Experiments on Error Growth Associated with Some Linear Least-Squares Procedures

Some numerical experiments were performed to compare the performance of procedures for solving the linear least-squares problem based on GramSchmidt, Modified Gram-Schmidt, and Householder transformations, as well as the classical method of forming and solving the normal equations. In addition, similar comparisons were made of the first three procedures and a procedure based on Gaussian elimination for solving an $n \times n$ system of equations. The results of these experiments suggest that: (1) the Modified Gram-Schmidt procedure is best for the least-squares problem and that the procedure based on Householder transformations performed competitively; (2) all the methods for solving least-squares problems suffer the effects of the condition number of $\begin {array}{*{20}{c}} A & {^T} & A \\ \end {array}$, although in a different manner for the first three procedures than for the fourth; and (3) the procedure based on Gaussian elimination is the most economical and essentially, the most accurate for solving $n \times n$ systems of linear equations. Some effects of pivoting in each of the procedures are included.