# Direct Comparison between Two Third Convergence Order Schemes for Solving Equations

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

**Chebyshev-Type Scheme:**

**Simplified Chebyshev-Type Scheme:**

**Two-Step-Newton-Type Scheme:**

## 2. Ball Convergence

- (A1)
- $G:D\u27f6{B}_{2}$ is differentiable; there exists a simple zero p of equation $G\left(x\right)=0.$
- (A2)
- There exists a continuous and increasing function ${\omega}_{0}$ defined on $[0,\infty )$ with values in $[0,\infty )$ such that for all $x\in D$$$\parallel {G}^{\prime}{\left(p\right)}^{-1}({G}^{\prime}\left(x\right)-{G}^{\prime}\left(p\right))\parallel \le {\omega}_{0}(\parallel x-p\parallel ),$$
- (A3)
- There exist continuous and increasing functions $\omega $ and ${\omega}_{1}$ on the interval $[0,{R}_{0})$ with values in $[0,\infty )$ such that for all $x,y\in {D}_{0}$$$\parallel {G}^{\prime}{\left(p\right)}^{-1}({G}^{\prime}\left(y\right)-{G}^{\prime}\left(x\right))\parallel \le \omega (\parallel y-x\parallel )$$$$\parallel {G}^{\prime}{\left(p\right)}^{-1}{G}^{\prime}\left(x\right)\parallel \le {\omega}_{1}(\parallel x-p\parallel ).$$
- (A4)
- (A5)
- There exists ${S}_{\alpha}\ge {R}_{\alpha}$ such that$${\int}_{0}^{1}{\omega}_{0}\left(\tau {S}_{\alpha}\right)d\tau <1.$$Set ${D}_{1}=D\cap U(p,{S}_{\alpha}).$

**Theorem**

**1.**

**Proof.**

**Remark**

**1.**

- 1.
- In view of ($A3$) and the estimate$$\begin{array}{ccc}\hfill \parallel {G}^{\prime}{\left(p\right)}^{-1}{G}^{\prime}\left(x\right)\parallel & =& \parallel {G}^{\prime}{\left(p\right)}^{-1}({G}^{\prime}\left(x\right)-{G}^{\prime}\left(p\right))+I\parallel \hfill \\ & \le & 1+\parallel {G}^{\prime}{\left(p\right)}^{-1}({G}^{\prime}\left(x\right)-{G}^{\prime}\left(p\right))\parallel \le 1+{\omega}_{0}(\parallel x-p\parallel )\hfill \end{array}$$$${\omega}_{1}\left(t\right)=1+{\omega}_{0}\left(t\right)\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}or\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}{\omega}_{1}\left(t\right)=1+{\omega}_{0}\left({R}_{0}\right).$$
- 2.
- The results obtained here can be used for operators F satisfying autonomous differential equations [3] of the form$${G}^{\prime}\left(x\right)=P\left(G\left(x\right)\right)$$
- 3.
- If ${\omega}_{0}$ and ${\omega}_{1}$ are constant functions, say ${\omega}_{0}\left(t\right)={L}_{0}t,\omega \left(t\right)=Lt,$ for some ${L}_{0}>0$ and $L>0,$ then the radius ${r}_{1}=\frac{2}{2{L}_{0}+L}$ was shown by us to be the convergence radius of Newton’s method [5,6]$${x}_{n+1}={x}_{n}-{G}^{\prime}{\left({x}_{n}\right)}^{-1}G\left({x}_{n}\right)\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{for}\mathrm{each}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}n=0,1,2,\cdots $$$${r}_{R}=\frac{2}{3{L}_{1}},$$$${r}_{R}<r$$$$\frac{{r}_{R}}{{r}_{1}}\to \frac{1}{3}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}as\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\frac{{L}_{0}}{{L}_{1}}\to 0.$$That is our convergence ball ${r}_{1}$ is at most three times larger than Rheinboldt’s. The same value for ${r}_{R}$ was given by Traub [15].
- 4.
- It is worth noticing that method (2) is not changing when we use simpler methods the conditions of Theorem 1 instead of the stronger conditions used in [10]. Moreover, we can compute the computational order of convergence (COC) defined by$$\xi =ln\left(\frac{\parallel {x}_{n+1}-p\parallel}{\parallel {x}_{n}-p\parallel}\right)/ln\left(\frac{\parallel {x}_{n}-p\parallel}{\parallel {x}_{n-1}-p\parallel}\right)$$$${\xi}_{1}=ln\left(\frac{\parallel {x}_{n+1}-{x}_{n}\parallel}{\parallel {x}_{n}-{x}_{n-1}\parallel}\right)/ln\left(\frac{\parallel {x}_{n}-{x}_{n-1}\parallel}{\parallel {x}_{n-1}-{x}_{n-2}\parallel}\right).$$This way we obtain in practice the order of convergence in a way that avoids the bounds involving estimates using estimates higher than the first Fréchet derivative of operator $F.$
- 5.
- Method (2) can be generalized as$$\begin{array}{ccc}\hfill {y}_{n}& =& G\left({x}_{n}\right)\hfill \end{array}$$$$\begin{array}{ccc}\hfill {x}_{n+1}& =& H\left({y}_{n}\right),\hfill \end{array}$$
- (i)
- For all $x\in U({x}_{\ast},R),$ we have $y=G\left(x\right)\in U({x}_{\ast},R)$ for $R>0,{x}_{\ast}\in D$ such that $U({x}_{\ast},R)\subset D$ and $G\left({x}_{\ast}\right)=0;$
- (ii)
- $\parallel H\left(y\right)-{x}_{\ast}\parallel \le d\parallel x-{x}_{\ast}\parallel $ for some $d\in [0,1).$Then, it follows ${lim}_{n\u27f6\infty}{x}_{n}={x}_{\ast}$ provided that ${x}_{0}\in U({x}_{\ast},R).$ Indeed, we have$$\parallel {x}_{n+1}-{x}_{\ast}\parallel =\parallel H\left({y}_{n}\right)-{x}_{\ast}\parallel \le d\parallel {x}_{n}-{x}_{\ast}\parallel \le {d}^{n+1}\parallel {x}_{0}-{x}_{\ast}\parallel <R$$

- 6.
- The ball convergence result for scheme (3) clearly is obtained from Theorem 1 for $\alpha =-1.$
- 7.
- In our earlier works with other schemes we assumed ${\omega}_{0}\left(0\right)={\omega}_{1}\left(0\right)=0$ to show the existence of ${R}_{\alpha}^{1}$ and ${R}_{\alpha}^{2}$ using the intermediate value theorem. Bur these initial conditions on ${\omega}_{0}$ and ${\omega}_{1}.$ This is not necessary with our approach. This way we further expand the applicability of scheme (2) and (3). The same is true for scheme (4) whose ball convergence follows.

**Theorem**

**2.**

**Theorem**

**3.**

**Remark**

**2.**

## 3. Numerical Examples

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Amat, S.; Busquier, S.; Negra, M. Adaptive approximation of nonlinear operators. Numer. Funct. Anal. Optim.
**2004**, 25, 397–405. [Google Scholar] [CrossRef] - Amat, S.; Argyros, I.K.; Busquier, S.; Magreñán, A.A. Local convergence and the dynamics of a two-point four parameter Jarratt-like method under weak conditions. Numer. Alg.
**2017**. [Google Scholar] [CrossRef] - Argyros, I.K. Computational Theory of Iterative Methods, Series: Studies in Computational Mathematics; Chui, C.K., Wuytack, L., Eds.; Elsevier Publ. Company: New York, NY, USA, 2007. [Google Scholar]
- Argyros, I.K.; George, S. Mathematical Modeling for the Solution of Equations and Systems of Equations with Applications; Volume III: Newton’s Method Defined on Not Necessarily Bounded Domain; Nova Publishes: New York, NY, USA, 2019. [Google Scholar]
- Argyros, I.K.; George, S. Mathematical Modeling for the Solution of Equations and Systems of Equations with Applications; Volume-IV: Local Convergence of a Class of Multi-Point Super–Halley Methods; Nova Publishes: New York, NY, USA, 2019. [Google Scholar]
- Argyros, I.K.; George, S.; Magreñán, A.A. Local convergence for multi-point- parametric Chebyshev-Halley- type method of higher convergence order. J. Comput. Appl. Math.
**2015**, 282, 215–224. [Google Scholar] [CrossRef] - Argyros, I.K.; Magreñán, A.A. Iterative Method and Their Dynamics with Applications; CRC Press: New York, NY, USA, 2017. [Google Scholar]
- Argyros, I.K.; Magreñán, A.A. A study on the local convergence and the dynamics of Chebyshev-Halley-type methods free from second derivative. Numer. Alg.
**2015**, 71, 1–23. [Google Scholar] [CrossRef] - Argyros, I.K.; Regmi, S. Undergraduate Research at Cameron University on Iterative Procedures in Banach and Other Spaces; Nova Science Publishers: New York, NY, USA, 2019. [Google Scholar]
- Babajee, D.K.R.; Davho, M.E.; Darvishi, M.T.; Karami, A.; Barati, A. Analysis of two Chebyshev-like third order methods free from second derivatives for solving systems of nonlinear equations. J. Comput. Appl. Math.
**2010**, 233, 2002–2012. [Google Scholar] [CrossRef] [Green Version] - Alzahrani, A.K.H.; Behl, R.; Alshomrani, A.S. Some higher-order iteration functions for solving nonlinear models. Appl. Math. Comput.
**2018**, 334, 80–93. [Google Scholar] [CrossRef] - Behl, R.; Cordero, A.; Motsa, S.S.; Torregrosa, J.R. Stable high-order iterative methods for solving nonlinear models. Appl. Math. Comput.
**2017**, 303, 70–88. [Google Scholar] [CrossRef] - Choubey, N.; Panday, B.; Jaiswal, J.P. Several two-point with memory iterative methods for solving nonlinear equations. Afrika Matematika
**2018**, 29, 435–449. [Google Scholar] [CrossRef] - Cordero, A.; Torregrosa, J.R. Variants of Newton’s method for functions of several variables. Appl. Math. Comput.
**2006**, 183, 199–208. [Google Scholar] [CrossRef] - Cordero, A.; Hueso, J.L.; Martínez, E.; Torregrosa, J.R. A modified Newton-Jarratt’s composition. Numer. Alg.
**2010**, 55, 87–99. [Google Scholar] [CrossRef] - Cordero, A.; Torregrosa, J.R. Variants of Newton’s method using fifth-order quadrature formulas. Appl. Math. Comput.
**2007**, 190, 686–698. [Google Scholar] [CrossRef] - Cordero, A.; Hueso, J.L.; Martínez, E.; Torregrosa, J.R. Increasing the convergence order of an iterative method for nonlinear systems. Appl. Math. Lett.
**2012**, 25, 2369–2374. [Google Scholar] [CrossRef] [Green Version] - Darvishi, M.T.; Barati, A. Super cubic iterative methods to solve systems of nonlinear equations. Appl. Math. Comput.
**2007**, 188, 1678–1685. [Google Scholar] [CrossRef] - Esmaeili, H.; Ahmadi, M. An efficient three-step method to solve system of non linear equations. Appl. Math. Comput.
**2015**, 266, 1093–1101. [Google Scholar] - Fang, X.; Ni, Q.; Zeng, M. A modified quasi-Newton method for nonlinear equations. J. Comput. Appl. Math.
**2018**, 328, 44–58. [Google Scholar] [CrossRef] - Fousse, L.; Hanrot, G.; Lefvre, V.; Plissier, P.; Zimmermann, P. MPFR: A multiple-precision binary floating-point library with correct rounding. ACM Trans. Math. Softw.
**2007**, 33, 15. [Google Scholar] [CrossRef] - Homeier, H.H.H. A modified Newton method with cubic convergence: The multivariate case. J. Comput. Appl. Math.
**2004**, 169, 161–169. [Google Scholar] [CrossRef] [Green Version] - Iliev, A.; Kyurkchiev, N. Nontrivial Methods in Numerical Analysis: Selected Topics in Numerical Analysis; LAP LAMBERT Academic Publishing: Saarbrucken, Germany, 2010; ISBN 978-3-8433-6793-6. [Google Scholar]
- Lotfi, T.; Bakhtiari, P.; Cordero, A.; Mahdiani, K.; Torregrosa, J.R. Some new efficient multipoint iterative methods for solving nonlinear systems of equations. Int. J. Comput. Math.
**2015**, 92, 1921–1934. [Google Scholar] [CrossRef] [Green Version] - Magreñán, A.A. Different anomalies in a Jarratt family of iterative root finding methods. Appl. Math. Comput.
**2014**, 233, 29–38. [Google Scholar] - Magreñán, A.A. A new tool to study real dynamics: The convergence plane. Appl. Math. Comput.
**2014**, 248, 29–38. [Google Scholar] [CrossRef] [Green Version] - Noor, M.A.; Waseem, M. Some iterative methods for solving a system of nonlinear equations. Comput. Math. Appl.
**2009**, 57, 101–10619. [Google Scholar] [CrossRef] [Green Version] - Ortega, J.M.; Rheinboldt, W.C. Iterative Solutions of Nonlinear Equations in Several Variables; Academic Press: New York, NY, USA, 1970. [Google Scholar]
- Ostrowski, A.M. Solution of Equation and Systems of Equations; Academic Press: New York, NY, USA, 1960. [Google Scholar]
- Rheinboldt, W.C. An adaptive continuation process for solving systems of nonlinear equations. In Mathematical Models and Numerical Solvers; Tikhonov, A.N., Ed.; Banach Center: Warsaw, Poland, 1977; pp. 129–142. [Google Scholar]
- Petkovic, M.S.; Neta, B.; Petkovic, L.j.D.; Dzunic, J. Multi-Point Methods for Solving Nonlinear Equations; Elsevier/Academic Press: Amsterdam, The Netherlands; Boston, MA, USA; Heidelberg, Germany; London, UK; New York, NY, USA, 2013. [Google Scholar]
- Sharma, J.R.; Sharma, R.; Bahl, A. An improved Newton-Traub composition for solving systems of nonlinear equa- tions. Appl. Math. Comput.
**2016**, 290, 98–110. [Google Scholar] - Sharma, J.R.; Arora, H. Improved Newton-like methods for solving systems of nonlinear equations. SeMA
**2017**, 74, 147–163. [Google Scholar] [CrossRef] - Sharma, J.R.; Arora, H. Efficient derivative-free numerical methods for solving systems of nonlinear equations. Comput. Appl. Math.
**2016**, 35, 269–284. [Google Scholar] [CrossRef] - Traub, J.F. Iterative Methods for the Solution of Equations; Chelsea Publishing Company: New York, NY, USA, 1982. [Google Scholar]

$\mathit{\alpha}=-1$ | (2) & (3) | (4) |
---|---|---|

${R}_{\alpha}^{1}=0.38269191223238574472986783803208$ | ${R}_{-1}^{1}=0.38269191223238574472986783803208$ | |

${R}_{\alpha}^{2}=0.21292963191331051864274570561975$ | ${\overline{R}}_{2}=1.6195946293252685421748537919484$ | |

radius | ${R}_{\alpha}={R}_{\alpha}^{2}$ | $\overline{R}={R}_{-1}^{1}$ |

$\mathit{\alpha}=-1$ | (2) & (3) | (4) |
---|---|---|

${R}_{\alpha}^{1}=0.066666666666666666666666666666667$ | ${R}_{-1}^{1}=0.066666666666666666666666666666667$ | |

${R}_{\alpha}^{2}=0.040057008172481922692043099232251$ | ${\overline{R}}_{2}=0.014407266709891463490889051968225$ | |

radius | ${R}_{\alpha}={R}_{\alpha}^{2}$ | $\overline{R}={\overline{R}}_{2}$ |

$\mathit{\alpha}=-1$ | (2) & (3) | (4) |
---|---|---|

${R}_{\alpha}^{1}=0.0068968199414654552878434223828208$ | ${R}_{-1}^{1}=0.0068968199414654552878434223828208$ | |

${R}_{\alpha}^{2}=0.00377575830521247697221798311773$ | ${\overline{R}}_{2}=0.000205625336098942375698261919581$ | |

radius | ${R}_{\alpha}={R}_{\alpha}^{2}$ | $\overline{R}={\overline{R}}_{2}$ |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Regmi, S.; Argyros, I.K.; George, S.
Direct Comparison between Two Third Convergence Order Schemes for Solving Equations. *Symmetry* **2020**, *12*, 1080.
https://doi.org/10.3390/sym12071080

**AMA Style**

Regmi S, Argyros IK, George S.
Direct Comparison between Two Third Convergence Order Schemes for Solving Equations. *Symmetry*. 2020; 12(7):1080.
https://doi.org/10.3390/sym12071080

**Chicago/Turabian Style**

Regmi, Samundra, Ioannis K. Argyros, and Santhosh George.
2020. "Direct Comparison between Two Third Convergence Order Schemes for Solving Equations" *Symmetry* 12, no. 7: 1080.
https://doi.org/10.3390/sym12071080