## Version 2.4.3

1st November 2023

# HSL_MC64: Permute and scale a sparse unsymmetric or rectangular matrix to put large entries on the diagonal

Given a sparse unsymmetric or rectangular matrix $$\mathbf{A}$$ = $${\{a _{ij}\}} _{m \times n}$$, $$m \ge n$$, this subroutine attempts to find a row and column permutation that makes the permuted matrix have $$n$$ entries on its diagonal. If the matrix is structurally nonsingular, the subroutine optionally returns a 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 $b _{ij} = a _{ij} \exp (u _i + v _j ).$

In this Fortran 95 version, there are added facilities from the original MC64 code for working on rectangular and symmetric matrices. For the rectangular case, a row and column permutation are returned so that the user can permute the matching to the diagonal and identify the rows in the structurally nonsingular block. For the symmetric case, the user must only supply the lower triangle and, if a scaling is computed, it will be a symmetric scaling with the same property as in the unsymmetric case. Structually non-singular matrices are supported using the maximum product matching only.