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MF29: Sparse unsymmetric matrix: calculate scaling factors

This subroutine calculates scaling factors for a complex sparse matrix \(\mathbf{A} = {\{a _{ij}\}} _{m \times n}\).They may be used, for instance, to scale the matrix prior to solving a corresponding set of linear equations, and are chosen so that the scaled matrix has its entries near to unity in the sense that the sum of the squares of the logarithms of the moduli of the entries is minimized. The natural logarithms of the scaling factors \(r _i\), \(i = 1, 2,..., m\) for the rows and \(c _j\), \(j = 1, 2,..., n\) for the columns are returned so that the scaled matrix has entries

\[b _{ij} = a _{ij} \exp (r _i + c _j ).\]