Volume 25, Issue 5
A Numerically Stable Block Modified Gram-Schmidt Algorithm for Solving Stiff Weighted Least Squares Problems
DOI:

J. Comp. Math., 25 (2007), pp. 595-619

Published online: 2007-10

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• Abstract

Recently, Wei in \cite{we6} 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 $\overline A$ satisfy several row rank preservation conditions. According to these conditions, in this paper we show that in general, ordinary 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 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.

• Keywords

Weighted least squares Stiff Row block MGS QR Numerical stability Rank preserve

65F20 65F35 65G50.

@Article{JCM-25-595, author = {}, 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 = { Recently, Wei in \cite{we6} 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 $\overline A$ satisfy several row rank preservation conditions. According to these conditions, in this paper we show that in general, ordinary 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 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.}, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/8716.html} }
TY - JOUR T1 - A Numerically Stable Block Modified Gram-Schmidt Algorithm for Solving Stiff Weighted Least Squares Problems 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 KW - Stiff KW - Row block MGS QR KW - Numerical stability KW - Rank preserve AB - Recently, Wei in \cite{we6} 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 $\overline A$ satisfy several row rank preservation conditions. According to these conditions, in this paper we show that in general, ordinary 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 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.