2022/03/21 by Manuel Bogoya, Bogoya, Manuel, Stefano Serra-Cappizano +3
Computer Science · Mathematics · Physics and Astronomy · #Electromagnetic Scattering and Analysis #FOS: Mathematics #Finite Group Theory Research #Matrix Theory and Algorithms #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.2203.11338
openalex publication_date 2022/03/21 · openalex created_date 2022/05/05 · openalex updated_date 2026/07/28
Under appropriate technical assumptions, the simple-loop theory allows to deduce 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 l,g 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 precision till machine precision and the same linear computation cost, when compared with the matrix-less procedures already proposed in the literature.