Abstract
A general optimal iterative method, for approximating the solution of nonlinear equations, of (n+1) steps with 2n+1 order of convergence is presented. Cases n=0 and n=1 correspond to Newton’s and Ostrowski’s schemes, respectively. The basins of attraction of the proposed schemes on different test functions are analyzed and compared with the corresponding to other known methods. The dynamical planes showing the different symmetries of the basins of attraction of new and known methods are presented. The performance of different methods on some test functions is shown.
Highlights
We note that most of the planes are similar, except those associated with methods M4 and M8, since they have a basin of attraction for the 0 point much larger than in the rest of the cases, being a little greater that of method M4; for this reason, it is more convenient in this case to take12one of of 17 these two methods since we have a greater number of initial estimations that converge to the solution that we are looking for
We solve some nonlinear equations to compare our proposed methods convenient are the M4 and M8 methods, since they have a larger basin of attraction in the with other schemes of the same order of convergence and to confirm the information cases we are interested in, they do not generate other basins of attraction from strange fixed obtained in the previous section
If we look at the eighth-order methods, we can observe that the M8 method performs fewer iterations and has the shortest computational time, what stands out most in this table is the value of the approximate computational convergence order (ACOC) of the M8 method, which in this case increases to 11 instead of 8, it is true that the J8 method increases to 9 and has the smallest value of the norm of the equation evaluated in the last iteration, by performing one more iteration than the M8 method, it is logical for this to happen
Summary
This behavior is characterized with efficiency and stability criteria provided by the tools of complex dynamics. The efficiency of these schemes has been widely considered, usually based on the efficiency index presented by Ostrowski [3] The expression of this index, I = p1/d , involves the order of convergence of the method p and the number of functional evaluations in each iteration, d. Many optimal schemes have been designed in the last years with different order of convergence (see, e.g., [5,6,7,8,9,10] and the references therein). With some conclusions and the references used in this manuscript, we finish the paper
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