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MC67 Reﬁne a proﬁle-reducing permutation of a symmetric matrix

Given the sparsity pattern of an $n×n$ symmetric matrix $A$ and a symmetric permutation that reduces the proﬁle of $A$, this routine computes a new symmetric permutation with a smaller proﬁle. The exchange algorithms of Hager are used to reﬁne the given permutation.

Any zeros on the diagonal of $A$ are regarded as nonzero. If ${m}_{i}$ is the column index of the ﬁrst nonzero in row $i$ (${m}_{i}\le i$), the length of row $i$ is $i-{m}_{i}+1$ and the proﬁle of $A$ is the sum of the lengths of the rows.

MC61 (or MC60) may be used to obtain an initial symmetric permutation.