# A New Three-Step Class of Iterative Methods for Solving Nonlinear Systems

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Design and Convergence Analysis of the Proposed Class

- (a)
- ${H}^{\prime}(u)(v)={H}_{1}uv,\phantom{\rule{0.277778em}{0ex}}\mathrm{where}\phantom{\rule{0.277778em}{0ex}}{H}^{\prime}:X\to \mathcal{L}(X)\phantom{\rule{0.277778em}{0ex}}\mathrm{and}\phantom{\rule{0.277778em}{0ex}}{H}_{1}\in \mathbb{R}$,
- (b)
- ${H}^{\u2033}(u,v)(v)={H}_{2}uvw,\phantom{\rule{0.277778em}{0ex}}\mathrm{where}\phantom{\rule{0.277778em}{0ex}}{H}^{\u2033}:X\times X\to \mathcal{L}(X)\phantom{\rule{0.277778em}{0ex}}\mathrm{and}\phantom{\rule{0.277778em}{0ex}}{H}_{2}\in \mathbb{R}$.

**Theorem**

**1.**

**Proof.**

- Family 1
- The weight function defined by$${H}_{1}(t)=I+2t+\frac{1}{2}\alpha {t}^{2}$$$$\begin{array}{ccc}\hfill {y}^{(k)}& =& {x}^{(k)}-{[{F}^{\prime}({x}^{(k)})]}^{-1}F({x}^{(k)}),\hfill \\ \hfill {z}^{(k)}& =& {y}^{(k)}-\left[I+2{t}^{(k)}+\frac{1}{2}\alpha {{t}^{(k)}}^{2}\right]{[{F}^{\prime}({x}^{(k)})]}^{-1}F({y}^{(k)}),\hfill \\ \hfill {x}^{(k+1)}& =& {z}^{(k)}-\left[I+2{t}^{(k)}+\frac{1}{2}\alpha {{t}^{(k)}}^{2}\right]{[{F}^{\prime}({x}^{(k)})]}^{-1}F({z}^{(k)}),\phantom{\rule{0.277778em}{0ex}}k\ge 0.\hfill \end{array}$$
- Family 2
- The weight function defined by$${H}_{2}(t)=I+2{(I+\alpha t)}^{-1}t$$$$\begin{array}{ccc}\hfill {y}^{(k)}& =& {x}^{(k)}-{[{F}^{\prime}({x}^{(k)})]}^{-1}F({x}^{(k)}),\hfill \\ \hfill {z}^{(k)}& =& {y}^{(k)}-\left[I+2{(I+\alpha {t}^{(k)})}^{-1}{t}^{(k)}\right]{[{F}^{\prime}({x}^{(k)})]}^{-1}F({y}^{(k)}),\hfill \\ \hfill {x}^{(k+1)}& =& {z}^{(k)}-\left[I+2{(I+\alpha {t}^{(k)})}^{-1}{t}^{(k)}\right]{[{F}^{\prime}({x}^{(k)})]}^{-1}F({z}^{(k)}),\phantom{\rule{0.277778em}{0ex}}k\ge 0.\hfill \end{array}$$

## 3. Computational Efficiency

## 4. Numerical Results

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

**Example**

**4.**

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**$Computationalefficiencyindex\mathit{CI}$ indices for PSH6${}_{1}$ and comparison methods.

Method | n.F | n.${\mathit{F}}^{\prime}$ | n.$[\xb7,\xb7;\mathit{F}]$ | FE | I |
---|---|---|---|---|---|

PSH6${}_{1}$ | 3 | 1 | 1 | $2{n}^{2}+2n$ | ${6}^{\frac{1}{2{n}^{2}+2n}}$ |

PSH6${}_{2}$ | 3 | 1 | 1 | $2{n}^{2}+2n$ | ${6}^{\frac{1}{2{n}^{2}+2n}}$ |

$\mathrm{C}{6}_{1}$ | 3 | 2 | 0 | $2{n}^{2}+3n$ | ${6}^{\frac{1}{2{n}^{2}+3n}}$ |

$\mathrm{C}{6}_{2}$ | 2 | 2 | 0 | $2{n}^{2}+2n$ | ${6}^{\frac{1}{2{n}^{2}+2n}}$ |

$\mathrm{XH}6$ | 2 | 2 | 0 | $2{n}^{2}+2n$ | ${6}^{\frac{1}{2{n}^{2}+2n}}$ |

$\mathrm{B}6$ | 2 | 2 | 0 | $2{n}^{2}+2n$ | ${6}^{\frac{1}{2{n}^{2}+2n}}$ |

Method | FE | LS(${\mathit{F}}^{\prime}(\mathit{x})$) | LS($\mathit{Others}$) | $\mathit{M}\times \mathit{V}$ | $\mathbf{CI}$ |
---|---|---|---|---|---|

$\mathrm{PSH}{6}_{1}\{\alpha \ne 0\}$ | $2{n}^{2}+2n$ | 7 | 0 | 4 | ${6}^{\frac{1}{\frac{1}{3}{n}^{3}+13{n}^{2}+\frac{5}{3}n}}$ |

$\mathrm{PSH}{6}_{1}\{\alpha =0\}$ | $2{n}^{2}+2n$ | 5 | 0 | 2 | ${6}^{\frac{1}{\frac{1}{3}{n}^{3}+5{n}^{2}+\frac{2}{3}n}}$ |

$\mathrm{PSH}{6}_{2}\{\alpha \ne 0\}$ | $2{n}^{2}+2n$ | 5 | 4 | 2 | ${6}^{\frac{1}{\frac{2}{3}{n}^{3}+11{n}^{2}-\frac{2}{3}n}}$ |

$\mathrm{PSH}{6}_{2}\{\alpha =0\}$ | $2{n}^{2}+2n$ | 5 | 0 | 2 | ${6}^{\frac{1}{\frac{1}{3}{n}^{3}+5{n}^{2}+\frac{2}{3}n}}$ |

$\mathrm{C}{6}_{1}$ | $2{n}^{2}+3n$ | 3 | 1 | 1 | ${6}^{\frac{1}{\frac{2}{3}{n}^{3}+7{n}^{2}+\frac{7}{3}n}}$ |

$\mathrm{C}{6}_{2}$ | $2{n}^{2}+2n$ | 1 | 3 | 2 | ${6}^{\frac{1}{\frac{2}{3}{n}^{3}+8{n}^{2}+\frac{4}{3}n}}$ |

$\mathrm{XH}6$ | $2{n}^{2}+2n$ | 3 | 2 | 2 | ${6}^{\frac{1}{\frac{2}{3}{n}^{3}+9{n}^{2}+\frac{4}{3}n}}$ |

$\mathrm{B}6$ | $2{n}^{2}+2n$ | 2 | 4 | 3 | ${6}^{\frac{1}{{n}^{3}+11{n}^{2}+n}}$ |

**Table 3.**Numerical results of the examined methods for ${F}_{1}({x}_{1},{x}_{2})$ and ${x}^{(0)}={(0.8,\phantom{\rule{0.166667em}{0ex}}0.8)}^{T}$.

Method | k | $\parallel {\mathit{x}}^{(\mathit{k}+1)}-{\mathit{x}}^{(\mathit{k})}\parallel $ | $\parallel \mathit{F}({\mathit{x}}^{(\mathit{k}+1)})\parallel $ | $\mathit{\rho}$ |
---|---|---|---|---|

$\mathrm{PSH}{6}_{1}\{\alpha =0.0\}$ | 4 | $5.7517\times {10}^{-60}$ | 0.0 | 5.9906 |

$\mathrm{PSH}{6}_{1}\{\alpha =5.5\}$ | 4 | $2.0238\times {10}^{-64}$ | 0.0 | 5.9962 |

$\mathrm{PSH}{6}_{1}\{\alpha =10\}$ | 4 | $2.9651\times {10}^{-78}$ | 0.0 | 6.0264 |

$\mathrm{PSH}{6}_{2}\{\alpha =0.0\}$ | 4 | $5.7517\times {10}^{-60}$ | 0.0 | 5.9906 |

$\mathrm{PSH}{6}_{2}\{\alpha =5.5\}$ | 4 | $1.0081\times {10}^{-46}$ | $3.6422\times {10}^{-275}$ | 5.9701 |

$\mathrm{PSH}{6}_{2}\{\alpha =10\}$ | 4 | $6.6149\times {10}^{-43}$ | $6.8963\times {10}^{-252}$ | 5.9523 |

$\mathrm{C}{6}_{1}$ | 4 | $1.5912\times {10}^{-73}$ | 0.0 | 5.9973 |

$\mathrm{C}{6}_{2}$ | 10 | $6.3065\times {10}^{-72}$ | 0.0 | 5.9975 |

$\mathrm{XH}6$ | 4 | $8.6943\times {10}^{-66}$ | 0.0 | 5.9953 |

$\mathrm{B}6$ | 4 | $5.0674\times {10}^{-80}$ | 0.0 | 6.0030 |

**Table 4.**Numerical results of the examined methods for ${F}_{3}({x}_{1},{x}_{2},{x}_{3})$ and ${x}^{(0)}={(2.0,\phantom{\rule{0.166667em}{0ex}}0.5,\phantom{\rule{0.166667em}{0ex}}1.0)}^{T}$.

Method | k | $\parallel {\mathit{x}}^{(\mathit{k}+1)}-{\mathit{x}}^{(\mathit{k})}\parallel $ | $\parallel \mathit{F}({\mathit{x}}^{(\mathit{k}+1)})\parallel $ | $\mathit{\rho}$ |
---|---|---|---|---|

$\mathrm{PSH}{6}_{1}\{\alpha =0.0\}$ | 5 | $1.1553\times {10}^{-91}$ | 0.0 | - |

$\mathrm{PSH}{6}_{1}\{\alpha =5.5\}$ | 5 | $1.3862\times {10}^{-138}$ | 0.0 | - |

$\mathrm{PSH}{6}_{1}\{\alpha =10\}$ | 5 | $3.1738\times {10}^{-101}$ | $0.0$ | - |

$\mathrm{PSH}{6}_{2}\{\alpha =0.0\}$ | 5 | $1.1553\times {10}^{-91}$ | 0.0 | - |

$\mathrm{PSH}{6}_{2}\{\alpha =5.5\}$ | 6 | $6.4700\times {10}^{-85}$ | 0.0 | - |

$\mathrm{PSH}{6}_{2}\{\alpha =10\}$ | 6 | $2.7383\times {10}^{-132}$ | $0.0$ | - |

$\mathrm{C}{6}_{1}$ | 4 | $5.5171\times {10}^{-38}$ | $7.1730\times {10}^{-225}$ | 6.0424 |

$\mathrm{C}{6}_{2}$ | 4 | $2.1522\times {10}^{-93}$ | 0.0 | 6.0006 |

$\mathrm{XH}6$ | 4 | $6.1878\times {10}^{-50}$ | $5.5325\times {10}^{-297}$ | 5.9482 |

$\mathrm{B}6$ | 4 | $5.1979\times {10}^{-168}$ | 0.0 | 6.0365 |

**Table 5.**Numerical results of the examined methods for ${F}_{4}({x}_{1},{x}_{2},{x}_{3},{x}_{4})$ and ${x}^{(0)}={(2.5,\phantom{\rule{0.166667em}{0ex}}2.5,\phantom{\rule{0.166667em}{0ex}}2.5,\phantom{\rule{0.166667em}{0ex}}2.5)}^{T}$.

Method | k | $\parallel {\mathit{x}}^{(\mathit{k}+1)}-{\mathit{x}}^{(\mathit{k})}\parallel $ | $\parallel \mathit{F}({\mathit{x}}^{(\mathit{k}+1)})\parallel $ | $\mathit{\rho}$ |
---|---|---|---|---|

$\mathrm{PSH}{6}_{1}\{\alpha =0.0\}$ | 5 | $1.7213\times {10}^{-82}$ | 0.0 | 5.8841 |

$\mathrm{PSH}{6}_{1}\{\alpha =5.5\}$ | 5 | $6.2032\times {10}^{-101}$ | 0.0 | 6.0319 |

$\mathrm{PSH}{6}_{1}\{\alpha =10\}$ | 5 | $5.9604\times {10}^{-139}$ | $0.0$ | 7.0104 |

$\mathrm{PSH}{6}_{2}\{\alpha =0.0\}$ | 5 | $1.7213\times {10}^{-82}$ | 0.0 | 5.8841 |

$\mathrm{PSH}{6}_{2}\{\alpha =5.5\}$ | 5 | $2.4280\times {10}^{-56}$ | 0.0 | 5.4681 |

$\mathrm{PSH}{6}_{2}\{\alpha =10\}$ | 5 | $2.2166\times {10}^{-50}$ | $0.0$ | 5.2317 |

$\mathrm{C}{6}_{1}$ | 4 | $2.8009\times {10}^{-167}$ | 0.0 | 6.1732 |

$\mathrm{C}{6}_{2}$ | 4 | $6.0097\times {10}^{-36}$ | $9.3590\times {10}^{-222}$ | 6.7740 |

$\mathrm{XH}6$ | 5 | $1.0184\times {10}^{-173}$ | 0.0 | 6.1665 |

$\mathrm{B}6$ | 4 | $9.0970\times {10}^{-198}$ | 0.0 | 5.6982 |

**Table 6.**Numerical results of proposed methods for ${F}_{5}({x}_{1},{x}_{2},\dots ,{x}_{n})$, $n=20$ and ${x}^{(0)}={(0.75,\dots ,0.75)}^{T}$.

Method | k | $\parallel {\mathit{x}}^{(\mathit{k}+1)}-{\mathit{x}}^{(\mathit{k})}\parallel $ | $\parallel \mathit{F}({\mathit{x}}^{(\mathit{k}+1)})\parallel $ | $\mathit{\rho}$ |
---|---|---|---|---|

$\mathrm{PSH}{6}_{1}\{\alpha =0.0\}$ | 4 | $1.8871\times {10}^{-184}$ | 0.0 | 6.0 |

$\mathrm{PSH}{6}_{1}\{\alpha =5.5\}$ | 4 | $1.1531\times {10}^{-189}$ | 0.0 | 6.0 |

$\mathrm{PSH}{6}_{1}\{\alpha =10\}$ | 4 | $2.8662\times {10}^{-195}$ | 0.0 | 6.0 |

$\mathrm{PSH}{6}_{2}\{\alpha =0.0\}$ | 4 | $1.8871\times {10}^{-184}$ | 0.0 | 6.0 |

$\mathrm{PSH}{6}_{2}\{\alpha =5.5\}$ | 4 | $2.0650\times {10}^{-171}$ | 0.0 | 6.0 |

$\mathrm{PSH}{6}_{2}\{\alpha =10\}$ | 4 | $4.6908\times {10}^{-165}$ | 0.0 | 6.0 |

$\mathrm{C}{6}_{1}$ | 3 | $9.2604\times {10}^{-39}$ | $7.5226\times {10}^{-233}$ | 5.7540 |

$\mathrm{C}{6}_{2}$ | 4 | $9.7326\times {10}^{-195}$ | 0.0 | 6.0 |

$\mathrm{XH}6$ | 4 | $2.4997\times {10}^{-191}$ | 0.0 | 6.0 |

$\mathrm{B}6$ | 6 | $5.7210\times {10}^{-197}$ | 0.0 | 6.0 |

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**MDPI and ACS Style**

Capdevila, R.R.; Cordero, A.; Torregrosa, J.R. A New Three-Step Class of Iterative Methods for Solving Nonlinear Systems. *Mathematics* **2019**, *7*, 1221.
https://doi.org/10.3390/math7121221

**AMA Style**

Capdevila RR, Cordero A, Torregrosa JR. A New Three-Step Class of Iterative Methods for Solving Nonlinear Systems. *Mathematics*. 2019; 7(12):1221.
https://doi.org/10.3390/math7121221

**Chicago/Turabian Style**

Capdevila, Raudys R., Alicia Cordero, and Juan R. Torregrosa. 2019. "A New Three-Step Class of Iterative Methods for Solving Nonlinear Systems" *Mathematics* 7, no. 12: 1221.
https://doi.org/10.3390/math7121221