  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 A. We analyze two algorithm that are based a factorization of A
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 equa- tion. 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.
