An Efficient Family of Root-Finding Methods with Optimal Eighth-Order Convergence

Table 2

Comparison of methods using same total number of function evaluations for all methods (TFE = 12).

	NM	C8	L8	P8	T8	B81	B82	B83	M81	M82	M83
		𝛽1=𝛽3=0,𝛽2=1	𝛼1=𝛼2=1		𝛼=0	𝛼=1	𝛼=𝛾=1	𝛼=1	𝛼=𝛾=1	𝛼=𝛾=1	𝛼=𝛾=1
𝑓1, 𝑥0 = 1.6
𝑥𝑖−𝑟	9.75𝑒−41	4.35𝑒−338	4.51𝑒−300	0.00𝑒+00	3.67𝑒−305	6.55𝑒−304	8.24𝑒−304	9.60𝑒−304	0.00𝑒+00	1.40𝑒−349	1.70𝑒−350
𝑓(𝑥𝑖)	3.61𝑒−39	1.61𝑒−336	1.67𝑒−298	0.00𝑒+00	1.36𝑒−303	2.43𝑒−302	3.05𝑒−302	3.55𝑒−302	6.96𝑒−350	5.19𝑒−348	6.28𝑒−349
ρ	2.0000000	7.9999999	7.9999999	8.0000000	7.9999999	7.9999989	7.9999989	7.9999989	7.9999997	7.9999997	7.9999997
𝑓2, 𝑥0 = 2.0
𝑥𝑖−𝑟	4.27𝑒−57	0.00𝑒+00	4.76𝑒−334	0.00𝑒+00	2.88𝑒−333	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00
𝑓(𝑥𝑖)	4.20𝑒−57	0.00𝑒+00	4.68𝑒−334	0.00𝑒+00	2.83𝑒−333	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00
ρ	2.0000000	8.0000000	7.9999993	8.0000000	7.9999994	8.0000000	8.0000000	8.0000000	8.0000000	8.0000000	8.0000000
𝑓3, 𝑥0 = 1.8
𝑥𝑖−𝑟	4.41𝑒−58	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00
𝑓(𝑥𝑖)	1.22𝑒−57	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00
ρ	2.0000000	8.0000000	8.0000000	8.0000000	8.0000000	8.0000000	8.0000000	8.0000000	7.9999997	7.9999997	7.9999997
𝑓4, 𝑥0 = 1.0
𝑥𝑖−𝑟	1.80𝑒−83	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00
𝑓(𝑥𝑖)	3.00𝑒−83	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00
ρ	2.0000000	8.0000000	8.0000000	8.0000000	8.0000000	8.0000000	8.0000000	8.0000000	8.0000000	8.0000000	8.0000000
𝑓5, 𝑥0 = −0.5
𝑥𝑖−𝑟	3.46𝑒−27	7.71𝑒−202	8.70𝑒−163	5.99𝑒−246	3.75𝑒−162	3.12𝑒−222	1.20𝑒−221	2.98𝑒−221	4.57𝑒−195	7.97𝑒−194	1.98𝑒−194
𝑓(𝑥𝑖)	1.04𝑒−26	2.31𝑒−201	2.61𝑒−162	1.80𝑒−245	1.13𝑒−161	9.36𝑒−222	3.61𝑒−221	8.93𝑒−221	1.37𝑒−194	2.39𝑒−193	5.95𝑒−194
ρ	2.0000000	7.9999339	7.9996776	7.9999847	7.9996400	8.0000580	8.0000594	8.0000603	7.9995556	7.9995321	7.9995437
𝑓6, 𝑥0 = 2.0
𝑥𝑖−𝑟	7.97𝑒−85	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00
𝑓(𝑥𝑖)	9.24𝑒−85	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00
ρ	2.0000000	8.0000000	8.0000000	8.0000000	8.0000000	8.0000000	8.0000000	8.0000000	8.0000000	8.0000000	8.0000000
𝑓7, 𝑥0 = 3.2
𝑥𝑖−𝑟	4.66𝑒−74	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00
𝑓(𝑥𝑖)	2.81𝑒−74	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00
ρ	2.0000000	8.0000000	8.0000000	8.0000000	8.0000000	8.0000000	8.0000000	8.0000000	8.0000000	8.0000000	8.0000000
𝑓8, 𝑥0 = 1.0
𝑥𝑖−𝑟	1.68𝑒−54	4.42𝑒−342	4.21𝑒−309	0.00𝑒+00	1.20𝑒−309	0.00𝑒+00	5.83𝑒−346	1.68𝑒−341	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00
𝑓(𝑥𝑖)	1.78𝑒−54	4.68𝑒−342	4.46𝑒−309	0.00𝑒+00	1.27𝑒−309	0.00𝑒+00	6.17𝑒−346	1.78𝑒−341	0.00𝑒+00	0.00𝑒+00	0.00𝑒+00
ρ	2.0000000	8.0000004	8.0000018	8.0000001	8.0000021	8.0000012	8.0000009	8.0000014	8.0000001	8.0000001	8.0000001
𝑓9, 𝑥0 = −1
𝑥𝑖−𝑟	8.63𝑒−33	0.00𝑒+00	1.76𝑒−339	0.00𝑒+00	0.00𝑒+00	4.78𝑒−218	3.12𝑒−214	8.95𝑒−212	1.83𝑒−216	3.85𝑒−218	2.63𝑒−217
𝑓(𝑥𝑖)	1.75𝑒−31	0.00𝑒+00	3.58𝑒−338	0.00𝑒+00	0.00𝑒+00	9.71𝑒−217	6.34𝑒−213	1.82𝑒−210	3.72𝑒−215	7.81𝑒−217	5.34𝑒−216
ρ	2.0000000	8.0000001	8.0000002	8.0000000	8.0000000	7.9999987	7.9999985	7.9999984	8.0001475	8.0001379	8.0001426