Reorthogonalization and stable algorithms for updating the Gram-Schmidt 𝑄𝑅 factorization

Abstract

Numerically stable algorithms are given for updating the Gram-Schmidt QR factorization of an m × n m \times n matrix A ( m ⩾ n ) A\;(m \geqslant n) when A is modified by a matrix of rank one, or when a row or column is inserted or deleted. The algorithms require O ( m n ) O(mn) operations per update, and are based on the use of elementary two-by-two reflection matrices and the Gram-Schmidt process with reorthogonalization. An error analysis of the reorthogonalization process provides rigorous justification for the corresponding ALGOL procedures.