Symmetric indefinite triangular factorization revealing the rank profile matrix

TL;DR

This paper introduces a new recursive algorithm for symmetric indefinite triangular factorization that efficiently reveals the rank profile matrix, matching or exceeding the speed of existing methods.

Contribution

The paper presents a novel recursive algorithm for symmetric indefinite triangular factorization that reveals the rank profile matrix with improved efficiency and adaptability to different field characteristics.

Findings

Algorithm requires $O(n^2r^{\omega-2})$ operations.

Achieves approximately the same speed as state-of-the-art Gaussian elimination.

Can recover the rank profile matrix from permutation and block support.

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 $\mathbf{P}^T\mathbf{A}\mathbf{P} = \mathbf{L}\mathbf{D}\mathbf{L}^T$ 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. Furthermore, experimental results demonstrate that our algorithm can even be slightly more than twice as fast as the state of the art unsymmetric Gaussian elimination in most cases, that is it achieves approximately the same computational speed. 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. 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 $\mathbf{P}\mathbf{L}\mathbf{D}\mathbf{L}^T\mathbf{P}^T$ decomposition at a negligible cost.