## Version 1.1.0

12th April 2013

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### MA75 Sparse over-determined system: weighted least squares

This subroutine solves weighted sparse least-squares problems. Given an $m×n$ ($m\ge n$) sparse matrix $A=\left\{{a}_{ij}\right\}$ of rank $n$, an $m×m$ diagonal matrix $W$ of weights, and an $m$-vector $b$, the routine calculates the solution vector $x$ that minimizes the Euclidean norm of the weighted residual vector $r=W\left(Ax-b\right)$ by solving the normal equations ${A}^{T}{W}^{2}Ax={A}^{T}{W}^{2}b$.

Three forms of data storage are permitted for the input matrix: storage by columns, where row indices and column pointers describe the matrix; storage by rows, where column indices and row pointers describe the matrix; and the coordinate scheme, where both row and column indices describe the position of entries in the matrix. For the statistical analysis of the weighted least-squares problem, there are two entries: one to obtain a column and one to obtain the diagonal of the covariance matrix ${\left({A}^{T}{W}^{2}A\right)}^{-1}$.