Rank Revealing Gaussian Elimination by the Maximum Volume Concept
For numerical linear algebra practitioners, this provides a new rank-revealing factorization method that avoids forming the normal matrix and is amenable to blocked and sparse implementations.
The paper presents a Gaussian elimination algorithm that reveals numerical rank by using the maximum volume concept to find a maximal nonsingular submatrix, achieving bounds similar to rank-revealing LU without using the normal matrix, with computational cost roughly twice that of LU with complete pivoting.
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.