A Numerically Stable Block Modified Gram-Schmidt Algorithm for Solving Stiff Weighted Least Squares Problems
Year: 2007
Journal of Computational Mathematics, Vol. 25 (2007), Iss. 5 : pp. 595–619
Abstract
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 $\overline A$ 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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/2007-JCM-8716
Journal of Computational Mathematics, Vol. 25 (2007), Iss. 5 : pp. 595–619
Published online: 2007-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 25
Keywords: Weighted least squares Stiff Row block MGS QR Numerical stability Rank preserve.