Some new variants of Newton's method

Multiplicative Newton s Methods with Cubic Convergence

“…Now, as in [7], if we consider this approximation as the arithmetic mean of ln f * (x n ) and ln f * (z n+1 ) and take the harmonic mean instead, we get the harmonic mean Multiplicative Newton (HMN) Method as…”

“…These iterative methods are obtained using different approaches to definite integral in Newton theorem. For instance, arithmetic mean Newton method [6], harmonic mean Newton method [7] and geometric mean Newton method [8] by the help of trapezoidal rule, midpoint Newton method [7] and iterative method in [9] via midpoint rule, iterative methods in [10] and [11] using Simpson rule and different quadrature formulae are obtained. Besides, the method which based on usage of the Newton method for the inverse function has cubically convergence and is developed in [12].…”

A Variant of Sharma-Arora's Optimal Eighth-Order Family of Methods for Finding A Simple Root of Nonlinear Equation

“…In literature one can find many papers presenting multipoint iterative methods that contain Newton's step (1) (or some of its modifications) as the first step, with greater efficiency indices (see for example [2], [3], [4], [5], [6], [7], [8]). In [9] Kung and Traub conjectured that the iterative method which requires n + 1 function evaluations per iteration can reach at most 2 n convergence order in general.…”

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

“…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].…”

“…It should be pointed out that this method has also been derived in [5] independently and it is now known as harmonic mean Newton method. It should also be noted that many of the known iterative methods developed in recent years including the third-order methods given above can be regarded as rediscovered methods, see [18] for more details.…”