Fast Toeplitz eigenvalue computations, joining interpolation-extrapolation matrix-less algorithms and simple-loop conjectures: the preconditioned setting

Under appropriate technical assumptions, the simple-loop theory allows to deduce various types of asymptotic expansions for the eigenvalues of Toeplitz matrices $T_{n}(f)$ generated by a function $f$, unfortunately, such a theory is not available in the preconditioning setting, that is for matrices of the form $T_{n}^{-1}(g)T_{n}(l)$ with $l,g$ real-valued, $g$ nonnnegative and not identically zero almost everywhere. Independently and under the milder hypothesis that $f=\frac{l}{g}$ is even and monotonic over $[0,\pi]$, matrix-less algorithms have been developed for the fast eigenvalue computation of large preconditioned matrices of the type above, within a linear complexity in the matrix order: behind the high efficiency of such algorithms there are the expansions as in the case $g\equiv 1$, combined with the extrapolation idea, and hence we conjecture that the simple-loop theory has to be extended in such a new setting, as the numerics strongly suggest.Here we focus our attention on a change of variable, followed by the asymptotic expansion of the new variable, and we consider new matrix-less algorithms ad hoc for the current case. Numerical experiments show a much higher precision till machine precision and the same linear computation cost, when compared with the matrix-less procedures already proposed in the literature.