Research Article

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 3

Approximate solution of scalar nonlinear equation (when guest point = 10) 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 ()CNMCNM + NNT with condition [A1]CNM + NNT with condition [A2]TMNM

010.00000000010.00000000010.00000000010.000000000
18.98117665357.9623533071 ()7.9623533071 ()8.4655416573
27.94567407855.8372149905 ()5.8372149905 ()6.8799380466
36.88256721283.5600762341 ()3.5600762341 ()5.2048513181
45.78038830411.4821734803 ()1.4821734803 ()3.4865001565
54.64044938060.9609028311 ()0.4402289741 ()2.0197617679
63.50560701820.6928062817 ()0.6483081663 ()1.0685736902
72.47442595730.6233620739 ()0.6199084714 ()0.6632122684
81.64610673250.6192585092 ()0.6192453069 ()0.6193191836
91.06354496020.6192449541 ()0.6192449540 ()0.6192449540
100.73544060020.6192449510 ()0.6192449540