Block Jacobi/Gauss-Seidel preconditioning for GLT sequences, and GLH sequences

arXiv:2606.01888v1 Announce Type: new Abstract: The theory of generalized locally Toeplitz (GLT) sequences is an apparatus for computing the spectral and singular value distribution of sequences of matrices that possess a (possibly hidden) Toeplitz-like structure. These sequences, which are known as GLT sequences, arise in several applications, including the discretization of differential equations. Associated with any GLT sequence is a special function called symbol. In this paper, we prove that, if $\{A_n\}_n$ is a GLT sequence with symbol $\kappa$ and $P_n$ is any block Jacobi or block Gauss-Seidel preconditioner for $A_n$ with a fixed number of blocks independent of $n$, then $\{P_n\}_n$ is a GLT sequence with symbol $\kappa$, just like $\{A_n\}_n$. This result allows us to predict a remarkable efficiency of block Jacobi/Gauss-Seidel preconditioning for GLT sequences, which is in fact illustrated through numerical experiments. It also allows us to extend the Fasino-Tilli theorem on the zero distribution of Hankel matrix sequences generated by $L^1$ functions to a larger class of matrix sequences called generalized locally Hankel (GLH) sequences.