# Extending the Convergence Domain of Methods of Linear Interpolation for the Solution of Nonlinear Equations

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Divided Differences and Their Properties

**Definition**

**1**

**.**Let F be a nonlinear operator defined on a subset Ω of a Banach space ${B}_{1}$ with values in a Banach space ${B}_{2}$ and let $x,y$ be two points of Ω. A linear operator from ${B}_{1}$ to ${B}_{2}$ which is denoted by $[x,y;F]$ and satisfies the conditions:

**Definition**

**2**

**.**The operator $[x,y,z;F]$ is called divided difference of the second order of function F at the points x, y and z, if

**Remark**

**1.**

**Theorem**

**1.**

**Proof.**

**Corollary**

**1.**

**Proof.**

**Proposition**

**1.**

**Proof.**

**Remark**

**2.**

**Theorem**

**2.**

**Remark 3.**

## 3. A Posteriori Estimation of Error of the Secant Method

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

## 4. Semilocal Convergence of the Kurchatov’s Method

**Theorem**

**5.**

**Proof.**

**Proof.**

**Remark**

**4.**

## 5. A Posteriori Estimation of Error of the Kurchatov’s Method

**Theorem**

**6.**

**Proof.**

**Proposition**

**2.**

**Proof.**

**Remark**

**5.**

## 6. Numerical Experiments

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Shakhno, S.M. Nonlinear majorants for investigation of methods of linear interpolation for the solution of nonlinear equations. In Proceedings of the ECCOMAS 2004—European Congress on Computational Methods in Applied Sciences and Engineering, Jyväskylä, Finland, 24–28 July 2004; Available online: http://www.mit.jyu.fi/eccomas2004/proceedings/pdf/424.pdf (accessed on 26 May 2020).
- Amat, S. On the local convergence of secant-type methods. Int. J. Comput. Math.
**2004**, 81, 1153–1161. [Google Scholar] [CrossRef] - Hernandez, M.A.; Rubio, M.J. The Secant method for nondifferentiable operators. Appl. Math. Lett.
**2002**, 15, 395–399. [Google Scholar] [CrossRef] [Green Version] - Shakhno, S.; Gnatyshyn, O. Iterative-Difference Methods for Solving Nonlinear Least-Squares Problem. In Progress in Industrial Mathematics at ECMI 98; Arkeryd, L., Bergh, J., Brenner, P., Pettersson, R., Eds.; Verlag B. G. Teubner GMBH: Stuttgart, Germany, 1999; pp. 287–294. [Google Scholar]
- Ul’m, S. Algorithms of the generalized Steffensen method. Izv. Akad. Nauk ESSR Ser. Fiz.-Mat.
**1965**, 14, 433–443. (In Russian) [Google Scholar] - Ul’m, S. On generalized divided differences I, II. Izv. Akad. Nauk ESSR Ser. Fiz.-Mat.
**1967**, 16, 13–26. (In Russian) [Google Scholar] - Kantorovich, L.V.; Akilov, G.P. Functional Analysis; Pergamon Press: Oxford, UK, 1982. [Google Scholar]
- Potra, F.A. On an iterative algorithm of order 1.839... for solving nonlinear operator equations. Numer. Funct. Anal. Optim.
**1985**, 7, 75–106. [Google Scholar] [CrossRef] - Ul’m, S. Iteration methods with divided differences of the second order. Doklady Akademii Nauk SSSR
**1964**, 158, 55–58. [Google Scholar] - Schwetlick, H. Numerische Lösung Nichtlinearer Gleichungen; VEB Deutscher Verlag der Wissenschaften: Berlin, Germany, 1979. [Google Scholar]
- Argyros, I.K. A Kantorovich-type analysis for a fast iterative method for solving nonlinear equations. J. Math. Anal. Appl.
**2007**, 332, 97–108. [Google Scholar] [CrossRef] [Green Version] - Argyros, I.K.; George, S. On a two-step Kurchatov-type method in Banach space. Mediterr. J. Math.
**2019**, 16. [Google Scholar] [CrossRef] - Argyros, I.K.; Magreñán, A.A. Iterative Methods and Their Dynamics with Applications: A Contemporary Study; CRC Press: Boca Raton, FL, USA, 2017. [Google Scholar]
- Kurchatov, V.A. On a method of linear interpolation for the solution of functional equations. Dokl. Akad. Nauk SSSR
**1971**, 198, 524–526. [Google Scholar] - Shakhno, S.M. Kurchatov method of linear interpolation under generalized Lipschitz conditions for divided differences of first and second order. Visnyk Lviv. Univ. Ser. Mech. Math.
**2012**, 77, 235–242. (In Ukrainian) [Google Scholar] - Shakhno, S.M. On the difference method with quadratic convergence for solving nonlinear operator equations. Matematychni Studii
**2006**, 26, 105–110. (In Ukrainian) [Google Scholar]

**Figure 1.**Values of (

**a**) $\parallel {x}_{n}-{x}_{n-1}\parallel $ and (

**b**) $\parallel {x}_{n}-{x}_{*}\parallel $ at each iteration.

Secant Method (Theorem 1) | Secant Method (Theorem 2) | Kurchatov’s Method (Theorem 5) | |
---|---|---|---|

$\overline{r}$ | 0.85933 | 0.70430 | 0.88501 |

${\overline{r}}_{0}$ | 0.18703 | 0.18807 | 0.18055 |

$U({x}_{0},{\overline{r}}_{0})$ | (0.31297, 0.68703) | (0.31193, 0.68807) | (0.31945, 0.68055) |

**Table 2.**New and old error estimates (13).

n | $\parallel {\mathit{x}}_{\mathit{n}}-{\mathit{x}}_{*}\parallel $ | ${\mathit{t}}_{\mathit{n}}^{\mathit{NEW}}$ | ${\mathit{t}}_{\mathit{n}}^{\mathit{OLD}}$ |
---|---|---|---|

0 | 1.62391 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | 1.87033 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | 1.94079 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ |

1 | 4.27865 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | 2.03636 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 2.74082 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ |

2 | 9.60298 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 2.45405 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | 4.40268 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ |

3 | 4.78431 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ | 3.65459 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 1.18474 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ |

4 | 5.37514 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-13}$ | 6.65774 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ | 5.26214 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ |

5 | 0 | 1.80951 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-12}$ | 6.31743 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ |

**Table 3.**New and old error estimates (27).

n | $\parallel {\mathit{x}}_{\mathit{n}}-{\mathit{x}}_{*}\parallel $ | ${\mathit{t}}_{\mathit{n}}^{\mathit{NEW}}$ | ${\mathit{t}}_{\mathit{n}}^{\mathit{OLD}}$ |
---|---|---|---|

0 | 1.62391 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | 1.88065 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | 1.95339 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ |

1 | 4.27865 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | 2.13956e $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 2.86694 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ |

2 | 9.60298 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 2.79473 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | 4.93386 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ |

3 | 4.78431 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-8}$ | 4.93737 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 1.54194 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ |

4 | 5.37514 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-13}$ | 1.16446 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ | 8.59640 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ |

5 | 0 | 4.86585 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-12}$ | 1.50730 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-10}$ |

**Table 4.**New and old error estimates (33).

n | $\parallel {\mathit{x}}_{\mathit{n}}-{\mathit{x}}_{*}\parallel $ | ${\mathit{t}}_{\mathit{n}}^{\mathit{NEW}}$ | ${\mathit{t}}_{\mathit{n}}^{\mathit{OLD}}$ |
---|---|---|---|

0 | 1.62391 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | 1.80552 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ | 1.95314 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-1}$ |

1 | 4.27562 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ | 1.38863 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ | 2.86498 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-2}$ |

2 | 1.16228 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-5}$ | 3.06832 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-4}$ | 1.30247 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-3}$ |

3 | 4.04780 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 6.34947 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-7}$ | 7.68893 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-6}$ |

4 | 0 | 2.81890 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-11}$ | 1.78179 $\times \phantom{\rule{3.33333pt}{0ex}}{10}^{-9}$ |

© 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**

Argyros, I.K.; Shakhno, S.; Yarmola, H.
Extending the Convergence Domain of Methods of Linear Interpolation for the Solution of Nonlinear Equations. *Symmetry* **2020**, *12*, 1093.
https://doi.org/10.3390/sym12071093

**AMA Style**

Argyros IK, Shakhno S, Yarmola H.
Extending the Convergence Domain of Methods of Linear Interpolation for the Solution of Nonlinear Equations. *Symmetry*. 2020; 12(7):1093.
https://doi.org/10.3390/sym12071093

**Chicago/Turabian Style**

Argyros, Ioannis K., Stepan Shakhno, and Halyna Yarmola.
2020. "Extending the Convergence Domain of Methods of Linear Interpolation for the Solution of Nonlinear Equations" *Symmetry* 12, no. 7: 1093.
https://doi.org/10.3390/sym12071093