Another Simple Way of Deriving Several Iterative Functions to Solve Nonlinear Equations

Table 2

Total number of iterations to approximate the zero of a function and the total number of function evaluations for various multipoint iterative methods.

Problem	Initial; guess	NM	TOM	JM	KM; (𝛾=1/2)	KM; (𝛾=1)	Method; (4.4)	Method; (4.9)	Method; (4.15)	Method; (4.22)
𝑓1(𝑥)	1.5	(4, 8)	(2, 6)	(3, 9)	(3, 9)	(4, 12)	(2, 6)	(2, 6)	(3, 9)	(3, 9)
	2	(3, 6)	(2, 6)	(2, 6)	(2, 6)	(3, 9)	(2, 6)	(2, 6)	(4, 12)	(2, 6)
𝑓2(𝑥)	0	(4, 8)	(2, 6)	(3, 9)	(3, 9)	(3, 9)	(2, 6)	(2, 6)	(3, 9)	(3, 9)
	2	(3, 6)	(2, 6)	(2, 6)	(2, 6)	(2, 6)	(2, 6)	(2, 6)	(2, 6)	(2, 6)
𝑓3(𝑥)	2	(3, 6)	(2, 6)	(3, 9)	(2, 6)	(2, 6)	(2, 6)	(2, 6)	(2, 6)	(2, 6)
	4	(5, 10)	(3, 9)	(3, 9)	(3, 9)	(3, 9)	(3, 9)	(3, 9)	(3, 9)	(3, 9)
𝑓4(𝑥)	1	(4, 8)	(3, 9)	(3, 9)	(3, 9)	(3, 9)	(3, 9)	(3, 9)	(3, 9)	(3, 9)
	2	(5, 10)	(3, 9)	(3, 9)	(3, 9)	(3, 9)	(2, 6)	(3, 9)	(3, 9)	(3, 9)
𝑓5(𝑥)	1.5	(6, 12)	(3, 9)	(3, 9)	(4, 12)	(5, 15)	(3, 9)	(4, 12)	(4, 12)	(4, 12)
	3.5	(6, 12)	(3, 9)	(3, 9)	(3, 9)	(3, 9)	(3, 9)	(3, 9)	(3, 9)	(4, 12)
𝑓6(𝑥)	1.5	(3, 6)	(2, 6)	(2, 6)	(2, 6)	(3, 9)	(2, 6)	(2, 6)	(2, 6)	(3, 9)
	3	(3, 6)	(2, 6)	(2, 6)	(2, 6)	(2, 6)	(2, 6)	(2, 6)	(2, 6)	(2, 6)
𝑓7(𝑥)	0	(3, 6)	(2, 6)	(2, 6)	(2, 6)	(2, 6)	(2, 6)	(2, 6)	(2, 6)	(2, 6)
	1	(3, 6)	(3, 9)	(2, 6)	(3, 9)	(3, 9)	(2, 6)	(2, 6)	(3, 9)	(3, 9)
𝑓8(𝑥)	−0.5	(2, 4)	(3, 9)	(3, 9)	(3, 9)	(3, 9)	(3, 9)	(3, 9)	(3, 9)	(3, 9)
	1	(3, 6)	(3, 9)	(3, 9)	(3, 9)	(3, 9)	(3, 9)	(3, 9)	(3, 9)	(3, 9)
𝑓9(𝑥)	−2	(5, 10)	(3, 9)	(3, 9)	(3, 9)	(3, 9)	(3, 9)	(4, 12)	(4, 12)	(3, 9)
	−1	(4, 8)	(2, 6)	(2, 6)	(2, 6)	(2, 6)	(2, 6)	(3, 9)	(2, 6)	(2, 6)
𝑓10(𝑥)	2.9	(6, 12)	(3, 9)	(6, 18)	(5, 15)	(8, 24)	(3, 9)	(3, 9)	(3, 9)	(3, 9)
	3.5	(11, 22)	(5, 15)	(11, 33)	(6, 18)	(6, 18)	(5, 15)	(6, 18)	(5, 15)	(6, 18)