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That is, the algorithm computes a factorization  $\\mathbf{P}^T\\mathbf{A}\\mathbf{P} = \\mathbf{L}\\mathbf{D}\\mathbf{L}^T$  where $\\mathbf{P}$ is a permutation matrix, $\\mathbf{L}$ is lower triangular with a unit diagonal and  $\\mathbf{D}$ is symmetric block diagonal with $1{\\times}1$ and $2{\\times}2$ antidiagonal blocks. The novel algorithm requires $O(n^2r^{\\omega-2})$ arithmetic operations. 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