  Under appropriate technical assumptions, the simple-loop theory allows to
deduce various types of asymptotic expansions for the eigenvalues of Toeplitz
matrices generated by a function $f$. Independently and under the milder
hypothesis that $f$ is even and monotonic over $[0,\pi]$, matrix-less
algorithms have been developed for the fast eigenvalue computation of large
Toeplitz matrices, within a linear complexity in the matrix order: behind the
high efficiency of such algorithms there are the expansions predicted by the
simple-loop theory, combined with the extrapolation idea.
  Here we focus our attention on a change of variable, followed by the
asymptotic expansion of the new variable, and we adapt the matrix-less
algorithm to the considered new setting.
  Numerical experiments show a higher precision (till machine precision) and
the same linear computation cost, when compared with the matrix-less procedures
already presented in the relevant literature. Among the advantages, we
concisely mention the following: a) when the coefficients of the simple-loop
function are analytically known, the algorithm computes them perfectly; b)
while the proposed algorithm is better or at worst comparable to the previous
ones for computing the inner eigenvalues, it is extremely better for the
computation of the extreme eigenvalues.
