TY - JOUR
T1 - A Numerically Stable Block Modified Gram-Schmidt Algorithm for Solving Stiff Weighted Least Squares Problems
AU - Musheng Wei & Qiaohua Liu
JO - Journal of Computational Mathematics
VL - 5
SP - 595
EP - 619
PY - 2007
DA - 2007/10
SN - 25
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/jcm/8716.html
KW - Weighted least squares, Stiff, Row block MGS QR, Numerical stability, Rank
preserve.
AB - <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>