Block Jacobi/Gauss-Seidel preconditioning for GLT sequences, and GLH sequences
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916072479784960 |
|---|---|
| author | Garoni, Carlo Serra-Capizzano, Stefano |
| author_facet | Garoni, Carlo Serra-Capizzano, Stefano |
| contents | 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 $κ$ 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 $κ$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2606_01888 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Block Jacobi/Gauss-Seidel preconditioning for GLT sequences, and GLH sequences Garoni, Carlo Serra-Capizzano, Stefano Numerical Analysis 15B05, 47B35, 65F08, 15A18, 47B06 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 $κ$ 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 $κ$, 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. |
| title | Block Jacobi/Gauss-Seidel preconditioning for GLT sequences, and GLH sequences |
| topic | Numerical Analysis 15B05, 47B35, 65F08, 15A18, 47B06 |
| url | https://arxiv.org/abs/2606.01888 |