Fast Toeplitz eigenvalue computations, joining interpolation-extrapolation matrix-less algorithms and simple-loop theory: The preconditioned setting - Bogoya, M.; Serra-Capizzano, S.; Vassalos, P. (Applied mathematics and computation, 2024)
Under appropriate technical assumptions, the simple-loop theory allows to derive various types of asymptotic expansions for the eigenvalues of Toeplitz matrices generated by a function f. Unfortunately, such a theory is not available in the preconditioning setting, that is for matrices of the form  with  real-valued, g nonnnegative and not identically zero almost everywhere. Independently and under the milder hypothesis that  is even and monotonic over , 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 , 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 accuracy till machine precision and the same linear computational cost, when compared with the matrix-less procedures already proposed in the literature.