Symplectic Householder transformations for a QR-like decomposition, a geometric and algebraic approaches

The aim of this paper is to show how geometric and algebraic approaches lead us to a new symplectic elementary transformations: the 2-D symplectic Householder transformations. Their features are studied in details. Their interesting properties allow us to construct a new algorithm for computing a SR factorization. This algorithm is based only on these 2-D symplectic Householder transformations. Its new features are highlighted. The study shows that, in the symplectic case, the new algorithm is the corresponding one to the classical QR factorization algorithm, via the Householder transformations. Some numerical experiments are given.