A time-optimal algorithm for solving (block-)tridiagonal linear systems of dimension N on a distributed computer of N nodes

Abstract

We are concerned with the fastest possible direct numerical solution algorithm for a thin-banded or tridiagonal linear system of dimension N on a distributed computing network of N nodes that is connected in a binary communication tree. Our research is driven by the need for faster ways of numerically solving discretized systems of coupled one-dimensional blackbox boundary-value problems.Our paper presents two major results: First, we provide an algorithm that achieves the optimal parallel time complexity for solving a tridiagonal linear system and thin-banded linear systems. Second, we prove that it is impossible to improve the time complexity of this method by any polynomial degree.To solve a system of dimension m • N and bandwidth m ∈ Ω(N 1/6 ) on 2 • N − 1 computing nodes, our method needs time complexity O(log(N ) 2 • m 3 ).