  The present work is devoted to the eigenvalue asymptotic expansion of the
Toeplitz matrix $T_{n}(a)$ whose generating function $a$ is complex valued and
has a power singularity at one point. As a consequence, $T_{n}(a)$ is
non-Hermitian and we know that the eigenvalue computation is a non-trivial task
in the non-Hermitian setting for large sizes. We follow the work of Bogoya,
B\"ottcher, Grudsky, and Maximenko and deduce a complete asymptotic expansion
for the eigenvalues. After that, we apply matrix-less algorithms, in the spirit
of the work by Ekstr\"om, Furci, Garoni, Serra-Capizzano et al, for computing
those eigenvalues. Since the inner and extreme eigenvalues have different
asymptotic behaviors, we worked on them independently, and combined the results
to produce a high precision global numerical and matrix-less algorithm.
  The numerical results are very precise and the computational cost of the
proposed algorithms is independent of the size of the considered matrices for
each eigenvalue, which implies a linear cost when all the spectrum is computed.
From the viewpoint of real world applications, we emphasize that the matrix
class under consideration includes the matrices stemming from the numerical
approximation of fractional diffusion equations. In the final conclusion
section a concise discussion on the matter and few open problems are presented.
