Abstract The Gram-Schmidt (GS) orthogonalization is one of the fundamental procedures in
linear algebra. In matrix terms it is equivalent to the factorization A Q 1 R, where Q 1∈ R …

SUMMARY In 1907, Erhard Schmidt published a paper in which he introduced an
orthogonalization algorithm that has since become known as the classical Gram‐Schmidt …

In the nearly seven years since I finished writing the first edition of this book research on the
accuracy and stability of numerical algorithms has continued to flourish and mature. Our …

Excerpt More than 25 years have passed since the first edition of this book was published in
1996. Least squares and least-norm problems have become more significant with every …

Krylov subspace methods (KSMs) are iterative algorithms for solving large, sparse linear
systems and eigenvalue problems. Current KSMs rely on sparse matrix-vector multiply …

Abstract Block Gram-Schmidt algorithms serve as essential kernels in many scientific
computing applications, but for many commonly used variants, a rigorous treatment of their …

The traditional metric for the efficiency of a numerical algorithm has been the number of
arithmetic operations it performs. Technological trends have long been reducing the time to …

This paper provides two results on the numerical behavior of the classical Gram-Schmidt
algorithm. The first result states that, provided the normal equations associated with the …

First, we consider the problem of orthonormalizing skinny (long) matrices. We propose an
alternative orthonormalization method that computes the orthonormal basis from the right …

In [Numer. Math., 23 (2013), pp. 395–423], Barlow and Smoktunowicz propose the
reorthogonalized block classical Gram–Schmidt algorithm BCGS2. New conditions for the …