MC64: Permute and scale a sparse unsymmetric matrix to put large entries on the diagonal

Given a sparse matrix \(\mathbf{A}\) = \({\{a _{ij}\}} _{n \times n}\), this subroutine attempts to find a column permutation vector that makes the permuted matrix have \(n\) entries on its diagonal. If the matrix is structurally nonsingular, the subroutine optionally returns a column permutation that maximizes the smallest element on the diagonal, maximizes the sum of the diagonal entries, or maximizes the product of the diagonal entries of the permuted matrix. For the latter option, the subroutine also finds scaling factors that may be used to scale the original matrix so that the nonzero diagonal entries of the permuted and scaled matrix are one in absolute value and all the off-diagonal entries are less than or equal to one in absolute value. The natural logarithms of the scaling factors \(u _i\), \(i = 1, ..., n\), for the rows and \(v _j\), \(j = 1, ..., n\), for the columns are returned so that the scaled matrix \(\mathbf{B}\) = \({\{b _{ij} \}} _ {n \times n}\) has entries

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