# Derivative-Free Families of With- and Without-Memory Iterative Methods for Solving Nonlinear Equations and Their Engineering Applications

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## Abstract

**:**

## 1. Introduction

## 2. Construction of New Iterative Schemes and Their Convergence Analysis

**Theorem 1.**

**Proof of Theorem 1.**

#### Parametric Family of Three-Point With-Memory Method and Its Convergence Analysis

**Remark 1.**

**Lemma 1.**

**Theorem 2.**

**Proof of Theorem 2.**

## 3. Numerical Discussion

**Example 1.**

**Example 2.**

**Example 3.**

**Example 4.**

**Example 5.**

**Example 6.**

**Example 7.**

**Example 8.**

## 4. Conclusions

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

GM | Grau fifth-order method |

NM | Newton–Raphson method |

NDM1 | Newly developed method 1 (without memory, Equation (3)) |

NDM2 | Newly developed method 2 (with memory, Equation (19)) |

NRTM | Nouri fifth-order method (Equation (35)) |

OM | Ostrowski’s method |

COC | Computational rate of convergence |

The following constants were used in this manuscript: | |

c | speed of light |

P | Planck constant |

For the authors, the following abbreviations were used: | |

E.S. | Ekta Sharma |

L.J. | Lorentz Jäntschi |

S.K.M. | Shubham Kumar Mittal |

S.P. | Sunil Panday |

D.-M.J. | Dan-Marian Joița |

L.L.P. | Lavinia Lorena Pruteanu |

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**Figure 1.**Comparison of the methods based on the error in consecutive iterations, $|{s}_{n}-{s}_{n-1}|$, after the first three iterations.

**Table 1.**Comparisons of without-memory and with-memory methods after first three (n = 3) iterations for ${\Theta}_{1}\left(s\right)$.

Method | $\left|\right({\mathit{s}}_{1}-{\mathit{s}}_{0}\left)\right|$ | $\left|\right({\mathit{s}}_{2}-{\mathit{s}}_{1}\left)\right|$ | $\left|\right({\mathit{s}}_{3}-{\mathit{s}}_{2}\left)\right|$ | $\left|\mathbf{\Theta}\right({\mathit{s}}_{3}\left)\right|$ | COC |
---|---|---|---|---|---|

NM | $0.11779$ | $1.7434\times {10}^{-1}$ | $3.5679\times {10}^{-4}$ | $2.0508\times {10}^{-7}$ | $2.0000$ |

OM | $0.10048$ | $4.8190\times {10}^{-4}$ | $1.4662\times {10}^{-13}$ | $1.6842\times {10}^{-51}$ | $4.0000$ |

NRTM | $0.099122$ | $8.7833\times {10}^{-4}$ | $8.2482\times {10}^{-15}$ | $7.9485\times {10}^{-70}$ | $5.0000$ |

GM | $0.098037$ | $1.9631\times {10}^{-3}$ | $1.2383\times {10}^{-12}$ | $1.6115\times {10}^{-58}$ | $5.0000$ |

NDM1 ($\alpha $ = 1, $\beta $ = 1) | $0.10105$ | $1.0533\times {10}^{-3}$ | $1.3616\times {10}^{-15}$ | $7.0363\times {10}^{-75}$ | $5.0000$ |

NDM2 | $0.99749$ | $2.5101\times {10}^{-4}$ | $6.0861\times {10}^{-35}$ | $5.7271\times {10}^{-341}$ | $10.0000$ |

**Table 2.**Comparisons of without-memory and with-memory methods after first three (n = 3) iterations for ${\Theta}_{2}\left(s\right)$.

Method | $\left|\right({\mathit{s}}_{1}-{\mathit{s}}_{0}\left)\right|$ | $\left|\right({\mathit{s}}_{2}-{\mathit{s}}_{1}\left)\right|$ | $\left|\right({\mathit{s}}_{3}-{\mathit{s}}_{2}\left)\right|$ | $\left|\mathbf{\Theta}\right({\mathit{s}}_{3}\left)\right|$ | COC |
---|---|---|---|---|---|

NM | $0.075667$ | $5.6350\times {10}^{-3}$ | $3.1750\times {10}^{-5}$ | $1.0081\times {10}^{-9}$ | $2.0000$ |

OM | $0.070030$ | $3.0385\times {10}^{-5}$ | $8.5226\times {10}^{-19}$ | $5.2758\times {10}^{-73}$ | $4.0000$ |

NRTM | $0.069979$ | $2.1309\times {10}^{-5}$ | $2.7830\times {10}^{-23}$ | $1.0574\times {10}^{-112}$ | $5.0000$ |

GM | $0.069961$ | $3.9227\times {10}^{-5}$ | $1.1149\times {10}^{-21}$ | $2.0673\times {10}^{-104}$ | $5.0000$ |

NDM1 ($\alpha $ = 1, $\beta $ = 1) | $0.070006$ | $6.1200\times {10}^{-6}$ | $1.3737\times {10}^{-24}$ | $7.8257\times {10}^{-118}$ | $5.0000$ |

NDM2 | $0.069984$ | $1.5860\times {10}^{-5}$ | $3.9187\times {10}^{-56}$ | $3.9040\times {10}^{-577}$ | $10.0000$ |

**Table 3.**Comparisons of without-memory and with-memory methods after first three (n = 3) iterations for ${\Theta}_{3}\left(s\right)$.

Method | $\left|\right({\mathit{s}}_{1}-{\mathit{s}}_{0}\left)\right|$ | $\left|\right({\mathit{s}}_{2}-{\mathit{s}}_{1}\left)\right|$ | $\left|\right({\mathit{s}}_{3}-{\mathit{s}}_{2}\left)\right|$ | $\left|\mathbf{\Theta}\right({\mathit{s}}_{3}\left)\right|$ | COC |
---|---|---|---|---|---|

NM | $0.020781$ | $4.1098\times {10}^{-4}$ | $1.6149\times {10}^{-7}$ | $6.8908\times {10}^{-14}$ | $2.0000$ |

OM | $0.020370$ | $1.4498\times {10}^{-7}$ | $3.5630\times {10}^{-28}$ | $3.5916\times {10}^{-110}$ | $4.0000$ |

NRTM | $0.020369$ | $1.8841\times {10}^{-8}$ | $1.0645\times {10}^{-38}$ | $1.6942\times {10}^{-189}$ | $5.0000$ |

GM | $0.020369$ | $3.7534\times {10}^{-8}$ | $6.9025\times {10}^{-37}$ | $4.0127\times {10}^{-180}$ | $5.0000$ |

NDM1 ($\alpha $ = 1, $\beta $ = 1) | $0.020369$ | $1.0344\times {10}^{-10}$ | $1.1052\times {10}^{-52}$ | $4.2540\times {10}^{-262}$ | $5.0000$ |

NDM2 | $0.020369$ | $2.7829\times {10}^{-9}$ | $2.4637\times {10}^{-84}$ | $2.0132\times {10}^{-834}$ | $10.0000$ |

**Table 4.**Comparisons of without-memory and with-memory methods after first three (n = 3) iterations for ${\Theta}_{4}\left(s\right)$.

Method | $\left|\right({\mathit{s}}_{1}-{\mathit{s}}_{0}\left)\right|$ | $\left|\right({\mathit{s}}_{2}-{\mathit{s}}_{1}\left)\right|$ | $\left|\right({\mathit{s}}_{3}-{\mathit{s}}_{2}\left)\right|$ | $\left|\mathbf{\Theta}\right({\mathit{s}}_{3}\left)\right|$ | COC |
---|---|---|---|---|---|

NM | $0.0010012$ | $1.1684\times {10}^{-6}$ | $1.5927\times {10}^{-12}$ | $8.8779\times {10}^{-24}$ | $2.0000$ |

OM | $0.0010000$ | $1.0065\times {10}^{-12}$ | $1.0311\times {10}^{-48}$ | $3.4070\times {10}^{-192}$ | $4.0000$ |

NRTM | $0.0010000$ | $9.7700\times {10}^{-15}$ | $8.6879\times {10}^{-70}$ | $1.4493\times {10}^{-344}$ | $5.0000$ |

GM | $0.0010000$ | $1.4655\times {10}^{-14}$ | $1.0014\times {10}^{-68}$ | $4.4761\times {10}^{-339}$ | $5.0000$ |

NDM1 ($\alpha $ = 1, $\beta $ = 1) | $0.0010000$ | $1.2108\times {10}^{-16}$ | $3.2123\times {10}^{-81}$ | $1.2667\times {10}^{-403}$ | $5.0000$ |

NDM2 | $0.0010000$ | $1.5040\times {10}^{-15}$ | $9.7884\times {10}^{-147}$ | $4.0028\times {10}^{-1458}$ | $10.0000$ |

**Table 5.**Comparisons of without-memory and with-memory methods after first three (n = 3) iterations for ${\Theta}_{5}\left(s\right)$.

Method | $\left|\right({\mathit{s}}_{1}-{\mathit{s}}_{0}\left)\right|$ | $\left|\right({\mathit{s}}_{2}-{\mathit{s}}_{1}\left)\right|$ | $\left|\right({\mathit{s}}_{3}-{\mathit{s}}_{2}\left)\right|$ | $\left|\mathbf{\Theta}\right({\mathit{s}}_{3}\left)\right|$ | COC |
---|---|---|---|---|---|

NM | $0.41667$ | $8.0128\times {10}^{-2}$ | $3.2000\times {10}^{-3}$ | $1.0240\times {10}^{-4}$ | $1.9990$ |

OM | $0.33654$ | $3.2051\times {10}^{-3}$ | $1.3107\times {10}^{-11}$ | $7.3787\times {10}^{-45}$ | $4.0000$ |

NRTM | $0.31962$ | $1.3711\times {10}^{-2}$ | $2.2606\times {10}^{-10}$ | $5.1662\times {10}^{-49}$ | $5.0000$ |

GM | $0.30867$ | $2.4662\times {10}^{-2}$ | $7.7382\times {10}^{-9}$ | $4.1619\times {10}^{-41}$ | $5.0000$ |

NDM1 ($\alpha $ = $0.1$, $\beta $ = $0.01$) | $0.33269$ | $1.3872\times {10}^{-3}$ | $7.1489\times {10}^{-16}$ | $1.3430\times {10}^{-78}$ | $5.1293$ |

NDM2 | $0.33216$ | $1.8261\times {10}^{-3}$ | $4.8530\times {10}^{-29}$ | $1.6984\times {10}^{-284}$ | $10.0000$ |

**Table 6.**Comparisons of without-memory and with-memory methods after first three (n = 3) iterations for ${\Theta}_{6}\left(s\right)$.

Method | $\left|\right({\mathit{s}}_{1}-{\mathit{s}}_{0}\left)\right|$ | $\left|\right({\mathit{s}}_{2}-{\mathit{s}}_{1}\left)\right|$ | $\left|\right({\mathit{s}}_{3}-{\mathit{s}}_{2}\left)\right|$ | $\left|\mathbf{\Theta}\right({\mathit{s}}_{3}\left)\right|$ | COC |
---|---|---|---|---|---|

NM | $1.1339$ | $5.4470\times {10}^{-1}$ | $1.9126\times {10}^{-1}$ | $3.6580\times {10}^{-2}$ | $2.0935$ |

OM | $1.6641$ | $1.8131\times {10}^{-1}$ | $2.6397\times {10}^{-4}$ | $1.6190\times {10}^{-15}$ | $4.0000$ |

NRTM | $1.6753$ | $1.7655\times {10}^{-1}$ | $2.2683\times {10}^{-4}$ | $8.0963\times {10}^{-19}$ | $5.0000$ |

GM | $1.6385$ | $2.1410\times {10}^{-1}$ | $1.0058\times {10}^{-3}$ | $2.3909\times {10}^{-15}$ | $5.0003$ |

NDM1 ($\alpha $ = $0.1$, $\beta $ = $0.01$) | $1.7591$ | $1.3677\times {10}^{-1}$ | $1.9137\times {10}^{-5}$ | $1.0706\times {10}^{-24}$ | $5.1009$ |

NDM2 | $1.7514$ | $1.3873\times {10}^{-1}$ | $1.8773\times {10}^{-9}$ | $3.2210\times {10}^{-88}$ | $10.0000$ |

**Table 7.**Comparisons of without-memory and with-memory methods after first three (n = 3) iterations for ${\Theta}_{7}\left(s\right)$.

Method | $\left|\right({\mathit{s}}_{1}-{\mathit{s}}_{0}\left)\right|$ | $\left|\right({\mathit{s}}_{2}-{\mathit{s}}_{1}\left)\right|$ | $\left|\right({\mathit{s}}_{3}-{\mathit{s}}_{2}\left)\right|$ | $\left|\mathbf{\Theta}\right({\mathit{s}}_{3}\left)\right|$ | COC |
---|---|---|---|---|---|

NM | $0.00011423$ | $2.3586\times {10}^{-10}$ | $1.0054\times {10}^{-21}$ | $3.5263\times {10}^{-45}$ | $2.0000$ |

OM | $0.00011423$ | $2.8491\times {10}^{-16}$ | $7.5636\times {10}^{-67}$ | $7.2512\times {10}^{-270}$ | $4.0000$ |

NRTM | $0.00011423$ | $2.8491\times {10}^{-16}$ | $1.1839\times {10}^{-82}$ | $2.8305\times {10}^{-415}$ | $5.0000$ |

GM | $0.00011423$ | $2.8491\times {10}^{-16}$ | $2.5115\times {10}^{-82}$ | $2.5802\times {10}^{-413}$ | $5.0000$ |

NDM1 ($\alpha $ = 1, $\beta $ = 1) | $0.00011423$ | $1.2212\times {10}^{-19}$ | $1.7056\times {10}^{-94}$ | $1.7490\times {10}^{-469}$ | $5.0000$ |

NDM2 | $0.00011423$ | $1.0938\times {10}^{-23}$ | $1.4380\times {10}^{-233}$ | $2.4917\times {10}^{-2341}$ | $10.0000$ |

**Table 8.**Comparisons of without-memory and with-memory methods after first three (n = 3) iterations for ${\Theta}_{8}\left(s\right)$.

Method | $\left|\right({\mathit{s}}_{1}-{\mathit{s}}_{0}\left)\right|$ | $\left|\right({\mathit{s}}_{2}-{\mathit{s}}_{1}\left)\right|$ | $\left|\right({\mathit{s}}_{3}-{\mathit{s}}_{2}\left)\right|$ | $\left|\mathbf{\Theta}\right({\mathit{s}}_{3}\left)\right|$ | COC |
---|---|---|---|---|---|

NM | $0.000093269$ | $2.4434\times {10}^{-9}$ | $1.6769\times {10}^{-18}$ | $6.6965\times {10}^{-37}$ | $2.0000$ |

OM | $0.000093272$ | $7.7299\times {10}^{-18}$ | $4.9129\times {10}^{-71}$ | $6.7973\times {10}^{-284}$ | $4.0000$ |

NRTM | $0.000093272$ | $7.7299\times {10}^{-18}$ | $5.7906\times {10}^{-88}$ | $1.1583\times {10}^{-438}$ | $5.0000$ |

GM | $0.000093272$ | $7.7299\times {10}^{-18}$ | $1.3540\times {10}^{-87}$ | $1.8930\times {10}^{-436}$ | $5.0000$ |

NDM1 ($\alpha $ = 1, $\beta $ = 1) | $0.000093272$ | $4.0787\times {10}^{-19}$ | $6.5223\times {10}^{-91}$ | $5.7827\times {10}^{-450}$ | $5.0000$ |

NDM2 | $0.000093272$ | $2.7995\times {10}^{-22}$ | $5.0470\times {10}^{-217}$ | $6.7502\times {10}^{-2169}$ | $10.0000$ |

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## Share and Cite

**MDPI and ACS Style**

Sharma, E.; Panday, S.; Mittal, S.K.; Joița, D.-M.; Pruteanu, L.L.; Jäntschi, L.
Derivative-Free Families of With- and Without-Memory Iterative Methods for Solving Nonlinear Equations and Their Engineering Applications. *Mathematics* **2023**, *11*, 4512.
https://doi.org/10.3390/math11214512

**AMA Style**

Sharma E, Panday S, Mittal SK, Joița D-M, Pruteanu LL, Jäntschi L.
Derivative-Free Families of With- and Without-Memory Iterative Methods for Solving Nonlinear Equations and Their Engineering Applications. *Mathematics*. 2023; 11(21):4512.
https://doi.org/10.3390/math11214512

**Chicago/Turabian Style**

Sharma, Ekta, Sunil Panday, Shubham Kumar Mittal, Dan-Marian Joița, Lavinia Lorena Pruteanu, and Lorentz Jäntschi.
2023. "Derivative-Free Families of With- and Without-Memory Iterative Methods for Solving Nonlinear Equations and Their Engineering Applications" *Mathematics* 11, no. 21: 4512.
https://doi.org/10.3390/math11214512