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

Table 4

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)
𝑓2(𝑥)	1.4	C	C	C	C	C	6.04𝑒−449	6.45𝑒−445
	1.5	8.22𝑒−48	2.29𝑒−59	C	C	C	9.54𝑒−301	9.53𝑒−299
	2.5	6.73𝑒−290	6.37𝑒−277	1.13𝑒−379	1.15𝑒−234	1.22𝑒−296	2.21𝑒−412	1.46𝑒−374
	4.0	5.05𝑒−172	1.68𝑒−154	1.94𝑒−247	1.32𝑒−305	1.21𝑒−189	8.07𝑒−433	4.60𝑒−359
Comparison of different optimal eight-order methods with respect to number of iterations
𝑓2(𝑥)	1.4	C	C	C	C	C	3	3
	1.5	4	4	C	C	C	3	3
	2.5	3	3	3	3	3	3	3
	4.0	4	4	4	4	4	3	3