Some Class of Third- and Fourth-Order Iterative Methods for Solving Nonlinear Equations

Table 5

Comparison of the norm of the error occurring in the estimates of the exact root with some existing third-order methods.

Example	Method	‖𝑋1^(𝑘)−𝛼‖	‖𝑋2^(𝑘)−𝛼‖	‖𝑋3^(𝑘)−𝛼‖
Example 1	NR1	3.0274𝑒−5	5.4762𝑒−16	4.3730𝑒−48
	NR2	3.0274𝑒−5	5.4762𝑒−16	4.3730𝑒−48
	NT	5.9292𝑒−5	8.2124𝑒−15	2.9488𝑒−44
	ON	3.0274𝑒−5	5.4762𝑒−16	4.3730𝑒−48
	HZ	3.0274𝑒−5	5.4762𝑒−16	4.3730𝑒−48
	NS	3.0274𝑒−5	5.4762𝑒−16	4.3730𝑒−48
	M3 I	2.6220𝑒−5	2.6977𝑒−16	4.2171𝑒−49
Example 2	NR1	1.7406𝑒−1	7.1470𝑒−4	4.5647𝑒−11
	NR2	1.6363𝑒−1	5.7582𝑒−4	2.3870𝑒−11
	NT	2.4632𝑒−1	3.8429𝑒−3	1.4191𝑒−8
	ON	1.7034𝑒−1	6.5958𝑒−4	3.5877𝑒−11
	HZ	1.7034𝑒−1	6.5958𝑒−4	3.5877𝑒−11
	NS	1.6817𝑒−1	6.3444𝑒−4	3.1928𝑒−11
	M3 I	1.0152𝑒−1	7.0395𝑒−5	7.8189𝑒−14
Example 3	NR1	1.3267𝑒−3	4.0708𝑒−10	2.9007𝑒−30
	NR2	1.3267𝑒−3	4.0708𝑒−10	2.9007𝑒−30
	NT	3.0573𝑒−4	9.6593𝑒−9	7.2203𝑒−26
	ON	1.3267𝑒−3	4.0708𝑒−10	2.9007𝑒−30
	HZ	1.3267𝑒−3	4.0708𝑒−10	2.9007𝑒−30
	NS	1.3267𝑒−3	4.0708𝑒−10	2.9007𝑒−30
	M3 I	1.3282𝑒−3	4.0708𝑒−10	2.9007𝑒−30
Example 4	NR1	4.9646𝑒−2	2.0377𝑒−4	1.9717𝑒−11
	NR2	4.9260𝑒−2	1.9653𝑒−4	1.7731𝑒−11
	NT	6.8356𝑒−2	9.2740𝑒−4	3.6811𝑒−9
	ON	4.9437𝑒−2	1.9989𝑒−4	1.8634𝑒−11
	HZ	4.9453𝑒−2	1.9989𝑒−4	1.8634𝑒−11
	NS	4.9460𝑒−2	2.0022𝑒−4	1.8727𝑒−11
	M3 I	1.1965𝑒−1	5.9892𝑒−4	2.1213𝑒−10
Example 5	NR1	4.6620𝑒−5	8.9529𝑒−14	6.1663𝑒−40
	NR2	4.6601𝑒−5	8.9516𝑒−14	6.1632𝑒−40
	NT	9.7286𝑒−5	1.6394𝑒−12	7.5686𝑒−36
	ON	4.6611𝑒−5	8.9521𝑒−14	6.1645𝑒−40
	HZ	4.6611𝑒−5	8.9523𝑒−14	6.1648𝑒−40
	NS	4.6611𝑒−5	6.9523𝑒−14	6.1648𝑒−40
	M3 I	2.2192𝑒−5	1.3119𝑒−14	1.6090𝑒−42
Example 6	NR1	1.4068𝑒−3	2.3999𝑒−10	1.3581𝑒−30
	NR2	1.4068𝑒−3	2.3999𝑒−10	1.3581𝑒−30
	NT	3.1325𝑒−3	5.4369𝑒−9	3.1678𝑒−26
	ON	1.4068𝑒−3	2.3999𝑒−10	1.3581𝑒−30
	HZ	1.4068𝑒−3	2.3999𝑒−10	1.3581𝑒−30
	NS	1.4068𝑒−3	2.3999𝑒−10	1.3581𝑒−30
	M3 I	1.4391𝑒−3	2.6482𝑒−10	1.8147𝑒−30