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This work performs a comparative numerical study of the impurity averageself-frequency $\tilde{\omega}$ in an unconventional superconducting alloy withnon-magnetic impurities. Two methods are used: the Levenberg-Marquardtalgorithm as a non-linear minimization problem, and a fixed-point iterationprocedure. The unconventional superconducting renormalized by impurities \bw isa self-consistent complex non-linear equation with two varying parameters: theimpurity concentration $\Gamma^+$ and the strength of the impurities c, forwhich its numerical solution is a computational challenge. Throughout thisstudy $\tilde{\omega}$ is the renormalized frequency, \bg represents theinverse of the residual average lifetime $\tau$ at zero frequency, and $N/N_0$is the normalized superconducting density of states (DOS). This study uses anorder parameter that corresponds to the high-temperature superconductingceramics (HTS) with a well-established gap symmetry. The results reveal thecomputational efficiency of the non-linear minimization technique by improvingthe calculations of the $\tilde{\omega}$ computation when using a 2D parameterspace ($\Gamma^+$, c), particularly in the unitary regime, where the imaginarypart of $\tilde{\omega}$ is a complicated expression of those parameters; thisallows to enhance the study of the universal behavior of this particularquantum mechanical state.

A non-linear minimization calculation of the renormalized frequency $\tilde{\omega}$ in dirty d-wave superconductors