A flexible block classical Gram–Schmidt skeleton with reorthogonalization

We investigate a variant of the reorthogonalized block classical Gram–Schmidt method for computing the QR factorization of a full column rank matrix. Our aim is to bound the loss of orthogonality even when the first local QR algorithm is only conditionally stable. In particular, this allows the use of modified Gram–Schmidt instead of Householder transformations as the first local QR algorithm. Numerical experiments confirm the stable behavior of the new variant. We also examine the use of non-QR local factorization and show by example that the resulting variants, although less stable, may also be applied to ill-conditioned problems.