 ## Version 1.1.1

15th February 2023

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

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

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 $$( \mathbf{A} ^ T \mathbf{W} ^2 \mathbf{A} ) ^{-1}$$.