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.