An Efficient and Straightforward Numerical Technique Coupled to Classical Newton’s Method for Enhancing the Accuracy of Approximate Solutions Associated with Scalar Nonlinear Equations

Table 4

Approximate solution of scalar nonlinear equation (when guest point = −4) obtained by Classical Newton’s Method (CNM) (cf. Section 2.2), Classical Newton’s Method coupled with New Numerical Technique (CNM + NNT) for conditions [A1] and [A2] (cf. Section 2.3), and Third-order Modified Newton’s Method (TMNM) (cf. Section 3.1). Notations: () (resp., ()) denotes that approximate solution is provided using (21a) (resp., (21b)) in Section 2.3.1.

Iteration ()	CNM	CNM + NNT with condition [A1]	CNM + NNT with condition [A2]	TMNM
0	−4.0000000000	−4.0000000000	−4.0000000000	−4.0000000000
1	−2.5987493208	−1.1976336149 ()	−1.1976336149 ()	−2.1088093909
2	−1.5803235380	−0.2513579056 ()	0.6858578709 ()	−0.7930763263
3	−0.6950023819	1.1303884977 ()	0.6226379112 ()	1.4022343921
4	0.5489290102	0.7665675348 ()	0.6192541648 ()	0.7698587570
5	0.6234126286	0.6349082108 ()	0.6192449540 ()	0.6217981372
6	0.6192588436	0.6194395336 ()	0.6192449540 ()	0.6192449694
7	0.6192449541	0.6192449843 ()		0.6192449540
8	0.6192449540	0.6192449540 ()		0.6192449540
9
10