# Method of Third-Order Convergence with Approximation of Inverse Operator for Large Scale Systems

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## Abstract

**:**

## 1. Introduction

## 2. LC study for the Method in Equation (7)

**Theorem**

**1.**

- (1)
- Operator ${A}_{*}={\left[{F}^{\prime}\left(p\right)\right]}^{-1}$ exists;$$\parallel {A}_{*}\parallel \le B$$and$$\parallel {F}^{\prime}\left(p\right)\parallel \le C,$$where p is the solution of Equation (1).
- (2)
- In ball $U=\{x\in D:\phantom{\rule{0.277778em}{0ex}}\parallel x-p\parallel \le {R}_{0}\}$, the following estimate is satisfied$$\parallel {F}^{\u2033}\left(x\right)\parallel \le L,$$where ${R}_{0}=max\{{r}_{0},{a}_{1}{r}_{0}^{2},{a}_{2}{r}_{0}^{3}\},\phantom{\rule{0.277778em}{0ex}}{r}_{0}=max\{\parallel {x}_{0}-p\parallel ,\phantom{\rule{0.277778em}{0ex}}\parallel {A}_{0}-{A}_{*}\parallel \},$$a}_{1}=C+\frac{3}{2}BL+\frac{3}{2}L{r}_{0$;$a}_{2}=(C+\frac{3}{2}BL{a}_{1}{r}_{0}+\frac{3}{2}L{a}_{1}{r}_{0}^{2}){a}_{1$.
- (3)
- Initial approximations ${x}_{0}$, ${A}_{0}$ are such that$$q{r}_{0}<1,$$where $q={(max\{{a}_{2},\phantom{\rule{0.277778em}{0ex}}{a}_{3}\})}^{\frac{1}{2}}$, ${a}_{3}=C{\gamma}^{2}{r}_{0}+L{a}_{2}{(C+\gamma {r}_{0}^{2})}^{2}$, $\gamma =C+L{(B+{r}_{0})}^{2}{a}_{2}{r}_{0}$.Then, sequences $\left\{{x}_{k}\right\},\phantom{\rule{0.166667em}{0ex}}\left\{{A}_{k}\right\},\phantom{\rule{0.166667em}{0ex}}k\ge 0$, generated by Equation (7), converge to $p,\phantom{\rule{0.277778em}{0ex}}{A}_{*}$, respectively, and$${r}_{k}=max\{\parallel {A}_{k}-{A}_{*}\parallel ,\phantom{\rule{0.166667em}{0ex}}\parallel {x}_{k}-p\parallel \}\le {\left(q{r}_{0}\right)}^{{3}^{k}-1}{r}_{0},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}k\ge 0.$$

**Proof.**

**Proposition**

**1.**

**Proof.**

## 3. Numerical Experiments

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Argyros, I.K.; Shakhno, S.; Yarmola, H.
Method of Third-Order Convergence with Approximation of Inverse Operator for Large Scale Systems. *Symmetry* **2020**, *12*, 978.
https://doi.org/10.3390/sym12060978

**AMA Style**

Argyros IK, Shakhno S, Yarmola H.
Method of Third-Order Convergence with Approximation of Inverse Operator for Large Scale Systems. *Symmetry*. 2020; 12(6):978.
https://doi.org/10.3390/sym12060978

**Chicago/Turabian Style**

Argyros, Ioannis K., Stepan Shakhno, and Halyna Yarmola.
2020. "Method of Third-Order Convergence with Approximation of Inverse Operator for Large Scale Systems" *Symmetry* 12, no. 6: 978.
https://doi.org/10.3390/sym12060978