Optimal High-Order Methods for Solving Nonlinear Equations

Table 1

Comparison of sixteenth-order schemes.

Test functions	𝑥0		T16	S16	M16
𝑓1(𝑥)	0.3	|𝑥1−𝜉|	2.079𝑒−5	5.991𝑒−8	5.987𝑒−5
		|𝑥2−𝜉|	5.195𝑒−67	3.791𝑒−112	3.613𝑒−58
		|𝑥3−𝜉|	1.202𝑒−1052	1.74𝑒−1779	1.125𝑒−909
		COC	16.0	16.0	16.0
𝑓1(𝑥)	1	|𝑥1−𝜉|	0.05729	7.143𝑒−4	1.549𝑒−2
		|𝑥2−𝜉|	7.02𝑒−14	6.056𝑒−47	4.122𝑒−20
		|𝑥3−𝜉|	1.483𝑒−202	4.408𝑒−736	9.269𝑒−301
		COC	16.0	16.0	16.0
𝑓2(𝑥)	−2	|𝑥1−𝜉|	0.1511	6.887𝑒−3	7.588𝑒−5
		|𝑥2−𝜉|	4.801	2.374𝑒−38	5.508𝑒−65
		|𝑥3−𝜉|	0.8086	1.299𝑒−605	3.5019𝑒−1023
		COC	—	16.0	16.0
𝑓2(𝑥)	−3	|𝑥1−𝜉|	0.1002	0.3238	8.93𝑒−3
		|𝑥2−𝜉|	1.153𝑒−5	3.062𝑒−8	8.602𝑒−32
		|𝑥3−𝜉|	6.83𝑒−75	7.209𝑒−224	7.042𝑒−496
		COC	16.0	16.0	15.99
𝑓3(𝑥)	2.1	|𝑥1−𝜉|	2.365𝑒−5	4.299𝑒−11	3.28𝑒−6
		|𝑥2−𝜉|	2.087𝑒−51	1.015𝑒−159	4.371𝑒−74
		|𝑥3−𝜉|	2.859𝑒−788	9.445𝑒−2538	4.319𝑒−1160
		COC	16.0	16.0	16.0