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

Table 6

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)
𝑓3(𝑥)	−0.8	5.15𝑒−120	3.08𝑒−82	2.64𝑒−146	2.17𝑒−156	9.58𝑒−124	1.21𝑒−162	2.48𝑒−178
	0.9	D	D	C	C	D	4.23𝑒−7	3.34𝑒−76
	1.0	F	F	F	F	F	1.56𝑒−1	2.10𝑒−55
Comparison of different optimal eight-order methods with respect to number of iterations
𝑓3(𝑥)	−0.8	4	4	4	4	4	4	4
	0.9	D	D	C	C	D	5	4
	1.0	F	F	F	F	F	6	4