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In a series of recent papers the spectral behavior of the matrix sequence$\{Y_nT_n(f)\}$ is studied in the sense of the spectral distribution, where$Y_n$ is the main antidiagonal (or flip matrix) and $T_n(f)$ is the Toeplitzmatrix generated by the function $f$, with $f$ being Lebesgue integrable andwith real Fourier coefficients. This kind of study is also motivated bycomputational purposes for the solution of the related large linear systemsusing the (preconditioned) MINRES algorithm. Here we complement the spectralstudy with more results holding both asymptotically and for a fixed dimension$n$, and with regard to eigenvalues, singular values, and eigenvectors of$T_n(f), Y_nT_n(f)$ and to several relationships among them: beside fast linearsolvers, a further target is the design of ad hoc procedures for thecomputation of the related spectra via matrix-less algorithms, with a costbeing linear in the number of computed eigenvalues. We emphasize that thechallenge of the case of non-monotone generating functions is considered in thecurrent work, for which the previous matrix-less algorithms fail. Numericalexperiments are reported and commented, with the aim of showing in a visual waythe theoretical analysis.

Theoretical results for eigenvalues, singular values, and eigenvectors of (flipped) Toeplitz matrices and related computational proposals