A variant of Newton's method with accelerated third-order convergence

By S. Weerakoon, T. Fernando. Applied Mathematics Letters, 2000

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On Newton-type methods with cubic convergence (2005, methodology): “Then, Weerakoon and Fernando [8] rederived the Newton method from the Newton theorem f (x)= f (xn)+ ∫ x”
Some variants of Ostrowski's method with seventh-order convergence (2007, methodology): “We use the following functions, many of which are the same as in [3,7], respectively: f1(x)= x3 + 4x2 − 15, x∗ = 1.”
Review of some iterative root-finding methods from a dynamical point of view (2004, methodology): “These three third{order root{flnding iterative methods are studied in [25], [41], [45], [ 59 ].”
New iterations of R-order four with reduced computational cost (2009, background): “over, if we consider the computational order of convergence ρ (see [ 14 ]), we obtain ρ ≈ 3.7908 ... , which is even more than 10/3, the R-order of convergence.”
Modified Newton's method for systems of nonlinear equations with singular Jacobian (2009, neutral): “(See [7]).”