TY - JOUR
T1 - Band-Times-Circulant Preconditioners for Non-Symmetric Real Toeplitz Systems with Unknown Generating Function
AU - Chaysri , Thaniporn
AU - Hadjidimos , Apostolos
AU - Noutsos , Dimitrios
AU - Tachyridis , Grigorios
JO - East Asian Journal on Applied Mathematics
VL - 2
SP - 285
EP - 322
PY - 2022
DA - 2022/02
SN - 12
DO - http://doi.org/10.4208/eajam.230721.251121
UR - https://global-sci.org/intro/article_detail/eajam/20255.html
KW - Non-symmetric, Toeplitz, band-times-circulant, preconditioner.
AB - <p style="text-align: justify;">In this paper we study the preconditioning of $n×n$ non-symmetric, real
Toeplitz systems, when the generating function of the coefficient matrix $T_n$ is not known
a priori, but we know that a generating function $f$ exists related to the matrix sequence $\{T_n\}$, $T_n = T_n
(f)$, with $f$ smooth enough. The proposed preconditioner is derived as
a combination of a band Toeplitz and a circulant matrix. We give details for the construction of the proposed preconditioner, by the entries of $T_n$ and we study the cluster
of the eigenvalues, as well as of the singular values, of the sequences of the coefficient
matrices related to the preconditioned systems. Theoretical results prove the efficiency
of the Preconditioned Generalized Minimal Residual method (PGMRES) and the Preconditioned Conjugate Gradient method of Normal Equations (PCGN). Such efficiency
is also shown in demonstrating numerical examples, using the proposed preconditioning
technique.</p>