Geometrically Constructed Families of Newton's Method for Unconstrained Optimization and Nonlinear Equations

Table 1

Test problem (nonlinear equations).

No.	Examples	[𝑎,𝑏]	Initial guesses	Root (𝑟)
(1)	𝑥𝑒^−𝑥−0.1=0	[0,2]	0.8	0.111832559108734
(2)	sin𝑥=0		1.0
		[−1,1.6]	1.51	0.000000000000000
			1.52
(3)	𝑒^−𝑥−sin𝑥=0	[5,7]	5.0	6.285049438476562
(4)	𝑒^𝑥2+7𝑥−30−1=0	[2,3.5]	2.0	3.000000000000000
			2.8
(5)	(𝑥−1)^6−1=0	[1,3]	1.0
			1.5	2.000000000000000
			2.5