The loss of orthogonality in the Gram-Schmidt orthogonalization process

In this paper, we study numerical behavior of several computational variants of the Gram-Schmidt orthogonalization process. We focus on the orthogonality of computed vectors which may be significantly lost in the classical or modified Gram-Schmidt algorithm, while the Gram-Schmidt algorithm with reorthogonalization has been shown to compute vectors which are orthogonal to machine precision level. The implications for practical implementation and its impact on the efficiency in the parallel computer environment are considered.