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KKTDirect implements an ordering method and accompanying factorization for
the direct solution of saddle-point matrices (also known as KKT or equilibrium
matrices). A simple constraint on ordering together with an assumption on
the rank of parts of the matrix are sufficient to guarantee the existence of
the LDL^T factorization, stability concerns aside. In fact, D may be taken
to be a diagonal matrix with +/-1 along the diagonal, and be fully determined
prior to factorization, giving rise to a "signed Cholesky" factorization. A
modified minimum-degree-like algorithm which incorporates this constraint is
used, along with a simple algorithm to modify an existing fill-reducing
ordering to respect the constraint. While a stability analysis is lacking,
numerical experiments indicate that this is generally sufficient to avoid the
need for numerical pivoting during factorization, with clear possible benefits
for performance.
Note this is only alpha-quality proof-of-concept code: for example,
out-of-memory errors are not handled gracefully, and the provided Minimum
Degree routine is not yet competitive with other packages.
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