A smoothing spline algorithm to interpolate and predict the eigenvalues of matrices extracted from the sequence of preconditioned banded symmetric Toeplitz matrices

Understanding the eigenvalue distribution of sequence Toeplitz matrices has advanced significantly in recent years. Notable contributors include Bogoya, Grudsky, Böttcher, and Maximenko, who have derived precise asymptotic expansions for these eigenvalues under certain conditions related to the generating function as the matrix size increases. Building on this foundation, the Stefano Serra-Capizzano conjectured that, under certain assumptions about $\mathrm{\Omega }$ and $\mathrm{\Phi }$, a similar expansion may hold for the eigenvalues of a sequence of preconditioned Toeplitz matrices $T_n^{-1}(\mathrm{\Phi })T_n(\mathrm{\Omega })$, given a monotonic ratio $r=\mathrm{\Omega }/\mathrm{\Phi }$. In contrast to current eigenvalue solvers, this work presents a novel method for efficiently calculating the eigenvalues of a sequence of large preconditioned banded symmetric Toeplitz matrices (PBST). Our algorithm uses a higher-order spline fitting extrapolation technique to gather spectral data from a smaller sequence of PBST matrices in order to forecast the spectrum of bigger matrices.