A Numerically Stable Block Modified Gram-Schmidt Algorithm for Solving Stiff Weighted Least Squares Problems

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.