Rank Revealing Gaussian Elimination by the Maximum Volume Concept

TL;DR

This paper introduces a Gaussian elimination method that reveals a matrix's numerical rank by selecting a maximum volume submatrix, avoiding the normal matrix, and suitable for dense and sparse matrices.

Contribution

It presents a novel rank revealing Gaussian elimination algorithm based on the maximum volume concept, differing from existing methods by not using the normal matrix.

Findings

Achieves bounds similar to rank revealing LU factorization

Computational cost roughly twice that of LU with complete pivoting

Flexible for dense and sparse matrix implementations

Abstract

A Gaussian elimination algorithm is presented that reveals the numerical rank of a matrix by yielding small entries in the Schur complement. The algorithm uses the maximum volume concept to find a square nonsingular submatrix of maximum dimension. The bounds on the revealed singular values are similar to the best known bounds for rank revealing LU factorization, but in contrast to existing methods the algorithm does not make use of the normal matrix. An implementation for dense matrices is described whose computational cost is roughly twice the cost of an LU factorization with complete pivoting. Because of its flexibility in choosing pivot elements, the algorithm is amenable to implementation with blocked memory access and for sparse matrices.