# A Newton-like Midpoint Method for Solving Equations in Banach Space

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## Abstract

**:**

## 1. Introduction

- 1
- The bisection method: a simple yet robust method that involves repeatedly bisecting an interval and determining which subinterval a root lies in.
- 2
- The Newton–Raphson method: this method uses an initial guess and an iterative process to converge on a root and requires the ability to compute the derivative of the function.
- 3
- The Secant method: this method is similar to the Newton–Raphson method but uses the slope of the secant line between two points rather than the derivative of the function.
- 4
- Fixed-point iteration: this method involves finding the fixed point of a function using an iterative process. It requires the function to be in a specific form.
- 5
- Muller’s method: this method is an extension of the secant method and is used for complex roots.
- 6
- Bairstow’s method: this method is used for finding the roots of polynomials with real coefficients, and it is used to find the roots of polynomials of degree greater than two.
- 7
- Aitken’s delta-squared method: this method is used for speeding up the convergence of fixed-point iteration method.
- 8
- The Hybrid method: as the name suggests, this method combines two or more methods to find the root of the nonlinear equation.

- (1)
- The uniqueness of the solution region is not provided.
- (2)
- The choice of the starting point ${x}_{0}\in D$ is a “shot in the dark ”.
- (3)
- There are no estimates on $\parallel {x}_{n+1}-{x}_{n}\parallel $ or $\parallel {x}^{*}-{x}_{n}\parallel $ that can be computed in advance based on the properties of the operator F.
- (4)
- The semilocal convergence of the method has not been studied.
- (5)
- The derivative higher than one used in the local convergence is not on the method.

## 2. Convergence I: Local

- There exists a nondecreasing and continuous function (NCF) ${w}_{0}:M\to \mathbb{R}$ such that the function ${w}_{0}(t)-1$ has a smallest positive root denoted by s.Set ${M}_{1}=[0,s)$.
- NCF $w:{M}_{1}\to \mathbb{R}$ exists such that the function ${g}_{1}(t)-1$ has a smallest root ${r}_{1}\in {M}_{1}-\left\{0\right\}$, where$$\begin{array}{c}\hfill {g}_{1}(t)=\frac{{\int}_{0}^{1}w((1-\theta )t)d\theta}{1-{w}_{0}(t)}.\end{array}$$

- $({E}_{1})$
- $\parallel {F}^{\prime}{({x}^{*})}^{-1}({F}^{\prime}(u)-{F}^{\prime}({x}^{*}))\parallel \le {w}_{0}(\parallel u-{x}^{*}\parallel )$ for each $u\in D$.Set ${D}_{1}=D\cap \mathcal{S}({x}^{*},r).$
- $({E}_{2})$
- $\parallel {F}^{\prime}{({x}^{*})}^{-1}({F}^{\prime}({u}_{1})-{F}^{\prime}({u}_{2}))\parallel \le w(\parallel {u}_{1}-{u}_{2}\parallel )$ for each ${u}_{1},{u}_{2}\in {D}_{1}$.and
- $({E}_{3})$
- $\mathcal{S}[{x}^{*},r]\subset D$.

**Theorem**

**1.**

**Proof.**

**Proposition**

**1.**

- (1)
- A solution ${u}^{*}\in \mathcal{S}({x}^{*},{\rho}_{3})$ of the equation $F(x)=0$ exists for some ${\rho}_{3}>0$.
- (2)
- The condition $({E}_{1})$ holds on the ball $\mathcal{S}({x}^{*},{\rho}_{3})$.
- (3)
- ${\rho}_{4}\ge {\rho}_{3}$ exists such that$${\int}_{0}^{1}{w}_{0}(\theta {\rho}_{4})d\theta <1.$$

**Proof.**

**Remark**

**1.**

## 3. Convergence II: Semilocal

**Lemma**

**1.**

**Proof.**

**Remark**

**2.**

- (a)
- If the function ${v}_{0}$ is strictly increasing on the interval $[0,\rho )$; then, we can choose $d={v}_{0}^{-1}(1)$.
- (b)
- If the smallest positive root ${\rho}_{0}$ of the function ${v}_{0}(t)-1$ exists then we can set $d={\rho}_{0}$.

- $({H}_{1})$
- $\parallel {F}^{\prime}{({x}_{0})}^{-1}({F}^{\prime}(u)-{F}^{\prime}({x}_{0}))\parallel \le {v}_{0}(\parallel u-{x}_{0}\parallel )$ for each $u\in D$.Set ${D}_{3}=D\cap \mathcal{S}({x}_{0},{\rho}_{0})$, where ${\rho}_{0}$ is the smallest positive root of the function ${v}_{0}(t)-1$.
- $({H}_{2})$
- $\parallel {F}^{\prime}{({x}_{0})}^{-1}({F}^{\prime}({u}_{1})-{F}^{\prime}({u}_{2}))\parallel \le v(\parallel {u}_{1}-{u}_{2}\parallel )$ for each ${u}_{1},{u}_{2}\in {D}_{3}$.
- $({H}_{3})$
- The condition (21) holdsand
- $({H}_{4})$
- $\mathcal{S}[{x}_{0},{d}^{*}]\subset D$.

**Theorem**

**2.**

**Proof.**

**Proposition**

**2.**

- (1)
- There exists a solution ${u}^{*}\in \mathcal{S}({x}_{0},{d}_{1})$ of the Equation (1) for some ${d}_{1}>0$.
- (2)
- The condition $({H}_{1})$ holds on the ball $\mathcal{S}({x}_{0},{d}_{1})$.
- (3)
- There exists ${d}_{2}\ge {d}_{1}$ such that$$\begin{array}{c}\hfill {\int}_{0}^{1}{v}_{0}((1-\theta ){d}_{1}+\theta {d}_{2})d\theta <1.\end{array}$$

**Proof.**

**Remark**

**3.**

- (i)
- Under all the conditions of Theorem 2, we can let ${d}_{1}={d}^{*}$ and ${u}^{*}={x}^{*}$.
- (ii)
- The condition $({H}_{4})$ can be replaced by ${({H}_{4})}^{\prime}$$\mathcal{S}[{x}_{0},{\rho}_{0}]\subset D$, where ${\rho}_{0}$ is given in closed form.

## 4. Examples and Numerical Calculations

- (i)
- $\parallel {x}_{k+1}-{x}_{k}\parallel \le \u03f5,$ and
- (ii)
- $\parallel F({x}_{k})\parallel <\u03f5$,

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

**Example**

**4.**

**Example**

**5.**

**Example**

**6.**

## 5. Concluding Remarks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Argyros, I.; Magreñán, Á.A. Iterative Methods and Their Dynamics with Applications; CRC Press: New York, NY, USA, 2017. [Google Scholar]
- Argyros, I.K. The Theory and Applications of Iteration Methods; Taylor and Francis: Abingdon, UK; CRC Press: New York, NY, USA, 2022. [Google Scholar]
- Ortega, J.M.; Rheinboldt, W.C. Iterative Solution of Nonlinear Equations in Several Variables; Academic Press: New York, NY, USA, 1970. [Google Scholar]
- Sharma, J.R.; Guha, R.K.; Sharma, R. An efficient fourth order weighted-Newton method for systems of nonlinear equations. Numer. Algo.
**2013**, 62, 307–323. [Google Scholar] [CrossRef] - Ezquerro, J.A.; Hernández, M.A. Newton’s Method: An Updated Approach of Kantorovich’s Theory; Springer: Cham, Switzerland, 2018. [Google Scholar]
- Argyros, I.K.; Shakhno, S.; Regmi, S.; Yarmola, H. Newton-Type Methods for Solving Equations in Banach spaces: A Unified Approach. Symmetry
**2023**, 15, 15. [Google Scholar] [CrossRef] - Argyros, I.K.; Deep, G.; Regmi, S. Extended Newton-like Midpoint Method for Solving Equations in Banach Space. Foundations
**2023**, 3, 82–98. [Google Scholar] [CrossRef] - Darvishi, M.T.; Barati, A. A fourth-order method from quadrature formulae to solve systems of nonlinear equations. Appl. Math. Comput.
**2007**, 188, 257–261. [Google Scholar] [CrossRef] - Frontini, M.; Sormani, E. Third-order methods from quadrature formulae for solving systems of nonlinear equations. Appl. Math. Comput.
**2004**, 149, 771–782. [Google Scholar] [CrossRef] - Gutiérrez, J.M. A new semilocal convergence for Newton’s method. J. Comput. Appl. Math.
**1997**, 79, 131–145. [Google Scholar] [CrossRef][Green Version] - Gutiérrez, J.M.; Hernández, M.A. Third-order iterative methods for operators with bounded second derivative. J. Comput. Appl. Math.
**1997**, 82, 171–183. [Google Scholar] [CrossRef][Green Version] - Herceg, D.; Herceg, D.J. Means based modifications of Newton’s method for solving nonlinear equations. Appl. Math. Lett.
**2013**, 219, 6126–6133. [Google Scholar] [CrossRef] - Kou, J. A third-order modification of Newton method for systems of non-linear equations. Appl. Math. Comput.
**2007**, 191, 117–121. [Google Scholar] [CrossRef] - Singh, S.; Gupta, D.; Badoni, R.; Martínez, E.; Hueso, J.L. Local convergence of a parameter based iteration with Hölder continuous derivative in Banach spaces. Calcolo
**2017**, 54, 527–539. [Google Scholar] [CrossRef] - Singh, S.; Gupta, D.K.; Martínez, E.; Hueso, J.L. Semilocal and local convergence of a fifth order iteration with Fréchet derivative satisfying Hölder condition. Appl. Math. Comput.
**2016**, 276, 266–277. [Google Scholar] - Traub, J.F. Iterative Methods for the Solution of Equations; Chelsea Publishing Company: New York, NY, USA, 1982. [Google Scholar]
- Wang, X.; Gu, C.; Kou, J. Semilocal convergence of a multipoint fourth-order super-Halley method in Banach spaces. Numer. Algo.
**2011**, 56, 497–516. [Google Scholar] [CrossRef] - Kamran, I.M.; Alotaibi, F.M.; Haque, S.; Mlaiki, N.; Shah, K. RBF-Based Local Meshless Method for Fractional Diffusion Equations. Fractal Fract.
**2023**, 7, 143. [Google Scholar] [CrossRef] - Khan, A.; Shah, K.; Abdeljawad, T.; Sher, M. On Fractional Order Sine-Gordon Equation Involving Nonsingular Derivative. Fractals
**2022**. [Google Scholar] [CrossRef] - Saifullah, S.; Ali, A.; Khan, A.; Shah, K.; Abdeljawad, T. A Novel Tempered Fractional Transform: Theory, Properties and Appli- cations to Differential Equations. Fractals
**2022**. [Google Scholar] [CrossRef] - Shah, K.; Sinan, M.; Abdeljawad, T.; El-Shorbagy, M.A.; Abdalla, B.; Abualrub, M.S. A Detailed Study of a Fractal-Fractional Transmission Dynamical Model of Viral Infectious Disease with Vaccination. Complexity
**2022**, 2022, 7236824. [Google Scholar] [CrossRef]

Cases | ${\mathit{x}}_{0}$ | $|\mathit{F}({\mathit{x}}_{\mathit{n}})|$ | $|{\mathit{x}}_{\mathit{n}+1}-{\mathit{x}}_{\mathit{n}}|$ | n | $\mathsf{\mu}$ | CPU Timing |
---|---|---|---|---|---|---|

Method (3) | ${(\frac{39}{100},\frac{39}{100},\frac{39}{100},\frac{39}{100},\frac{39}{100})}^{tr}$ | $8.5\times {10}^{-827}$ | $1.2\times {10}^{-826}$ | 3 | 4 | $7.56632$ |

Cases | ${\mathit{x}}_{0}$ | $|\mathit{F}({\mathit{x}}_{\mathit{n}})|$ | $|{\mathit{x}}_{\mathit{n}+1}-{\mathit{x}}_{\mathit{n}}|$ | n | $\mathsf{\mu}$ | CPU Timing |
---|---|---|---|---|---|---|

Method (3) | ${(\frac{34}{100},\frac{62}{100},\frac{8}{10},\frac{9}{10},\frac{12}{10},\frac{11}{10},\frac{13}{10})}^{tr}$ | $8.7\times {10}^{-944}$ | $5.7\times {10}^{-944}$ | 3 | 4 | $3.63682$ |

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**MDPI and ACS Style**

Regmi, S.; Argyros, I.K.; Deep, G.; Rathour, L. A Newton-like Midpoint Method for Solving Equations in Banach Space. *Foundations* **2023**, *3*, 154-166.
https://doi.org/10.3390/foundations3020014

**AMA Style**

Regmi S, Argyros IK, Deep G, Rathour L. A Newton-like Midpoint Method for Solving Equations in Banach Space. *Foundations*. 2023; 3(2):154-166.
https://doi.org/10.3390/foundations3020014

**Chicago/Turabian Style**

Regmi, Samundra, Ioannis K. Argyros, Gagan Deep, and Laxmi Rathour. 2023. "A Newton-like Midpoint Method for Solving Equations in Banach Space" *Foundations* 3, no. 2: 154-166.
https://doi.org/10.3390/foundations3020014