A Two-Parameter Family of Fourth-Order Iterative Methods with Optimal Convergence for Multiple Zeros

Table 5

Comparison of |𝑥_𝑛−𝛼| for high-order iterative methods.

𝑓(𝑥)	𝑥0	|𝑥𝑛−𝛼|	S	J	Y1	Y2	Y3	Y4
		|𝑥1−𝛼|	5.37𝑒−7^†	4.00𝑒−7	6.26𝑒−8	5.37𝑒−7	1.96𝑒−7	2.14𝑒−6
𝑓1	1.32	|𝑥2−𝛼|	1.20𝑒−26	1.27𝑒−27	7.31𝑒−31	1.2𝑒−26	9.24𝑒−30	9.29𝑒−24
		|𝑥3−𝛼|	3.02𝑒−105	1.27𝑒−109	𝟏.𝟑𝟔𝐞−𝟏𝟐𝟐	3.02𝑒−104	4.53𝑒−119	3.24𝑒−93
		|𝑥4−𝛼|	0.𝑒−299	0.𝑒−299	0.𝑒−299	0.𝑒−299	0.𝑒−299	0.𝑒−299
		|𝑥1−𝛼|	0.001168	0.00144	0.00124	0.00097	0.00136	0.0018
		|𝑥2−𝛼|	4.99𝑒−10	1.60𝑒−9	7.38𝑒−10	1.45𝑒−10	1.19𝑒−9	5.18𝑒−9
𝑓2	−1.14−0.92𝑖	|𝑥3−𝛼|	1.65𝑒−35	2.46𝑒−33	8.97𝑒−35	7.24𝑒−38	7.15𝑒−34	3.46𝑒−31
		|𝑥4−𝛼|	2.02𝑒−137	1.37𝑒−128	1.95𝑒−134	𝟒.𝟒𝟒𝐞−𝟏𝟒𝟕	9.06𝑒−131	6.89𝑒−120
		|𝑥5−𝛼|	0.𝑒−299	0.𝑒−299	0.𝑒−299	0.𝑒−299	0.𝑒−299	0.𝑒−299
		|𝑥1−𝛼|	0.000428741	0.0002722	0.000399	0.00492273	0.0003078	0.000191074
		|𝑥2−𝛼|	1.77𝑒−12	3.38𝑒−13	1.37𝑒−12	5.14𝑒−8	5.32𝑒−13	9.02𝑒−14
𝑓3	1.39	|𝑥3−𝛼|	5.19𝑒−46	8.05𝑒−49	1.89𝑒−46	1.19𝑒−28	4.73𝑒−48	4.48𝑒−51
		|𝑥4−𝛼|	3.79𝑒−180	2.57𝑒−191	6.92𝑒−182	3.46𝑒−111	2.96𝑒−188	𝟐.𝟕𝟑𝐞−𝟐𝟎𝟎
		|𝑥5−𝛼|	0.𝑒−299	0.𝑒−299	0.𝑒−299	0.𝑒−299	0.𝑒−299	0.𝑒−299
		|𝑥1−𝛼|	0.00016	0.000159	0.00016	0.000153	0.0001602	0.000159
		|𝑥2−𝛼|	3.55𝑒−16	3.39𝑒−16	3.55𝑒−16	2.81𝑒−16	3.43𝑒−16	3.32𝑒−16
𝑓4	0.47−1.79𝑖	|𝑥3−𝛼|	8.36𝑒−63	6.94𝑒−63	8.42𝑒−63	3.14𝑒−63	7.28𝑒−63	6.33𝑒−63
		|𝑥4−𝛼|	2.56𝑒−249	1.21𝑒−249	2.63𝑒−249	4.89𝑒−249	1.46𝑒−249	𝟖.𝟑𝟕𝐞−𝟐𝟓𝟎
		|𝑥5−𝛼|	0.𝑒−299	0.𝑒−299	0.𝑒−299	0.𝑒−299	0.𝑒−299	0.𝑒−299
		|𝑥1−𝛼|	0.0000222	0.0000184	0.000023	0.0000118	0.0000193	0.0000164
𝑓5	8.79	|𝑥2−𝛼|	1.43𝑒−21	5.53𝑒−22	1.90𝑒−21	5.51𝑒−23	7.12𝑒−22	3.06𝑒−22
		|𝑥3−𝛼|	2.47𝑒−86	4.46𝑒−88	8.07𝑒−86	𝟐.𝟔𝟎𝐞−𝟗𝟐	1.29𝑒−87	3.71𝑒−89
		|𝑥4−𝛼|	0.𝑒−299	0.𝑒−299	0.𝑒−299	0.𝑒−299	0.𝑒−299	0.𝑒−299
		|𝑥1−𝛼|	2.84𝑒−7	3.45𝑒−7	2.36𝑒−7	4.14𝑒−7	3.30𝑒−7	3.65𝑒−7
𝑓6	3.09	|𝑥2−𝛼|	2.51𝑒−28	6.55𝑒−28	1.00𝑒−28	1.62𝑒−27	5.28𝑒−28	8.68𝑒−28
		|𝑥3−𝛼|	1.52𝑒−112	8.50𝑒−111	𝟑.𝟐𝟒𝐞−𝟏𝟏𝟒	3.86𝑒−109	3.44𝑒−111	2.76𝑒−110
		|𝑥4−𝛼|	0.𝑒−299	0.𝑒−299	0.𝑒−299	0.𝑒−299	0.𝑒−299	0.𝑒−299
		|𝑥1−𝛼|	0.00109	0.001241	0.00088	0.00137	0.0012	0.00128
		|𝑥2−𝛼|	2.04𝑒−11	1.97𝑒−11	3.14𝑒−11	1.00𝑒−10	6.32𝑒−12	4.04𝑒−11
𝑓7	−2.88	|𝑥3−𝛼|	1.77𝑒−42	1.67𝑒−42	4.48𝑒−41	2.90𝑒−39	9.48𝑒−45	4.52𝑒−41
		|𝑥4−𝛼|	1.01𝑒−166	8.05𝑒−167	1.84𝑒−160	2.05𝑒−153	𝟒.𝟕𝟕𝐞−𝟏𝟕𝟔	7.12𝑒−161
		|𝑥5−𝛼|	0.𝑒−299	0.𝑒−299	0.𝑒−299	0.𝑒−299	0.𝑒−299	0.𝑒−299

  ^†5.37𝑒−7=5.37×10⁻⁷.