Reasons for stability in the construction of derivative-free multistep
iterative methods
Abstract
In this paper, a deep dynamical analysis is made using tools from
multidimensional real discrete dynamics of some derivative-free
iterative methods with memory. They all have good qualitative
properties, but one of them (due to Traub) shows the same behavior as
Newton’s method on quadratic polynomials. Then, the same techniques are
employed to analyze the performance of several multipoint schemes with
memory, whose first step is Traub’s method, but their construction was
made using different procedures. Therefore, their stability is analyzed,
showing which is the best in terms of the wideness of basins of
convergence or the existence of free critical points that would yield
convergence towards different elements from the desired zeros of the
nonlinear function. Therefore, the best stability properties are linked
with the best estimations made in the iterative expressions rather than
their simplicity. These results have been checked with a numerical and
graphical comparison with many other known methods with and without
memory, with different orders of convergence, with excellent
performance.