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Mathematics of Computation

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Experiments on error growth associated with some linear least-squares procedures


Author: T. L. Jordan
Journal: Math. Comp. 22 (1968), 579-588
MSC: Primary 65.35
MathSciNet review: 0229373
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Abstract: 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{displaymath}\begin{array}{*{20}{c}} A & {^T} & A \\ \end{array} \end{displaymath}, 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.


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DOI: https://doi.org/10.1090/S0025-5718-1968-0229373-X
Article copyright: © Copyright 1968 American Mathematical Society