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 2

Approximate solution of scalar nonlinear equation (when guest point = 10⁻³) 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	0.0010000000	0.0010000000	0.0010000000	0.0010000000
1	0.0082410886	0.0154821756 ()	0.0154821756 ()	0.0344577770
2	0.0505339927	0.1548613133 ()	0.1548613133 ()	0.3980194000
3	0.2182202963	0.4938846484 ()	0.8311197179 ()	0.9031891893
4	0.6163033790	0.8261543800 ()	0.7760693275 ()	0.7755098327
5	0.7975366151	0.7754244611 ()	0.7720442199 ()	0.7720250858
6	0.7727863680	0.7720385875 ()	0.7720250260 ()	0.7720250256
7	0.7720257060	0.7720250258 ()	0.7720250256 ()
8	0.7720250256	0.7720250256 ()
9
10