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

Table 5

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)
𝑓3(𝑥)	−0.8	3.04𝑒−84	1.74𝑒−52	4.00𝑒−91	5.21𝑒−61	1.45𝑒−74	1.83𝑒−85	1.41𝑒−104
	0.9	1.83𝑒−2	D	5.10𝑒−5	1.71𝑒+3475	C	1.50𝑒−2	1.10𝑒−46
	1.0	F	F	2.48𝑒−1	F	F	F	1.57𝑒−35
Comparison of different optimal fourth-order methods with respect to number of iterations
𝑓3(𝑥)	−0.8	5	5	5	5	5	5	5
	0.9	8	D	7	4048	C	8	5
	1.0	F	F	8	F	F	F	5

D stands for diverge. F stands for failure.