J Supercomput (2008) 44: 237–256
DOI 10.1007/s11227-007-0157-x
A multilevel parallel algorithm to solve symmetric
Toeplitz linear systems
Miguel O. Bernabeu · Pedro Alonso ·
Antonio M. Vidal
Published online: 14 December 2007
© Springer Science+Business Media, LLC 2007
Abstract This paper presents a parallel algorithm to solve a structured linear system
with a symmetric-Toeplitz matrix. Our main result concerns the use of a combination
of shared and distributed memory programming tools to obtain a multilevel algo-
rithm that exploits the actual different hierarchical levels of memory and computa-
tional units present in parallel architectures. This gives, as a result, a so-called paral-
lel hybrid algorithm that is able to exploit each of these different configurations. Our
approach has been done not only by means of combining standard implementation
tools like OpenMP and MPI, but performing the appropriate mathematical derivation
that allows this derivation. The experimental results over different representations of
available parallel hardware configurations show the usefulness of our proposal.
Keywords Toeplitz matrix · Cauchy-like matrix · Rank displacement · Multilevel
parallel programming · MPI · OpenMP
1 Introduction
In this paper, we present a parallel algorithm for the solution of the linear system
Tx = b, (1)
where T ∈ R
n×n
is a symmetric Toeplitz matrix T = (t
ij
)
n−1
i,j =0
= (t
|i −j |
)
n−1
i,j =0
and
b,x ∈ R
n
are the independent and the solution vectors, respectively.
M.O. Bernabeu ( ) · P. Alonso · A.M. Vidal
Departament de Sistemes Informàtics i Computació, Universitat Politècnica de València,
46022 Valencia, Spain
e-mail: mbernabeu@dsic.upv.es
P. Alonso
e-mail: palonso@dsic.upv.es
A.M. Vidal
e-mail: avidal@dsic.upv.es