New Variants of Newton's Method for Solving Nonlinear Equations

Two Newton-type iterative techniques have been created in this work to locate the true root of univariate nonlinear equations. One of these can be acquired by modifying the double Newton's method in a straightforward manner, while the other can be gotten by modifying the midpoint Newton's method. The iterative approach developed by McDougall and Wortherspoon is employed for the change. The study demonstrates that the modified double Newton's approach outperforms the current one in terms of both convergence order and efficiency index, even though both methods assess the same amount of functions and derivatives every iteration. In comparison to the midpoint Newton's technique, which has a convergence order of 3, the modified midpoint Newton's method has a convergence order of 5.25 and requires two extra functions to be evaluated per iteration. In order to evaluate the effectiveness of recently introduced approaches with current methods, some numerical examples are shown in the final section.