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

Table 7

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 = 10)
𝑓4(𝑥)	−1.9	1.34𝑒−51	4.52𝑒−20	4.68𝑒−28	2.65𝑒−21	8.80𝑒−23	4.94𝑒−59	5.06𝑒−59
	−0.8	1.64𝑒−139	8.21𝑒−15	2.97𝑒−106	2.08𝑒−43	6.54𝑒−50	3.08𝑒−176	1.19𝑒−235
	0.0	3.24𝑒+4	D	8.07𝑒−2	D	D	2.80𝑒+4	7.44𝑒−98
	0.2	1.05𝑒+426	D	1.19𝑒+5	D	D	9.32𝑒+425	2.85𝑒−71
Comparison of different optimal fourth-order methods with respect to number of iterations
𝑓4(𝑥)	−1.9	5	6	6	6	6	5	5
	−0.8	4	6	5	5	5	4	4
	0.0	12	D	7	D	D	12	5
	0.2	425	D	14	D	D	417	5