Direct Solution of Sets of Linear Equations whose Matrix is Sparse, Symmetric and Indefinite

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Abstract
We consider the use of 1×1 and 2×2 pivots for direct solution of sets of linear equations whose matrix is sparse and symmetric. Inclusion of 2×2 pivots permits a stable decomposition to be obtained in the indefinite case and we demonstrate that in practice there is little loss of speed even in positive definite cases. A pivotal strategy suitable for the sparse case is proposed and compared experimentally with alternatives. We present an analysis of error, explain how the stability may be monitored cheaply, discuss automatic scaling and consider implementation details.