Report Number: CS-TR-69-122
Institution: Stanford University, Department of Computer Science
Title: Heuristic analysis of numerical variants of the Gram-Schmidt
Author: Mitchell, William C.
Author: McCraith, Douglas L.
Date: February 1969
Abstract: The Gram-Schmidt orthonormalization process is a fundamental
formula of analysis which is notoriously unstable
computationally. This report provides a heuristic analysis of
the process, which shows why the method is unstable. Formulas
are derived which describe the propagation of round-off error
through the process. These formulas are supported by
numerical experiments. These formulas are then applied to a
computational variant of a basic method proposed by John R.
Rice, and this method is shown to offer significant
improvement over the basic algorithm. This finding is also
supported by numerical experiment.
The formulas for the error propagation are then used to
produce a linear corrector for the basic Gram-Schmidt
process, which shows significant improvement over both
previous methods, but at the cost of slightly more