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

Table 3

𝑓(𝑥)	𝑥0	JM4	KM4	MKM4	CM^14	CM^24	OM4	MOM4
Comparison of different optimal fourth-order methods with the same total number of functional evaluations (TNFE = 12)
𝑓2(𝑥)	1.4	C	C	2.02𝑒−216	9.76𝑒−93	C	C	3.14𝑒−212
	1.5	C	C	1.31𝑒−147	C	C	C	1.90𝑒−145
	2.5	1.24𝑒−188	2.33𝑒−177	1.62𝑒−203	1.12𝑒−131	1.17𝑒−104	6.84𝑒−187	1.45𝑒−141
	4.0	2.20𝑒−140	7.87𝑒−122	5.98𝑒−232	6.43𝑒−80	1.51𝑒−40	6.36𝑒−140	3.62𝑒−163
Comparison of different optimal fourth-order methods with respect to number of iterations
𝑓2(𝑥)	1.4	C	C	4	5	C	C	4
	1.5	C	C	4	C	C	C	4
	2.5	4	4	4	5	5	4	4
	4.0	4	5	4	5	5	4	4