Wilkinson's Inertia-Revealing Factorization and Its Application to Sparse Matrices
For researchers and practitioners working with sparse symmetric matrices, this provides a potentially more efficient alternative to existing inertia-revealing methods, though the work is incremental as it adapts an existing scheme.
The paper revives a 1960s inertia-revealing factorization for sparse symmetric matrices, showing its fill is bounded by sparse QR factorization fill and typically smaller, with a proof-of-concept implementation demonstrating numerical stability and performance.
We propose a new inertia-revealing factorization for sparse symmetric matrices. The factorization scheme and the method for extracting the inertia from it were proposed in the 1960s for dense, banded, or tridiagonal matrices, but they have been abandoned in favor of faster methods. We show that this scheme can be applied to any sparse symmetric matrix and that the fill in the factorization is bounded by the fill in the sparse QR factorization of the same matrix (but is usually much smaller). We describe our serial proof-of-concept implementation, and present experimental results, studying the method's numerical stability and performance.