  This work performs a comparative numerical study of the impurity average
self-frequency $\tilde{\omega}$ in an unconventional superconducting alloy with
non-magnetic impurities. Two methods are used: the Levenberg-Marquardt
algorithm as a non-linear minimization problem, and a fixed-point iteration
procedure. The unconventional superconducting renormalized by impurities \bw is
a self-consistent complex non-linear equation with two varying parameters: the
impurity concentration $\Gamma^+$ and the strength of the impurities c, for
which its numerical solution is a computational challenge. Throughout this
study $\tilde{\omega}$ is the renormalized frequency, \bg represents the
inverse of the residual average lifetime $\tau$ at zero frequency, and $N/N_0$
is the normalized superconducting density of states (DOS). This study uses an
order parameter that corresponds to the high-temperature superconducting
ceramics (HTS) with a well-established gap symmetry. The results reveal the
computational efficiency of the non-linear minimization technique by improving
the calculations of the $\tilde{\omega}$ computation when using a 2D parameter
space ($\Gamma^+$, c), particularly in the unitary regime, where the imaginary
part of $\tilde{\omega}$ is a complicated expression of those parameters; this
allows to enhance the study of the universal behavior of this particular
quantum mechanical state.
