Title: Orthonormal Completion of an array of Unit Length Vectors

Abstract:
Some interesting new results on orthogonality,
and their contribution to the slowly developing understanding
of a class of very useful numerical algorithms
which have very strange numerical behaviours.
These include Modified Gram-Schmidt (MGS)
for orthogonalization and linear least squares problems,
MGS-GMRES for solution of large sparse systems of equations,
the Conjugate Gradients Method for that problem and for optimization,
Lanczos' tridiagonalization of a matrix for the eigenproblem,
the Golub-Kahan bidiagonalization for many classes of problem,
and some other crucial algorithms for large sparse matrix problems.