# An Efficient Class of Weighted-Newton Multiple Root Solvers with Seventh Order Convergence

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Formulation of Method

**Theorem**

**1.**

**Proof.**

#### 2.1. Some Concrete Forms of $H\left(u\right)$

**Case 1.**Considering $H\left(u\right)$ a polynomial function, i.e.,

**Case 2.**When $H\left(u\right)$ is a rational function, i.e.,

**Case 3.**Consider $H\left(u\right)$ as another rational weight function, e.g.,

**Case 4.**When $H\left(u\right)$ is a yet another rational function of the form

#### 2.2. Some Concrete Forms of $G(u,w)$

**Case 5.**Considering $G(u,w)$ a polynomial function, e.g.,

**Case 6.**Considering $G(u,w)$ a sum of two rational functions, that is

**Case 7.**When $G(u,w)$ is a product of two rational functions, that is

## 3. Complex Dynamics of Methods

**Problem**

**1.**

**Problem**

**2.**

**Problem**

**3.**

**Problem**

**4.**

## 4. Numerical Tests

- (a)
- ${Q}_{f}(u,s)=m(1+2(m-1)(u-s)-4us+{s}^{2}).$
- (b)
- ${Q}_{f}(u,s)=m(1+2(m-1)(u-s)-{u}^{2}-2us).$
- (c)
- ${Q}_{f}(u,s)=\frac{m+au}{1+bu+cs+dus},$ where $a=\frac{2m}{m-1}$, $b=2-2m$, $c=\frac{2(2-2m+{m}^{2})}{m-1}$, $d=-2m(m-1)$.
- (d)
- ${Q}_{f}(u,s)=\frac{m+{a}_{1}u}{1+{b}_{1}u+{c}_{1}{u}^{2}}\frac{1}{1+{d}_{1}s},$ where ${a}_{1}=\frac{2m(4{m}^{4}-16{m}^{3}+31{m}^{2}-30m+13}{(m-1)(4{m}^{2}-8m+7)}$, ${b}_{1}=\frac{4(2{m}^{2}-4m+3)}{(m-1)(4{m}^{2}-8m+7)}$,${c}_{1}=-\frac{4{m}^{2}-8m+3}{4{m}^{2}-8m+7}$, ${d}_{1}=2(m-1)$.

- (a)
- ${Q}_{f}\left(u\right)=\frac{1+{u}^{2}}{1-u}$, ${K}_{f}(u,v)=\frac{1+{u}^{2}-v}{1-u+(u-2)v}$.
- (b)
- ${Q}_{f}\left(u\right)=1+u+2{u}^{2}$, ${K}_{f}(u,v)=1+u+2{u}^{2}+(1+2u)v$.
- (c)
- ${Q}_{f}\left(u\right)=\frac{1+{u}^{2}}{1-u}$, ${K}_{f}(u,v)=1+u+2{u}^{2}+2{u}^{3}+2{u}^{4}+(2u+1)v$.
- (d)
- ${Q}_{f}\left(u\right)=\frac{(2u-1)(4u-1)}{1-7u+13{u}^{2}}$, ${K}_{f}(u,v)=\frac{(2u-1)(4u-1)}{1-7u+13{u}^{2}-(1-6u)v}$.

**Example**

**1**(Eigen value problem)

**.**

**Example**

**2**(Manning equation for fluid dynamics)

**.**

**Example**

**3**(Beam designing model)

**.**

**Example**

**4**(van der Waals equation)

**.**

**Example**

**5.**

**Example**

**6.**

**Example**

**7.**

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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Methods | n | $|{\mathit{e}}_{\mathit{n}-3}|$ | $|{\mathit{e}}_{\mathit{n}-2}|$ | $|{\mathit{e}}_{\mathit{n}-1}|$ | COC | CPU-t (s) |
---|---|---|---|---|---|---|

GKN-I(a) | 4 | $1.06\times {10}^{-9}$ | $3.86\times {10}^{-56}$ | $9.03\times {10}^{-335}$ | 6.0000 | 0.1567 |

GKN-I(b) | 4 | $1.06\times {10}^{-9}$ | $3.91\times {10}^{-56}$ | $9.85\times {10}^{-335}$ | 6.0000 | 0.1583 |

GKN-I(c) | 4 | $1.06\times {10}^{-9}$ | $4.34\times {10}^{-56}$ | $2.02\times {10}^{-334}$ | 6.0000 | 0.1525 |

GKN-I(d) | 4 | $1.07\times {10}^{-9}$ | $1.17\times {10}^{-55}$ | $2.02\times {10}^{-331}$ | 6.0000 | 0.1600 |

GKN-II(a) | 4 | $1.19\times {10}^{-6}$ | $5.39\times {10}^{-38}$ | $4.56\times {10}^{-226}$ | 5.9999 | 0.1835 |

GKN-II(b) | 4 | $1.20\times {10}^{-6}$ | $1.61\times {10}^{-37}$ | $9.49\times {10}^{-223}$ | 5.9999 | 0.1640 |

GKN-II(c) | 4 | $1.20\times {10}^{-6}$ | $1.12\times {10}^{-37}$ | $7.51\times {10}^{-224}$ | 5.9999 | 0.1718 |

GKN-II(d) | 4 | $1.20\times {10}^{-6}$ | $1.87\times {10}^{-37}$ | $2.76\times {10}^{-222}$ | 5.9999 | 0.1680 |

NM-I(a) | 3 | $9.83\times {10}^{-8}$ | $4.34\times {10}^{-51}$ | 0 | 7.0000 | 0.1562 |

NM-I(b) | 3 | $1.16\times {10}^{-9}$ | $1.38\times {10}^{-64}$ | 0 | 7.0000 | 0.1170 |

NM-I(c) | 3 | $6.30\times {10}^{-10}$ | $7.75\times {10}^{-67}$ | 0 | 7.0000 | 0.1485 |

NM-II(a) | 3 | $9.83\times {10}^{-8}$ | $4.41\times {10}^{-51}$ | 0 | 7.0000 | 0.1367 |

NM-II(b) | 3 | $1.16\times {10}^{-9}$ | $1.40\times {10}^{-64}$ | 0 | 7.0000 | 0.1562 |

NM-II(c) | 3 | $6.30\times {10}^{-10}$ | $8.07\times {10}^{-67}$ | 0 | 7.0000 | 0.1405 |

Methods | n | $|{\mathit{e}}_{\mathit{n}-3}|$ | $|{\mathit{e}}_{\mathit{n}-2}|$ | $|{\mathit{e}}_{\mathit{n}-1}|$ | COC | CPU-t (s) |
---|---|---|---|---|---|---|

GKN-I(a) | 4 | $2.17\times {10}^{-8}$ | $4.61\times {10}^{-25}$ | $1.01\times {10}^{-152}$ | 6.0000 | 1.4218 |

GKN-I(b) | 4 | $2.17\times {10}^{-8}$ | $4.60\times {10}^{-25}$ | $2.27\times {10}^{-151}$ | 6.0000 | 1.4923 |

GKN-I(c) | 4 | $2.11\times {10}^{-8}$ | $4.21\times {10}^{-25}$ | $1.03\times {10}^{-150}$ | 6.0000 | 1.4532 |

GKN-I(d) | 4 | $1.77\times {10}^{-8}$ | $2.48\times {10}^{-25}$ | $2.68\times {10}^{-151}$ | 6.0000 | 1.4960 |

GKN-II(a) | 4 | $4.83\times {10}^{-7}$ | $1.36\times {10}^{-41}$ | $6.84\times {10}^{-249}$ | 6.0000 | 1.3867 |

GKN-II(b) | 4 | $4.90\times {10}^{-7}$ | $2.89\times {10}^{-41}$ | $1.21\times {10}^{-246}$ | 6.0000 | 1.3790 |

GKN-II(c) | 4 | $4.88\times {10}^{-7}$ | $2.22\times {10}^{-41}$ | $1.98\times {10}^{-247}$ | 6.0000 | 1.4110 |

GKN-II(d) | 4 | $4.89\times {10}^{-7}$ | $3.22\times {10}^{-41}$ | $2.62\times {10}^{-246}$ | 6.0000 | 1.3982 |

NM-I(a) | 3 | $1.65\times {10}^{-8}$ | $2.82\times {10}^{-58}$ | 0 | 7.0000 | 1.1367 |

NM-I(b) | 3 | $7.69\times {10}^{-9}$ | $1.35\times {10}^{-60}$ | 0 | 7.0000 | 1.1915 |

NM-I(c) | 3 | $3.65\times {10}^{-9}$ | $3.19\times {10}^{-63}$ | 0 | 7.0000 | 1.1407 |

NM-II(a) | 3 | $1.65\times {10}^{-9}$ | $2.86\times {10}^{-58}$ | 0 | 7.0000 | 1.1290 |

NM-II(b) | 3 | $7.69\times {10}^{-9}$ | $1.36\times {10}^{-60}$ | 0 | 7.0000 | 1.2540 |

NM-II(c) | 3 | $3.65\times {10}^{-9}$ | $3.27\times {10}^{-63}$ | 0 | 7.0000 | 1.1445 |

Methods | n | $|{\mathit{e}}_{\mathit{n}-3}|$ | $|{\mathit{e}}_{\mathit{n}-2}|$ | $|{\mathit{e}}_{\mathit{n}-1}|$ | COC | CPU-t (s) |
---|---|---|---|---|---|---|

GKN-I(a) | 4 | $1.29\times {10}^{-3}$ | $5.18\times {10}^{-20}$ | $2.19\times {10}^{-118}$ | 6.0000 | 0.0313 |

GKN-I(b) | 4 | $1.48\times {10}^{-3}$ | $1.63\times {10}^{-19}$ | $2.19\times {10}^{-115}$ | 5.9998 | 0.0390 |

GKN-I(c) | 4 | $1.45\times {10}^{-3}$ | $1.76\times {10}^{-19}$ | $5.56\times {10}^{-115}$ | 5.9997 | 0.0352 |

GKN-I(d) | 4 | $1.97\times {10}^{-3}$ | $1.80\times {10}^{-18}$ | $1.07\times {10}^{-108}$ | 5.9996 | 0.0428 |

GKN-II(a) | 4 | $5.67\times {10}^{-4}$ | $1.20\times {10}^{-22}$ | $1.06\times {10}^{-134}$ | 5.9999 | 0.0314 |

GKN-II(b) | 4 | $2.39\times {10}^{-3}$ | $5.78\times {10}^{-18}$ | $1.16\times {10}^{-105}$ | 5.9996 | 0.0396 |

GKN-II(c) | 4 | $1.70\times {10}^{-3}$ | $4.26\times {10}^{-19}$ | $1.08\times {10}^{-112}$ | 5.9997 | 0.0392 |

GKN-II(d) | 4 | $1.55\times {10}^{-2}$ | $5.18\times {10}^{-13}$ | $7.23\times {10}^{-76}$ | 6.0000 | 0.0354 |

NM-I(a) | 4 | $1.13\times {10}^{-4}$ | $6.52\times {10}^{-23}$ | $1.41\times {10}^{-157}$ | 6.9998 | 0.0275 |

NM-I(b) | 4 | $9.26\times {10}^{-4}$ | $1.63\times {10}^{-23}$ | $8.75\times {10}^{-162}$ | 6.9998 | 0.0313 |

NM-I(c) | 4 | $4.64\times {10}^{-4}$ | $4.44\times {10}^{-26}$ | $3.23\times {10}^{-180}$ | 6.9998 | 0.0275 |

NM-II(a) | 4 | $1.13\times {10}^{-4}$ | $6.83\times {10}^{-23}$ | $2.00\times {10}^{-157}$ | 6.9998 | 0.0316 |

NM-II(b) | 4 | $9.33\times {10}^{-4}$ | $1.77\times {10}^{-23}$ | $1.58\times {10}^{-161}$ | 6.9998 | 0.0275 |

NM-II(c) | 4 | $4.78\times {10}^{-4}$ | $5.86\times {10}^{-26}$ | $2.43\times {10}^{-179}$ | 6.9998 | 0.0354 |

Methods | n | $|{\mathit{e}}_{\mathit{n}-3}|$ | $|{\mathit{e}}_{\mathit{n}-2}|$ | $|{\mathit{e}}_{\mathit{n}-1}|$ | COC | CPU-t (s) |
---|---|---|---|---|---|---|

GKN-I(a) | 5 | $1.90\times {10}^{-5}$ | $9.03\times {10}^{-22}$ | $1.05\times {10}^{-119}$ | 6.0000 | 0.0471 |

GKN-I(b) | 5 | $2.31\times {10}^{-5}$ | $3.69\times {10}^{-21}$ | $6.14\times {10}^{-116}$ | 6.0000 | 0.0472 |

GKN-I(c) | 5 | $2.18\times {10}^{-5}$ | $3.18\times {10}^{-21}$ | $3.14\times {10}^{-116}$ | 6.0000 | 0.0465 |

GKN-I(d) | 5 | $3.58\times {10}^{-5}$ | $1.01\times {10}^{-19}$ | $5.02\times {10}^{-107}$ | 6.0000 | 0.0483 |

GKN-II(a) | 5 | $3.00\times {10}^{-6}$ | $4.91\times {10}^{-27}$ | $9.51\times {10}^{-152}$ | 6.0000 | 0.0474 |

GKN-II(b) | 5 | $4.78\times {10}^{-5}$ | $5.42\times {10}^{-19}$ | $1.17\times {10}^{-102}$ | 6.0000 | 0.0472 |

GKN-II(c) | 5 | $2.51\times {10}^{-5}$ | $6.82\times {10}^{-21}$ | $2.75\times {10}^{-114}$ | 6.0000 | 0.0481 |

GKN-II(d) | 7 | $3.85\times {10}^{-11}$ | $1.78\times {10}^{-55}$ | $1.75\times {10}^{-321}$ | 6.0000 | 0.0625 |

NM-I(a) | 5 | $1.06\times {10}^{-5}$ | $4.09\times {10}^{-26}$ | $5.33\times {10}^{-169}$ | 7.0000 | 0.0368 |

NM-I(b) | 5 | $5.10\times {10}^{-6}$ | $2.51\times {10}^{-28}$ | $1.73\times {10}^{-184}$ | 7.0000 | 0.0322 |

NM-I(c) | 5 | $1.15\times {10}^{-6}$ | $2.55\times {10}^{-33}$ | $6.75\times {10}^{-220}$ | 7.0000 | 0.0327 |

NM-II(a) | 5 | $1.05\times {10}^{-5}$ | $4.13\times {10}^{-23}$ | $5.89\times {10}^{-169}$ | 7.0000 | 0.0316 |

NM-II(b) | 5 | $5.16\times {10}^{-6}$ | $2.76\times {10}^{-23}$ | $3.48\times {10}^{-184}$ | 7.0000 | 0.0323 |

NM-II(c) | 5 | $1.20\times {10}^{-6}$ | $3.65\times {10}^{-26}$ | $9.09\times {10}^{-219}$ | 7.0000 | 0.0314 |

Methods | n | $|{\mathit{e}}_{\mathit{n}-3}|$ | $|{\mathit{e}}_{\mathit{n}-2}|$ | $|{\mathit{e}}_{\mathit{n}-1}|$ | COC | CPU-t (s) |
---|---|---|---|---|---|---|

GKN-I(a) | 4 | $1.12\times {10}^{-4}$ | $5.78\times {10}^{-24}$ | $1.10\times {10}^{-139}$ | 6.0000 | 0.2772 |

GKN-I(b) | 4 | $1.55\times {10}^{-4}$ | $7.30\times {10}^{-23}$ | $8.07\times {10}^{-133}$ | 6.0000 | 0.2462 |

GKN-I(c) | 4 | $1.39\times {10}^{-4}$ | $4.40\times {10}^{-23}$ | $4.43\times {10}^{-134}$ | 6.0000 | 0.2497 |

GKN-I(d) | 4 | $2.32\times {10}^{-4}$ | $1.95\times {10}^{-21}$ | $6.85\times {10}^{-124}$ | 6.0000 | 0.2812 |

GKN-II(a) | 4 | $3.36\times {10}^{-5}$ | $8.72\times {10}^{-28}$ | $2.66\times {10}^{-163}$ | 6.0000 | 0.3397 |

GKN-II(b) | 4 | $3.39\times {10}^{-5}$ | $2.19\times {10}^{-20}$ | $1.57\times {10}^{-117}$ | 6.0000 | 0.2695 |

GKN-II(c) | 4 | $2.16\times {10}^{-5}$ | $7.70\times {10}^{-22}$ | $1.58\times {10}^{-126}$ | 6.0000 | 0.2460 |

GKN-II(d) | 4 | $3.51\times {10}^{-3}$ | $3.25\times {10}^{-14}$ | $2.03\times {10}^{-80}$ | 6.0000 | 0.2342 |

NM-I(a) | 4 | $1.52\times {10}^{-4}$ | $8.45\times {10}^{-26}$ | $1.41\times {10}^{-174}$ | 6.9999 | 0.1445 |

NM-I(b) | 4 | $1.25\times {10}^{-4}$ | $2.22\times {10}^{-26}$ | $1.23\times {10}^{-178}$ | 6.9999 | 0.1522 |

NM-I(c) | 4 | $5.26\times {10}^{-4}$ | $1.58\times {10}^{-29}$ | $3.54\times {10}^{-201}$ | 6.9999 | 0.1640 |

NM-II(a) | 4 | $1.52\times {10}^{-4}$ | $9.05\times {10}^{-26}$ | $2.36\times {10}^{-174}$ | 6.9999 | 0.1482 |

NM-II(b) | 4 | $1.27\times {10}^{-4}$ | $2.49\times {10}^{-26}$ | $2.84\times {10}^{-178}$ | 6.9999 | 0.1492 |

NM-II(c) | 4 | $5.54\times {10}^{-4}$ | $2.51\times {10}^{-29}$ | $9.82\times {10}^{-200}$ | 6.9999 | 0.1642 |

Methods | n | $|{\mathit{e}}_{\mathit{n}-3}|$ | $|{\mathit{e}}_{\mathit{n}-2}|$ | $|{\mathit{e}}_{\mathit{n}-1}|$ | COC | CPU-t (s) |
---|---|---|---|---|---|---|

GKN-I(a) | 4 | $1.20\times {10}^{-5}$ | $6.82\times {10}^{-31}$ | $2.31\times {10}^{-182}$ | 6.0000 | 0.6797 |

GKN-I(b) | 4 | $1.20\times {10}^{-5}$ | $6.86\times {10}^{-31}$ | $2.40\times {10}^{-182}$ | 6.0000 | 0.6680 |

GKN-I(c) | 4 | $1.21\times {10}^{-5}$ | $7.72\times {10}^{-31}$ | $5.18\times {10}^{-182}$ | 6.0000 | 0.6992 |

GKN-I(d) | 4 | $1.58\times {10}^{-5}$ | $1.00\times {10}^{-29}$ | $6.51\times {10}^{-175}$ | 6.0000 | 0.6720 |

GKN-II(a) | 4 | $3.17\times {10}^{-5}$ | $1.64\times {10}^{-28}$ | $3.21\times {10}^{-168}$ | 6.0000 | 0.8047 |

GKN-II(b) | 4 | $3.50\times {10}^{-5}$ | $6.90\times {10}^{-28}$ | $4.05\times {10}^{-164}$ | 6.0000 | 0.8280 |

GKN-II(c) | 4 | $3.41\times {10}^{-5}$ | $4.42\times {10}^{-28}$ | $2.09\times {10}^{-165}$ | 6.0000 | 0.7967 |

GKN-II(d) | 4 | $3.54\times {10}^{-5}$ | $8.45\times {10}^{-28}$ | $1.56\times {10}^{-163}$ | 6.0000 | 0.8242 |

NM-I(a) | 4 | $5.14\times {10}^{-6}$ | $4.35\times {10}^{-38}$ | $1.35\times {10}^{-262}$ | 7.0000 | 0.5625 |

NM-I(b) | 4 | $3.45\times {10}^{-6}$ | $2.68\times {10}^{-39}$ | $4.53\times {10}^{-271}$ | 7.0000 | 0.5782 |

NM-I(c) | 4 | $2.05\times {10}^{-6}$ | $2.95\times {10}^{-41}$ | $3.76\times {10}^{-285}$ | 7.0000 | 0.5277 |

NM-II(a) | 4 | $5.14\times {10}^{-6}$ | $4.42\times {10}^{-38}$ | $1.53\times {10}^{-262}$ | 7.0000 | 0.4805 |

NM-II(b) | 4 | $3.45\times {10}^{-6}$ | $2.73\times {10}^{-39}$ | $5.24\times {10}^{-271}$ | 7.0000 | 0.4725 |

NM-II(c) | 4 | $2.05\times {10}^{-6}$ | $3.07\times {10}^{-41}$ | $5.17\times {10}^{-285}$ | 7.0000 | 0.4610 |

Methods | n | $|{\mathit{e}}_{\mathit{n}-3}|$ | $|{\mathit{e}}_{\mathit{n}-2}|$ | $|{\mathit{e}}_{\mathit{n}-1}|$ | COC | CPU-t (s) |
---|---|---|---|---|---|---|

GKN-I(a) | 4 | $2.53\times {10}^{-6}$ | $3.79\times {10}^{-35}$ | $4.32\times {10}^{-208}$ | 6.0000 | 1.1564 |

GKN-I(b) | 4 | $2.53\times {10}^{-6}$ | $3.92\times {10}^{-35}$ | $5.33\times {10}^{-208}$ | 6.0000 | 1.1577 |

GKN-I(c) | 4 | $2.68\times {10}^{-6}$ | $6.07\times {10}^{-35}$ | $8.23\times {10}^{-207}$ | 6.0000 | 1.1415 |

GKN-I(d) | 4 | $4.80\times {10}^{-6}$ | $5.34\times {10}^{-33}$ | $1.01\times {10}^{-194}$ | 6.0000 | 1.0473 |

GKN-II(a) | 4 | $5.04\times {10}^{-6}$ | $1.82\times {10}^{-33}$ | $4.04\times {10}^{-198}$ | 6.0000 | 1.0212 |

GKN-II(b) | 4 | $7.15\times {10}^{-6}$ | $4.23\times {10}^{-32}$ | $1.81\times {10}^{-189}$ | 6.0000 | 1.1215 |

GKN-II(c) | 4 | $6.39\times {10}^{-6}$ | $1.51\times {10}^{-32}$ | $2.64\times {10}^{-192}$ | 6.0000 | 1.2035 |

GKN-II(d) | 4 | $8.22\times {10}^{-6}$ | $1.41\times {10}^{-31}$ | $8.09\times {10}^{-187}$ | 6.0000 | 1.1416 |

NM-I(a) | 4 | $1.08\times {10}^{-6}$ | $6.96\times {10}^{-43}$ | $3.13\times {10}^{-296}$ | 7.0000 | 0.5787 |

NM-I(b) | 4 | $9.01\times {10}^{-7}$ | $1.91\times {10}^{-43}$ | $3.71\times {10}^{-300}$ | 7.0000 | 0.5632 |

NM-I(c) | 4 | $4.64\times {10}^{-7}$ | $7.44\times {10}^{-46}$ | $2.01\times {10}^{-317}$ | 7.0000 | 0.5586 |

NM-II(a) | 4 | $1.09\times {10}^{-6}$ | $7.21\times {10}^{-43}$ | $4.10\times {10}^{-296}$ | 7.0000 | 0.5478 |

NM-II(b) | 4 | $9.04\times {10}^{-7}$ | $2.00\times {10}^{-43}$ | $5.10\times {10}^{-300}$ | 7.0000 | 0.5946 |

NM-II(c) | 4 | $4.68\times {10}^{-7}$ | $8.21\times {10}^{-46}$ | $4.20\times {10}^{-317}$ | 7.0000 | 0.5644 |

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## Share and Cite

**MDPI and ACS Style**

Sharma, J.R.; Kumar, D.; Cattani, C.
An Efficient Class of Weighted-Newton Multiple Root Solvers with Seventh Order Convergence. *Symmetry* **2019**, *11*, 1054.
https://doi.org/10.3390/sym11081054

**AMA Style**

Sharma JR, Kumar D, Cattani C.
An Efficient Class of Weighted-Newton Multiple Root Solvers with Seventh Order Convergence. *Symmetry*. 2019; 11(8):1054.
https://doi.org/10.3390/sym11081054

**Chicago/Turabian Style**

Sharma, Janak Raj, Deepak Kumar, and Carlo Cattani.
2019. "An Efficient Class of Weighted-Newton Multiple Root Solvers with Seventh Order Convergence" *Symmetry* 11, no. 8: 1054.
https://doi.org/10.3390/sym11081054