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

Given a sparse matrix $A$
= ${\left\{{a}_{ij}\right\}}_{n\times n}$, 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
$B$ =
${\left\{{b}_{ij}\right\}}_{n\times n}$ has
entries

$${b}_{ij}={a}_{ij}exp\left({u}_{i}+{v}_{j}\right).$$