Version 1.1.2
1st November 2023 User documentation
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MA75: Sparse overdetermined system: weighted least squares
This subroutine solves weighted sparse leastsquares 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 leastsquares problem, there are two entries: one to obtain a column and one to obtain the diagonal of the covariance matrix \(( \mathbf{A} ^ T \mathbf{W} ^2 \mathbf{A} ) ^{1}\).