An upper $J$- Hessenberg reduction of a matrix through symplectic   Householder transformations
  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. 
Ahmed Salam, Haithem Ben Kahla
math.NA