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

Table 1

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)
𝑓1(𝑥)	−1.51	C	C	1.72𝑒−84	C	C	C	8.91𝑒−81
	−1.51	C	C	1.72𝑒−84	C	C	C	8.91𝑒−81
Comparison of different optimal fourth-order methods with respect to number of iterations
𝑓1(𝑥)	−1.51	C	C	5	C	C	C	5
	−1.51	C	C	5	C	C	C	5

C stands for converge to undesired root.