A Two-Point Newton Method Suitable for Nonconvergent Cases and with Super-Quadratic Convergence

Table 1

Comparison of results of iterations of the two-point Newton method with Newton and secant methods.

Function	Root	Starting point	Secant method	Newton method	Two-point Newton method
𝑦=𝑥^3+4𝑥^2−10	1.365230013414100	0.5	10	8	6
		1	9	6	5
𝑦=[sin(𝑥)]^2−𝑥^2+1	−1.404491648215340	−1	9	7	5
		−3	9	7	6
𝑦=(𝑥−2)(𝑥+2)^4	−2.0000000000000	−3	168	119	83
		1.4	116	81	60
𝑦=(𝑥−1)^6−1	2.00000000000000	1.5	25	17	8
		2.5	12	8	6
		3.5	16	11	8
𝑦 = sin(𝑥) · 𝑒^𝑥 + ln(𝑥^2+1)	−0.603231971557215	−0.8	8	7	5
		−0.65	8	5	4
𝑦=𝑒^𝑥2+7𝑥−30−1	3.000000000000000	4	29	20	14
		4.5	39	28	18
𝑦=𝑥−3ln(𝑥)	1.857183860207840	2	8	5	4
		0.5	11	8	5