Rank-profile revealing Gaussian elimination and the CUP matrix decomposition

TL;DR

This paper introduces a new CUP matrix decomposition algorithm that efficiently reveals the rank profile, compares it with existing methods, and discusses its advantages in terms of complexity and in-place computation.

Contribution

The paper presents a novel CUP decomposition algorithm, demonstrates its reductions from other Gaussian elimination methods, and analyzes its efficiency and in-place computation benefits.

Findings

CUP decomposition is rank sensitive with favorable asymptotic complexity.

The CUP algorithm can compute matrix invariants efficiently and in place.

It outperforms or matches existing algorithms in time and space complexity.

Abstract

Transforming a matrix over a field to echelon form, or decomposing the matrix as a product of structured matrices that reveal the rank profile, is a fundamental building block of computational exact linear algebra. This paper surveys the well known variations of such decompositions and transformations that have been proposed in the literature. We present an algorithm to compute the CUP decomposition of a matrix, adapted from the LSP algorithm of Ibarra, Moran and Hui (1982), and show reductions from the other most common Gaussian elimination based matrix transformations and decompositions to the CUP decomposition. We discuss the advantages of the CUP algorithm over other existing algorithms by studying time and space complexities: the asymptotic time complexity is rank sensitive, and comparing the constants of the leading terms, the algorithms for computing matrix invariants based on the CUP decomposition are always at least as good except in one case. We also show that the CUP algorithm, as well as the computation of other invariants such as transformation to reduced column echelon form using the CUP algorithm, all work in place, allowing for example to compute the inverse of a matrix on the same storage as the input matrix.