Comparison Theorems for Splittings of M-matrices in (block) Hessenberg Form

TL;DR

This paper compares convergence rates of different iterative methods for solving linear systems with M-matrices in Hessenberg form, introducing new comparison results and a stair partitioning technique for parallel computation.

Contribution

It establishes novel comparison theorems for iterative methods on M-matrices in Hessenberg form and introduces stair partitioning for designing parallel-friendly algorithms.

Findings

Gauss--Seidel iteration convergence rate bounds are established.

Stair partitioning enhances parallel computation of iterative methods.

New comparison results are derived for non-comparable splittings.

Abstract

Some variants of the (block) Gauss--Seidel iteration for the solution of linear systems with $M$-matrices in (block) Hessenberg form are discussed. Comparison results for the asymptotic convergence rate of some regular splittings are derived: in particular, we prove that for a lower-Hessenberg M-matrix $\rho(P_{GS})\geq \rho(P_S)\geq \rho(P_{AGS})$, where $P_{GS}, P_S, P_{AGS}$ are the iteration matrices of the Gauss--Seidel, staircase, and anti-Gauss--Seidel method. This is a result that does not seem to follow from classical comparison results, as these splittings are not directly comparable. It is shown that the concept of stair partitioning provides a powerful tool for the design of new variants that are suited for parallel computation.