Parallel two-stage reduction to Hessenberg-triangular form

TL;DR

This paper introduces a parallel two-stage algorithm for reducing a matrix pencil to Hessenberg-triangular form, achieving high performance in both stages and outperforming existing methods in shared memory environments.

Contribution

It extends existing two-stage reduction techniques to also deliver high performance in the second stage, improving overall efficiency.

Findings

Outperforms state-of-the-art implementations in shared memory environments

Achieves high performance in both reduction stages

Demonstrates efficiency through experimental results

Abstract

We present a two-stage algorithm for the parallel reduction of a pencil to Hessenberg-triangular form. Traditionally, two-stage Hessenberg-triangular reduction algorithms achieve high performance in the first stage, but struggle to achieve high performance in the second stage. Our algorithm extends techniques described by Karlsson et al. to also achieve high performance in the second stage. Experiments in a shared memory environment demonstrate that the algorithm can outperform state-of-the-art implementations.