Some New Variants of Cauchy's Methods for Solving Nonlinear Equations

Table 1

Comparison of various third-order methods and newton's method.

𝑓(𝑥𝑛)	𝑥0					𝑛
		NM	WF	HM	CHM	SHM	LM1	LM2	LM3	LM4	LM5
𝑓1	−0.5	131	6	45	5	40	9	9	7	8	10
	2	5	4	3	4	3	3	4	4	4	4
𝑓2	0	4	3	3	3	3	2	3	3	3	3
	0.5	4	3	3	3	3	2	3	3	3	3
𝑓3	−3	6	4	4	4	4	3	5	4	4	4
	−5	7	5	4	5	4	4	4	5	5	5
𝑓4	1	7	4	5	5	4	4	4	4	5	4
	2	6	4	6	7	5	4	4	4	4	4
𝑓5	2.5	6	4	4	4	3	4	4	4	4	4
	3.5	7	5	4	5	4	4	4	4	5	5
𝑓6	−0.3	5	3	4	—	4	3	4	4	6	4
	2.7	6	4	4	5	4	3	4	4	3	4
𝑓7	1	6	4	4	5	3	3	4	4	16	4
	3	6	3	4	4	4	3	4	4	4	4
𝑓8	3.25	8	6	5	6	—	5	4	4	6	5
	3.45	11	8	6	7	—	6	5	5	8	7