Numerical stability of orthogonalization methods with a non-standard inner product

In this paper we study the numerical properties of several orthogonalization schemes where the inner product is induced by a nontrivial symmetric and positive definite matrix. We analyze the effect of its conditioning on the factorization and the loss of orthogonality between vectors computed in finite precision arithmetic. We consider the implementation based on the backward stable eigendecomposition, modified and classical Gram–Schmidt algorithms, Gram–Schmidt process with reorthogonalization as well as the implementation motivated by the AINV approximate inverse preconditioner.