Simultaneous computation of the row and column rank profiles

TL;DR

This paper introduces a new pivoting strategy in Gaussian elimination that simultaneously reveals both row and column rank profiles of a matrix, improving computational efficiency and providing more comprehensive matrix analysis.

Contribution

A novel pivoting strategy and a rank-sensitive, quad-recursive algorithm for computing a PLUQ decomposition that reveals both rank profiles simultaneously.

Findings

Computes both row and column rank profiles at once.

Achieves a time complexity of O(mnr^{8-2}) for rank r matrices.

Improves practical efficiency over previous implementations in finite fields.

Abstract

Gaussian elimination with full pivoting generates a PLUQ matrix decomposition. Depending on the strategy used in the search for pivots, the permutation matrices can reveal some information about the row or the column rank profiles of the matrix. We propose a new pivoting strategy that makes it possible to recover at the same time both row and column rank profiles of the input matrix and of any of its leading sub-matrices. We propose a rank-sensitive and quad-recursive algorithm that computes the latter PLUQ triangular decomposition of an m \times n matrix of rank r in O(mnr^{\omega-2}) field operations, with \omega the exponent of matrix multiplication. Compared to the LEU decomposition by Malashonock, sharing a similar recursive structure, its time complexity is rank sensitive and has a lower leading constant. Over a word size finite field, this algorithm also improveLs the practical efficiency of previously known implementations.