@Article{JCM-25-595,
author = {Musheng Wei and Qiaohua Liu},
title = {A Numerically Stable Block Modified Gram-Schmidt Algorithm for Solving Stiff Weighted Least Squares Problems},
journal = {Journal of Computational Mathematics},
year = {2007},
volume = {25},
number = {5},
pages = {595--619},
abstract = {<p style="text-align: justify;">Recently, Wei in [18] proved that perturbed stiff weighted pseudoinverses and stiff weighted least squares problems are stable, if and only if the original and perturbed coefficient matrices $A$ and $\overline A$ &nbsp;satisfy several row rank preservation conditions. According to these conditions, in this paper we show that in general, ordinary &nbsp;modified Gram-Schmidt with column pivoting is not numerically stable for solving the stiff weighted least squares problem. We then propose a row block modified Gram-Schmidt algorithm with column pivoting, and show that with appropriately chosen tolerance, this algorithm can correctly determine the numerical ranks of these row partitioned sub-matrices, and the computed QR factor $\overline R$ contains small roundoff error which &nbsp;is row stable. Several numerical experiments are also provided to compare the results of the ordinary Modified Gram-Schmidt algorithm with column pivoting and the row block Modified Gram-Schmidt algorithm with column pivoting.</p>},
issn = {1991-7139},
doi = {https://doi.org/},
url = {http://global-sci.org/intro/article_detail/jcm/8716.html}
}