${\tilde{\alpha }}_n$

$f\left(x\right)$

$f_2\left(x\right)$

$f_3\left(x\right)$

Methods

$x_0$

−0.6

−1.5

$a\mathrm{,}b\mathrm{,}c$

n

$\left|f\left(x_n\right)\right|$

n

$\left|f\left(x_n\right)\right|$

(32)

$a=1$

$b=1$

4

1.60 (−691)

4

1.37 (−349)

$c=-1$

$a=1$

$b=1$

5

2.44 (−395)

4

2.20 (−260)

$c=-1$

$a=0$

$b=1$

-

5

6.80 (−1233)

(4)

$c=0$

$a=0$

$b=1$

4

9.48 (−541)

4

9.00 (−300)

$c=0$

$a=-1$

$b=1$

-

5

8.55 (−988)

$c=1$

$a=-1$

$b=1$

4

7.55 (−316)

5

2.25 (−1777)

$c=1$

$a=1$

$b=1$

4

2.72 (−484)

-

-

Soleymani [23]

$c=-1$

$a=1$

$b=1$

4

1.75 (−449)

4

7.99 (−702)

Thukral [7]

$c=-1$

(45)

$\eta =\frac{1}{1+\gamma {\varphi }_n}$

4

1.75 (−449)

4

9.49 (−267)

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

$\beta =0$

5

1.05 (−875)

5

4.59 (−1301)

P1 Thukral [24]

$\beta =\frac{{\tilde{d}}_n}{1+\gamma {\varphi }_n}$

5

1.09 (−707)

5

1.14 (−1709)

P2 Thukral [24]

$\eta =0$

5

2.98 (−1069)

4

2.33 (−373)

Sharma et al. [19]