M. Rozloznik:  Symmetric indefinite factorization and orthogonalization 
with respect to bilinear form

In this contribution we study the numerical behavior of 
orthogonalization schemes for computing vectors that are mutually
orthogonal with respect to the bilinear form  induced by asymmetric 
indefinite  but nonsingular matrix. Under assumption
on strong nonsingularity of this matrix  we develop bounds for the 
extremal singular values of the triangular factor
that comes from is symmetric indefinite factorization. It appears that 
they depend on the
  the extremal singular values of the matrix and of only those principal 
submatrices where there is a change
of sign in the associated subminors. Using these results we analyze two 
types of schemes used for
orthogonalization  and we  give the worst-case
bounds for quantities computed in finite precision
arithmetic. In particular, we analyze the QR
implementation based on the symmetric indefinite factorization and the 
Gram-Schmidt process with respect to
this bilinear form.
We consider also their versions  with reorthogonalization and with one 
step of iterative refinement.