27th Conference of the International Linear Algebra Society (ILAS 2026)

On symplectic reduction of a matrix to upper $J$-Hessenberg form

May 21, 2026, 11:25 AM

25m

McBryde Hall 113

Minisymposium Talk Symplectic Linear Algebra and Applications Symplectic Linear Algebra and Applications

Speaker

Ahmed Salam (Université du Littoral-Côte d'Opale)

Description

In the context of computations of eignevalues and eignevectors, structure-preserving of a class of specific structured matrices, the reduction of a matrix to a $J$-Hessenberg condensed form is needed.\
Such reduction is based on symplectic similarity transformations.
It is a crucial step in the $SR$-algorithm (which is a $QR$-like algorithm), structure-preserving, for computing eigenvalues and vectors of such structured matrices.
The algorithm JHESS, or its variant JHMSH are the main algorithms used for such reduction.

These algorithms may meet fatal breakdowns, causing brutal stops of the computations or encounter near-breakdowns, which are source of serious numerical instability.

In this talk, we point out the source of these breakdowns and present strategies
for curing them.