Householder Orthogonalization with a Nonstandard Inner Product

Abstract. Householder orthogonalization plays an important role in numerical linear algebra. It attains perfect orthogonality regardless of the conditioning of the input. However, in the context of a nonstandard inner product, it becomes difficult to apply Householder orthogonalization, partly due to the lack of an initial orthonormal basis. We propose strategies to overcome this obstacle and discuss algorithms and variants of Householder orthogonalization with a nonstandard inner product. Theoretical analysis and numerical experiments demonstrate that our approach is numerically stable under mild assumptions.