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Given a full column rank matrix $A \in \mathbb{R}^{m\times n}$ ($m\geq n$),we consider a special class of linear systems of the form $A^\top Ax=A^\topb+c$ with $x, c \in \mathbb{R}^{n}$ and $b \in \mathbb{R}^{m}$. The occurrenceof $c$ in the right-hand side of the equation prevents the direct applicationof standard methods for least squares problems. Hence, we investigatealternative solution methods that, as in the case of normal equations, takeadvantage of the peculiar structure of the system to avoid unstablecomputations, such as forming $A^\top A$ explicitly. We propose two iterativemethods that are based on specific reformulations of the problem and we provideexplicit closed formulas for the structured condition number related to eachproblem. These formula allow us to compute a more accurate estimate of theforward error than the standard one used for generic linear systems, that doesnot take into account the structure of the perturbations. The relevance of ourestimates is shown on a set of synthetic test problems. Numerical experimentshighlight both the increased robustness and accuracy of the proposed methodscompared to the standard conjugate gradient method. It is also found that thenew methods can compare to standard direct methods in terms of solutionaccuracy.

On the iterative solution of systems of the form $A^T A x=A^Tb+c$