On Newton-type methods with cubic convergence

Certain improvements of Newton’s method with fourth-order convergence

“…The method (34) requires one evaluation of the function and two of its first derivative per iteration, so it has the same efficiency as the third-order methods given in Weerakoon-Fernando [2], Frontini-Sormani [3], Homeier [4] and others in the literature. Note that not all third-order methods in the lietrature are as efficient.…”

“…Traub [1]) that this method is quadratically convergent. Some modifications of Newton's method to achieve higher order and better efficiency have been suggested and analyzed using several different techniques such as quadrature rules [2][3][4][5][6][7][8][9][10][11][12][13], decomposition [14,15] and homotopy techniques [16,17].…”

“…where y n is defined by (4). This method is widely used and extended in more general setting for applications.…”

New Family of Iterative Methods for Solving Non-Linear Equations Using New Adomian Polynomials

“…We compare the Newton method (NM), the Double Newton method (DNM), the method of Noor (NR) [12] and the method (AP) , introduced in the Algorithm 3 (see Table 1 ). Now we compare the Newton method (NM), the Double Newton method (DNM), the method of Weerakoon and Fernando [14], the method of Frontini and Sormani [7], the method of Homeier [9], the method of Noor (NR) [12] and the method (A), introduced in the Algorithm 5 (see Table 2). …”

“…In recent years, much attention has been given to develop several iterative methods for solving nonlinear equations (see for example [1,[6][7][8][9][10][11][12]14]). These methods can be classified as one-step and two-step methods.…”