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

Under appropriate technical assumptions, the simple-loop theory allows to derive various types of asymptotic expansions for the eigenvalues of Toeplitz matrices Tn(f) generated by a function f. Unfortunately, such a theory is not available in the preconditioning setting, that is for matrices of the form Tn−1(g)Tn(ℓ) with g,ℓ real-valued, g nonnnegative and not identically zero almost everywhere. Independently and under the milder hypothesis that f=ℓ/g is even and monotonic over [0,π], 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≡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 accuracy till machine precision and the same linear computational cost, when compared with the matrix-less procedures already proposed in the literature.