Application of the Parallel Dichotomy Algorithm for solving Toeplitz tridiagonal systems of linear equations with one right-hand side

TL;DR

This paper introduces a parallel version of the Dichotomy Algorithm tailored for Toeplitz tridiagonal systems, significantly reducing preliminary calculations and enabling efficient solutions for multiple systems with nearly linear speedup.

Contribution

A modified parallel Dichotomy Algorithm exploiting Toeplitz structure, improving efficiency for solving multiple tridiagonal systems in parallel.

Findings

Comparable accuracy to sequential Thomas method

Nearly linear speedup with increasing processors

Effective for series and single system solutions

Abstract

Basing on a modification of the "Dichotomy Algorithm" (Terekhov, 2010), we propose a parallel procedure for solving tridiagonal systems of equations with Toeplitz matrices. Taking the structure of the Toeplitz matrices, we may substantially reduce the number of the "preliminary calculations" of the Dichotomy Algorithm, which makes it possible to effectively solve a series as well as a single system of equations. On the example of solving of elliptic equations by the Separation Variable Method, we show that the computation accuracy is comparable with the sequential version of the Thomas method, and the dependence of the speedup on the number of processors is almost linear. The proposed modification is aimed at parallel realization of a broad class of numerical methods including the inversion of Toeplitz and quasi-Toeplitz tridiagonal matrices.