Generating function method for constructing new iterations

On the development and extensions of some classes of optimal three-point iterations for solving nonlinear equations

“…It is worth to note that similar results for derivative-free case and for some choices of τn were obtained by Thukral in [18] and by Khattri et al in [9]. We also note that the iteration (1), (21) for β = 4 was considered by Sharma et al in [15]. (6) Let…”

“…Table 3 gives some numerical results in order to show convergence behaviour of method (1) with α n parameter given by ( 4), ( 17)- (22). We observe from Table 3 that the methods (1) with parameters τn given by case iv and α n given by ( 4), (21) produce approximations of higher accuracy compared to the eight-order methods SAWN8, ZO8.…”

“…The optimal methods (1) distinguish each other only by choices of parameters τn and α n . It should be pointed out that to establish the convergence order of iterative methods often used either the error equation, see for example [1-5, 8-15, 17-19], or the nonlinear residuals [20][21][22][23]. An more detailed explanation of various aspects of convergence order based on error analysis, corrections and nonlinear residuals was given in the excellent surveys [6,7].…”

Advanced Iterative Procedures for Solving the Implicit Colebrook Equation for Fluid Flow Friction

“…Iterative solutions and approximations for calculation of flow friction factor are implemented in software packages which are in common use in everyday engineering practice {Brkić 2016b}. So in this paper we analyzed selected iterative procedures in order to solve the Colebrook equation {Chun and Neta 2017, Zhanlav et al 2017} and we found that two or three iterations of Halley and the Schröder methods are suitable for the required accuracy needed for the engineering practice, when the fixed initial starting point of Section 2.2.2 is applied. On the other hand, using three-point iterative method with the same initial conditions, the required high accuracy can be reached after only one iteration but using three internal steps {Džunić et al , Petković et al 2014 Moreover, to simplified calculation for engineering use we can recommend:…”

Advanced Iterative Procedures for Solving the Implicit Colebrook Equation for Fluid Flow Friction

“…Iterative solutions and approximations for the calculation of the flow friction factor are implemented in software packages which are in common use in everyday engineering practice [88]. So in this paper, we analyzed selected iterative procedures in order to solve the Colebrook equation [93,94], and we found that up 2 to 3 iterations of the Halley and the Schröder method are suitable for the accuracy required by engineering practice, when the fixed initial starting point described in Section 2.2.2 is applied. On the other hand, using a threepoint iterative method with the same initial conditions, the required high accuracy can be reached after only 1 iteration (2 in the worst case) but using three internal steps [26][27][28].…”