Several New Third-Order and Fourth-Order Iterative Methods for Solving Nonlinear Equations

Table 4

Norm of the functions by different methods after first, second, third, and fourth iteration.

Example	Guess	Method	||𝐹(𝑥^(1))||	||𝐹(𝑥^(2))||	||𝐹(𝑥^(3))||	||𝐹(𝑥^(4))||
Example 9	(5.1, 6.1)	NR1	3.8774𝑒−4	9.2700𝑒−15	7.7652𝑒−47	4.0858𝑒−143
		NR2	3.8774𝑒−4	9.2700𝑒−15	7.7652𝑒−47	4.0858𝑒−143
		MM3	1.2657𝑒−4	1.0705𝑒−16	3.9789𝑒−53	1.8320𝑒−162
		BB4	2.1416𝑒−5	1.1267𝑒−24	4.6477𝑒−102	1.2561𝑒−411
		SH4	1.2923𝑒−5	9.2420𝑒−26	1.2710𝑒−106	4.2184𝑒−430
		MM4	3.0768𝑒−6	3.9419𝑒−29	8.7758𝑒−121	2.1039𝑒−487
Example 10	(1, 0.5, 1.5)	NR1	3.0006𝑒−2	1.3681𝑒−4	1.3174𝑒−11	1.1754𝑒−32
		NR2	2.9765𝑒−2	1.3230𝑒−4	1.1848𝑒−11	8.5484𝑒−33
		MM3	9.9051𝑒−3	3.7473𝑒−6	9.1835𝑒−17	1.3035𝑒−48
		BB4	2.1133𝑒−2	6.9602𝑒−6	7.1401𝑒−20	7.7987𝑒−76
		SH4	1.5676𝑒−2	1.1309𝑒−6	2.4814𝑒−23	5.5195𝑒−90
		MM4	6.1451𝑒−3	8.1169𝑒−8	2.3264𝑒−28	6.9642𝑒−110
Example 11	(0.5, 0.5, 0.5, −0.2)	NR1	2.3097𝑒−3	7.5761𝑒−10	5.6516𝑒−30	2.8662𝑒−91
		NR2	2.3097𝑒−3	7.5761𝑒−10	5.6516𝑒−30	2.8662𝑒−91
		MM3	9.4336𝑒−4	1.6380𝑒−11	1.7401𝑒−35	2.5268𝑒−108
		BB4	9.1400𝑒−4	1.8627𝑒−14	8.7599𝑒−59	6.9713𝑒−238
		SH4	5.3618𝑒−4	1.4537𝑒−15	2.1746𝑒−63	1.7744𝑒−256
		MM4	7.7084𝑒−5	2.1932𝑒−20	3.4979𝑒−84	3.6487𝑒−341
Example 12	(1.0, 2.0)	NR1	2.1427𝑒−3	1.7987𝑒−10	1.0504𝑒−31	2.0958𝑒−95
		NR2	2.1498𝑒−3	1.8262𝑒−10	1.1001𝑒−31	2.4077𝑒−95
		MM3	7.6174𝑒−4	2.3592𝑒−12	7.9632𝑒−38	3.0435𝑒−114
		BB4	5.3124𝑒−4	8.2104𝑒−16	3.8411𝑒−63	2.8216𝑒−252
		SH4	2.9895𝑒−4	6.5567𝑒−17	1.8332𝑒−67	1.5913𝑒−269
		MM4	1.0131𝑒−4	2.8562𝑒−19	3.4019𝑒−77	2.4842𝑒−308
Example 13	(−0.8, 1.1, 1.1)	NR1	2.9692𝑒−4	5.5149𝑒−14	3.8063𝑒−40	1.2456𝑒−118
		NR2	2.9718𝑒−5	5.5137𝑒−14	3.8044𝑒−40	1.2438𝑒−118
		MM3	9.8775𝑒−6	6.7719𝑒−16	2.3491𝑒−46	9.7596𝑒−138
		BB4	4.0723𝑒−6	1.6873𝑒−21	9.1974𝑒−83	7.8287𝑒−328
		SH4	2.1907𝑒−6	8.6294𝑒−23	3.6734𝑒−87	1.1711𝑒−349
		MM4	1.0838𝑒−6	8.4404𝑒−25	1.2327𝑒−96	4.3437𝑒−384
Example 14	(0.5, 1.5)	NR1	1.8661𝑒−1	7.1492𝑒−4	4.5647𝑒−11	1.1889𝑒−32
		NR2	1.7417𝑒−1	5.7596𝑒−4	2.3870𝑒−11	1.7000𝑒−33
		MM3	1.2770𝑒−1	4.6794𝑒−5	4.1708𝑒−15	3.0222𝑒−45
		BB4	9.8299𝑒−2	9.2624𝑒−6	8.3046𝑒−22	5.3716𝑒−86
		SH4	1.0359𝑒−1	5.4166𝑒−6	4.6302𝑒−23	2.4821𝑒−91
		MM4	1.4490𝑒−1	1.0558𝑒−5	6.7122𝑒−22	1.0964𝑒−86