Ordered modified Gram-Schmidt orthogonalization revised

Nour-Omid et al. (1991) proposed an ordered modified Gram-Schmidt (MGS) algorithm, which was supposed to improve the orthogonality state of the solution. Thorough analysis, even of a simple, planar case, shows (Štuller, 1994) that, yet in the exact arithmetic, one cannot expect to obtain — independently of the forward, reverse, or any other type, including “ordered”, orthogonalization — the desired solution: a vector in the orthogonal complement of the given vectors. For the planar case, some simple rules of the thumb can be given to minimize the error, but we are not sure they can be directly generalized to higher dimensions as has been done by Nour-Omid et al. Naturally, in finite precision, the situation is even worse. The unique way out we see in the iterative (Bjorck, 1994) Gram-Schmidt methods, for which we (1994) presented several algorithms with different criterions of efficiency applied to.