# Modified Potra–Pták Multi-step Schemes with Accelerated Order of Convergence for Solving Systems of Nonlinear Equations

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## Abstract

**:**

## 1. Introduction

## 2. The Method and Analysis of Convergence

**Theorem**

**1.**

**Proof.**

#### 2.1. Multi-step Method with Order $3r+6$

**Theorem**

**2.**

**Proof.**

**Remark**

**1.**

## 3. Computational Efficiency

#### 3.1. Comparison among the Efficiencies

**Theorem**

**3.**

## 4. Numerical Results

#### 4.1. Example 1

#### 4.2. Example 2

#### 4.3. Example 3

#### 4.4. Example 4

#### 4.5. Example 5

## 5. Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Basins of attraction of known and proposed methods on $F({x}_{1},{x}_{2})=0$. (

**a**) H${}_{6,1}$; (

**b**) H${}_{6,2}$; (

**c**) H${}_{6,3}$; (

**d**) H${}_{6,4}$; (

**e**) H${}_{9,1}$.

**Figure 2.**Basins of attraction of known and proposed methods on $G({x}_{1},{x}_{2})=0$. (

**a**) H${}_{6,1}$; (

**b**) H${}_{6,2}$; (

**c**) H${}_{6,3}$; (

**d**) H${}_{6,4}$; (

**e**) H${}_{9,1}$.

Method | H${}_{6,1}$ | H${}_{6,2}$ | H${}_{6,3}$ | H${}_{6,4}$ | H${}_{9,1}$ | |
---|---|---|---|---|---|---|

ACOC | 5.5833 | 6.0210 | - | - | 6.2081 | |

iter | 3 | 3 | 4 | 4 | 3 | |

$m=20$ | $\parallel {x}^{(k+1)}-{x}^{(k)}\parallel $ | $2.78\times {10}^{-35}$ | $1.37\times {10}^{-34}$ | $2.69\times {10}^{-98}$ | $3.59\times {10}^{-96}$ | $8.63\times {10}^{-59}$ |

$\parallel F({x}^{(k+1)})\parallel $ | $6.10\times {10}^{-125}$ | $1.17\times {10}^{-101}$ | $4.55\times {10}^{-229}$ | $8.10\times {10}^{-225}$ | $1.87\times {10}^{-210}$ | |

CPU time (seconds) | 0.025 | 0.024 | 0.028 | 0.026 | 0.022 | |

ACOC | 3.1024 | 2.1869 | - | - | 6.0462 | |

iter | 3 | 3 | 4 | 4 | 3 | |

$m=50$ | $\parallel {x}^{(k+1)}-{x}^{(k)}\parallel $ | $3.81\times {10}^{-34}$ | $2.08\times {10}^{-34}$ | $6.93\times {10}^{-97}$ | $9.18\times {10}^{-95}$ | $2.24\times {10}^{-57}$ |

$\parallel F({x}^{(k+1)})\parallel $ | $2.76\times {10}^{-121}$ | $8.62\times {10}^{-101}$ | $1.03\times {10}^{-225}$ | $1.80\times {10}^{-221}$ | $7.16\times {10}^{-206}$ | |

CPU time (seconds) | 0.042 | 0.042 | 0.043 | 0.042 | 0.044 |

Method | H${}_{6,1}$ | H${}_{6,2}$ | H${}_{6,3}$ | H${}_{6,4}$ | H${}_{9,1}$ | |
---|---|---|---|---|---|---|

ACOC | 5.9898 | 5.9078 | 5.9248 | 5.9442 | 8.4359 | |

iter | 3 | 3 | 3 | 3 | 3 | |

$m=20$ | $\parallel {x}^{(k+1)}-{x}^{(k)}\parallel $ | $3.10\times {10}^{-45}$ | $6.62\times {10}^{-46}$ | $1.67\times {10}^{-46}$ | $3.55\times {10}^{-47}$ | $8.19\times {10}^{-78}$ |

$\parallel F({x}^{(k+1)})\parallel $ | $3.45\times {10}^{-155}$ | $1.94\times {10}^{-127}$ | $1.24\times {10}^{-128}$ | $5.59\times {10}^{-130}$ | $6.49\times {10}^{-271}$ | |

CPU time (seconds) | 0.031 | 0.028 | 0.030 | 0.028 | 0.029 | |

ACOC | 4.3931 | 2.0177 | 5.8962 | 5.9425 | 7.0463 | |

iter | 3 | 3 | 3 | 3 | 3 | |

$m=50$ | $\parallel {x}^{(k+1)}-{x}^{(k)}\parallel $ | $1.04\times {10}^{-49}$ | $1.10\times {10}^{-52}$ | $2.90\times {10}^{-53}$ | $8.37\times {10}^{-54}$ | $2.50\times {10}^{-83}$ |

$\parallel F({x}^{(k+1)})\parallel $ | $9.16\times {10}^{-170}$ | $6.01\times {10}^{-142}$ | $4.15\times {10}^{-143}$ | $3.46\times {10}^{-144}$ | $5.37\times {10}^{-289}$ | |

CPU time (seconds) | 0.060 | 0.058 | 0.058 | 0.061 | 0.059 |

Method | H${}_{6,1}$ | H${}_{6,2}$ | H${}_{6,3}$ | H${}_{6,4}$ | H${}_{9,1}$ | |
---|---|---|---|---|---|---|

ACOC | 3.0100 | 2.0107 | 5.6132 | 5.8724 | 5.2651 | |

iter | 3 | 3 | 3 | 3 | 3 | |

$\parallel {x}^{(k+1)}-{x}^{(k)}\parallel $ | $4.51\times {10}^{-40}$ | $1.82\times {10}^{-48}$ | $3.74\times {10}^{-47}$ | $3.56\times {10}^{-46}$ | $6.95\times {10}^{-67}$ | |

$\parallel F({x}^{(k+1)})\parallel $ | $6.27\times {10}^{-138}$ | $4.19\times {10}^{-129}$ | $1.67\times {10}^{-126}$ | $1.52\times {10}^{-124}$ | $2.45\times {10}^{-234}$ | |

CPU time (seconds) | 0.035 | 0.036 | 0.035 | 0.035 | 0.035 |

H${}_{6,1}$ | H${}_{6,2}$ | H${}_{6,3}$ | H${}_{6,4}$ | H${}_{9,1}$ | |
---|---|---|---|---|---|

$\parallel {x}^{(1)}-{x}^{(0)}\parallel $ | $1.90\times {10}^{-1}$ | $1.55\times {10}^{-1}$ | $1.45\times {10}^{-1}$ | $1.27\times {10}^{-1}$ | $2.01\times {10}^{-1}$ |

$\parallel {x}^{(2)}-{x}^{(1)}\parallel $ | $1.44\times {10}^{-2}$ | $5.69\times {10}^{-2}$ | $7.36\times {10}^{-2}$ | $1.18\times {10}^{-1}$ | $2.62\times {10}^{-3}$ |

$\parallel {x}^{(3)}-{x}^{(2)}\parallel $ | $1.07\times {10}^{-9}$ | $5.15-7$ | $1.25\times {10}^{-5}$ | $1.74\times {10}^{-4}$ | $1.69\times {10}^{-18}$ |

$\parallel F({x}^{(1)})\parallel $ | $4.12\times {10}^{-2}$ | $1.21\times {10}^{-1}$ | $1.41\times {10}^{-1}$ | $1.85\times {10}^{-1}$ | $7.56\times {10}^{-3}$ |

$\parallel F({x}^{(2)})\parallel $ | $2.41\times {10}^{-9}$ | $2.65\times {10}^{-6}$ | $3.05\times {10}^{-5}$ | $2.99\times {10}^{-4}$ | $6.77\times {10}^{-18}$ |

$\parallel F({x}^{(3)})\parallel $ | $3.21\times {10}^{-44}$ | $1.54\times {10}^{-23}$ | $6.19\times {10}^{-20}$ | $6.93\times {10}^{-16}$ | $5.39\times {10}^{-86}$ |

H${}_{6,1}$ | H${}_{6,2}$ | H${}_{6,3}$ | H${}_{6,4}$ | H${}_{9,1}$ | |
---|---|---|---|---|---|

$\parallel {x}^{(1)}-{x}^{(0)}\parallel $ | $5.10\times {10}^{-1}$ | $5.15\times {10}^{-1}$ | $5.125\times {10}^{-1}$ | $5.10\times {10}^{-1}$ | $5.16\times {10}^{-1}$ |

$\parallel {x}^{(2)}-{x}^{(1)}\parallel $ | $7.96\times {10}^{-3}$ | $2.38\times {10}^{-3}$ | $5.63\times {10}^{-3}$ | $8.30\times {10}^{-3}$ | $1.46\times {10}^{-3}$ |

$\parallel {x}^{(3)}-{x}^{(2)}\parallel $ | $6.03\times {10}^{-12}$ | $3.54\times {10}^{-16}$ | $3.60\times {10}^{-13}$ | $8.89\times {10}^{-12}$ | $1.14\times {10}^{-23}$ |

$\parallel F({x}^{(1)})\parallel $ | $1.13\times {10}^{-2}$ | $3.37\times {10}^{-3}$ | $8.00\times {10}^{-3}$ | $1.18\times {10}^{-2}$ | $2.07\times {10}^{-3}$ |

$\parallel F({x}^{(2)})\parallel $ | $8.53\times {10}^{-12}$ | $5.00\times {10}^{-16}$ | $5.10\times {10}^{-13}$ | $1.26\times {10}^{-11}$ | $1.61\times {10}^{-23}$ |

$\parallel F({x}^{(3)})\parallel $ | $2.56\times {10}^{-56}$ | $8.87\times {10}^{-62}$ | $8.99\times {10}^{-57}$ | $5.02\times {10}^{-54}$ | $6.87\times {10}^{-161}$ |

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**MDPI and ACS Style**

Arora, H.; Torregrosa, J.R.; Cordero, A. Modified Potra–Pták Multi-step Schemes with Accelerated Order of Convergence for Solving Systems of Nonlinear Equations. *Math. Comput. Appl.* **2019**, *24*, 3.
https://doi.org/10.3390/mca24010003

**AMA Style**

Arora H, Torregrosa JR, Cordero A. Modified Potra–Pták Multi-step Schemes with Accelerated Order of Convergence for Solving Systems of Nonlinear Equations. *Mathematical and Computational Applications*. 2019; 24(1):3.
https://doi.org/10.3390/mca24010003

**Chicago/Turabian Style**

Arora, Himani, Juan R. Torregrosa, and Alicia Cordero. 2019. "Modified Potra–Pták Multi-step Schemes with Accelerated Order of Convergence for Solving Systems of Nonlinear Equations" *Mathematical and Computational Applications* 24, no. 1: 3.
https://doi.org/10.3390/mca24010003