Rank revealing Gaussian elimination by the maximum volume concept

Abstract

A Gaussian elimination algorithm is presented that reveals the numerical rank of a matrix and identifies a square nonsingular submatrix of maximum dimension. The bounds on the singular values of the submatrix and its Schur complement 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.