HSL_MA85: Sparse diagonally-weighted linear least squares solver

HSL_MA85 uses a direct method to solve large-scale diagonally-weighted linear least squares problems. Given an \(m \times n\) (\(m \ge n\)) matrix \({A} = \{ a_{ij} \}\), an \(m \times m\) diagonal matrix of weights \(W\), and an \(m-\)vector \(b\), HSL_MA85 solves either the least squares problem \[\label{eq:ls} \min_x \| W(Ax - b) \|^2_2 ,\] or the regularized least squares problem \[\label{eq:ls_reg} \min_x \| W(Ax - b) \|^2_2 + \alpha\|x\|^2_2,\] where \(\alpha > 0\) is a regularization parameter chosen by the user. The matrix \(A\) may contain one or more rows that are to be treated as dense but must otherwise be sparse. Rows of \(A\) that lead to a large amount of fill in the normal matrix should be treated as dense (they may contain fewer than \(n\) non zero entries but generally have more non zeroes than the other rows of \(A\)). The package offers the option of (i) a Cholesky-based approach or (ii) an approach that uses a symmetric indefinite solver applied to the (regularized) augmented system.