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 5

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
19.01042676078.0208535216 ()8.0208535216 ()8.5171506533
28.01468591305.9615544966 ()5.9615544966 ()6.9859301990
36.98475037203.9493415483 ()3.9493415483 ()5.3851997390
45.91891104352.4996523082 ()2.4996523082 ()4.0778432768
54.91808767081.7226271036 ()0.9467658294 ()3.0067861343
64.07388753111.3881795157 ()1.4273487664 ()1.8098899052
73.34045880951.3598140013 ()1.3607723321 ()1.3648044141
82.60913631411.3595986336 ()1.3595989925 ()1.3595986468
91.81944056791.3595986211 ()1.3595986211 ()1.3595986211
101.4045600984