Symmetric Indefinite Triangular Factorization Revealing the Rank Profile Matrix

Abstract

We present a novel recursive algorithm for reducing a symmetric matrix to a triangular factorization which reveals the rank profile matrix. That is, the algorithm computes a factorization P TA P = L D L T where P is a permutation matrix, L is lower triangular with a unit diagonal and D is symmetric block diagonal with 1 x 1 and 2 x 2 antidiagonal blocks. This algorithm requires O(n2rømega-2) arithmetic operations, with n the dimension of the matrix, r its rank and ømega an admissible exponent for matrix multiplication. Furthermore, experimental results demonstrate that our algorithm has very good performance: its computational speed matches that of its numerical counterpart and is twice as fast as the unsymmetric exact Gaussian factorization. By adapting the pivoting strategy developed in the unsymmetric case, we show how to recover the rank profile matrix from the permutation matrix and the support of the block-diagonal matrix. We also note that there is an obstruction in characteristic 2 for revealing the rank profile matrix, which requires to relax the shape of the block diagonal by allowing the 2-dimensional blocks to have a non-zero bottom-right coefficient. This relaxed decomposition can then be transformed into a standard PLDLTP T decomposition at a negligible cost.