  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.
