(-10)$ | $7.4(-10)$ | $2.3(-8)$ | $5.0(-14)$ | $5.6(-10)$ |

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**MDPI and ACS Style**

Kanwar, V.; Sharma, P.; Argyros, I.K.; Behl, R.; Argyros, C.; Ahmadian, A.; Salimi, M.
Geometrically Constructed Family of the Simple Fixed Point Iteration Method. *Mathematics* **2021**, *9*, 694.
https://doi.org/10.3390/math9060694

**AMA Style**

Kanwar V, Sharma P, Argyros IK, Behl R, Argyros C, Ahmadian A, Salimi M.
Geometrically Constructed Family of the Simple Fixed Point Iteration Method. *Mathematics*. 2021; 9(6):694.
https://doi.org/10.3390/math9060694

**Chicago/Turabian Style**

Kanwar, Vinay, Puneet Sharma, Ioannis K. Argyros, Ramandeep Behl, Christopher Argyros, Ali Ahmadian, and Mehdi Salimi.
2021. "Geometrically Constructed Family of the Simple Fixed Point Iteration Method" *Mathematics* 9, no. 6: 694.
https://doi.org/10.3390/math9060694