The Symmetric–Toeplitz Linear System Problem in Parallel

Abstract

Many algorithms exist that exploit the special structure of Toeplitz matrices for solving linear systems. Nevertheless, these algorithms are difficult to parallelize due to its lower computational cost and the great dependency of the operations involved that produces a great communication cost. The foundation of the parallel algorithm presented in this paper consists of transforming the Toeplitz matrix into a another structured matrix called Cauchy-like. The particular properties of Cauchy-like matrices are exploited in order to obtain two levels of parallelism that makes possible to highly reduce the execution time. The experimental results were obtained in a cluster of PC's.

A Parallel Algorithm for the Solution of the Deconvolution Problem on Heterogeneous Networks

“…The use of Cauchy-like matrices to design efficient parallel algorithms for the solution of standard numerical linear algebra problems such as the linear systems solution or the minimisation of the least squares problem with structured matrices has been successfully applied in the complex, hermitian, real and symmetric real cases [3,4,7,1,5].…”

“…A more detailed description of triangularization of symmetric Cauchy-like matrices can be found, i.e. [7].…”

A multilevel parallel algorithm to solve symmetric Toeplitz linear systems

“…Recently, Thirumalai [27] proposed a parallel algorithm to solve symmetric Toeplitz linear systems but only for two processors. This last work was successful followed in depth to develop efficient parallel algorithms for different kinds of Toeplitz matrices [28][29][30] even with refinement techniques to improve the accuracy of the solution [31].…”

A Parallel Solution of Hermitian Toeplitz Linear Systems,

“…The more general non-hermitian case can be found in [3]. Other related works based on the same idea applied to the real symmetric case was presented in [4].…”