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On the augmented system approach to sparse least-squares problems. (English) Zbl 0678.65024
When the least-squares problem \(\| Ax-b\|_ 2\to \min !\) is to be solved, one may consider the augmented system \[ \left( \begin{matrix} I\\ A^ T\end{matrix} \begin{matrix} A\\ 0\end{matrix} \right)\left( \begin{matrix} r\\ x\end{matrix} \right)=\left( \begin{matrix} b\\ 0\end{matrix} \right) \] instead of the normal equations. In order to reduce the condition number of the augmented matrix, the identity matrix I may be replaced by \(\alpha\) I with \(\alpha\) being a suitable scaling factor. Iterative refinements and perturbation theoretic arguments are discussed in the framework of an error analysis. Ten tables with numerical results for several test matrices are given.
Reviewer: D.Braess

65F20 Numerical solutions to overdetermined systems, pseudoinverses
65F35 Numerical computation of matrix norms, conditioning, scaling
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