New Highly Efficient Families of Higher-Order Methods for Simple Roots, Permitting 𝑓^′(𝑥_𝑛)=0

Table 8

𝑓(𝑥)	𝑥0	LW8; (𝛼=0,𝛽=0)	TM8	SS8; (𝛼=1)	KW8; (𝛼=3)	SM8	MKM8; |𝜆11|=2	MOM8; |𝜆11|=2
Comparison of different optimal eight-order methods with the same total number of functional evaluations (TNFE = 12)
𝑓4(𝑥)	−1.9	2.37𝑒−71	3.77𝑒−29	4.23𝑒−97	3.58𝑒−96	1.13𝑒−63	7.56𝑒−43	2.04𝑒−63
	−0.8	5.22𝑒−229	D	1.38𝑒−318	4.00𝑒−353	3.79𝑒−123	3.06𝑒−194	3.59𝑒−339
	0.0	D	D	D	D	D	1.76𝑒−5	6.59𝑒−121
	0.2	D	D	D	D	D	1.04𝑒+00	4.52𝑒−83
Comparison of different optimal eight-order methods with respect to number of iterations
𝑓4(𝑥)	−1.9	4	5	4	4	4	4	4
	−0.8	4	D	3	3	4	4	3
	0.0		D	D	D	D	5	4
	0.2	D	D	D	D	D	8	4