A Householder-based algorithm for Hessenberg-triangular reduction

TL;DR

This paper introduces a Householder-based algorithm for Hessenberg-triangular reduction that aims to improve computational efficiency and scalability over traditional Givens rotation methods, with promising initial experimental results.

Contribution

The paper presents a novel Householder reflector-based approach for HT reduction, differing from the Givens rotation method, and discusses its potential for better parallel scalability.

Findings

The new algorithm is competitive in sequential settings despite more floating point operations.

Early experiments suggest improved parallel scalability with multi-threaded BLAS.

The approach may lead to more efficient eigenvalue computations for matrix pencils.

Abstract

The QZ algorithm for computing eigenvalues and eigenvectors of a matrix pencil $A - \lambda B$ requires that the matrices first be reduced to Hessenberg-triangular (HT) form. The current method of choice for HT reduction relies entirely on Givens rotations regrouped and accumulated into small dense matrices which are subsequently applied using matrix multiplication routines. A non-vanishing fraction of the total flop count must nevertheless still be performed as sequences of overlapping Givens rotations alternately applied from the left and from the right. The many data dependencies associated with this computational pattern leads to inefficient use of the processor and poor scalability. In this paper, we therefore introduce a fundamentally different approach that relies entirely on (large) Householder reflectors partially accumulated into block reflectors, by using (compact) WY representations. Even though the new algorithm requires more floating point operations than the state of the art algorithm, extensive experiments on both real and synthetic data indicate that it is still competitive, even in a sequential setting. The new algorithm is conjectured to have better parallel scalability, an idea which is partially supported by early small-scale experiments using multi-threaded BLAS. The design and evaluation of a parallel formulation is future work.