Fast solver for band Toeplitz-block-band Toeplitz system

Houssam Khalil (1, 2)

1 MAPLY - Laboratoire de Mathématiques Appliquées de Lyon
2 ICJ - Institut Camille Jordan

Let $T$ be a Toeplitz block Toeplitz matrix with coefficients in a field $K$ of size $N$. Assume that the blocks are of size $n\times n$ and that there are $n\times n$ blocks. Assume moreover that the matrix is block banded and that the blocks themselves have a block structure. This means that outside of the $2k_1+1$ central block diagonals, the blocks vanish, and that outside of the central $2k_2+1$ central diagonals, the coefficients in each block vanish. In this article, we give three direct methods of resolution with a count of $O(N^{3/2})$ operations to resolve the problem $Tx=b$. We give also a statistical study which prouve that the matrix $T$ becomes increasingly ill conditioned when the band widths decrease.

Domaines: Analyse numérique [math.NA]

Houssam Khalil. Fast solver for band Toeplitz-block-band Toeplitz system. 2008. ⟨hal-00366297v2⟩