Calculating the basic solution to the least squres problem (Algorithm 5.7.1 from Matrix Computations - Golub/VanLoan)
$min_x |\mathbf{Ax}-\mathbf{b}|$
with $A \in \mathbb{R}^{m \times n }$ and $m <n$, using a pivoted QR decomposition,
$\mathbf{Q}^T \mathbf{A} \mathbf{P}= \left[\mathbf{R_1}\, \mathbf{R_2}\right]$,
end in a solution for the coefficients:
$\mathbf{x}_b = \mathbf{P} (\mathbf{R_{1}\backslash c \,\,0})$
with $\mathbf{c} = \mathbf{Q}^T \mathbf{b}$.
Is there an iterative refinement procedure for this basic solution, keeping the sparity of $\mathbf{x_b}$, while enhancing the accuracy of the solution?