An upper $J$- Hessenberg reduction of a matrix through symplectic Householder transformations

TL;DR

This paper presents a method to reduce matrices to upper J-Hessenberg form using symplectic Householder transformations, with improved stability variants demonstrated through numerical experiments.

Contribution

It introduces a new reduction technique to upper J-Hessenberg form using symplectic Householder transformations and proposes more stable variants.

Findings

Two numerically more stable variants are developed.

Numerical experiments demonstrate the efficiency of the proposed methods.

The reduction process is achieved via elementary symplectic Householder transformations.

Abstract

In this paper, we introduce a reduction of a matrix to a condensed form, the upper $J$- Hessenberg form, via elementary symplectic Householder transformations, which are rank-one modification of the identity . Features of the reduction are highlighted. Two variants numerically more stables are then derived. Some numerical experiments are given, showing the efficiency of these variants.