Heuristic analysis of numerical variants of the Gram-Schmidt orthonormalization process

[摘要] 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 computations.

[学科分类] 计算机科学（综合）