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

Table 2

Results of application of the proposed method for cases in which Newton or secant method fails to converge.

Function	Root	Comparison of number of iterations required
		Starting point	Secant method	Newton method	Two-point Newton method
𝑦=−𝑥^4+3𝑥^2+2	1.887207676120680	1	11	Oscillates	7
		0.5	23	Oscillates	6
𝑦=log(𝑥)	1.000000000000000	3.0	Fails	Fails	5
𝑦 = arctan(𝑥)	0.0000000000000	3.0	Diverges	Diverges	6
		−3.0	Diverges	Diverges	6
𝑦=𝑥^5−𝑥+1	−1.167303978261420	2.0	Oscillates	Oscillates	12
		3.0	14	Oscillates	15
𝑦=0.5𝑥^3−6𝑥^2+21.5𝑥−22	4.00000000000000	3.0	10	Oscillates	7
		5.0	8	Oscillates	6
𝑦=𝑥^1/3	0.00000000000000	1.0	Oscillates	Diverges	101
		−1.0	Oscillates	Diverges	101
𝑦=10𝑥𝑒^−𝑥2−1	1.679630610428450	3.0	Diverges	Diverges	8
	0.101025848315685	−1.0	Diverges	Diverges	11