Wilkinson's inertia-revealing factorization and its application to sparse matrices

We propose a new inertia-revealing factorization for sparse 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 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 present experimental results, studying the method’s numerical stability and performance. Our implementation of the method is somewhat naive, but still demonstrates its potential.