  For a given matrix, we are interested in computing GR decompositions $A=GR$,
where $G$ is an isometry with respect to given scalar products. The orthogonal
QR decomposition is the representative for the Euclidian scalar product. For a
signature matrix, a respective factorization is given as the hyperbolic QR
decomposition. Considering a skew-symmetric matrix leads to the symplectic QR
decomposition. The standard approach for computing GR decompositions is based
on the successive elimination of subdiagonal matrix entries. For the hyperbolic
and symplectic case, this approach does in general not lead to a satisfying
numerical accuracy. An alternative approach computes the QR decomposition via a
Cholesky factorization, but also has bad stability. It is improved by repeating
the procedure a second time. In the same way, the hyperbolic and the symplectic
QR decomposition are related to the $LDL^T$ and a skew-symmetric Cholesky-like
factorization. We show that methods exploiting this connection can provide
better numerical stability than elimination-based approaches.
