Parallel dichotomy algorithm for solving tridiagonal SLAEs

TL;DR

This paper introduces a parallel algorithm for efficiently solving multiple tridiagonal systems with the same matrix but different right-hand sides, optimizing communication and demonstrating high scalability on supercomputers.

Contribution

It presents a novel parallel sweep algorithm based on a new theorem, reducing communication and improving efficiency for solving SLAEs on distributed-memory systems.

Findings

High efficiency demonstrated on multicomputer systems

Scalability confirmed for supercomputing environments

Effective solution of 2D Poisson equations using the proposed method

Abstract

A parallel algorithm for solving a series of matrix equations with a constant tridiagonal matrix and different right-hand sides is proposed and studied. The process of solving the problem is represented in two steps. The first preliminary step is fixing some rows of the inverse matrix of SLAEs. The second step consists in calculating solutions for all right-hand sides. For reducing the communication interactions, based on the formulated and proved main parallel sweep theorem, we propose an original algorithm for calculating share components of the solution vector. Theoretical estimates validating the efficiency of the approach for both the common- and distributed-memory supercomputers are obtained. Direct and iterative methods of solving a 2D Poisson equation, which include procedures of tridiagonal matrix inversion, are realized using the mpi technology. Results of computational experiments on a multicomputer demonstrate a high efficiency and scalability of the parallel sweep algorithm.