There exists a number of Fortran codes for the systems of linear equations which are either sparse or do not have a positive definite matrix. Perversely, many practical applications call for a coincidence of both these features. The paper tries to meet the need and offers a new computational strategy.
With emphasis on the efficiency (in both the accuracy and economy sense) and robust generality, a detailed proposal is based on a combined pivoting (mediated by a two-by-two block-diagonalizing rotation) and factorization (in the, combined again, up and down direction).
The details are inspiring and include the re-orderings and simultaneity of the symbolic and numerical factorizations. Together with the restarts of memory arrangements they are designed to minimize the complexity of the – variable – fill-in pattern. The real impact of these technical ingredients is illustrated and validated by a few tests.