Sets of vectors that are mutually orthogonal with respect to an oblique inner product are often used in iterative methods such as the conjugate gradient method. However, these vectors cannot be computed exactly since floating point arithmetic introduces error into the computation. Large error in the computation my slow the convergence of the associated iterative method, a topic we do not address in this paper. Rather, we wish to improve our understanding of the error introduced in computing such vectors. For this study we focus on QR factorization algorithms.
In this paper, we consider the stability of the QR factorization in an
oblique inner product.
The oblique inner product is defined by a symmetric positive definite matrix 
.
We analyze two algorithm that are based a factorization of
and converting the
problem to the Euclidean case. The two algorithms we consider use the Cholesky
decomposition and the eigenvalue decomposition. We also analyze algorithms that
are based on computing the Cholesky factor of the normal equation. We present numerical
experiments to show the error bounds are tight. Finally we present performance
results for these algorithms as well as Gram-Schmidt methods on parallel architecture.
The performance experiments demonstrate the benefit of the communication avoiding
algorithms.