A multilevel parallel algorithm to solve symmetric Toeplitz linear systems

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 algorithm that exploits the actual different hierarchical levels of memory and computational units present in parallel architectures. This gives, as a result, a so-called parallel 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.

Block pivoting implementation of a symmetric Toeplitz solver

“…The algorithm appeared in [9] also used both levels of parallelism, by partitioning the work into two MPI processes, each one in charge of factorizing one of the two rank-2 Cauchylike matrices (C 1 and C 2 , respectively) resulting from the splitting process shown in (3). Each one of these two matrices were factorized in turn by an OpenMP parallel loop.…”

“…Vector λ has analytically known entries that just depend on the size of the problem. Details of all of these operations can be found in [9]. We used a Fortran 90 module to apply the DST [2] which, based on the problem size, automatically chooses the best routine between dfftpack [32] and Intel MKL [27].…”

“…A more detailed description of the backgrounds of the computations of all of these operations can be found in [9].…”

“…For example, this was used in [33] to propose a shared memory algorithm. It was also used in [9] leading to a multilevel parallel MPI-OpenMP algorithm. In that paper, the Toeplitz matrix was transformed into a Cauchy-like matrix and split into two independent matrices, each one assigned to an MPI process which could be mapped onto different nodes in a network, or onto the same node thus fixing the problem of the lack of compilers that support OpenMP nested parallelism.…”