2006.170: Improved error bounds for underdetermined system solvers

2006.170: James W. Demmel and Nicholas J. Higham (1993) Improved error bounds for underdetermined system solvers. SIAM Journal On Matrix Analysis And Applications, 14 (1). pp. 1-14. ISSN 1095-7162

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Abstract

The minimal 2-norm solution to an underdetermined system $Ax = b$ of full rank can be computed using a QR factorization of $A^T $ in two different ways. One method requires storage and reuse of the orthogonal matrix $Q$, while the method of seminormal equations does not. Existing error analyses show that both methods produce computed solutions whose normwise relative error is bounded to first order by $c\kappa_2 ( A )u$, where $c$ is a constant depending on the dimensions of $A$, $\kappa_2 ( A ) = \| A^ + \|_2 \| A \|_2 $ is the 2-norm condition number, and $u$ is the unit roundoff. It is shown that these error bounds can be strengthened by replacing $\kappa_2(A)$ by the potentially much smaller quantity ${\operatorname{cond}}_2 ( A ) = \| \,| A^ + | \cdot | A |\, \|_2 $, which is invariant under row scaling of $A$. It is also shown that ${\operatorname{cond}}_2 ( A )$ reflects the sensitivity of the minimum norm solution $x$ to row-wise relative perturbations in the data $A$ and $b$. For square linear systems $Ax = b$ row equilibration is shown to endow solution methods based on LU or QR factorization of $A$ with relative error bounds proportional to ${\operatorname{cond}}_\infty ( A )$, just as when a QR factorization of $A^T $ is used. The advantages of using fixed precision iterative refinement in this context instead of row equilibration are explained.

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