A modified Gram–Schmidt algorithm with iterative orthogonalization and column pivoting

Abstract

Iterative orthogonalization is aimed to ensure small deviation from orthogonality in the Gram–Schmidt process. Former applications of this technique are restricted to classical Gram–Schmidt (CGS) and column-oriented modified Gram–Schmidt (MGS). The major aim of this paper is to explain how iterative orthogonalization is incorporated into row-oriented MGS. The interest that we have in a row-oriented iterative MGS comes from the observation that this method is capable of performing column pivoting. The use of column pivoting delays the deteriorating effects of rounding errors and helps to handle rank-deficient least-squares problems. A second modification proposed in this paper considers the use of Gram–Schmidt QR factorization for solving linear least-squares problems. The standard solution method is based on one orthogonalization of the r.h.s. vector b against the columns of Q. The outcome of this process is the residual vector, r ∗ , and the solution vector, x ∗ . The modified scheme is a natural extension of the standard solution method that allows it to apply iterative orthogonalization. This feature ensures accurate computation of small residuals and helps in cases when Q has some deviation from orthogonality.