Methods

${\tilde{\tau }}_n$

$H\left({\theta }_n\right)=\frac{c+\left({\tilde{d}}_{n}c+d\right){\theta }_n+\omega {\theta }_n^2}{c+d{\theta }_n+b{\theta }_n^2}$

$M_1\mathrm{,}{M}_2\mathrm{,}{M}_3$ . Thukral [24]

$\frac{1+\gamma {\varphi }_n}{\left(1+\gamma {\varphi }_n-{\theta }_n\right)\left(1-{\theta }_n\right)}$

$c=1,d=-{\tilde{d}}_n,b=\frac{1}{1+\gamma {\varphi }_n},\omega =0$

Kung-Traub’s method

Thukral [7]

P1 Thukral [24]

$1+{\tilde{d}}_n{\theta }_n$

$c=1,d=b=\omega =0$

Soleymani, Khattri [5]

P2 Thukral [24]

$\frac{1+{\theta }_n}{1-\frac{{\theta }_n}{1+\gamma {\varphi }_n}}$

$c=1,d=-\frac{1}{1+\gamma {\varphi }_n},b=\omega =0$ ,

Sharma [14]

M2 [24]

$\left(1+\frac{{\theta }_n}{1+\gamma {\varphi }_n}+\frac{{\theta }_n^2}{{\left(1+\gamma {\varphi }_n\right)}^2}\right)\cdot \frac{1}{1-{\theta }_n}$

$b=0,\mathrm{<mi>\; </mi>}c=1,\mathrm{<mi>\; </mi>}d=-1,\mathrm{<mi>\; </mi>}\omega =\frac{1}{{\left(1+\gamma {\varphi }_n\right)}^2}$

method in [3]

$h\left({\theta }_n\mathrm{,}{s}_n\right)=1+{\theta }_n+s_n+\frac{a}{2}{\theta }_n^2+b{\theta }_n{s}_n+\frac{c}{2}{s}_n^2$

$c=1,\mathrm{<mi>\; </mi>}d=b=0,\mathrm{<mi>\; </mi>}\omega =\left(\frac{a+c}{2}+\frac{a}{1+\gamma \varphi }\right)$

Soleymani [23]

$\frac{1}{1-{\tilde{d}}_n{\theta }_n}$

$c=1,d=-{\tilde{d}}_n,b=\omega =0$

Zheng et al. [12]

Soleymani [8]

$1+\frac{{\theta }_n}{1+\gamma {\varphi }_n}+\frac{{\theta }_n^2}{{\left(1+\gamma {\varphi }_n\right)}^2}$

$c=1,d=b=0,\omega =\frac{1}{{\left(1+\gamma {\varphi }_n\right)}^2}$

Cordero et al. [17]

${\psi }_4\left(x_n,y_n,{\omega }_n\right),\left(\gamma =0\right)$

Sharifi et al. [16]

$\frac{1+\beta {\theta }_n}{1+\left(\beta -2\right){\theta }_n}\frac{1}{1-\frac{{\theta }_n}{1+\gamma {\varphi }_n}}G\left({\theta }_n\right)$

$c=1,b=\frac{2-\beta }{1+\gamma {\varphi }_n},\omega =-\beta ,d=\beta -{\tilde{d}}_n-1$

Chebyshew-Halley

type method [4]

$\frac{1}{1-2\alpha {\theta }_n}\frac{1}{1-\frac{{\theta }_n}{1+\gamma {\varphi }_n}}H\left({\theta }_n\right)$

$H\left(0\right)=1,H^{\prime }\left(0\right)=1-2\alpha$

$c=1,d=-\left(2\alpha +\frac{1}{1+\gamma {\varphi }_n}\right),b=\frac{2\alpha }{1+\gamma {\varphi }_n},\omega =H\left({\theta }_n\right)$

Lotfi et al. [15]

$\frac{1+{\theta }_n+a{\tilde{d}}_n\frac{{\theta }_n^2}{2}}{1-\frac{{\theta }_n}{1+\gamma {\varphi }_n}}$

$c=1,b=0,d=-\frac{1}{1+\gamma {\varphi }_n},\omega =\frac{a{\tilde{d}}_n}{2}$

Behl. [18]

${\psi }_4\left(x_n\mathrm{,}{y}_n\mathrm{,}{\omega }_n\right)$