Manuel Bogoya | Searcharxiv

arXiv2026

Condition numbers of block Toeplitz matrices and stability of space-time IgA approximations for the wave and Schr\"odinger equations

In previous work by several authors, the behavior of the condition numbers of banded Toeplitz matrices was studied as the matrix size tends to infinity. In the present contribution, two main directions are pursued. As a first step, we extend this study to block Toeplitz matrices with blocks of fixed size $N$. As in the scalar case, we show that even when the symbol generates a Fredholm infinite Toeplitz operator, the condition numbers of the finite matrices may grow at least exponentially. Upper and lower bounds for the condition numbers are obtained, and examples showing that they may grow arbitrarily fast are presented. Then, as a second step, we apply the developed theory to the stability analysis of space-time Galerkin methods, where in time an Isogeometric approach is used with regularity $r$, $1\le r\le p-1$, $p$ being the employed polynomial degree. These stability issues are related exactly to the conditioning of block Toeplitz-like matrices with blocks of size $N=p-r$. Specific examples are treated in detail and related numerical experiments are presented and critically discussed. We finally present a short list of relevant open problems.

math.NA↗

arXiv2022

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

Under appropriate technical assumptions, the simple-loop theory allows to deduce various types of asymptotic expansions for the eigenvalues of Toeplitz matrices $T_{n}(f)$ generated by a function $f$, unfortunately, such a theory is not available in the preconditioning setting, that is for matrices of the form $T_{n}^{-1}(g)T_{n}(l)$ with $l,g$ real-valued, $g$ nonnnegative and not identically zero almost everywhere. Independently and under the milder hypothesis that $f=\frac{l}{g}$ 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 $g\equiv 1$, 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.

math.NA↗